Concepts of Physics Exercises Electric Field & Potential

Concepts of Physics

Exercises

Electric Field amp Potential

1 Find the dimensional formula of 1205760

Solution

Electrostatic force between two point charges F = 1

41205871205760

11990211199022

1199032 rArr 1205760 = 11990211199022

41205871198651199032

Dimensional formula of charge q = IT

Dimensional formula of force F = MLT-2

Dimensional formula of distance r = L

Dimensional formula of permittivity of free space 1205760 = (119868119879) (119868119879)

119872119871119879minus2119871= M-1L-2T4I2


2 A charge of 10 C is placed at the top of your college building and another equal charge at the top of your house

Take the separation between the two charges to be 20 km Find the force exerted by the charges on each other How

many times of your weight is this force

Solution

Charge on each particle q = 1 C

Separation between the charges r = 2 km = 2000 m

Electrostatic force between the charge F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 1 119909 1

4 119909 106 = 225 x 103 N

3 At what separation should two equal charges 10 C each be placed so that the force between them equals the weight

of a 50 kg person

Solution

Charge on each particle q1 = q2 = 1 C

Force between the particles F = mg = 50 x 98 = 490 N

For separation between the particle F = 1

41205871205760

11990211199022

1199032 rArr 490 = 9 x 109 1 119909 1

1199032 rArr r = 9 119909 109

490=

3

7x 104 = 43 x 103 m


4 Two equal charges are placed at a separation of 10 m What should be the magnitude of the charges so that the force

between them equals the weight of a 50 kg person

Solution

Let the charge on each particle be lsquoqrsquo

Separation between the charges r = 1 m

Magnitude of force between the charges F = mg = 50 x 98 = 490 N

For magnitude of charges F = 1

41205871205760

11990211199022

1199032 rArr 490 = 9 x 109 1199022

1rArr q =

490

9 119909 109 = 7

3 119909 104 = 23 x 10-4 C

5 Find the electric force between two protons separated by a distance of 1 fermi (1 fermi = 10minus15m) The protons in a

nucleus remain at a separation of this order

Solution

Charge on each proton q = 16 x 10-19 C

Separation between the protons r = 1 fm = 10-15 m

Electrostatic force between the protons F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus30 = 9 x 1039 x 256 x 10-38 = 2304 N


6 Two charges 20 times 10minus6C and 10 times 10minus6C are placed at a separation of 10 cm Where should a third charge be

placed such that it experiences no net force due to these charges

Solution

Charge on the first particle q1 = 2 x 10-6 C

Charge on the second particle q2 = 1 x 10-6 C

Separation between the particles r = 10 cm = 01 m

Let the third charge be placed at lsquoxrsquo from the larger charge

Force between the larger charge and test charge F1 = 1

41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

1199092 [repulsive]

Force between the smaller charge and test charge F2 = 1

41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)

(10minus119909)2 [repulsive]

Given net force on test charge is zero F1 ndash F2 = 0 rArr F1 = F2 rArr 9 x 109 2 119909 10minus6 (119902)

1199092 = 9 x 109 1 119909 10minus6 (119902)

(10 minus 119909)2

2 119909 10minus6 (119902)

1199092 = 1 119909 10minus6

(10 minus 119909)2 rArr 2

1199092 = 1

(10 minus 119909)2 rArr 2

119909=

1

10 minus 119909rArr (1 + 2) x = 10 2 rArr x =

10 2

1 + 2= 59 cm


7 Suppose the second charge in the previous problem is minus10 times 10minus6C Locate the position where a third charge will

not experience a net force

Solution

When one of the charges is negative null point is not possible between the two charges

Let the null point is located at lsquoxrsquo from smaller charge outside the two charges

Force on the test charge due to larger charge F1 = 1

41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

(10+119909)2 [repulsive]

Force on the test charge due to smaller charge F2 = 1

41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)

(119909)2 [attractive]


(10+119909)2 = 9 x 109 1 119909 10minus6 (119902)

(119909)2

2 119909 10minus6 (119902)

(10+119909)2 = 1 119909 10minus6 (119902)

(119909)2 rArr 2

(10+119909)2 = 1

(119909)2 rArr 2

10 + 119909=

1

119909rArr 2 minus 1 119909 = 10 rArr x =

10

2minus1=

10

0414= 241 cm

So the test charge is a distance of 10 + 241 = 341 cm from larger charge


8 Two charged particles are placed at a distance 10 cm apart What is the minimum possible magnitude of the electric

force acting on each charge

Solution

Separation between the particles r = 1 cm

Minimum magnitude of electric charge charge on electron q = 16 x 10-19 C

Force between the two particle

F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus4 = 9 x 1013 x 256 x 10-38 = 2304 x 10-25 N = 23 x 10-24 N


9 Estimate the number of electrons in 100 g of water How much is the total negative charge on these electrons

Solution

Mass of water m = 100 g

Molecular weight of water M = 18 gmole

Number of molecules N = nNA = 119898

119872NA =

100

18x 6023 x 1023 = 335 x 1024

Number of electrons in each water molecule Ne = 2 + 16 = 18

Number of electrons in given sample of water n = 10 x 335 x 1024 = 335 x 1025

Total negative charge on the electrons q = ne = 335 x 1025 x 16 x 10-19 = 535 x 106


10 Suppose all the electrons of 100 g water are lumped together to form a negatively charged particle and all the

nuclei are lumped together to form a positively charged particle If these two particles are placed 100 cm away from

each other find the force of attraction between them Compare it with your weight

Solution

Total negative charge available q1 = 535 x 106 C

Total positive charge available q2 = 535 x 106 C

Separation between the charges r = 10 cm = 01 m

Force of attraction between the charges F = 1

41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N


11 Consider a gold nucleus to be a sphere of radius 69 fermi in which protons and neutrons are distributed Find the

force of repulsion between two protons situated at largest separation Why do these protons not fly apart under this

repulsion

Solution

Radius of the gold nucleus R = 69 fm = 69 x 10-15 m

Largest separation between protons r = 2R = 138 x 10-15 m


Force of repulsion between the protons F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N


12 Two insulting small spheres are rubbed against each other and placed 1 cm apart If they attract each other with a

force of 01 N how many electrons were transferred from one sphere to the other during rubbing

Solution

Let the charge acquired by each sphere q = ne

Separation between the spheres r = 1 cm = 10-2 m

Force of attraction between the spheres F = 01 N

Force of attraction between the spheres F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011


13 NaCI molecule is bound due to the electric force between the sodium and the chlorine ions when one electron of

sodium is transferred to chlorine Taking the separation between the ions to be 275 times 10minus8 cm find the force of

attraction between them State the assumptions (if any) that you have made

Solution

Separation between the ions r = 275 x 10-8 cm

Charge on sodium and chlorine ions q1 = q2 = q = 16 x 10-19 C

Force of attraction between the ions F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N


14 Find the ratio of the electric and gravitational forces between two protons

Solution

Mass of each proton m1 = m2 = m = 167 x 10-27 kg

Charge on each proton q1 = q2 = q = 16 x 10-19 C

Let the separation between the protons be lsquorrsquo

Electric force between the protons Fe = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032

Gravitational force between the protons Fg = 11986611989811198982

1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032

Ratio of electric and gravitational forces 119865119890

119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036


15 Suppose an attractive nuclear force acts between two protons which may be written as F = Ceminuskrr2

(a) write down the dimensional formulae and appropriate SI units of C and k (b) suppose that k = 1 fermiminus1 and that

the repulsive electric force between the protons is just balanced by the attractive nuclear force when the separation is

5 fermi Find the value of C

Solution

(a) Nuclear force between protons F = Ceminuskrr2

Dimensions of exponential function M0L0T0

Dimensional formula for kr M0L0T0 rArr kL = M0L0T0 rArr k = M0L-1T0 (SI unit m-1)

Dimensional formula of force MLT-2

Dimensional formula of Cr2 = MLT-2 rArr C = MLT-2L2 = ML3T-2 (SI unit Nm2)

(b) Repulsive force between protons Fe = 1

41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N

Nuclear attractive force between protons Fn = C(eminuskr)r2 = C 119890

minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25

Given Fe = Fn rArr 92 = 1030

251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2


16 Three equal charges 20 times 10minus6C each are held fixed at the three corners of an equilateral triangle of side 5 cm

Find the Coulomb force experienced by one of the charges due to the rest two

Solution

Charge on each particle q1 = q2 = q3 = 2 x 10-6 C

Side of the triangle r = 5 cm = 5 x 10-2 m

Force between A amp C FAC = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N

Force between B amp C FBC = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N

Net force on C Fc = 2FAC cos 602 = 2 x 144 x cos 30 = 249 N [along the angular bisector 300]


