SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

CASE STUDY BASED QUESTION An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge upto a radius

R. The atom as a whole is neutral. According to Gauss

theorem the net electric flux associated with a surface is 1/ϵ0 times the net charge

enclosed within the Gaussian surface. 1. What will be the net flux associated with the nucleus?

(a) e/ ϵ0

(b) 2e/ ϵ0

(c) Ze/ ϵ0

(d) None of the above. 2. What should be the electric field due to the atom as a whole at an outside

point? (a) Zero.

(b)

(c) 1

4𝜋𝜀0

𝑞

𝛾2

(d) None of the above 3. Which of the following is incorrect?

(a) Electric flux is due to both the charges inside as well as outside. (b) Electric flus does not depend on shape and size of the surface. (c) Validity of Gauss theorem depends on inverse square nature of Coulomb’s

law. (d) None is correct.

4. If the net flux associated through a surface is positive, it means (a) Net charge inside is positive. (b) No. of lines leaving the surface is greater than no. of lines entering the

surface. (c) Both (a) & (b) (d) Neither (a) nor (b)

5. An arbitrary surface encloses 100 dipoles of dipole moment 2 Cm and dipole length 1 mm. What shall be the net flux associated with the surface?

(a) 2/ ϵ0

(b) Zero

(c) 1/ ϵ0

(d) None of the above. 6. What is the electric field at a distance r from the nucleus?

(a) Zero.

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

(b)

(c) 1

4𝜋𝜀0

𝑞

𝛾2

(d) None of the above

Assertion-Reason Type Questions 1. Assertion:- If the distance of a point from an infinitely long linear charge

distribution is doubled then the electric field at that point becomes half. Reason:- Electric field due to a linear charge distribution is inversely proportional

to distance. 2. Assertion:- Gauss Theorem states that the net electric flux is equal to 1 upon

epsilon not times the net charge enclosed within the surface.

Reason:- The electric flux due to an outside charge is always zero 3. Assertion:- If the radius of a Gaussian sphere is doubled then the flux also gets

doubled.

Reason:- Flux is directly proportional to the surface area enclosing the charge. 4. Assertion:- The validity of Gauss theorem depends on the inverse square nature of

Coulomb's law. Reason:- Otherwise the flux will also depend on shape and size.

5. Assertion:- The electric field due to an infinitely charged plane sheet is independent of distance.

Reason:- It is given by σ/ ϵ0.

6. Assertion:- The electric flux through a Gaussian surface enclosing a dipole is always zero. Reason:- The net charge on a dipole is zero.

7. Assertion:- Area vector of a square of side 10 m lying in XY plane is 100 k or 100 (-k) where k is a unit vector along Z axis. Reason:- The direction of area vector is normally outwards to the surface.

8. Assertion:- If the medium within the Gaussian surface is replaced by a medium of dielectric constant K then the flux will also become 1/K times. Reason:- The permittivity increases in presence of medium.

9. Assertion:- Flux coming out through to one of the faces of a cube enclosing charge q is 1/6 time the total flux. Reason:- A cube has 6 faces in all.

10. Assertion:- If a charge is moved from one corner of a cube to another corner the flux associated with the changes. Reason:- Flux does not depend upon the location of charge inside the Gaussian surface

NUMERICALS:- TYPE-1

1. A cube of side 16 cm is lying with its base along + x axis. Electric field along x-axis is given by 500√x, whereas the components of field along z axis and y axis are zero. What is the flux through the cube and also the charge in the cube if left face of the cube is at (16,0,0) cm. [2.12 Nm2 C -1, 1.88 x 10-11 C]

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

2. Given the components of an electric field as Ex = αx, Ey = 0 and Ez = 0, where α is a dimensional constant. Calculate the flux through each face of the cube of side ‘a’ as shown in the figure and the effective charge inside the cube.

3. Use Gauss’s Law to obtain an expression for the electric field due to an

infinitely long straight uniformly charged wire. Electric field in the above figure is directed along +x axis and given by Ex = 5Ax + 2B, where E is in N/C and x is in meter, A and B are constants with dimensions. Taking A = 10 NC-1 m-1 and B = 5N/C calculate (i) The electric flux through the cube. (ii) Net charge enclosed within the cube. [2015]

4. The electric field components due to a charge inside the cube of side 0.1 m

are as shown: Ex = ax, where a = 500 N/C-m, Ey = 0, Ez = 0. Calculate the electric flux through the cube and (ii) the charge inside the cube. [AI

2008] 5. Given the electric field in the region E = 2x i ,find the net electric flux through

the cube and the charge enclosed by it.

