AP Physics C Exam Questions 1991-2005Coulomb's Law, E-fields & Electric Potential

2005E1. Consider the electric field diagramabove.

a. Points A, B, and C are all located aty:0.06 m.i. At which of these three points is the

magnitude of the electric field thegreatest? Justify your answer.

e. fre(j /i-*.s &ce ftrzffa-sar6tu{ralkg

ii. At which of these three points is theelecfric potential the greatest? Justifyyour answer.

frb. An electron is released from rest at point B.

gi4-€* rJtt", f ae-t-u-

0.1

0.o4

0.0?

,r(m)

i. Qualitatively describe the electron's motion in terms of direction, speed, and acceleration.; ;:;;;;" ; "#;;F""

:;; ;;.{" A;# ;',y;:;xiw*x H7=; ; -Y'.t-./J{-."- /&- ;;!G-*+t* et t / c*&tu"<

a6 e H;/,W /#* dg,,ou,u",r"..*€.* e*$'##mtero.* #-i fn f*,.ff.

c. Points B and C are separated by a potential difference of 20 V. Estimate the magnitude of the electric field

,"ots4tl

d.

AP Physics C Exam Questions 1991-2005Coulomb's Law. E-fields & Electric Potential

200481. In the left of thefigure is a hollow, inf,rnite,cylindrical, unchargedconducting shell of innerradius 11 ?nd outer radius12. An infinite line chargeof linear charge density +2

is parallel to its axis but offcenter. An enlarged crosssection of the cylindricalshell is shown at the right.

a. On the cross sectionabove right,i. sketch the electricfield lines, if any, ineach of regions I, II, and III and

ii. use * and - signs to indicate any charge induced on the conductor.

b. In the spaces below, rank the electric potentials at points e, b, c, d, and e from highest to lowest (1 : highestpotential). If two points are at the same potential, give them the same number.

/ v,, L vu 3 v"- *._rru 1p r"

,{

I

---

-t *b

*a-*

Cross Section \b

c. The shell is replaced by anothercylindrical shell that has the samedimensions but is nonconducting andcarries a uniform volume charge density+p. The infinite line charge, still of chargedensity *,1, is located at the center of theshell as shown above. Using Gauss's law,calculate the magnitude of the electricfield as a function of the distance r fromthe center of the shell for each of thefollowing regions. Express your answersin terms of the given quantities andfundamental constants.

i.r1rt'r 1( zrl r L) > at,

Cross Section

{, ^' ^ v*'

Av - fivu*Trrtt:)

f-t:!-::nZ€o

+ 9(r"-d)Z€o f

I5 1d4'-E,zwrL

&n* #t4& *Tzb\

?' --,e€ r

ii.rl<r1rz4(zt"L) : b"(,\l+ f Lv\:e(

!,(A(h+Pn(""-''1 )

> TI ( r*-r,*"'l t-

;Lt grr(r""nt\

tre- "-v

lLL. f > f2

r .*,ui\Y u;

\:L+7:r{;,:r

,\L t-ArILr f-rT('t-,:{E L--*l'* r^ "-' LIT f L

AP Physics C Exam Questions 1991-2005

Coulornb's Law, E-fields & Electric Potential

2003E1. A spherical cloud of charge of radius R contains

a total charge +Q with a nonuniform volume charge

density that varies according to the equation

ilr) : p.(l - r/R) for r < R and

P:0forr)R,

where r is the distance from the center of the cloud'

Express all algebraic answers in terms of Q, R, and

fundamental constants.

- a +'lZ(r:L3 {{L Jo

e.

=3fr€o

)(!

)P

a. Determine the following as a function of r for r > R '

i. The magnitude E of the electric field l_7 , .a ., - €Fra^ &e"r*^-t\ {*.*' !87Art - #?rL& > T*

ii. The electric Potential Z

6*9-ll* {

b.AprotonisplacedatpointPshownabove.andreleased.Describeitsmotionforalongtimeafteritsrelep.-5'€5

/as/eaJe ^-';i44'1; tTLcttt b<* az-<s-/6a***+' aez

,z;K d,/a'-cs.}iJ pL. -iy* 4a%c. L electron of charge magnitude a is now placed at point P, which is a distance r ftom the center ofthe sphere,

and released. D"r.ririo. th" kio"ti. energy ofthe elecqon as a function oflas it strikes the cloud., ^ /- ,z I

T-:;t;':;dt;*:aI'#i a)- aA4(4-r) = c.Lq(6)r: : t<+-k= {) * # = lc * *# :_y k : *a.e ( f- * # )

Derive an exprcssion for P' . , f | .