17 Four equal charges 20 times 10minus6C each are fixed at the four corners of a square of side 5 cm Find the Coulomb

force experienced by one of the charges due to the rest three

Solution

Charge on each particle q1 = q2 = q3 = q4 = q = 2 x 10-6 C

Side of the square r = 5 cm = 5 x 10-2 m

Diagonal of the square R = 2r = 5 2 x 10-2 m


41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032

Force between D amp C FDC 1

41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032

Resultant of the above two forces FBD = 21

41205871205760

1199022

1199032 [along the diagonal angular bisector 450]


41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032

Resultant force on C FC = FAC + FBD = 1

41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N


18 A hydrogen atom contains one proton and one electron It may be assumed that the electron revolves in a circle of

radius 053 angstrom (1 angstrom = 10minus10119898 and is abbreviated as Å) with the proton at the centre The hydrogen atom

is said to be in the ground state in this case Find the magnitude of the electric force between the proton and the

electron of a hydrogen atom in its ground state

Solution

Charge on the proton q1 = 16 x 10-19 C

Charge on the electron q2 = 16 x 10-19 C

Separation between proton and electron radius of the orbit r = 053 A0 = 053 x 10-10 m

Electric force between proton and electron F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N

19 Find the speed of the electron in the ground state of a hydrogen atom The description of ground state is given in the previous problem

Solution

Centripetal force for circular motion of the electron is provided by electrostatic force between proton amp electron

1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms


20 Ten positively charged particles are kept fixed on the x-axis at points x = 10 cm 20 cm 30 cm 100 cm The

first particle has a charge 10 times 10minus8C the second 8 times 10minus8C the third 27 times 10minus8C and so on The tenth particle has a

charge 1000 times 10minus8C Find the magnitude of the electric force acting on a 1 C charge placed at the origin

Solution

Charges on the particles 1 119909 10minus2 3 2 119909 10minus2 3 3 119909 10minus2 3 4 119909 10minus2 3 5 119909 10minus2 3 hellip 10 119909 10minus2 3

Distances of the particles 10 cm 2 x 10 cm 3 x 10 cm helliphellip10 x 10 cm

Charge on the particle at origin q0 = 1 C

Electric force on the particle placed at origin F = 1

41205871205760

11990201199021

11990312 + helliphellip

1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N


21 Two charged particles having charge 20 times 10minus8119862 each are joined by an insulting string of length 1 m and the

system is kept on a smooth horizontal table Find the tension in the string

Solution

Charge on each particle q1 = q2 = q = 2 x 10-8 C

Separation between the particle length of the string r = 1 m

Tension in the string electric force between the particle F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N


22 Two identical balls each having a charge of 200 times 10minus7C and a mass of 100 g are suspended from a common

point by two insulating strings each 50 cm long The balls are held at a separation 50 cm apart and then released Find

(a) the electric force on one of the charged balls (b) the components of the resultant force on it along and perpendicular

to the string (c) the tension in the string (d) the acceleration of one of the balls Answers are to be obtained only for the

instant just after the release

Solution

Charge on each ball q1 = q2 = q = 2 x 10-7 C

Mass of each ball m1 = m2 = m = 100 g = 01 kg

Length of each string L = 50 cm = 05 m

Initial separation between the balls r = 5 cm = 005 m

Gravitational force on the ball Fg = mg = 100 x 10-3 x 98 = 098 N

(a) Electric force between the balls Fe = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N

From the diagram sin 120579 = 25

50=

1

20and cos θ asymp 1


Solution

(b) Component of force along the string F1 = 1198981199072

119897=

119898(0)

119897= 0 [the speed of the ball is zero just after release]

component of force perpendicular to string F2 = Fe cos θ ndash Fg sin θ = 0144 (1) ndash 098 (005) = 0095 N

(c) Tension in the string T = Fe sin θ + Fg cos 120579 = 0144 x 005 + 098 399

400= 0986 N

(d) Acceleration of the ball a = 1198652

119898=

0095

01= 095 ms2


23 Two identical pith balls are charged by rubbing against each other They are suspended from a horizontal rod

through two strings of length 20 cm each the separation between the suspension points being 5 cm In equilibrium the

separation between the balls is 3 cm Find the mass of each ball and the tension in the strings The charge on each ball

has a magnitude 20 times 10minus8C

Solution

Charge on each ball q = 2 x 10-8 C

Gravitational force (weight) on each ball Fg = mg

Electrostatic force on each ball Fe = 1

41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N

For equilibrium of the ball T cos θ = mg ---- (1)

For equilibrium of the ball T sin θ = Fe ---- (2)

From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm

(for small angles tan θ = sin θ)

Tension in the string T = 1198651198902 + 119865119892

2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N


24 Two small spheres each having a mass of 20 g are suspended from a common point by two insulating strings of

length 40 cm each The spheres are identically charged and the separation between the balls at equilibrium is found to

be 4 cm Find the charge on each sphere

Solution

Mass of each sphere m1 = m2 = m = 20 gm = 20 x 10-3 kg

Length of each string L1 = L2 = L = 40 cm = 04 m

Charge on each sphere q1 = q2 = q

Separation between the spheres at equilibrium r = 4 cm = 004 m

Gravitational force on each sphere Fg = mg = 002 x 98 = 0196 N

Electrostatic force on each sphere Fe = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2

For equilibrium of the system T cos θ = Fg and T sin θ = Fe rArr tan θ = 119865119890

119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C


25 Two identical pith balls each carrying a charge q are suspended from a common point by two strings of equal

length l Find the mass of each ball if the angle between the strings is 2θ in equilibrium

Solution

Let the mass of each ball be lsquomrsquo

Gravitational force on the ball Fg = mg

Electrostatic force on the ball Fe = 1

41205871205760

11990211199022

1199032 = 1

41205871205760

1199022

1199032 [r is separation between the balls at equilibrium]


119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)

From the diagram sin θ = Τ119903

2

119897rArr r = 2119897 sin θ ---- (2)

From (1) amp (2) tan θ = 1

41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637


26 A particle having a charge of 20 times 10minus4C is placed directly below and at separation of 10 cm from the bob of a

simple pendulum at rest The mass of the bob is 100 g What charge should the bob be given so that the string becomes

loose

Solution

Charge on the particle q = 20 x 10-4 C

Distance of the particle from the bob d = 10 cm = 01 m

Mass of the bob m = 100 g = 01 kg

Let the charge on the bob be lsquoQrsquo

The string becomes loose when the weight of the bob is balanced by the electrostatic force of repulsion

Weight of the bob Fg = mg = 01 x 98 = 098 N

Electrostatic force of repulsion Fe = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q

For equilibrium Fg = Fe rArr 098 = 18 x 107Q rArr Q = 54 x 10-9 C


27 Two particles A and B having charges q and 2q respectively are placed on a smooth table with a separation d A

third particle C is to be clamped on the table in such a way that the particles A and B remain at rest on the table under

electrical forces What should be the charge on C and where should it be clamped

Solution

Given the particle at C is clamped to the table

Particles A and B should remain at rest on the table under electrostatic forces

Resultant electrostatic force on particle at A FA = 1

41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092

For the particle at A to be at rest FA = 0 rArr 1

41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)

Resultant electrostatic force on particle at B FB = 1

41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2

For the particle at B to be at rest FB = 0 rArr 1

41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)

From (1) amp (2) minus119876

1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876

1198892 2minus12 = minusq (6 ndash 4 120784 )


28 Two identically charged particle are fastened to the two ends of a spring of spring constant 100 N mminus1 and natural

length 10 cm The system rests on a smooth horizontal table If the charge on each particle is 20 times 10minus8C find the

extension in the length of the spring Assume that the extension is small as compared to the natural length Justify this

assumption after you solve the problem

Solution

Force constant of the spring k = 100 Nm

Natural length of the spring L = 10 cm = 01 m

Charge on each particle q = 2 x 10-8 C

Let the extension in the spring be lsquoxrsquo

Restoring force in the sprint Fs = kx = 100x

Electrostatic force of repulsion between the charges Fe = 1

41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909

For equilibrium of the system 100x = 36 119909 10minus7

001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m

(as x is negligibly small x2 term is neglected in the simplification process)


29 A particle A having a charge of 20 times 10minus6C is held fixed on a horizontal table A second charged particle of mass

80 g stays in equilibrium on the table at a distance of 10 cm from the first charge The coefficient of friction between

the table and this second particle is μ = 02 Find the range within which the charge of this second particle may lie

Solution

Charge of the fixed particle q1 = 2 x 10-6 C

Let the charge on the second particle be lsquoqrsquo

Mass of the second particle m = 80 gm = 80 x 10-3 kg


Coefficient of friction between the particle and the table surface μ = 02

Electric force between the particles F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N

Force of friction between the particle and surface f = μN = μmg = 02 x 80 x 10-3 x 98 = 01568 N

For equilibrium of the particle f = F rArr 01568 = 18 x 105q rArr q = 871 x 10-8 C

