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AP Physics C: Electricity and Magnetism
2015 Free-Response Questions
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ADVANCED PLACEMENT PHYSICS C TABLE OF INFORMATION
CONSTANTS AND CONVERSION FACTORS
Proton mass, m p
1.67 – 10 27 kg
Neutron mass, mn
1.67 – 10 27 kg
Electron mass, me
9.11 – 10 31 kg
6.02 – 10 23 mol 1
Avogadro’s number, N 0
R
Universal gas constant,
8.31 J (mol <K)
1.60 – 10 19 C
1 electron volt, 1 eV
1.60 – 10 19 J
Speed of light,
Universal gravitational
constant,
Acceleration due to gravity
at Earth’s surface,
3.00 – 108 m s
c
6.67 – 10 11 N<m 2
G
kg 2
9.8 m s2
g
1.38 – 10 23 J K
Boltzmann’s constant, k B
1 unified atomic mass unit,
1u
1.66 – 10 27 kg
931 MeV c 2
Planck’s constant,
h
6.63 – 10 34 J <s
4.14 – 10 15 eV <s
hc
1.99 – 10 25 J <m
1.24 – 103 eV < nm
e0
8.85 – 10 12 C2 N < m 2
Vacuum permittivity,
1 4 pe0 Coulomb’s law constant, k
m0
Vacuum permeability,
m0 4 p Magnetic constant, k „
1 atmosphere pressure,
UNIT
SYMBOLS
e
Electron charge magnitude,
meter,
kilogram,
second,
ampere,
kelvin,
PREFIXES
Factor
Prefix
Symbol
m
kg
s
A
K
mole,
hertz,
newton,
pascal,
joule,
1 atm
mol
Hz
N
Pa
J
9.0 – 109 N< m 2
C2
4 p – 10 7 (T <m) A
1 – 10 7 (T< m) A
1.0 – 105 N m 2
watt,
coulomb,
volt,
ohm,
henry,
W
C
V
W
H
1.0 – 105 Pa
farad,
tesla,
degree Celsius,
electron volt,
F
T
’C
eV
VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES
D
q
0
30D
37D
45D
53D
60D
90D
109
giga
G
sin q
0
12
35
2 2
45
3 2
1
106
mega
M
cos q
1
3 2
45
2 2
35
12
0
103
kilo
k
tan q
0
3 3
34
1
43
3
‡
10 2
centi
c
10 3
milli
m
6
micro
m
10 9
nano
n
10 12
pico
p
10
The following assumptions are used in this exam.
I. The frame of reference of any problem is inertial unless otherwise
stated.
II. The direction of current is the direction in which positive charges
would drift.
III. The electric potential is zero at an infinite distance from an isolated
point charge.
IV. All batteries and meters are ideal unless otherwise stated.
V. Edge effects for the electric field of a parallel plate capacitor are
negligible unless otherwise stated.
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ADVANCED PLACEMENT PHYSICS C EQUATIONS
MECHANICS
Ãx
Ãx 0 a x t
x
x0 Ãx 0 t Ãx2

a

F

J

p
1 2
at
2 x
Ãx 0 2 a x x x0 2

Fnet
m

ÇF
m

dp
dt

Ô F dt

Dp

mv


F f … m FN
 
Ô F dr
DE
W
K
1 2
mÃ
2
P
P
dE
dt
 
F v
DUg
ac

t

a
I
mg Dh
Ã2
r
Us
 
r–F

Çt
I
x

t net
I
2
Ô r dm
Ç mr
Ã
rw

L
 
r – p
1 2
Iw
2
w0 at
q
q0 w0 t 1
2
k Dx 2
xmax cos( wt f

FE

E

FE
q
Ex
DV
V
Q
e0
dV
dx
 
Ô E dr
DV
Q
C
1 q1q2
4pe0 r
C
k e0 A
d
Cp
Ç Ci
1
Cs
ÇC
i
1
i
i
m
k

E
Tp

2p
g
I
I
1
f
R

FG
Gm1m2
r2
Gm1m2
r
Rs
1
Rp
P
-3-
P
Q
q
R
r
t
U
=
=
=
=
=
=
=
V=
v =
r =
F =
k =

FM

area
magnetic field
capacitance
distance
electric field
emf
force
current
current density
inductance
length
number of loops of wire
per unit length
number of charge carriers
per unit volume
power
charge
point charge
resistance
radius or distance
time
potential or stored energy
electric potential
velocity or speed
resistivity
flux
dielectric constant
 