6. The electric field components in the figure are Ex = ὰ√x, Ey = 0, Ez = 0 in

which ὰ = 800 N/C-m1/2, Consider the cube shown in the figure. Calculate (a) the flux φE through the cube. And (b) charge within the cube.

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

7. A right circular cylinder of length ‘a’ and radius ‘r’ has its center at the origin and its axis along the x-axis so that one face is at x = + a/2 and the other at x = -a/2, as shown in the figure. A uniform electric field is acting parallel to the x-axis such that E = E0ifor x>0 and E = - E0ifor x<0. Find out the flux (i) through flat faces, (ii) through the curved surface of the cylinder. What is the net outward flux through the cylinder and the net charge inside the

cylinder? 8. An electric field is uniform in the positive x direction for positive x, and

uniform with the same magnitude but in the negative x-direction for negative x. It is given that E = 200 i N/C for x>0 and E = -200 i N/C for x<0. A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the x-axis so that one face is at x = +10 cm and the other is at x = -10 cm. (a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (c) What is the charge inside the cylinder? [NCERT Solved Example]

9. A point charge +100 nC is at a distance of 2 mm directly above the centre of

a square of side 5 cm of a cube as shown in the figure. What is the magnitude of electric flux through the square?

10. The net outward flux through the surface of a cube is 16k Nm2 C-1 (i)

What is the net charge inside the cube? (ii) If the net outward flux through the surface of the cube were zero, is it correct to say that there is no charge inside the box.

11. The electric field intensity at a distance of 4 cm due to infinite linear charge is 18 x 104 N/C. Calculate the linear charge density. [4 x107 C/m]

12. A uniformly charged conducting sphere of radius 1.2 m has a surface charge density of 16µC/m2. Find (a) the charge on the sphere and (ii) the total electric flux leaving the surface of the sphere. [2.9 x 10-4 C, 3.28 x 107 NC-1m2]

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

CONCEPTUAL PROBLEMS 1. A long charged cylinder of linear density +λ1 surrounded by a hollow

conducting coaxial cylinder of linear density –λ2. Use Gauss law to obtain expressions for the electric field at a point (i) in the space between the cylinders, and (ii) outside the larger cylinder. [FOREIGN 2017]

2. A point charge placed at any point on the axis of an electric dipole at some large distance experiences force F. What will be the force acting on the point charge when its distance from the dipole is doubled? [CPMT 91]

3. Charges of magnitudes 2Q and –Q are located at points (a,0,0) and (4a,0,0). Find the ratio of the flux of electric field due to these charges through concentric sphere of radii 2a and 8a centred at the origin.

4. Given figure shows three charges enclosed by a Gaussian surface. What is the flux of electric field through the Gaussian surface if q1 = +3.0 nC, q2 = - 6.0 nC, and q3 = -3.0 nC? [-6.78 x 102 Nm2 C-1]

5. If 1 Coulomb charge is placed at the centre of a cube of side 10 cm, calculate

the flux coming out of any face of the cube. [1.88 x 1010 Nm2 C-1] 6. What is the electric flux through a cube of side 1 cm which encloses an

electric dipole of dipole moment 20 x 10-6 C-m ? [D 2005, 15] 7. If Coulomb’s law involved (1/r3) dependence (instead of 1/r2), would Gauss

law be still true? 8. What is the significance of positive and negative electric flux? 9. There is a certain net flux φ through a Gaussian sphere of radius r enclosing

an isolated charged particle. Suppose the enclosing Gaussian surface is changed to (a) a large Gaussian sphere (b) A Gaussian cube with edge length equal to 2r. In each case, is the net flux through the new Gaussian surface larger than, smaller than or equal to φ?

10. Suppose a Gaussian surface does not include any net charge. Does it necessarily mean that E is equal to zero for all points on the surface? Ans:- If a Gaussian surface does not include any net charge, the electric field on the GS must be parallel to the surface. However this electric field will be due to a source charge outside the GS.