The same calculation can be repeated when the second particle is negative

So the range of charges for equilibrium plusmn 871 x 10-8 C


30 A particle A having a charge of 20 times 10minus6C and a mass of 100 g is placed at the bottom of a smooth inclined plane

of inclination 30o Where should another particle B having same charge and mass be placed on the incline so that it

may remain in equilibrium

Solution

Charge on the particle A qA = 2 x 10-6 C

Mass of the particle A mA = 100 g = 01 kg

Inclination of the plane θ = 300

Given mass and charge of the second particle are same as that of the first one

For equilibrium of the second particle mg sin θ = 1

41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm


31 Two particles A and B each having a charge Q are placed a distance d apart Where should a particle of charge q

be placed on the perpendicular bisector of AB so that it experiences maximum force What is the magnitude of this

maximum force

Solution

Charge on both particles is same qA = qB = Q and Charge on the third particle q

Separation between the particles d

Let the third particle be placed at a distance y on the perpendicular bisector

Net force on the particle F = 2Fe cos 180minus2120579

2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32

Apply maxima-minima condition 119889119865

119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784

Magnitude of maximum force Fmax = 119876119902

21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784


32 Two particles A and B each carrying a charge Q are held fixed with a separation d between them A particle C

having mass m and charge q is kept at the middle point of the line AB (a) if it is displaced through a distance x

perpendicular to AB what would be the electric force experienced by it (b) assuming xltltd show that this force is

proportional to x (c) under what conditions will the particle C execute simple harmonic motion if it is released after

such a small displacement Find the time period of the oscillations if these conditions are satisfied

Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4

and Force between B amp C FBC = 1

41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4

(a) Resultant force F = 2FAC cos θ = 2 1

41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784

(b) For x ltlt d F = Qqx

2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]

Angular frequency ω = 4119876119902

π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784


33 Repeat the previous problem if the particle C is displaced through a distance x along the line AB

Solution

Force between A amp C FAC = = 1

41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909

2 and Force between B amp C FBC = 1

41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2

Resultant force F = FBC ndash FAC = 1

41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]

Angular frequency of oscillation ω = 119876119902

1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784


34 The electric force experienced by a charge of 10 times 10minus6 C is 15 times 10minus3N Find the magnitude of the electric field

at the position of the charge

Solution


Force experienced by the particle F = 15 x 10-3 N

Magnitude of electric field E = 119865

119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC


35 Two particles A and B having charges of +200 times 10minus6C and of minus400 times 10minus6C respectively are held fixed at a

separation of 200 cm Locate the point(s) on the line AB where (a) the electric field is zero (b) the electric potential is

zero

Solution

Charge on the first particle qA = 2 x 10-6 C and Charge on the second particle qB = 4 x 10-6 C


Since the particles are of opposite sign zero field is not possible between the charges

(a) Let us consider a point P on the line AB outside the charges at a distance x from A

Field at P due to positive charge EA = 1

41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000

1199092 (away from A)

Field at P due to negative charge EB = 1

41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)

Net field at P = 0 rArr EA ndash EB = 0 rArr EA = EB rArr 18000

1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution

(b) For zero potential consider a point P at a distance x from A between the charges

Potential at P due to A VA = 1

41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909

Potential at P due to B VB = 1

41205871205760

119902119861

20minus119909= 9 x 109 x

minus4 119909 10minus6

20minus119909=

minus36000

20minus119909

Net potential at P V = VA + VB = 18000

119909minus

36000

20minus119909= 0 rArr

18000

119909=

36000

20minus119909rArr

1

119909=

2

20minus119909rArr 2x = 20 minus 119909 rArr x =

120784120782

120785cm

Consider a point P at a distance x from A outside the two charges


41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm


36 A point charge produces an electric field of magnitude 50 N Cminus1 at a distance of 40 cm from it What is the

magnitude of the charge

Solution

Magnitude of electric field E = 5 NC

Distance of point r = 40 cm = 04 m


41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C

37 A water particle of mass 100 mg and having a charge of 150 times 10minus6C stays suspended in a room What is the

magnitude of electric field in the room What is its direction

Solution

Mass of water particle m = 10 mg = 10 x 10-6 kg


Gravitational force on the water particle Fg = mg = 10-5 x 98 N (downward)

Electrostatic force on the particle Fe = Eq = E (15 x 10-6) (upward)

For equilibrium Fg = Fe rArr 98 x 10-5 = 15 x 10-6 E rArr E = 98

15= 653 NC


38 Three identical charges each having a value 10 times 10minus8C are placed at the corners of an equilateral triangle of

side 20 cm Find the electric field and potential at the centre of the triangle

Solution

Charge on each particle q1 = q2 = q3 = q = 1 x 10-8 C

Side of the equilateral triangle a = 20 cm = 02 m

Distance of center of the triangle from vertex r = 119886

3=

02

3m

Electric field at the center of the triangle due to one particle E = 1

41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC

Electric field at the center of the triangle due to all particles E1 = 0 (triangle law)

Electric potential at the center of the triangle due to one particle V = 1

41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt

Electric potential at the center of the triangle due to all particles V1 = 3V = 3 x 10 3

2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt


39 Positive charge Q is distributed uniformly over a circular ring of radius R A particle having a mass m and a

negative charge q is placed on its axis at a distance x from the centre Find the force on the particle Assuming xltltR

find the time period of oscillation of the particle if it is released from there

Solution

Electric field intensity on the axis of a charged ring E = 1

41205871205760

119876119909

1198772+1199092 32

Force on the particle placed on the axis of the ring F = Eq = 1

41205871205760

119876119902119909

1198772+1199092 32

Force on the particle for x ltlt R F = 1

41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902

412058712057601198981198773 (x) [compare with a = ω2x]

ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954


40 A rod of length L has a total charge Q distributed uniformly along its length It is bent in the shape of a semicircle

Find the magnitude of the electric field at the centre of curvature of the semicircle

Solution

Linear charge density λ = 119876

119871

Consider an elemental length lsquod119897rsquo of the semicircle at an angle θ with the vertical

Charge of this element dq = λ d119897 = 119876

119871d119897

Field intensity due to the element dE = 1

41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032

Field intensity due to the element along the vertical dE cos θ = 1

41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ

Total electric field E = 119889119864 cos 120579 = 1

41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876

119871119903sin θ limits minus

120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784


41 A 10-cm long rod carries a charge of +50 μC distributed uniformly along its length Find the magnitude of the

electric field at a point 10 cm from both the ends of the rod

Solution

Consider a rod of length L and charge q uniformly distributed on it

Linear charge density of the rod λ = 119902

119871

Consider a point at a distance d from the mid point of the rod at which we

need to find electric field intensity

Take an elemental length lsquodyrsquo of the rod at a distance lsquoyrsquo from the mid point

Charge of this element lsquodqrsquo (point charge) = λ dy = 119902

119871dy

Field intensity due to elemental charge dE = 1

41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy

Resultant field intensity due to entire rod E = 119889119864 cos 120579 = 1

41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32

Put y = d tan θ rArr dy = d sec2 θ dθ and d2 + y2 = d2 + d2 tan2 θ = d2 sec2 θ

E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910

1198892+1199102[limits minusL2 to L2]

E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC


42 Consider a uniformly charged ring of radius R Find the point on the axis where the electric field is maximum

Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784


43 A wire is bent in the form of a regular hexagon and a total charge q is distributed uniformly on it What is the

electric field at the centre You may answer this part without making any numerical calculations

Solution

Electric field intensity on the perpendicular bisector of a charge wire E = 1

41205871205760

119902

119889

1

1198892+1198712

4

[directed away from the wire along the bisector]

In the case of regular hexagon all the six electric fields at the center have equal magnitudes and separated by 600 as

shown in figure

From polygon law of vectors the resultant field is zero


44 A circular wire-loop of radius lsquoarsquo carries a total charge Q distributed uniformly over its length A small length dL of

the wire is cut off Find the electric field at the centre due to the remaining wire

Solution

Linear charge density of the wire λ = 119902

119897=

119876

2120587119886

Charge on small length of the wire dQ = λ dL = 119876

2120587119886dL

The electric field generated by any small part of the wire at the center is cancelled by diametrically opposite part

When dL part of the wire is cut off the field created by the diametrically opposite part remains

Field intensity at the center due to elemental charge dE = 1

41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785


45 A positive charge q is placed in front of a conducting solid cube at a distance d from its centre Find the electric

field at the centre of the cube due to the charges appearing on its surface

Solution

Since the cube is made of conducting material the electric field inside it is zero

