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Instructor: ALSIN, Michael Course: AP Physics Conductors & Capacitors Dielectrics Circuits RC Circuits Electric Field Magnetic Fields Inductance (LR, LC circuits) Electromagnetism Design Observe, Measure Analyze Data, Errors Gauss’s Law Lab Electrostatics E&M Question Day Date Name (LAST, First):_________________, __________________ Block (circle): 1 2 3 4 5 6 7 8 Date (MM/DD/YY): ___/___/___ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ 99.A.1 99.A.2 99.A.3 00.A.1 00.A.2 00.A.3 01.A.1 01.A.2 01.A.3 02.A.1 02.A.2 02.A.3 03.A.1 03.A.2 03.A.3 04.A.1 04.A.2 04.A.3 05.A.1 05.A.2 05.A.3 06.A.1 06.A.2 06.A.3 07.A.1 07.A.2 07.A.3 08.A.1 08.A.2 08.A.3 09.A.1 09.A.2 09.A.3 10.A.1 10.A.2 10.A.3 11.A.1 11.A.2 11.A.3 12.A.1 12.A.2 12.A.3 Test Analysis Form C 5/11/2012 10:29 AM 1
2 of 2 2
TABLE OF’ INFORMATION DEVELOPED FOR 2012
CONSTANTS AND CON\ ERSION FAQ WRS
Proton mass,
— 167
10
kg
I
it)
kg
1 electron s di. I eV
kg
Speed of light.
Neutron mass, or
Electron mass, In
As gadr’
=
number. .\
0
*11
l0
,(L
I th mal
-,
Bltzmann’ cnsmnt.
=
l3s
.
nmi.k
lo
.
1 u
=
I
Planck’s constant.
h
=
l
=
Vacuum perIllittis it.
constant.
k
=
1 atmosphere pressure.
PREFIXES
Factor
io
S
nev ton.
A
K
pascal.
joule,
G
mega
NI
cosO
kilo
k
tan6
milli
300
10’
I0
C
ni
-
:n
S m
It)
kg
=
03
.o3 x l0
J.5
=
4 14 ,. 1
1.09> 10
Jon
=
C
Nin
1(1
mol
Hz
N
Pa
J I
124
0(1 x
10
Nnr C
=
4r x 10
T.m A
u,, 4
=
1
10
1 atm
=
1.0
Y
watt,
coulomb,
solt,
ohm,
henrs,
MeV
=
10’ eV.mn
T.m,, A
l0 N/ nf
W
C
V
=
1.0 x 101 Pa
farad.
tesla.
degree Celsius.
elect! on—s nit.
F
‘F
C
e’
H
VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES
giga
10
r
mole.
hertz.
Snibo1
centi
I
ni
ke
Prefix
io-
1
(, —
=
I 4€
p
Magnetic constant. k’
UNIT
SYMBOLS
I0
1 60
-
5.85
Vacuum permeability.
meter.
kiloeram.
oml,
ampere.
kelvin.
<
J K
I unitted at’niic mas unit.
Coulomb’s law
—
=
I nis era1 era itational
en-.tant.
Aceciera’ den due to eras it
,t Earta ‘tiflJ.O,
-
S.’ I i
t nrs eral ca’ constant.
‘
160
Electron charge magnitude,
m
micro
It)
nano
n
10
pim
p
60
90
0
1
4 5
3/2
1
i
2
I
3 5
1 ‘2
4 3
3
L
0
oo
The fo1loing con> entions are used in this exam
1. Unless otherwise stated, the frame of reference of any pr&hlein is
assumed to be inertial.
