•
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flc1.lh?t~
Potential and Field AP Free Response problem
2 (}I 2. I. Two thin, concentric, conducting spherical shells. insulated from each other, have radii of 0.10 m and 0.20 m,
as shown above. The inner shell is set at an electric potential of -1 00 V , and the outer shell is set at an electric potential of
+ 100 V, with each potential defined relative to the conventional reference point. Let Q, and Q" represent the net charge on
the inner and outer shells. respectively, and let r be the radial distance from the center of the shells. Express all algebraic
answers in terms of Q" Q", r, and fundamental constants. as appropriate.
(a) Using Gauss's Law, derive an algebraic expression for the electric field E(r) for 0.1 0 m < r < 0.20 m.
E. A
E (4i\f t ') "Qi
-:0 ~
fo
U:>
17
-= Q\.
qfff~f''1-
(b) Determine an algebraic expression for the electric field E(r) for r > 0.20 m.
E ==
£L t £(>
[, -=
Q L f- 00
'-I (j''t~("1-
(d) Using the numerical information given, calculate the value of the total charge Q, on the two spherical
shells
(Q, = Q, + Q,,) .
i.
100 -:.
Potential and Field AP Free Response problems
.-
20 I0 I. A charge +Q is uniformly distributed over a quarter circle of radius R, as shown above. Points A, B.
and C are located as shown, with A and C located symmetrically relative to the x-axis. Express all algebraic
answers in terms of the given quantities and fundamental constants.
(a) Rank the magnitude of the electric potential at points A, B. and C from greatest to least, with number I
being greatest. If two points have the same potential, give them the same ranking.
------2V.4
_I_VB _~,'yC
Justify your rankings.
lAJ ~
B
go
\J
i/)
~o.Jv.-.
$~-kwd"
Point l' is at the origin, as shown below, and is the center of curvature of the charge distribution.
(b) Determine an expression for th!£electric potential at point P due to the charge Q.
V;= IL..~
r
fdo.
QU~
\f -= fZ
==' I( G.
~
o
~
(c) A positive point charge q with mass 111 is placed at point P and released from rest. Derive an expression for
the speed of the point charge when it is very far from the origin __
. __ ,..---
-~.~~T
\<Q~::
~=k
~
V~fi).KQ'D
J-r'Y\\J'l-
I
~
~M
(d) On the dot representing point l' below, indicate the direction of the electric field at point P due to the charge Q.
,"
!
:,t- ,
-----.~-x
I
(e) Derive an expression for the magnitude of the electric field at point P.
Wq
rr/~
E ~\(~ coo.e;;
, J-= W
j
/L 6.. COJe&&
~
-IT:-
"
10IZ
sIne
-1i'/ot
Potential and Field AP Free Response problems
2007E2.
nonc=?~~f:ztad
In the fi~ure ~bove, a
sphere o.fradius ~ with charge +Q unifor:mly distributed througho!l! jts volu~e ~s
concentnc with a-D9nCfjjijj! __~g--mf €-I shell ofmner radlU£2a and outer radlus,Jo that has a charge -0 uniformly dlstnbuted
throughout its volume. Express all answers in terms of tile given quantities and fundamental constants.
(3) Using Gauss's la\v, derive expressions for the magnitude of tile electric field as a function of radius r in the following regions.
e 'eM.
:Jvrt
i. Within the solid sphere (,. < iL)
l'
'b!!i
c
v
to
L
:=-\ Or
41to.3
.
:::""'I~"1 ~
-w..3
I.(fTr
ii. Between theQ~
G,o/A ~
sphere and the spherical shell (a <,. < 2a)
r:: -
eo
\..I ~
+Q
411 r')..
iii. 'ijithin the spherical shell (2a < ,. < 3a )
£:= ~~ 'I Iff,,(
~'>
4(j'£;/'l,C/ii~
Q.t.n ~ +Q
+ qll~r
Q~
-~
z.lq~
+-0
C-==1ft 0
(19 - a.~ t8\)
li'f.,C
1-
iv. Outside the spherical shell (,. > 3a)
(;=0
(b) What is the el~tric
kic.Q~-:O
potential at the outer surface of the spherical shell (,. ~ 30)7 Explain your reasoning.