Electric field due to point charge E = 120783

120786120645120634120782

119954

119941120784

Electric field due to induced charge Ei

Net electric field at the center E = E + Ei rArr 0 = E + Ei rArr Ei = minusE

So the field due to induced charge is equal in magnitude to that of point charge and in opposite direction of the field

due to point charge

The field is towards the point charge


46 A pendulum bob of mass 80 mg and carrying a charge of 2 times 10minus8C is at rest in a uniform horizontal electric field

of 20 kVmminus1 Find the tension in the thread

Solution

Mass of the bob m = 80 mg = 80 x 10-6 kg

Charge on the bob q = 2 x 10-8 C

Strength of electric field E = 20 kVm = 2 x 104 Vm

Gravitational force on the bob Fg = mg = 80 x 10-6 x 98 = 784 x 10-4 N

Electrostatic force on the bob Fe = Eq = 2 x 104 x 2 x 10-8 = 4 x 10-4 N

Tension in the thread T = 1198651198902 + 119865119892

2 = 10-4 42 + 7842 = 88 x 10-4 N


47 A particle of mass m and charge q is thrown at a speed u against a uniform electric field E How much distance will

it travel before coming to momentary rest

Solution

Mass of the particle m

Charge on the particle q

Initial speed of the particle u

Final speed of the particle v = 0

Strength of electric field E

Force on the particle F = Eq

Acceleration of the particle a = 119865

119898=

119864119902

119898

Distance travelled by the particle before momentarily coming to rest d = 1199062

2119886=

119950119958120784

120784119916119954


48 A particle of mass 1 g and charge 25 times 10minus4C is released from rest in an electric field of 12 times 104 N Cminus1 (a) find

the electric force and the force of gravity acting on this particle Can one of these forces be neglected in comparison

with the other for approximate analysis (b) How long will it take for the particle to travel a distance of 40 cm

(c) what will be the speed of the particle after travelling this distance (d) how much is the work done by the electric

force on the particle during this period

Solution

Mass of the particle m = 1 g = 10-3 kg and Charge on the particle q = 25 x 10-4 C

Strength of the electric field E = 12 x 104 NC

(a) Electric force on the particle Fe= Eq = 12 x 104 x 25 x 10-4 = 3 N

Gravitational force on the particle Fg = mg = 10-3 x 98 = 98 x 10-3 N

Gravitational force can be neglected as we can see from the numerical values above


119898=

3

0001= 3 x 103 ms2

(b) Time taken by the particle to travel 40 cm t = 2119904

119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec

(c) Speed of the particle v2 ndash u2 = 2as rArr v2 ndash 0 = 2 x 3 x 103 x 04 rArr v = 2400 = 49 ms

(d) Work done by electric force on the particle w = Fd = 3 x 04 = 12 J


49 A ball of mass 100 g and having a charge of 49 times 10minus5C is released from rest in a region where a horizontal

electric field of 20 times 104 N Cminus1 exists (a) find the resultant force acting on the ball (b) what will be the path of the

ball (c) where will the ball be at the end of 2 s

Solution

Mass of the ball m = 100 g = 01 kg

Charge on the ball q = 49 x 10-5 C

Strength of electric field E = 2 x 104 NC [horizontal]

Electrical force on the ball Fe = Eq = 2 x 104 x 49 x 10-5 = 098 N

Gravitational force on the ball Fg = mg = 01 x 98 = 098 N

(a) Resultant force on the ball F = 1198651198902 + 119865119892

2 = (098)2+(098)2 = 098 2 = 14 N (450 to the horizontal)

(b) The particle follows straight line path along the resultant direction

(c) Displacement of the particle s = ut + 1

2at2 = 0 +

1

2

119865

119898t2 = 05 x

14

01x 4 = 28 m along the straight line


50 The bob of a simple pendulum has a mass of 40 g and a positive charge of 40 times 10minus6C It makes 20 oscillations in

45 s A vertical electric field pointing upward and of magnitude 25 times 104 N Cminus1 is switched on How much time will

it now take to complete 20 oscillations

Solution

Time period of oscillation of the bob T1 = 45

20= 225 sec



Strength of electric field E = 25 x 104 NC

Effective acceleration of the bob geff = g ndash a = g minus119864119902

119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2

Ratio of time periods 1198791

1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec

Time taken for 20 oscillations t = 20T = 20 x 26 = 52 sec


51 A block mass m having a charge q is placed on a smooth horizontal table and is connected to a wall through an

unstressed spring of spring constant k as shown in figure A horizontal electric field E parallel to the spring is switched

on Find the amplitude of the resulting SHM of the block

Solution

Electrostatic force on the block Fe = Eq (towards right)

Restoring force in the spring Fs = kx (towards mean position)

Let the amplitude of oscillation be lsquoArsquo

At the extreme position kA = Eq rArr A = 119916119954

119948


52 A block of mass m containing a net positive charge q is placed on a smooth horizontal table which terminates in a

vertical wall as shown in figure The distance of the block from the wall is d A horizontal electric field E towards right

is switched on Assuming elastic collisions (if any) find the time period of the resulting oscillatory motion Is it a

simple harmonic motion

Solution

Charge on the block q

Mass of the block m

Electric field intensity E

Force on the block F = Eq

Acceleration of the block a = 119865

119898=

119864119902

119898

Time taken to travel upto the wall s = ut + 1

2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902

Time taken for round trip time period T = 2t = 22119889119898

119864119902=

120790119941119950

119916119954

since the acceleration is constant the oscillations are not simple harmonic


53 A uniform electric field of 10 N Cminus1 exists in the vertically down ward direction Find the increase in the electric

potential as one goes up through a height of 50 cm

Solution

Strength of the electric field E = 10 NC [vertically downward]

Distance travelled d = 50 cm = 05 m

Increase in potential ∆V = Ed = 10 x 05 = 5 volt

54 12 J work has to be done against an existing electric field to take a charge of 001 C from A to B How much is the

potential difference 119881119861 minus 119881119860

Solution

Charge on the particle q = 001 C

Work done against electric field w = 12 J

Since work is done against electric field B is at higher potential than A

Work done w = VBAq = VBA (001) = 12 rArr VB ndash VA = 1200 volt


55 Two equal charges 20 times 10minus7C each are held fixed at a separation of 20 cm A third charge of equal magnitude is

placed midway between the two charges It is now moved to a point 20 cm from both the charges How much work is

done by the electric field during the process

Solution



Charge on the third particle q3 = 2 x 10-7 C

Initial potential energy of the system Ui = 1

41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J

Final potential energy of the system Uf = 1

41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J

Work done by electric field w = Ui ndash Uf = (9 ndash 54) x 10-3 = 36 x 10-3 J


56 An electric field of 20 N Cminus1 exists along the x-axis in space Calculate the potential difference VB minus VA where the

points A and B are given by

(a) A = (0 0) B = (4 m 2 m) (b) A = (4 m 2 m) B = (6 m 5 m) (c) A = (0 0) B = (6 m 5 m)

Do you find any relation between the answers of parts (a) (b) and (c)

Solution

Electric field intensity E = 20 NC along x axis

Relation between potential difference and field intensity E = minus119889119881

119889119909rArr dV = minus E dx rArr 1198811

1198812 119889119881 = minus 1199091

1199092 119864 119889119909

(a) Potential difference 119860

119861119889119881 = minus 1199091

1199092 119864 119889119909 rArr VBminusVA = minus20x [limits 0 to 4] = minus(80 ndash 0) = minus80 volt

(b) Potential difference 119860

119861119889119881 = minus 1199091


(c) Potential difference 119860

119861119889119881 = minus 1199091



57 Consider the situation of the previous problem A charge of minus20 times 10minus4C is moved from the point A to the point

B Find the change in electrical potential energy UB minus UA for the cases (a) (b) and (c)

Solution

Change in potential energy ∆119880 = UB ndash UA = qVBA = q(VB ndash VA)

(a) Change in potential energy in first case UB ndash UA = minus20 times 10minus4 x minus80 = 0016 J

(b) Change in potential energy in second case UB ndash UA = minus20 times 10minus4 x minus40 = 0008 J

(c) Change in potential energy in third case UB ndash UA = minus20 times 10minus4 x minus120 = 0024 J


58 An electric field E = Ԧi20 + Ԧj30 N Cminus1 exists in the space If the potential at the origin is taken to be zero find the

potential at (2 m 2 m)

Solution

Electric field intensity in the region E = Ԧi20 + Ԧj30 N Cminus1

Potential at the origin V0 = 0 volt

Relation between potential difference and electric field intensity E = minus119889119881

119889119903rArr dV = minus ത119864 d ҧ119903

dV = minus 20 Ƹ119894 + 30 Ƹ119895 (119889119909 Ƹ119894 + 119889119910 Ƹ119895) = minus (20 dx + 30 dy)

119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910

(0 ndash V) = minus20x (limits 2 to 0) minus 30y (limits 2 to 0)

V = 20 (0 ndash 2) + 30 (0 ndash 2)

V = minus40minus60 = minus100 volt


59 An electric field E = Ԧ119894Ax exists in the space where A = 10 V mminus2 Take the potential at (10 m 20 m) to be zero

Find the potential at the origin

Solution

The electric field intensity in the given region E = 119860119909Ԧ119894

Potential at (10 m 20 m) V = 0 volt


119889119903

Since the field is changing only along x axis E = minus119889119881

119889119909rArr dV = minusE dx

0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt


60 The electric potential existing in space is V(x y z) = A(xy + yz + zx) (a) write the dimensional formula of A

(b) Find the expression for the electric field (c) If A is 10 SI units find the magnitude of the electric field at