11. The direction f an> electric current is the direction of tlk u. of positise
charge corn entional current I,
111. For any is lated electric charge, the electric potentid js defined as ze
an infinite dismnce from the charge
-22
at
3
ADVANCED PLACEMENT PHYSICS C EQUATIONS DES’ ELOPED FOR 2012
MECHANICS
a
I
at
c ‘v
,,at
1
a(r
2
r
—t
1)
ma
F
JFdr
3
p
\p
in
*
fr,
<it\
JF.dr
=
1
K
mu
—
=
cit
P
r)
accelerati n
tc ice
1= trequenc y
height
I— rotational mertia
impulse
K= kinetic, energy
spring .a ntant
length
angular mc mentum
In = mass
N = normal lorc
p wer
momentum
I’ = radius or distance
position sector
7’ = period
time
potential energy
velocity or speed
w = work done on a system
5— pasition
coefficient of friction
mgh
A
B
C
d
B
qq,
r
1
4irE
F
q
F
—
F
fr
1
J
B
E.JA
B—
Jr
=
—
—
=
1
n
‘ç’q
r
1
ql
Uj,
icA
d
qq
r
A
=
P
=
Q
1
—
q
R
r
=
t
=
V
v
p
1
—
=
(1
0= angle
torque
(0= angular speed
angular acceleration
phase angle
-
—
ELEC RIC1TY AND MAGNETISM
=
=
=
area
mgnetic field
capautance
distance
electric field
cml
trce
current
current density
inductance
length
number t I I ) ps ci sne
per unit length
number ol charge arriem
per unit solume
posser
charge
point charge
resistance
distance
time
potential or stored energy
electric potential
velocity oi speed
resistisity
magnetic flux
1
dielectric constant
dQ
cit
=
LI
=
=
B*d€=p I
—kg
=
t
—
t
r
—
a’
F
=
a
—
*
mr
L
T
rw
rxp
=
I
=
2,r
I
V
77)
-
—
-
F
pJ
Mu A
JR
IdJxr
fId€XB
—pnl
-
JB.dA
F
2ir—
K
1 ,n
Gm
t((t
r
1
R
P
R
IV
Gm,n
00 +144u1
B
p
4t
RR
Jo>
7I
(0—0)
JB=
R=1
A
F
wJ
11 —mrm
r
—
cos(iut + c
=
=
o
1
1(1
17
t,
=
1
U
=
a’ B
£
L’
11
cit
4
-33
4
ADVANCED PLACEMENT PHYSICS C EQUATIONS DEVELOPEI) FOR 2012
GEOMETRY AND TRIGONOMETR
Rectangle
A area
C = circumference
V solume
5 = surface area
b =base
Ii = height
/ = length
hh
1h
1
—
Circle
yrr
A
C
=
dl
di
—
Triangle
A
CAL CUL1 S
r
2,rr
—
ci’ do
do dr
d
x
ci
dx
)
fi
ax
-
d
(In v)
dx
radius
d
Rectangular Solid
k = iirh
cit
(sini)
ci
Cy finder
V
S
=
—
r2f
2rc
=
+
2
2r
di
7
Jc
Sphere
r cit
1’ —-irr
I—
x
J
S
=
sinO
=
=
ccsr
=
—sinx
1
x
nfl
,
a
—l
mt
sindr
Right Triangle
x
—
J cosxdx
4r
a +b
C
=
—
—cosx
=
a
—
C
b
cosfl
—
9O
12
C
tanO
a
—
-44
5
1999
PHYSICS C
SECTION II, ELECTRICITY AND MAGNETISM
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. Shov all your work in the pink
booklet in the spaces provided afler each part. NOT in this green insert.
E&M I.
An isolated conducting sphere of radius a
(a)
Determine the charge
0
Q
=
0.20 m is at a potential of —2,000 V.
on the sphere.
The charged sphere is then concentrically surrounded by two uncharged conducting hemispheres of
inner radius b
0.40 m and outer radius c
0.50 m, which are joined together as shown above,
forming a spherical capacitor. A wire is connected from the outer sphere to ground, and then removed.
(b)
Determine the magnitude of the electric field in the following regions as a function of the
distance r from the center of the inner sphere.
i.
ii.
ijj
iv
r
a
<
b
r
<
<
>
a
r
r
c
<
b
<
c
(c)
Determine the magnitude of the potential difference between the sphere and the
conducting shell.