qv – B

Ô B  d 
dQ
dt
2p
2p
w
UC
=
=
=
=
=
=
=
=
=
=
=
=
N =
q
Ç rii
i
1
4pe0
qV
I
F
I
J
L

n
UE
A
B
C
d
E
ε
Ts
UG
1 2
at
2
1 q1q2
4pe0 r 2
 
Ô E  dA
2

Iw
w
acceleration
energy
force
frequency
height
rotational inertia
impulse
kinetic energy
spring constant
length
angular momentum
mass
power
momentum
radius or distance
period
time
potential energy
velocity or speed
work done on a system
position
coefficient of friction
angle
torque
angular speed
angular acceleration
phase angle

k D x
1
QDV
2
r
A

rJ
T
Ç mi xi
Ç mi
xcm
K
w2r
a =
E=
F =
f =
h =
I =
J =
K =
k =
 =
L =
m=
P =
p =
r =
T =
t =
U=
v =
W=
x =
m =
q =
t =
w =
a =
f =

Fs
ELECTRICITY AND MAGNETISM
1
C DV 2
2

dB

F
m0 I

m0 I d  – r
4p r 2
 
Ô I d – B
Bs
m0 nI
Nevd A
FB
Ô B  dA
DV
R
e
Ô E  d 
e
L
Ç Ri
i
1
ÇR
i
I DV
i
UL




dI
dt
1 2
LI
2
d FB
dt
ADVANCED PLACEMENT PHYSICS C EQUATIONS
GEOMETRY AND TRIGONOMETRY
Rectangle
A bh
1
bh
2
Circle
A
pr2
C
2p r
s rq
Rectangular Solid
V wh
Cylinder
V
pr 
S
2p r  2p r
df
dx
A = area
C = circumference
V = volume
S = surface area
b = base
h = height
 = length
w = width
r = radius
s = arc length
q = angle
Triangle
A
CALCULUS
S
d ax
e dx
aeax
q
2
a cos ax d
>cos ax @
dx
a sin ax Ôe
n
dx
ax
dx
dx
Ôxa
4 3
pr
3
4p r
2
a 2 b2
1
x
d
>sin ax @
dx
Ôx
r
Right Triangle
1 n 1
x , n › 1
n 1
1 ax
e
a
ln x a
Ô cos ax dx
1
sin ax a
Ô sin ax dx
1
cos ax a
c2
sin q
a
c
cos q
b
c
tan q
nx n 1
d
ln ax dx
Sphere
V
d n
x dx
s
2
d f du
du dx
a
b
c
q
VECTOR PRODUCTS
 
A B AB cos q
 
A–B
AB sin q
a
90°
b
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2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
PHYSICS C: ELECTRICITY AND MAGNETISM
SECTION II
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this
booklet in the spaces provided after each part.
E&M.1.
A parallel-plate capacitor is constructed of two parallel metal plates, each with area A and separated by a
distance D. The plates of the capacitor are each given a charge of magnitude Q, as shown in the figure above.
Ignore edge effects.
(a)
i. On the figure above, draw an arrow to indicate the direction of the electric field between the plates.
ii. On the figure above, draw an appropriate Gaussian surface that will be used to derive an expression for
the magnitude of the electric field E between the plates.
iii. Using Gauss’s law and the Gaussian surface from part (a)-ii, derive an expression for the magnitude
of the electric field E between the plates. Express your answer in terms of A, D, Q, and physical
constants, as appropriate.
The space between the plates is now filled with a dielectric material that is engineered so that its dielectric
constant varies with the distance from the bottom plate to the top plate, defined by the x-axis indicated in the