11. Two charges of magnitudes -2Q and +Q are located at points (a,0) and (4a,0) respectively. What is the electric flux due to these charges through a sphere of radius ‘3a’ with its centre at the origin? [2013]

12. Two charges of magnitudes +4Q and –Q are located at points (a,0) and (-3a,0) respectively. What is the electric flux due to these charges through a sphere of radius ‘2a’ with its centre at the origin? [2013]

13. State Gauss’s law in electrostatics. A cube with each side a is kept in an electric field given by E = Cx i, where C is a positive dimensional constant. Find out (i) the net electric flux through the cube, and the net charge inside the cube. [Foreign 2012]

14. A point charge +Q is kept in the vicinity of an uncharged conducting plate. Sketch electric field lines between the charge and the plate.

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

15. Two infinitely large thin parallel sheets having surface charge densities σ1& σ2 (σ1 > σ2) are shown in the figure. Write the magnitudes and directions of net fields in the regions marked II and III. [Foreign

2014] 16. A sphere S1 of radius r1 encloses a total charge Q. If there is another

concentric sphere of radius r2 (> r1) and there be no additional charge between S1 and S2. Find the ratio of electric flux through S1 and S2.

17. Two plane sheets of charge having charge densities σ1 and σ2 are normal to each other. What will be the magnitude of electric flux at point O between

the plates? 18. A spherical rubber balloon carries a charge that is uniformly

distributed over its surface. As the balloon is blown up and increases in size, how does the total electric flux coming out of the surface change? Give reason. [Delhi 2007, 2008 C]

19. An infinitely long straight positively charged wire has linear charge density λ Cm-1. An electron is revolving around the wire as its centre with a constant velocity in a circular plane perpendicular to the wire. Deduce the expression for its Kinetic energy. (b) Plot a graph of the kinetic energy as a function of charge density λ. [CBSE F 2013]

20. Three charges Q1, Q2and Q3are placed inside and outside a closed Gaussian surface as shown in the figure. Answer the following:- (a) Which charges will contribute to the electric field at any point on the Gaussian surface? (b) Which charges contribute to the net flux through this surface? (c) If Q1= - Q2, will electric field on the surface be zero.?

21. Using Gauss’s law , obtain the electric flux due to a point charge ‘q’

enclosed in a cube of side ‘a’. 22. Given a uniform electric field E = 5 x 103 i N/C, find the flux of this field

through a square of 10 cm on a side whose plane is parallel to the y-z plane. What would be the flux through the same square if the plane makes a 30° angle with the x-axis? [2014]

23. Show that the electric field due to a uniformly charged infinite plane sheet at any point x from it, is independent of x. [2014]

LONG ANSWER QUESTIONS [5-MARKS ASKED FROM THIS TOPIC] ALL INDIA 2019

1. Two large charged plane sheets of charge densities σ and –2σ C/m2 are arranged vertically with a separation of d between them. Deduce expressions for the electric field at points (i) to the left of the first sheet,

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

(ii) to the right of the second sheet, and (iii) between the two sheets.

2. A spherical conducting shell of inner radius r1 and outer radius r2 has a charge Q. (a) A charge q is placed at the centre of the shell. Find out the surface charge density on the inner and outer surfaces of the shell. (b) Is the electric field inside a cavity (with no charge) zero; independent of the fact whether the shell is spherical or not ? Explain.

1. ALL INDIA 2018:- Q.24 (a) Define electric flux. Is it a scalar or a vector quantity ? A point charges q is at a distance of d/2 directly above the centre of a square of side a, as shown in the figure. Use Gauss' law to obtain the expression for the electric flux through

the square. (b) If the point charge is now moved to a distance 'd' from the centre of the square and the side of the square is doubled, explain how the electric flux will be affected. [5] OR (a) Use Gauss' law to derive the expression for the electric field (E) due to a straight uniformly charged infinite line of charge density λ C/m. (b) Draw a graph to show the variation of E with perpendicular distance 'r' from the line of charge. (c) Find the work done in bringing a charge q from perpendicular distance r1 to r2 (r2 > r1). [BASED ON CONCEPT OF ELECTRIC POTENTIAL] ALL INDIA 2017:-

2. Derive an expression for the electric field E due to a dipole of length ‘2a’ at a point distant r from the centre of the dipole on the axial line. (b) Draw a graph of E versus r for r >> a. (c) If this dipole were kept in a uniform external electric field E0, diagrammatically represent the position of the dipole in stable and unstable equilibrium and write the expressions for the torque acting on the dipole in both the cases.