(d)
Determine the capacitance of the spherical capacitor.
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5
6
1999 PHYSICS C—E & M
B (out of page)
P
E&M 2.
::::
N,
0
A uniform magnetic field B exists in a region of space defined by a circle of radius a
0.60 m
as shown above. The magnetic field is perpendicular to the page and increases out of the page at
a constant rate of 0.40T/s. A single circular loop of wire of negligible resistance and radius
r
0.90 m is connected to a lightbulb with a resistance R = 5.0 Q, and the assembly is placed
concentrically around the region of magnetic field.
(a)
Determine the emf induced in the loop.
(b)
Determine the magnitude of the current in the circuit. On the figure above, indicate the
direction of the current in the loop at point 0.
(c)
Determine the total energy dissipated in the lightbulb during a 15 s interval.
The experiment is repeated with a loop of radius b
0.40 in placed concentrically in the same
magnetic field as before. The same lighthulh is connected to the loop, and the manetic field again
increases out of the page at a rate of 0.40 Ts. Neglect any direct effects of the field on the lightbulb
itself.
(d)
State whether the brightness of the bulb will be greater than. less than, or equal to the
brightness of the bulb in part (a). Justify your answer.
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6
7
1999 PHYSICS C—E & M
V
E&M 3.
The nonconducting ring of radius R shown above lies in the vz-plane and carries a uniformly
distributed positive charge Q.
(a)
Determine the electric potential at points along the x-axis as a function of x.
(b) i. Show that the x-component of the electric field along the x-axis is given by
Qx
F
—
4c
(
0
R
ii. What are they- and z- components of the electric field along the v-axis?
(C)
Determine the following.
i. The value of x for which E is a maximum
ii. The maximum electric field E
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7
8
1999 PHYSICS C—E & M
(d)
On the axes below, sketch E versus x for points on the xaxis from x
tox = +2R.
=
—2R
I
—2R
(e)
—R
An electron is placed at x
R
=
2R
R/2 and released from rest. Qualitatively describe its
subsequent motion.
STOP
END OF SECTION II, ELECTRICITY AND MAGNETISM
8
9
EEEEEEEEEEEEEEEEE
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, ELECTRICITY AND MAGNETISM
Time—45 niinutes
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 the
pink booklet in the spaces provided after each part, NOT in this green insert.
A
l02
B&M I
Lightbulbs A, B, and C are connected in the circuit shown above,
(a) List the bulbs in order of their brightness, from brightest to least bright. If any bulbs have the same
brightness, state which ones. Justify your answer.
£ =42 V
A
10 2
Now a switch S and a 5.0 mH inductor are added to the circuit, as shown above. The switch is closed at
time t=O.
(b) Determine the currents
‘A’ ‘B’
and ‘c for the following times.
i. Immediately after the switch is closed
ii. A lone time after the switch is closed
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AP is a registered trademark of the College Entrance Examination Board.
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-29
10
EEEEEEEEEEEEEEEEE
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
(C)
On the axes below, sketch the magnitude of the potential difference VL across the inductor as a function
of time, from immediately after the switch is closed until a long time after the switch is closed.
0
E42V
j
1
° l2Qs.
5.0 l.LF
(d) Now consider a similar circuit with an uncharged 5.0 pP capacitor instead of the inductor, as shown above.
The switch is again closed at time t = 0. On the axes below, sketch the magnitude of the potential difference
Vcap across the capacitor as a function of time, from immediately after the switch is closed until a long time
after the switch is closed.
ap
1
0
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AP is a registered trademark of the College Entrance Examination Board.
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-310
11
EEEEEEEEEEEEEEEEE
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
E&M2.