(1 m 1 m 1 m)

Solution

The electric potential existing in space is V(x y z) = A(xy + yz + zx)

Here x y and z are distances (dimensional formula L)

Dimensional formula of potential V = 119908

119902=

1198721198712119879minus2

119868119879= ML2T-3I-1

(a) So the dimensional formula of A ML2Tminus3Iminus1

1198712 = MT-3I-1

(b) Electric field intensity E = minus120597119881

120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881

120597119911119896 = minus119860(y + 0 + z) Ƹ119894 minus119860 (x + z + 0) 119895 minus119860 (0 + y + x) 119896

E = minus119912 [(y + z) Ƹ119946 minus119912 (x + z) 119947 minus119912 (y + x) 119948]

(c) Magnitude of electric field E = 10 [(1 + 1) Ƹ119894 + (1 + 1) 119895 + (1 + 1) 119896] = 20 ( Ƹ119894 + 119895 + 119896) = 20 3 = 346 NC


61 Two charged particles having equal charges of 20 times 10minus5C each are brought from infinity to within a separation

of 10 cm Find the increase in the electric potential energy during the process

Solution


Initial separation between the particles ri = infinity

Final separation between the particles rf = 10 cm = 01 m


41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J

Increase in potential energy of the system ∆U = Uf ndash Ui = 36 ndash 0 = 36 J


62 Some equipotential surfaces are shown in figure What can you say about the magnitude and the direction of the

electric field

Solution

(a) Magnitude of electric field E = ∆119881

∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm

Electric field is directed from high potential to low potential and perpendicular to the equi-potentials 1200

(b) The field is directed radially outward (in the direction of decreasing potential)

From the diagram potential at any point V = 6

119903 (119888119898)

Electric field intensity E = minus119889119881

119889119903= minus minus

6

1199032 = 120788

119955120784 Vcm


63 Consider a circular ring of radius r uniformly charged with linear charge density λ Find the electric potential at a

point on the axis at a distance x from the centre of the ring Using this expression for the potential find the electric

field at this point

Solution

Radius of the ring r

Linear charge density λ

Consider an elemental part (point charge) of the ring of length d119897 (charge dq = λ d119897)

Potential due to elemental ring on the axis dV = 119896 119889119902

119877=

119896λ 119889119897

1199092+1199032

Total potential due to the ring V = 119889119881 = 119896λ

1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784

Electric field at the given point E = minus119889119881

119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784


64 An electric field of magnitude 1000 N Cminus1 is produced between two parallel plates having a separation of 20 cm

as shown in figure (a) what is the potential difference between the plates (b) with what minimum speed should an

electron be projected from the lower plate in the direction of the field so that it may reach the upper plate (c) suppose

the electron is projected from the lower plate with the speed calculated in part (d) the direction of projection makes an

angle of 60o with the field Find the maximum height reached by the electron

Solution

Magnitude of the electric field E = 1000 NC

Separation between the parallel plates d = 2 cm = 2 x 10-2 m

(a) Potential difference between the plates ∆V = Ed = 1000 x 2 x 10-2 = 20 volt

(b) Work done by electric field on the electron w = minusEqd = minus1000 x 16 x 10-19 x 2 x 10-2 = minus32 x 10-18 J

change in kinetic energy of the electron ∆k = kf ndash ki = 0 minus1

2mv2 = minus05 x 911 x 10-31 v2 = minus455 x 10-31 v2

work energy theorem w = ∆k rArr minus32 x 10-18 = minus455 x 10-31 v2 rArr v = 7 119909 1012 = 265 x 106 ms

(c) Acceleration provided by electric field a = 119865

119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms

Maximum height reached by the electron H = 11990721199041198941198992120579

2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm


65 A uniform field of 20 N Cminus1 exists in space in x-direction (a) taking the potential at the origin to be zero write an

expression for the potential at a general point (x y z) (b) at which points the potential is 25 V (c) if the potential at

the origin is taken to be 100 V what will be the expression for the potential at a general point (d) what will be the

potential at a general point (d) what will be the potential at the origin if the potential at infinity is taken to be zero Is

it practical to chose the potential at infinity to be zero

Solution


Given potential at the origin is zero

Relation between potential and field intensity E = minus∆119881

∆119909rArr V ndash 0 = minusE ∆119909 rArr V = minus2 (Vm) x

The points at which potential is 25 V 25 = minus2x rArr x = minus125 m

Given potential at the origin is 100 volt

Potential at any point V ndash 100 = minus2x rArr V = 100 ndash 2x

Given potential at infinity is zero

Potential at the origin 2 = 0minus119881

infinrArr V = infinity


66 How much work has to be done in assembling three charged particles at the vertices of an equilateral triangle as

shown in figure

Solution

Initially the particles are at infinite separation potential energy is zero Ui = 0

Potential energy in the final configuration Uf = 1

41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J

Work done in assembling the charges w = Uf ndash Ui = 234 ndash 0 = 234 J


67 The kinetic energy of a charged particle decreases by 10 J as it moves from a point at potential 100 V to a point at

potential 200 V Find the charge on the particle

Solution

Potential of initial position V1 = 100 volt

Potential of final position V2 = 200 volt

Decrease in kinetic energy of the particle ∆k =10 J

Work done by the field w = Fd = Eqd = ∆119881

119889qd =

200minus100

119889qd = 100q

Work energy theorem w = ∆k rArr 10 = 100q rArr q = 01 C


68 Two identical particles each having a charge of 20 times 10minus4C and mass of 10 g are kept a separation of 10 cm and

then released What would be the speeds of the particles when the separation becomes large

Solution


Mass of each particle m1 = m2 = m = 10 g = 10 x 10-3 kg

Initial separation between the particles r1 = 10 cm

Final separation between the particles r2 = large (infin)

Initial potential energy of the system U1 = 1

41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J

Final potential energy of the system U2 = 1

41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J

Initial kinetic energy of the system k1 = 0 J

Final kinetic energy of the system k2 = 2 1

2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2

Conservation of energy U1 + k1 = U2 + k2 rArr 3600 + 0 = 0 + 10-2 v2 rArr v = 600 ms


69 Two particles have equal masses of 50 g each and opposite charges of +40 times 10minus5C and minus40 times 10minus5C They are

released from rest with a separation of 10 m between them Find the speeds of the particles when the separation is

reduced to 50 cm

Solution



Charge on the second particle q2 = minus40 times 10minus5 C

Initial separation between the charges r1 = 1 m = 100 cm

Final separation between the charges r2 = 50 cm


41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2

Conservation of energy U1 + k1 = U2 + k2 rArr minus144 + 0 = minus288 + 5 x 10-3 v2 rArr v = 144 119909 103

5= 536 ms


70 A sample of HCI gas is placed in an electric field of 25 times 104 N Cminus1 The dipole moment of each HCI molecule is

34 times 10minus30Cm Find the maximum torque that can act on a molecule

Solution

Electric field intensity E = 25 x 104 NC

Dipole moment of the molecule P = 34 x 10-30 Cm

Torque on the dipole τ = PE sin θ

Maximum torque on the dipole τmax = PE = 34 x 10-30 x 25 x 104 = 85 x 10-26 Nm


71 Two particles A and B having opposite charges 20 times 10minus6C and minus20 times 10minus6C are placed at a separation of

10 cm (a) write down the electric dipole moment of this pair (b) calculate the electric field at a point on the axis of

the dipole 10 m away from the centre (c) calculate the electric field at a point on the perpendicular bisector of the

dipole and 10 m away from the centre

Solution

Charge on the first particle qA= 2 x 10-6 C

Charge on the second particle qB = minus20 times 10minus6 C


(a) Electric dipole moment of the combination P = q(d) = 2 x 10-6 x 001 = 2 x 10-8 Cm

(b) Electric field intensity on the axis of the dipole Eax = 1

41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC

(c) Electric field intensity on the equatorial line of the dipole Eeq = 1

41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC


72 Three charges are arranged on the vertices of an equilateral triangle as shown in figure Find the dipole moment of

the combination

Solution

Dipole moment of AB PAB = qd (from A to B)

Dipole moment of BC PBC = qd (from C to B)

Angle between the two moments 600

Resultant dipole moment P = 2PAB cos 300 = 2qd 3

2= 120785 qd (along the angular bisector)


73 Find the amplitude of the electric field at the point P in the configuration shown in figure for dgtgta Take 2qa = p

Solution

(a) Electric field intensity due to point charge E = 120783

120786120645120634120782

119954

119941120784

(b) Electric field intensity on the equatorial line of a dipole Eeq = 120783

120786120645120634120782

119927

119941120785 (d gtgt a)