V=Jc;Jr
601 30-=0 56
v==
0
-=:>
"l
-/~<dr
(c) Derive an expression f~ the electric potential difference V•.- ") ~t\\een
'J, -'Jj
e.J'
V,-"~•
U~- ~)
points.X and Y shown in the figure.
"V, -V1 ' '~
V X- '1y -: ~ITQ(lc,) ~ 8~A
r:l
Potential and Field AP Free Response problems
---------I
J
-Q
i
I
I
I
I
-Q
i
.
:
.
I
I
I
I
I
I
I
:l
a
I
I
I
.------------.
+Q
R
_Q
.
2006E I. The square of side a above contains a positive point charge +Q fixed at the lower left corner and negative
fixed at the other three corners of the square. Point P is located at the center of the square.
a.
b.
indicate with an arrow the direction of the net electric field at point P.
On the diagram,
Derive expressions
i. The magnitude
for each of the following in terms of the given quantities
of the electric field at point P
£.-
-c:rt
I<Q
potential
\}~
constants.
at point
&t
P
Q
I~G _
3L
']ZC\.
~
~
c.
and fundamental
-\- \<0
I
ii. The electric
1-
mOj
A positive charge is placed at point P. It is then moved from point P to point R, which is at the midpoint
square. As the charge is
is the work done on it by the electric field positive, negative, or zero?
Positive
Explain
point charges-Q
your reasoning.
Negative
fo<c..e.-:f
of the bottom
side of
Zero
~Q ~
~
d.
i:
Describe one way to replace a single charge
square equal to zero. Justify your answer.
-\-Q -\ 1> P
II.
Describe
one way to replace
is zero but the electric
in this configuration
/v'-
Qtit~~ur.:sO'
field at the center of the
c.{\d-
a single charge in this configuration
field is not ze~
that would make the electric
such that the electric
0:1 100 ihM
potential
at the center of the square
fit ~f.j-
Potential and Field AI' Free Response problems
y (m)
----hT---~r--h----i
------r-------]-
0.1
I
I
---------r
I
------~---I
,"
i
I
r
---r--------~---------I
I
0.08
I
I
I
I
I
•
"
,
A
"
I
I
I
I
~--------i-------.~-----,---~-------B.-,
, -.----
0.06
,,
,
,,
,
I
J.
_l-
,,
,
I
______
0.02 -------
0.04
0.06
,
,
,
1
I
,
I
J.
,
I
0.02
,,
I
,
,
,
I
,
0.08
.r (m)
0.1
2005E I. Consider the electric field diagram above.
a.
Points A. B. and Care all located at y ~ 0.06 m .
i. At which of these three points is the magnitude of the electric field the greatest? Justify your answer.
C
UJJQf;\ ~
ii. At \vhich of these three points is the electric potential the greatest? Justify your ans'\-'er.
A
b.
An electron is released from rest at point B.
i. Qualitatively describe the electron's
~vY\
ii. ~c:a;
~
c.
NI
in terlllS ofdirectioQ., speed, and acceleration.
()J!/I-WiJV-01
V:' it?-V 7--
electron's speed after (t&~;;~hrqJ'("/~r~ar(r~ce'I~~il)
q ~\I-; ~)'Y\Ij V
o
1110tion
'"2--
-
V~ 1.~1~D(Ptl11s
Points Band C are separated by a potential difference of20
and state any assumptions that you make.
v.
Estlma e t le magnitude of the electric field midway between them
d. On the diagram. draw an equipotential line that passes through point D and intersects at least three electric field lines.