;':""j;;;='j; {,*/*\{onkr'\ * *w f. )(,-fr)rb{r -4rfi

z +7f & EoL * -*I' 4o f',{e'{f,) : * rf'E'5

f. = z</r?\Determine ttie magnitude E of the electric field as a function of

2for r < R .

{t" €/#** 6ts'pr&*b*- sJ$*v - J {n {,-f*}.r{r*Jr*s"o 6*- 'l ) " L

,.".,*_,-J

\rl'"-e f -E+E- I -t'/

d.

rtFk G 4-lgvkl. r/

tI r "f , ,SQf +Wrf ?,i ;;** ii, TfZ->L&*it

AP Physics C Exam Questions 1991-2005

Coulomb's Law, E-fields & Electric Potential

4) Uggr,Ell A solid metal sphere of radius a is charged to a potential

ff" t O and then isolated from the charging source. It is then surrounded

Uy;oining two uncharged metal hemispherical shells of inner radius b

aoi o.rteiradius 2b, as shown above, without touching the inner sphere

or any source ofcharge.a. In terms of the given quantities and fundamental constants,

determine the initial charge Qo on the solid sphere before it was

surrounded by the outer shell. ! V *_ l,^*%+nt*t*^"!L i*ta a"f'wru- :'-

..-**J

b. Indicate the induced charge on the

i. The inner surface of the shellt\*&o

ii. The outer surface of the shell

{1

"r {t8

following after the outer shell is in place.

gf -- *is*'fu#

v*u1

L.,-l\t;t\

iI,t

ijo

LLtU..d-'-'*"j1f\It- r u/

r P-GlJ @

-)*Y

nfuqp4L b 6r4V

c. Indicate the magnitude of the electric field as a function of r and the direction (if any) of the field for the regions

indicated below. Write your answers on the appropriate lines.

ffier/,+i.r<a

ii.a<r<b

iii. b< r<2b

Magnitud" c , o

Magnitud " L"'4'- '?#'

Direction

Direction

Direction

Direction

flr.t;'",t, eo^A,-*'$'t/*

Does the inner sphere exert a force on the uncharged hemispheres while the shell is being assenrbl€d? Why otwhy not?

Vg;. "'.r-ru, ., ,5- ,,, ,".-'ffi ^ ,;,ab- (htat-/'t't')'

;^/'*n*#ffi: J f-.""-t.6::'-, "..Although the charge on the inndriolidiphere bas not changed, its potential has. In terms of v", a, and b,

determine tle new potential on the inner sphere. Be stlle to sOo* to*.!b).2 -<-., 4e ,- *_ t4P"t"-lJ gt +''tt"e 4 -7 r r

0.' ,*H/-^t 4 *& 4/'^"V'(r -zb) - tr, ,j . *4.-

,,#€***r&J 'l ' k

W"tuT4 r

d.

iv.2b<r Magnitude ortlu/y,-

(:. tL ) V L

Magnitude otIr,,*r"1M

e.

d**a,vo

\#

zb

I

{rt;t' t | 1 q..

*-\/,,f :*, I rt l*-"oJt ^6

1= vull*L _T .-r t.,Jl-

v'ah{\

'F,.dF f

tuf*;

fiI- f"- fL fu1r= feoelr > [r.r- - \r'cz/- - le-Jr' J J )=o u6

@eo

-;?Ef:-1* , [ 7 4'a(i]*L'\*-+- L'J + L'J,

AP Physics C Exam Questions 1991-2005Coulomb's Law, E-fields & Electric Potential

L992EL A positive charge distribution exists within a nonconducting spherical region of radius a. The volume

charge density p is Ugl_uniform but varies with the distance r from the center of the spherical charge distribution,

according to the relationship p: BrforO <r( a, where B is apositive constant, and p:O, forr>a.

a. Show that the total charge Q in the spherical region of radius a is Bnaa

b. In terms of B, r, a, and fundamental constants, determine the magnitude of the electric field at a point a distance

r from the center of the spherical charge distribution for each of the following cases.

i.r>a ii.r:a iii.O(r(ac. In terms of B, a, and fundamental constants, determine the electric potential at a point a distance r from the

center of the spherical charge distribution for each of the following cases.i.f:a

\ -/7/)-, 6/ 'Ir

ii. r: 0 f*rtrGLrrJ fdv:(-n

{

{II

\ E ilFJ^r

J Yt*,4*

Y*f?