(c) Field due to middle positive charge E1 = 1

41205871205760

119902

1198892 (vertically up)

field due to dipole E2 = 1

41205871205760

119875

1198893 (horizontal from +ve charge to negative charge left)

resultant field E = 11986412 + 1198642

2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784


74 Two particles carrying charges ndashq and +q and having equal masses m each are fixed at the ends of a light rod of

length lsquoarsquo to form a dipole The rod is clamped at an end and is placed in a uniform electric field E with the axis of the

dipole along the electric field The rod is slightly titled and then released Neglecting gravity find the time period of

small oscillations

Solution

Let the rod be clamped at the end of negative charge

Force acting on the positive charge F = Eq (in the direction of the field)

Let the angular displacement of the rod be lsquoθrsquo

Torque on the rod due to electrostatic force τ = F (perpendicular distance)

τ = Eq (a sin θ) = Eqa θ [for small angles sin θ = θ]

Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]

Angular frequency of the system ω = 119864119902

119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954


75 Assume that each atom in a copper wire contributes one free electron Estimate the number of free electrons in a

copper wire having a mass of 64 g (take the atomic weight of copper to be 64 g molminus1)

Solution

Mass of copper wire m = 64 g

Atomic weight of copper M = 64 gmol

Number of moles n = 119898

119872=

64

64= 01

Number of copper atoms N = nNA = 01 x 6023 x 1023 = 6023 x 1022

Given Each copper atoms contributes one free electron

Number of free electrons Ne = 6 x 1022


1 Find the dimensional formula of 1205760

Solution

Electrostatic force between two point charges F = 1

41205871205760

11990211199022

1199032 rArr 1205760 = 11990211199022

41205871198651199032

Dimensional formula of charge q = IT

Dimensional formula of force F = MLT-2

Dimensional formula of distance r = L

Dimensional formula of permittivity of free space 1205760 = (119868119879) (119868119879)

119872119871119879minus2119871= M-1L-2T4I2





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 1 119909 1

4 119909 106 = 225 x 103 N


of a 50 kg person

Solution




41205871205760

11990211199022

1199032 rArr 490 = 9 x 109 1 119909 1

1199032 rArr r = 9 119909 109

490=

3

7x 104 = 43 x 103 m




Solution





41205871205760

11990211199022

1199032 rArr 490 = 9 x 109 1199022

1rArr q =

490

9 119909 109 = 7

3 119909 104 = 23 x 10-4 C



Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus30 = 9 x 1039 x 256 x 10-38 = 2304 N




Solution






41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

1199092 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



1199092 = 9 x 109 1 119909 10minus6 (119902)

(10 minus 119909)2

2 119909 10minus6 (119902)

1199092 = 1 119909 10minus6

(10 minus 119909)2 rArr 2

1199092 = 1

(10 minus 119909)2 rArr 2

119909=

1


10 2

1 + 2= 59 cm




Solution




41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

(10+119909)2 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



(10+119909)2 = 9 x 109 1 119909 10minus6 (119902)

(119909)2

2 119909 10minus6 (119902)

(10+119909)2 = 1 119909 10minus6 (119902)

(119909)2 rArr 2

(10+119909)2 = 1

(119909)2 rArr 2

10 + 119909=

1

119909rArr 2 minus 1 119909 = 10 rArr x =

10

2minus1=

10

0414= 241 cm





Solution




F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus4 = 9 x 1013 x 256 x 10-38 = 2304 x 10-25 N = 23 x 10-24 N



Solution




119872NA =

100

18x 6023 x 1023 = 335 x 1024








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 1 119909 1

4 119909 106 = 225 x 103 N


of a 50 kg person

Solution




41205871205760

11990211199022

1199032 rArr 490 = 9 x 109 1 119909 1

1199032 rArr r = 9 119909 109

490=

3

7x 104 = 43 x 103 m




Solution





41205871205760

11990211199022

1199032 rArr 490 = 9 x 109 1199022

1rArr q =

490

9 119909 109 = 7

3 119909 104 = 23 x 10-4 C



Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus30 = 9 x 1039 x 256 x 10-38 = 2304 N




Solution






41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

1199092 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



1199092 = 9 x 109 1 119909 10minus6 (119902)

(10 minus 119909)2

2 119909 10minus6 (119902)

1199092 = 1 119909 10minus6

(10 minus 119909)2 rArr 2

1199092 = 1

(10 minus 119909)2 rArr 2

119909=

1


10 2

1 + 2= 59 cm




Solution




41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

(10+119909)2 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



(10+119909)2 = 9 x 109 1 119909 10minus6 (119902)

(119909)2

2 119909 10minus6 (119902)

(10+119909)2 = 1 119909 10minus6 (119902)

(119909)2 rArr 2

(10+119909)2 = 1

(119909)2 rArr 2

10 + 119909=

1

119909rArr 2 minus 1 119909 = 10 rArr x =

10

2minus1=

10

0414= 241 cm





Solution




F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus4 = 9 x 1013 x 256 x 10-38 = 2304 x 10-25 N = 23 x 10-24 N



Solution




119872NA =

100

18x 6023 x 1023 = 335 x 1024








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution





41205871205760

11990211199022

1199032 rArr 490 = 9 x 109 1199022

1rArr q =

490

9 119909 109 = 7

3 119909 104 = 23 x 10-4 C



Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus30 = 9 x 1039 x 256 x 10-38 = 2304 N




Solution






41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

1199092 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



1199092 = 9 x 109 1 119909 10minus6 (119902)

(10 minus 119909)2

2 119909 10minus6 (119902)

1199092 = 1 119909 10minus6

(10 minus 119909)2 rArr 2

1199092 = 1

(10 minus 119909)2 rArr 2

119909=

1


10 2

1 + 2= 59 cm




Solution




41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

(10+119909)2 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



(10+119909)2 = 9 x 109 1 119909 10minus6 (119902)

(119909)2

2 119909 10minus6 (119902)

(10+119909)2 = 1 119909 10minus6 (119902)

(119909)2 rArr 2

(10+119909)2 = 1

(119909)2 rArr 2

10 + 119909=

1

119909rArr 2 minus 1 119909 = 10 rArr x =

10

2minus1=

10

0414= 241 cm





Solution




F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus4 = 9 x 1013 x 256 x 10-38 = 2304 x 10-25 N = 23 x 10-24 N



Solution




119872NA =

100

18x 6023 x 1023 = 335 x 1024








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution






41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

1199092 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



1199092 = 9 x 109 1 119909 10minus6 (119902)

(10 minus 119909)2

2 119909 10minus6 (119902)

1199092 = 1 119909 10minus6

(10 minus 119909)2 rArr 2

1199092 = 1

(10 minus 119909)2 rArr 2

119909=

1


10 2

1 + 2= 59 cm




Solution




41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

(10+119909)2 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



(10+119909)2 = 9 x 109 1 119909 10minus6 (119902)

(119909)2

2 119909 10minus6 (119902)

(10+119909)2 = 1 119909 10minus6 (119902)

(119909)2 rArr 2

(10+119909)2 = 1

(119909)2 rArr 2

10 + 119909=

1

119909rArr 2 minus 1 119909 = 10 rArr x =

10

2minus1=

10

0414= 241 cm





Solution




F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus4 = 9 x 1013 x 256 x 10-38 = 2304 x 10-25 N = 23 x 10-24 N



Solution




119872NA =

100

18x 6023 x 1023 = 335 x 1024








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




41205871205760

1199021119902

1199032 = 9 x 109 2 119909 10minus6 (119902)

(10+119909)2 [repulsive]


41205871205760

1199022119902

1199032 = 9 x 109 1 119909 10minus6 (119902)



(10+119909)2 = 9 x 109 1 119909 10minus6 (119902)

(119909)2

2 119909 10minus6 (119902)

(10+119909)2 = 1 119909 10minus6 (119902)

(119909)2 rArr 2

(10+119909)2 = 1

(119909)2 rArr 2

10 + 119909=

1

119909rArr 2 minus 1 119909 = 10 rArr x =

10

2minus1=

10

0414= 241 cm





Solution




F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus4 = 9 x 1013 x 256 x 10-38 = 2304 x 10-25 N = 23 x 10-24 N



Solution




119872NA =

100

18x 6023 x 1023 = 335 x 1024








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




F = 1

41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

10minus4 = 9 x 1013 x 256 x 10-38 = 2304 x 10-25 N = 23 x 10-24 N



Solution




119872NA =

100

18x 6023 x 1023 = 335 x 1024








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






Solution




119872NA =

100

18x 6023 x 1023 = 335 x 1024








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution





41205871205760

11990211199022

1199032

F = 9 x 109 x 535 119909 106 2

001= 9 x 1011 x 286 x 1012 = 256 x 1025 N




repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







repulsion

Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

138 119909 10minus15 2 = 12 N




Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 1198992 16 119909 10minus19 2