OR (a) Use Gauss’s theorem to find the electric field due to a uniformly charged infinitely large plane thin sheet with surface charge density σ. (b) An infinitely large thin plane sheet has a uniform surface charge density + σ. Obtain the expression for the amount of work done in bringing a point charge q from infinity to a point, distant r, in front of the charged plane sheet. [BASED ON CONCEPT OF ELECTRIC POTENTIAL]

3. Define electric flux. Write its SI unit. “Gauss’s law in electrostatics is true for any closed surface, no matter what its shape or size is”. Justify this statement with the help of suitable example.

4. (b) Use Gauss’s law to prove that the electric field inside a uniformly charged spherical shell is zero. [Allahabad Region 2015]

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

5. Consider two hollow concentric spheres S1 and S2 , enclosing charges 2Q

and 4Q respectively are shown in the figure. (i) Find out the ratio of electric flux through them. (ii) How will the electric flux through the sphere S1 change if a medium of dielectric constant K is introduced in the space inside S1in place of air? Deduce the necessary expression. [AI 2014]

6. Using Gauss’ law deduce the expression for the electric field due to a uniformly charged spherical conducting shell of radius R at a point (i) outside and (ii) inside the shell. Plot a graph showing variation of electric field as a function of r > R and r < R. (r being the distance from the centre of the shell). [Delhi 2011, AI 2013] [NOT IN SYLLABUS FOR THIS YEAR]

7. Define electric flux. Write its SI units. (b) Using Gauss’s law, prove that the electric field at a point due to a uniformly charged infinite plane sheet is independent of the distance from it. (c)How is the field directed if (i) the sheet is positively charged (ii) negatively charged? [Delhi 2012]

8. Using Gauss’s law derive an expression for the electric field intensity at any point outside a uniformly charged thin spherical shell of radius R and charge density σ C/m2 . Draw the field lines when the charge density of the sphere is (i) positive (ii) negative. [NOT IN SYLLABUS FOR THIS YEAR]

9. (b) A uniformly charged conducting sphere of 2.5 m in diameter has a surface charge density of 100µC/m2. Calculate the (i) charge on the sphere (ii) total electric flux passing through the sphere. [Delhi 2008] [NOT IN SYLLABUS FOR THIS YEAR]

10. A thin metallic spherical shell of radius R carries a charge Q on its

surface. A point charge Q/2 is placed at its center C and another charge +2Q is placed outside the shell at a distance x from the center as shown in the figure. Find (i) the force on the charge at the center of the shell and at the point A, (ii) electric flux through the shell. [CBSE D 2015]

11. An early model for an atom considered it to have a positively charged point nucleus of charge Ze, surrounded by a uniform density of negative charge upto a radius R. The atom as a whole is neutral. For this model what is the electric field at a distance r from the nucleus? [NCERT Solved Example]

12.

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]

13.

14. 15. A particle, having a charge +5 μC, is initially at rest at the point x = 30

cm on the x axis. The particle begins to move due to the presence of a charge Q that is kept fixed at the origin. Find the kinetic energy of the particle at the instant it has moved 15 cm from its initial position if (a) Q =+15μC and (b) Q = -15μC

16. (a) An electric dipole is kept first to the left and then to the right of a negatively charged infinite plane sheet having a uniform surface charge density. The arrows p1 and p2 show the directions of its electric dipole moment in the two cases. Identify for each case, whether the dipole is in stable or unstable equilibrium. Justify each answer.

(b) Next, the dipole is kept in a similar way (as shown), near an infinitely long straight wire having uniform negative linear charge density.

17. “ The outward electric flux due to charge +Q is independent of the shape and size of the surface which encloses it.” Give two reasons to justify the statement. (b) Two identical circular loops ‘1’ and ‘2’ of radius R each have linear charge densities –λ and +λ C/m respectively. The loops are placed coaxially with their centres R√3 distance apart. Find the magnitude and direction of the net electric field at the centre of loop ‘1’. 18. Define electric dipole moment. Is it scalar or vector? Derive the expression for the electric field of a dipole at a point on the equatorial plane of the dipole. 19. Two thin concentric and coplanar spherical shells of radii a and b (b>a) carry charge q and Q respectively. Find the magnitude of electric field at a point distant x from their common centre for (i) 0<x<a (ii) a<x<b (iii) b<x<∞

SHEET:-3

TOPIC:- ELECTRIC FLUX & GAUSS THEOREM

[VERY IMPORTANT TOPIC FROM BOARD’S POINT OF VIEW- A QUESTION OF 3/5 MARKS IS ASKED ALMOST EVERY YEAR.]