,r -- :.rLLPL"" -

IL 6. 'c--lf '*)

(2e*a7

ff'ar.d \ Ttafi r%t

tJ

l" -*

td

\

*"-****4"'+7 t'nrP

\b) r>L:

f o dr-t

r ztu 9

(vpt,1-;E-tfl-

+L.+

2-

E%lr a

q r=4- :

f .o 3 V{r'$ ; }"

(:funI

v = \o-,ioln =Joa

E

I- rr;,h3

re{d_

I d Lbi[.iT\-rt j"- Fa- rV s- {"'- L r L:)v

a,s ?"Fl ILl \at \\.f, ,ft J

-ti

iIIl

AP Physics C Exam Questions 1991-2005Coulomb's Law, E-fields & Electric Potential

6) [1991E1] Two equal positive charges Q are fixedon the x-axis. one at *a and the other at -a, as shown

above. Point P,is a point on the y-axis withcoordinates (0, b ). Determine each of the followingin terms of the given quantities and fundamentalconstants.

f-t{ { 7'i t'p.J

a. The electric field E at the origin O---4:€ gute--v'y,*r* [. cr{-* {'o, V g.-.cfo-- gc.l-*,&'r"..*/

J

"tf 'ftf-" $'u"**"' "C--,{g- e* &{ *,';-'E'$AU, d}-*

.{ }

-r_*r. ryJ p.,, ftrx*y'€)*_.__ii*

ffi- x;fr>r" :.e.".)

t+r {4'

d dil-'(:- 4

4!t

LLE

'7y 1-{:{s

A small particle of charge q (q << Q ) and mass m isplaced at the origin, displaced slightly, and thenreleased. Assume that the only subsequent forces actingare the electric forces from the fwo fixed charges Q. atx: *a and x: -a. and that the particle moves only inthe xy -plane. In each of the following cases, describebriefly the motion of the charged particle after it isreleased. Write an expression for its speed when faraway if the resulting force pushes it away from the origin.

c.,ro /4t**n $€--*'$..n',p*t, d,c"r-e-." e p,ctr#r A t"r r*

t

k-,',,t f f Frtbt i"r .--,i,'0-*o J ;o**o ., %cl

b. The electric potential V at the origin O.

?on &- 9 r ',\C€- cLar^t e-- \/=-g-- , t \l -r* 6-hi#- *r rrn uut{- i 7ro ' { =- L *+*

ftl.(+a) ff"nr/5D V* :-- (;O lr* -*l { r*o.t

c. The magnitude of the electric field E atpoint P.

i5""rl:t;"f'J"H****L *{*,-se- qi *T"^l-T ."*-u

-+#ftaq-Sp X., W d dft"4-n*;6;:r*-'lt.e--t-^-r-d

u o ILL\s-;eeEp: zEJ = L\ f* J

.3,*f tfldrt:.t* oof ,""-*;. -*'&leu :; qidts"*"t*l3*H"{

d. q is positive and is displaced in the +x direction. (.s,*ni,1uq-. L"ru P*cQ o"s

u' ''t';,fir'.;'*i*b:* i:-:T#tr*i -8,-" *-."-"t*-' r".f,r|e. q is positiveuand is aisplaced iD the +v direction. t hUnL p. k-.-*,*-|r, a t>ill

o-i 6^)* i, 'i+l-r"-LQ'-' <4"i (;i:'ti":'- ,ul

a..,-,,..-i.^^.r-"r- {" "r;;.h;G: ea1yt1' = L--G, "-l'-'^-* :: "f. q is negative and is displaced in the +y direction. \ f . . .E.*e

i i5 *Y' ,YLbu'{tri p-J a ,.\y- - .f j, ('ff"r-)n 'L5 rr ij j vJ')i f-*b "Lr, ,f :-(#jlg)5 D t, Mlll *+ c-rl i*f* \.-

' !; w-, *;--

q, # Ao.-.*&\"a