10minus4

01 = 9 x 1013 x 256 x 10-38 n2

n2 = 1024

2304=

100

2304x 1022 rArr n = 2 x 1011





Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

275 119909 10minus10 2 = 305 x 10-9 N



Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

1199032 = 2304 119909 10minus29

1199032


1199032 = 667 119909 10minus11 119909 167 119909 10minus27 2

1199032 = 186 119909 10minus65

1199032


119865119892=

2304 119909 10minus29

186 119909 10minus65 = 123 x 1036






Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01









Solution







41205871205760

1199022

1199032 = 9 119909 109 119909 16 119909 10minus19 2

5 119909 10minus15 2 = 2304

25x 101 = 92 N


minus11 119909 5

5 119909 10minus15 2 = (Ce-5 x 1030 )25


251198905 C rArr C = 92 x 25e5 x 10-30 = 34 x 10-26 Nm2




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N


41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6 2

25 119909 10minus4 = 36 119909 10minus3

25 119909 10minus4 = 144 N





Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution





41205871205760

11990221199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

11990241199023

1199032 = 1

41205871205760

1199022

1199032


41205871205760

1199022



41205871205760

11990211199023

1198772 = 1

41205871205760

1199022

21199032


41205871205760

1199022

21199032 + 21

41205871205760

1199022

1199032 = 1

41205871205760

1199022

1199032 2 +1

2

FC = 9 x 109 x 4 119909 10minus12

25 119909 10minus4 2 +1

2= 275 N






Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01









Solution





41205871205760

11990211199022

1199032 = 9 x 109 x 16 119909 10minus19 2

053 119909 10minus10 2 = 82 x 10-8 N


Solution


1198981199072

119903=

1

41205871205760

11990211199022

1199032 rArr v2 = 119903

119898

1

41205871205760

11990211199022

1199032 = 053 119909 10minus10

911 119909 10minus31 x 82 x 10-8 = 53 119909 82

911x 1012 = 48 x 1012

v = 48 119909 1012 = 218 x 106 ms





Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution





41205871205760

11990201199021


1

41205871205760

119902011990210

119903102

F = 9 x 109 x 10minus8

01 2

13

12 +23

22 +33

32 + ⋯ hellip103

102 = 9000 [1 + 2 + 3 + helliphellip10] =

F = 9000 x 10 119909 (10+1)

2= 9000 x 55 = 495 x 105 N




Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus16

1= 36 x 10-6 N







Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01










Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 4 119909 10minus14

25 119909 10minus4 = 36

25x 10-1 = 0144 N


50=

1



Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01





Solution


119897=

119898(0)




400= 0986 N


119898=

0095

01= 095 ms2






Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01









Solution




41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

9 119909 10minus4 = 4 x 10-3 N



From (1) and (2) tan θ = 119865119890

119898119892rArr

1

20=

4 119909 10minus3

98119898rArr m =

80

98= 82 gm



2 = (4 119909 10minus3)2+(82 119909 10minus3 119909 98)2 = 805 x 10-3 N = 81 x 10-2 N





Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 1199022

16 119909 10minus4 = 9

16x 1013 q2


119865119892

1

20=

9

16x 1013 q2 1

0196rArr q2 = 174 x 10-16 rArr q = 417 x 10-8 C




Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




41205871205760

11990211199022

1199032 = 1

41205871205760

1199022



119865119892=

1

41205871205760

1199022

1199032 119898119892---- (1)


2

119897rArr r = 2119897 sin θ ---- (2)


41205871205760

1199022

2119897 sin 120579 2 119898119892rArr m =

119954120784 119836119848119853 120637

120783120788120645120634120782119944119949120784119956119946119951120784120637




loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







loose

Solution








41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus4119876

001= 18 x 107Q






Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution




41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

1199092 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

119902119876

1199092 rArr 2119902

1198892 = minus119876

1199092 ---- (1)


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2


41205871205760

21199022

1198892 + 1

41205871205760

119902119876

(119889minus119909)2 = 0 rArr 1

41205871205760

21199022

1198892 = minus1

41205871205760

2119902119876

(119889minus119909)2 rArr 2119902

1198892 = minus2119876

(119889minus119909)2 --- (2)


1199092 = minus2119876

(119889minus119909)2 rArr x = 119889

2+1= d 120784 minus 120783

From (1) 2119902

1198892 = minus119876







Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01









Solution







41205871205760

11990211199022

1199032 = 9 x 109 4 119909 10minus16

01+119909 2 = 36 119909 10minus7

001+02119909


001+02119909rArr x + 20x2 = 36 x 10-7 rArr x = 36 x 10-6 m






Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution







41205871205760

11990211199022

1199032 = 9 x 109 x 2 119909 10minus6119902

001= 18 x 105q N









Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution






41205871205760

11990211199022

1199032

01 x 98 x sin 30 = 9 x 109 x 4 119909 10minus12

1199032

049 = 36 x 10-3 1

1199032 rArr r2 = 7346 x 10-4 rArr r = 7346 x 10-2 = 27 cm




maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







maximum force

Solution





2= 2Fe cos (90 - 120579) = 2Fe sin θ

F = 2 1

41205871205760

119876119902

1199102+119889

2

22 sin θ = 2

1

41205871205760

119876119902

1199102+119889

2

22

119910

1199102+119889

2

2=

119876119902

21205871205760

119910

1199102+119889

2

2 32


119889119910= 0 rArr

119876119902

21205871205760

1 1199102+119889

2

2 32

minus 1199103

21199102+

119889

2

2 12

2119910

1199102+119889

2

2 3 = 0

1 1199102 +119889

2

2 32

minus 1199103

21199102 +

119889

2

2 12

2119910 = 0 rArr 1199102 +119889

2

2 32

= 1199103

21199102 +

119889

2

2 12

2119910 rArr y = 119941

120784 120784


21205871205760

119889

2 2

119889

2 2

2+

119889

2

2 32 = 308 119928119954

120786120645120634120782119941120784







Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01










Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

11990211199022

1199032 = 1

41205871205760

119876119902

1199092+1198892

4


41205871205760

119876119902

1199092+1198892

4

119909

1199092+1198892

4

= 119824119850119857

120784120529120518120782 119857120784+119837120784

120786

120785120784


2πε0d2

4

32

= 4119876119902

πε01198893 119909 rArr F prop x

(C) ma = 4119876119902

πε01198893 119909 rArr a = 4119876119902

π119898ε01198893 119909 [a = ω2x]


π119898ε01198893 rArr T = 2120587π119898ε01198893

4119876119902=

119950120645120785120634120782119941120785

119928119954

120783

120784



Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






Solution


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2+ 119909


41205871205760

11990211199022

1199032 = 1

41205871205760

119902119876

119889

2minus 119909

2


41205871205760

119902119876

119889

2minus 119909

2 minus1

41205871205760

119902119876

119889

2+ 119909

2 = 119876119902

41205871205760

1

119889

2minus 119909

2 minus1

119889

2+ 119909

2 = 119876119902

41205871205760

2119909119889

1198892

4minus 1199092

2

When x ltlt d F = 119876119902

41205871205760

2119909119889

1198892

4

2 = 119876119902

41205871205760

32119909

1198893 = 119876119902

1205871205760

8119909

1198893 = 119876119902

1205871205760

8

1198893 (119909) rArr ma = 119876119902

1205871205760

8

1198893 (119909) rArr a = 119876119902

1198981205871205760

8

1198893 (119909) [a = ω2x]


1198981205871205760

8

1198893 rArr 2120587

119879=

119876119902

1198981205871205760

8

1198893 rArr T = 212058711989812058712057601198893

8119902119876=

119950120645120785120634120782119941120785

120784119954119928

120783

120784




Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




119902=

15 119909 10minus3

1 119909 10minus6 = 15 x 103 = 1500 NC




zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







zero

Solution






41205871205760

119902119860

1199092 = 9 x 109 x 2 119909 10minus6

1199092 = 18000



41205871205760

119902119861

(119909+02)2 = 9 x 109 x 4 119909 10minus6

(119909+02)2 = 36000

(119909+02)2 (towards B)


1199092 = 36000

(119909+02)2 rArr 1

1199092 = 2

(119909+02)2 rArr 1

119909=

2

119909 + 02rArr x =

02

2minus1= 483 cm


Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01





Solution



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20minus119909= 9 x 109 x


20minus119909=

minus36000

20minus119909


119909minus

36000


18000

119909=

36000

20minus119909rArr

1

119909=

2


120784120782

120785cm



41205871205760

119902119860

119909= 9 x 109 x

2 119909 10minus6

119909=

18000

119909


41205871205760

119902119861

20+119909= 9 x 109 x


20+119909=

minus36000

20+119909


119909minus

36000

20+119909= 0 rArr

18000

119909=

36000

20+119909rArr

1

119909=

2

20+119909rArr 2x = 20 + 119909 rArr x = 20 cm




Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




41205871205760

119902

1199032 = 9 x 109 x 119902

016= 5 rArr q = 89 x 10-11 C



Solution






15= 653 NC




Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




3=

02

3m


41205871205760

119902

1199032 = 1

41205871205760

10minus8

02

3

2 = 1

41205871205760

3

4x 10-6 NC



41205871205760

119902

119903=

1

41205871205760

10minus8

02

3

= 10 3

2

1

41205871205760x 10-8 volt


2

1

41205871205760x 10-8

V1 = 15 31

41205871205760x 10-8 = 15 3 x 9 x 109 x 10-8 = 23 x 102 volt





Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution


41205871205760

119876119909

1198772+1199092 32


41205871205760

119876119902119909

1198772+1199092 32


41205871205760

119876119902119909

1198772 32 = 1

41205871205760

119876119902119909

1198773

ma = 119876119902

412058712057601198773 (x) rArr a = 119876119902


ω2 = 119876119902

412058712057601198981198773 rArr ω = 119876119902

412058712057601198981198773 rArr 2120587

119879=

119876119902

412058712057601198981198773 rArr T = 120783120788120645120785120634120782119950119929120785