Q
i , where k is a
diagram above. As a result, the electric field between the plates is given by E 0
x D
e0 k 0 e
A
positive constant. Express all algebraic answers to the remaining parts in terms of A, D, Q, k0 , x, and physical
constants, as appropriate.
(b) Determine an expression for the dielectric constant k as a function of x.
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2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(c)
i. Write, but do NOT solve, an equation that could be used to determine the potential difference V
between the plates of the capacitor.
ii. Using the equation from part (c)-i, derive an expression for the potential difference VD V0 , where
VD is the potential of the top plate and V0 is the potential of the bottom plate.
(d) Determine the capacitance of the capacitor.
(e) The energy stored in the capacitor that has a varying dielectric is UV . A second capacitor that has a constant
dielectric of value k0 is also given a charge Q. The energy stored in the second capacitor is UC . How do the
values of UV and UC compare?
____ UV UC
____ UV ! UC
____ UV
UC
Justify your answer.
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-6-
2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
E&M.2.
A student performs an experiment to determine the emf e and internal resistance r of a given battery. The
student connects the battery in series to a variable resistance R, with a voltmeter across the variable resistor, as
shown in the figure above, and measures the voltmeter reading V as a function of the resistance R. The data are
shown in the table below.
Trial #
1
2
3
4
5
6
Resistance W 0.50
1.0
2.0
3.0
5.0
10
Voltage V 5.6
7.4
9.4
10.6
10.9
11.4
1 R 1 W 1 V 1 V 2.00
1.00
0.50
0.33
0.20
0.10
0.179
0.135
0.106
0.094
0.092
0.088
(a)
i. Derive an expression for the measured voltage V. Express your answer in terms of R,
physical constants, as appropriate.
e,
r, and
ii. Rewrite your expression from part (a)-i to express 1 V as a function of 1 R .
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-7-
2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(b) On the grid below, plot data points for the graph of 1 V as a function of 1 R . Clearly scale and label all
axes, including units as appropriate. Draw a straight line that best represents the data.
(c) Use the straight line from part (b) to obtain values for the following.
i.
e
ii. r
(d) Using the results of the experiment, calculate the maximum current that the battery can provide.
(e) A voltmeter is to be used to determine the emf of the battery after removing the battery from the circuit.
Two voltmeters are available to take this measurement—one with low internal resistance and one with high
internal resistance. Indicate which voltmeter will provide the most accurate measurement.
____ The voltmeter with low resistance will provide the most accurate measurement.
____ The voltmeter with high resistance will provide the most accurate measurement.
____ The two voltmeters will provide equal accuracy.
Justify your answer.
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-8-
2015 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
E&M. 3.

A circular wire loop with radius 0.10 m and resistance 50 W is held in place horizontally in a magnetic field B
directed upward at an angle of 60° with the vertical, as shown in the figure above. The magnetic field in the
direction shown is given as a function of time t by B(t ) a 1 bt , where a = 4.0 T and b = 0.20 s 1 .
(a) Derive an expression for the magnetic flux through the loop as a function of time t.
(b) Calculate the numerical value of the induced emf in the loop.
(c)
i.
Calculate the numerical value of the induced current in the loop.
ii. What is the direction of the induced current in the loop as viewed from point P ?
____ Clockwise
____ Counterclockwise
Justify your answer.
(d) Assuming the loop stays in its current position, calculate the energy dissipated in the loop in 4.0 seconds.
(e) Indicate whether the net magnetic force and net magnetic torque on the loop are zero or nonzero while the
loop is in the magnetic field.
Net magnetic force:
____ Zero
____ Nonzero
Net magnetic torque:
____ Zero
____ Nonzero
Justify both of your answers.
STOP
END OF EXAM
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