\ Rcsponsc problcms
Potcntlal. an dF'Icld AP Frcc
I
~'
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/
,~~~
2004E 1. The figure above left shows a hollow. infinite, cylindrical. uncharged conducting shell of inner radius rl and outer radius r2.
AI] infinite line charge of linear charge density +). is parallel to its axis but off center. An enlarged cross section orthe cylindrical
shell is shown above right.
a. On the cross section above right.
i. sketch the electric field lines, ifany, in each 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 G, b, c, d, and e from highest to lowest (I ~ highest potential). If two
points are at the same potential. give them the same number.
'1
V"
::;J V,
i-v"
c.
-.:Lv,
__
The shell is replaced by another cylindrical shell that has the same dimensions but is nonconducting
and carries a uniform volume
charge density +p. The infinite line charge, still of charge density +,t. is located at the center of the shell as shown above. Using
Gauss's law, calculate the magnitude of the electric field as a function of the distance, from the center of the shell for each of the
following regions. Express your answers in terms of the given quantities and fundamental constants.
i. , < "
t ,J~~
3":.
""l"
ZO
E(
zny 1.)"= ~
iLrl:5r:51'2
..
I
Lv,-
Cj
D~
E:;
'"
dlT'l".
r
~
>- \.-
A '" ;;l\\r L
Potential and Field AP Free Response problems
!I/
P
1003E I. A spherical cloud of charge of radius R contains a total charge +Q with a nonullifonll
according to the equation
p(i) ~ p,( 1 - r/R) for r:S Rand
.~
p = 0 for r > 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. Determine the following as a function of r for .r..;;...R.
i. The magnitude £ of the electric field
C1--Y'
-r::.
C\
-.-=;-C -....-- Q
£.
ii.
volume charge density that varies
IA. - 0
lllHo r
f.o
The electric potential
\J ~
fi~dr
2.-
V
\1-=
C>
b.
A proton is placed at point P shown above and released. Describe its motion for a long time after its release.
i nCA.QQ~
c.
.;d-fJeM
An electron of charge magnitude e is now placed at point P, which is a distance r from the center of the sphere, and released.
Determine the kinetic energy of the electron as a function ofr as it strOkes tile loud.
W.:l.v.. w~~r
-~
t!-.
JV
J fo{J-f)(}
QJf
~~
r
t
(1- ~')
Po
Q:;.J.
J:
IJ
G"" ~o
~
~
'l1l
0
)Q
j
(l) "
'iTi'r'1-o-r
(1- ~
U<
r
R.
.
a
1/11
fir
l. -
('3
'Jo ( {[
/JfiT'
J'd r
(~e3_ 3P.3)
-=
:btLl
I.
r
": jJ~IT
--10
3
;?
I
0
rj[:?? \
2-
)r cit" -
~3
().V
1Jm-/?:;('l (
the mag,?itude £ of the electric field as a function of,. for.r:s R
Determine
Ed A ;- ~
-
/0
7<>
E-
r - ~ ':/11(1
I~ -=- ~.
-
Derive an (!,If.-SSion for p".
.r
e.
(~r
lffili. ) ( (i-
r
d.
": ~ D.-u.-
'If ~-
,I'
1
11
fa \.-"3
6-t.,
~
r) '1m~y
{I- (I..-
4\
fl\'.
4Q...
0
%"".
Wf f'f 0
- to
p
10
oA ~
'1-
lIT fo
(2 -!::.~ ~
J
(f -~j~~ [1~-~tfL)I0----J~;a>( ~- f \
111'-
fon~
\
Z
11:
rT' (2.3 ~
3
Q
:, ~I~
r
4~ ()
Potential and Field AP Free Response problems
?,,11
y
'3
,
R
\
+
I
I
" \ I1
R__~~~~
60
0\/1
I
1
1
1
~x
I
1
1
R=O.lOm
1
1
'I1~ff
2002E I. A rod of uniform linear charge density A ~ + 1.5 x 10" elm is bent into an arc of radius R ~ 0.10 m. The arc is placed
\'lith its center at the origin of the axes shown above.