119928119954




Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution


119871



119871d119897


41205871205760

119889119902

1199032 = 1

41205871205760

119876

119871119889119897

1199032


41205871205760

119876

119871119889119897

1199032 cos θ = 1

41205871205760

119876

1198711199032 119903 119889θ cos θ


41205871205760

119876

119871119903

minus120587

2

120587

2 cos 120579 119889120579 = = 1

41205871205760

119876


120587

2to

120587

2

E = 1

41205871205760

119876

119871119903(1 ndash (minus1)) =

1

21205871205760

119876

119871119903=

1

21205871205760

119876

119871119871

120587

= 119928

120784120634120782119923120784




Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution



119871





119871dy


41205871205760

119889119902

1199032 = 1

41205871205760

119902

119871 1198892+1199102 dy


41205871205760

119902

1198711198712minus

1198712 119889119910

1198892+1199102

119889

1198892+1199102=

1

41205871205760

119902119889

1198711198712minus

1198712 119889119910

1198892+1199102 32


E = 1

41205871205760

119902119889

1198711198712minus

1198712 d sec2 θ dθ11988931199041198901198883120579

= 1

41205871205760

119902

1198711198891198712minus

1198712cos θ 119889θ =

1

41205871205760

119902

119871119889sin θ =

1

41205871205760

119902

119871119889

119910


E = 1

41205871205760

119902

119889

1

1198892+1198712

4

= 9 x 109 x 50 119909 10minus6

0087

1

(0087)2+001

4

= 52 x 107 NC



Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






Solution


41205871205760

119876119909

1198772+1199092 32


119889119909= 0 rArr

119889

119889119909

1

41205871205760

119876119909

1198772+1199092 32 = 119876

41205871205760

119889

119889119909

119909

1198772+1199092 32

119889119864

119889119909=

119876

41205871205760

1 1198772+1199092 32minus119909

3

21198772+1199092

12 2119909

1198772+1199092 3 = 0

1198772 + 1199092 32 minus 1199093

21198772 + 1199092

1

2 2119909 = 0

1198772 + 1199092 32 = 3x2 1198772 + 11990921

2

1198772 + 1199092 = 3x2 rArr R2 = 2x2 rArr x = 119929

120784




Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution


41205871205760

119902

119889

1

1198892+1198712

4



shown in figure





Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution


119897=

119876

2120587119886


2120587119886dL




41205871205760

119889119876

1198862

dE = 1

41205871205760

119876

2120587119886dL

1198862 = 1

41205871205760

119876 119889119871

21205871198863 = 119954 119941119923

120790120645120784120634120782119938120785




Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution



120786120645120634120782

119954

119941120784




due to point charge





Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution







2 = 10-4 42 + 7842 = 88 x 10-4 N




Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution








119898=

119864119902

119898


2119886=

119950119958120784

120784119916119954







Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01










Solution







119898=

3

0001= 3 x 103 ms2


119886=

2 119909 04

3 119909 103 = 163 x 10-2 sec







Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution










2at2 = 0 +

1

2

119865

119898t2 = 05 x

14






Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution


20= 225 sec





119898= 10 minus

25 119909 104 119909 4 119909 10minus6

004= 10 ndash 25 = 75 ms2


1198792=

119892119890119891119891

119892rArr

225

1198792=

75

10rArr T2 = 225

10

75= 26 sec






Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution





119948






Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01









Solution


Mass of the block m




119898=

119864119902

119898


2at2 rArr d = 0 +

1

2

119864119902

119898t2 rArr t =

2119889119898

119864119902


119864119902=

120790119941119950

119916119954





Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution






Solution









Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








Solution





41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Ui = 9 x 109 4 119909 10minus14

02+

4 119909 10minus14

01+

4 119909 10minus14

01= 36 x 10-5 1

02+

1

01+

1

01= 36 x 10-5 (5 + 10 + 10) = 9 x 10-3 J


41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 x 109 x 3 x 4 119909 10minus14

02= 540 x 10-5 = 54 x 10-3 J







Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01









Solution




1198812 119889119881 = minus 1199091

1199092 119864 119889119909


119861119889119881 = minus 1199091



119861119889119881 = minus 1199091



119861119889119881 = minus 1199091





Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution








Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




119889119903rArr dV = minus ത119864 d ҧ119903


119881

0119889119881 = minus 2

020 119889119909 minus 2

030 119889119910


V = 20 (0 ndash 2) + 30 (0 ndash 2)





Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




119889119903



0

1198810 119889119881 = minus 10

0119860119909 rArr V0 = minus A

1199092

2[limits 10 to 0] =

10

2(100 ndash 0) = 500 volt




(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







(1 m 1 m 1 m)

Solution




119902=

1198721198712119879minus2

119868119879= ML2T-3I-1


1198712 = MT-3I-1


120597119909Ƹ119894 minus

120597119881

120597119910119895 minus

120597119881







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution





41205871205760

11990211199022

119903119894=

1

41205871205760

1199022

infin= 0 J


41205871205760

11990211199022

119903119891= 9 x 109 x

2 119909 10minus5 2

01= 9 x 1010 x 4 x 10-10 = 36 J




electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






electric field

Solution


∆119903=

20minus10

01 sin 30=

10

01 119909 05= 200 Vm




119903 (119888119898)



6

1199032 = 120788

119955120784 Vcm




field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







field at this point

Solution





119877=

119896λ 119889119897

1199092+1199032


1199092+1199032 119889119897 = 119896λ

1199092+1199032(2120587r) =

1

41205871205760

λ

1199092+1199032(2120587r) =

119955λ

120784120634120782 119961120784+119955120784


119889119909= minus

119889

119889119909

119903λ

21205760 1199092+1199032= minus

119889

119889119909

λ119903

212057601199032 + 1199092 minus12

E = minusλ119903

21205760minus

1

21199032 + 1199092 minus

3

2(2119909) = 119955119961120640

120784120634120782 119955120784+119961120784 120785120784







Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01










Solution









119898=

119864119902

119898= 103 x 176 x 1011 = 176 x 1014 ms


2119886=

7 119909 1012 119909 12 2

2 119909 176 119909 1014 = 05 cm







Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01










Solution













shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






shown in figure

Solution



41205871205760

11990211199022

11990312+

1

41205871205760

11990221199023

11990323+

1

41205871205760

11990231199021

11990331

Uf = 9 119909 109

01[2 x 10-5 x 4 x 10-5 + 4 x 10-5 x 3 x 10-5 + 2 x 10-5 x 3 x 10-5]

Uf = 9 x 1010 x 10-10 [8 + 12 + 6] = 234 J





Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution





119889qd =

200minus100

119889qd = 100q





Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution






41205871205760

11990211199022

1199031= 9 x 109 x

4 119909 10minus8

10 119909 10minus2 = 3600 J


41205871205760

11990211199022

1199032= 9 x 109 x

4 119909 10minus8

infin= 0 J



2mv2 = mv2 = 10 x 10-3 v2 = 10-2 v2





reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







reduced to 50 cm

Solution







41205871205760

11990211199022

1199031= 9 x 109 x


100 119909 10minus2 = 144 x 10-1 = minus144 J


41205871205760

11990211199022

1199032= 9 x 109 x


50 119909 10minus2 = minus288 J



2mv2 = mv2 = 5 x 10-3 v2


5= 536 ms




Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution










Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01









Solution






41205871205760

2119875

1198893 = 9 x 109 x 2 119909 2 119909 10minus8

1= 36 x 101 = 360 NC


41205871205760

119875

1198893 = 9 x 109 x 2 119909 10minus8

1= 18 x 101 = 180 NC



the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






the combination

Solution








Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01






Solution


120786120645120634120782

119954

119941120784


120786120645120634120782

119927

119941120785 (d gtgt a)


41205871205760

119902



41205871205760

119875



2 = 120783

120786120645120634120782

120783

119941120785 119954120784119941120784 + 119927120784





small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01








small oscillations

Solution






Iα = Eqa θ rArr α = 119864119902119886

119868θ =

119864119902119886

1198981198862 θ = 119864119902

119898119886θ [α = ω2 θ]


119898119886rArr

2120587

119879=

119864119902

119898119886rArr T = 120784120645

119950119938

119916119954




Solution




119872=

64

64= 01







Solution




119872=

64

64= 01





Concepts of Physics Exercises Electric Field & Potential

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Transcript of Concepts of Physics Exercises Electric Field & Potential