.
. a.- Determine the total-charge on the rod.
II
? I" Y:tO- ~ f'
'II 0
I
} ~ ;3
..)."T
\.....
>- -:.
-S" ) ( 10V~) ~
Q ~~,S
Q
--)...
b. Detennine the magnitude and direction of the elec~Ofield
t:~1:f!10l0-9
f!.'"
.=.
~j
p.
()Y.J
at the center
ede'
~1oo
c. Determine the electric potential at point
O.
-"
, .. ' )
C
.~,
.
V {kdrQ
::_ (~K/D"
),C ,IV'K/D
~_...:...--3\
,J ::
Ads
d.s -- If!.d9
~
(hlo" )(I')KJo'5) (/lSin{P()')
-,
~
)~
-::
a of the arc.
I
~
3
_
'>'17\
0
l,--_CX:_~{'-,"'~
v
V
IG-1ll-J2acl-~
---~
~ :1~d 10(0 V
J
o
A p..I2!9JJis now placed at point a and held in place. Ignore the effects of gravity in ihe rest of this problem.
d. Determine the magnitude and direction of the force that must be applied in order to keep the proton at rest.
f"" 6~ =~, ~~/d')( I, lP:<ID-,qJ
~
3.(&'I:,BN, vJ~
e. The proton is now released. Describe in words its motion for a long time after its release.
Vt'vUe-U\tVY\lf
V ) CU0WL~
D-
. ,
Potential and Field AP Free Response problems
--,
•
~.
1989E I. A negative charge - Q is uniformly distributed throughout the spherical volume of radius R shown above. A positive point
charge + Q is at the center of the sphere. Determine each of the following in terms of the quantities given and fundamental constants.
a. The electric field E outsidethe sphere at a distance r> R from the center
b.
The electric potential V outside the sphere at a distance r> R from the center
c.
The electric field inside the sphere at a distance r < R from the center
~~
d.
The electric potential inside the sphere at a distance r < R from the center
V
0
JE
r
'(l..
v~- \-'-11
(
J
- IL .
(-
_'2.
< .,-
_ yL.
~
J-'("
L
1..
{2-
- \LQ
f l'
V~ - 'KQ
.J ("
2j3
('2.
'3
- rz~
T
I
... -l<.Q '-f
'-
+~+~)
Z P-
-1-~
t,(t
Ze.'t
\(
-Ii;_
(2..
Potential and Field AP Free Response problem
1999E3.
3.
The nonconducting ring of radius R shown above lies in the yz-plane and carries a
uniformly distributed positive charge Q.
Determine
-----
the electric potential at points along the x-axis as a function ofx.
KQ
:::::.
~.
-,
b.
i. Show that the x-component
of the electric field along the x-axis is give
df,l-:: ~
dt
W-e
r
1-
y
Q
(J'X
e
Z f-)(t
)!>/t-
~S
o
ii. What are the y- and z~ components
V:::
of the electric field along the x-axis?
-;?e-VC?
c.
Determine
the following.
Q.- (~ 8-)(
0.)C
'\ _ kQ
'\..TF t 'I-~)~hJ-
d 6 :::0
y.
~
d. )( L\l.1\-ll3~0
dv
()~(~'t1-1/1- \-(~;l-'f)g2-
0
-l- ~ 3{~+
(g.1l-Y )
3 xl.
I
'Z-
d.
(1\0.'1': ().lW\
[
i. The value ofx for which E, is a maximum
(rPF/t!2-- ~ {ii+-./J51
1--
On the axes below, sketch E, versus x for points on the x-axis from x = -2R to x = +2R.
E,
x
2R
e.
= R/2 and released from rest. Qualitativel
\.;vvCM..'V~ V QN\.- .J-I'A.
An electron is placed at x
a ~
~~
wI
70
~t~
. describe its Jiubsequent
;vJ"""""OW.
4 (ostll
,~-fv\fM-tU]Yl..
e
m
""I=~1-
Tf2. Zf';(L
I'otcntial and Field AI' F,'cc Rcsponsc problcms
)'
+
Questions
A-l-q
-d ~
..o,
:0
,
B
2d
C
3d
,
...E
I
---
-d
---
Questions I -2 Two positive charges of magnillide q are
each a distance d from the origin A of a coordinate system,
as showil above.
I. ,At which of the following points is the electric field
least in magnitude?
(A) A
5-6
(8) 8
(C) C
(D) D
Positivc charge Q is uniformly distributed over a thin ring
of radius a that lics in a plane perpendicular to the x-axis
with its ccnter at the origin 0, as shown above.
5. The potential V at points on the x-axis is represented
which of the following functions?
(A)
Vex) =
kQ
x2 + 02
(E) E
(C)
Vex) =
2. At which of the following points is the electric potential
kG
2"
(D) Vex) = kQ
x
x
greatest in magnitude?
(A) A
(8) 8
(C) C
(D) 0
(E) E
Questions 3-4 refer to a sphere of radius R that has
positive charge Q uniformly distributed on its surface
3. Which of the following represents the magnitude of the
electric field E and the potential V as functions ofr,
the distance fromthc center of the sphcre, when r < R
?
J;;
y
(A) 0
kQ/R
(8) 0
kQ/r
(C) 0
0
(D) kQ/r'
0
(E) kQ/R'
0
4. Which of the following represents the magnitudc. of the
electric field E and the potential Vas functions of r.
the distance from the ccnter of sphere. whcn r> I{ ?
J;;
V
(A) kQ/I{'
kQ/1{
(8) kQ/1{
kQ/1{
(C) kQ/1{
kQ/r
(D) kQIr'
kQ/r
(E) kQ/r'
kQ/r'
(E) Vex) = kQ
a+x
6. Which of the following graphs best represents
electric field along the positive x-axis?
(8)
Ex
oP=-x
te) E.
(E):~.
o~
the
•
by
Potential
and Field AP Free Response
problems
10. A positive electric charge is moved at a constanl speed
between two locations in an electric field. with no
work done by or against the field at any time during
the motion. This situation can occur only if the
7. Concentric conducting spheres of radii a and 2a bear
equal but opposite charges +Q and -Q. respectively.
Which of the following graphs best represents the
electric potential Vas a function of r ?
'A:~,(B'L,
o
0
20
a
a
(A) charge is moved in the direction of the field
(B) charge is moved opposite to the direction of the
field
(C) charge is moved perpendicular to an equipotential
line
(D) charge is moved along an equipotential line
20
(E) electric field is uniform
II. A positive charge of 3.0 x 10" coulomb is placed in an
upward directed uniform electric field of 4.0 x 10'N/c.
When the charge is movcd 0.5 meter upward, the work
done by the electric force on the charge is
8. What is the radial component of the electric field
associated with the potential V = a(' where a is a
constant?
A) _2a(1
B) -2a('
D) 2a('
E) 2a(1
(A)6xl0"'J
(B) 12x 10".1
(D) 8 x 10' J
(E) 12 x 10' J
(C) 2 x 10' J
12. A potential difference V is maintained between two
large, parallel conducting plates. An electron sia'1s
from rest on the surface of one plate and accelerates
toward the other. Its speed as it reaches the second
plate is proportional to
(A) IIV
(B) 1/
JV
(C)
JV
(D) V
(E) V'
A
a
c
9.
The graph above shows the eleclric potential V in a
region of space as a function of position along the
x-axis. At which point would a charged particle
experience the force of greatest magnitude?
(A) A
(E) E:
(B) B
(C) C
(D) 0
~4 ~D SI)
CIt)
leD
11
<013
A- \1.("