Three particles, A, B, and C, have equal positive charges
Q
and are held in place at the vertices of an
equilateral triangle with sides of length f, as shown in the figures below, The dotted lines represent the
bisectors for each side. The base of the triangle lies on the x -axis, and the altitude of the triangle lies on
the y-axis.
y
V
Q
x
Figure I
Figure 2
(a)
, the intersection of the three bisectors, locates the geometric center of the triangle and is one
1
i. Point P
point where the electric field is zero. On Figure 1 above, draw the electric field vectors EA, EB, and E
1 due to each of the three charges. Be sure your arrows are drawn to reflect the relative magnitude
at P
of the fields.
ii. Another point where the electric field is zero is point P
2 at (0, y
). On Figure 2 above, draw electric
2
field vectors EA, EB, and E at P
2 due to each of the three point charges. Indicate below whether the
magnitude of each of these vectors is greater than, less than, or the same as for point P
.
1
1
Greater than at P
1
1
Less than at P
1
1
The same as at P
EA
‘B
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AP is a registered trademark of the College Entrance Examination Board,
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-411
12
EEEEEEEEEEEEEEEEE
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
(h) Explain why the x-component of the total electric field is zero at any point on the v-axis.
(c) Write a general expression for the electric potential V at any point on the y-axis inside the triangle in terms
of Q, 1, and v.
2 at
(d) Describe how the answer to part (c) could be used to determine the v-coordinates of points P
1 and P
which the electric field is zero. (You do not need to actually determine these coordinates.)
E & M 3.
A capacitor consists of two conducting, coaxial, cylindrical shells of radius a and b. respectively, and length
L>> b. The space between the cylinders is filled with oil that has a dielectric constant w. Initially both cylinders
are uncharged, but then a battery is used to charge the capacitor, leaving a charge +Q on the inner cylinder and
—Q on the outer cylinder, as shown above. Let r be the radial distance from the axis of the capacitor.
(a) Using Gauss’s law, determine the electric field midway along the length of the cylinder for the following
values of r. in terms of the given quantities and fundamental constants. Assume end effects are negligible.
i. a<r<b
ii. b
<
r
< <
L
(h) Determine the following in terms of the given quantities and fundamental constants.
i. The potential difference across the capacitor
ii. The capacitance of this capacitor
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13
EEEEEEEEEEEEEEEEE
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
(c) Now the capacitor is discharged and the oil is drained from it. As shown above, a battery of emf £ is
connected to opposite ends of the inner cylinder and a battery of emf 32 is connected to opposite ends
of the outer cylinder. Each cylinder has resistance R. Assume that end effects and the contributions to the
magnetic field from the wires are negligible. Using Ampere’s law, determine the magnitude B of the
magnetic field midway along the length of the cylinders due to the current in the cylinders for the following
values of r.
i. a<r<b
ii. b<r<<L
STOP
END OF SECTION II, ELECTRICITY AND MAGNETISM
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON SECTION II,
ELECTRICITY AND MAGNETISM, ONLY. DO NOT TURN TO ANY OTHER TEST MATERIALS.
Cop) right © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.
AP is a registered trademark of the College Entrance Examination Board.
-613
___
_______
14
2001 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, ELECTRICITY AND MAGNETISM
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 the
pink booklet in the spaces provided after each part, NOT in this green insert.
2km
Ground
Ground
•
2 km
3km
t+30C
Note: Figures not drawn to scale.
-‘-
—30 C
E&M 1.
A thundercloud has the charge distribution illustrated above left. Treat this distribution as two point charges, a
negative charge of —30 C at a height of 2 km above ground and a positive charge of +30 C at a height of 3 km.
The presence of these charges induces charges on the ground. Assuming the ground is a conductor, it can be
shown that the induced charges can be treated as a charge of +30 C at a depth of 2 km below ground and a
, which is just above the ground
1
charge of—30 Cat a depth of 3 1cm, as shown above right. Consider point P
.
1
directly below the thundercloud, and point P
, which is 1 km horizontally away from P
2
.
1
(a) Determine the direction and magnitude of the electric field at point P
(b)
.
2
i. On the diagram above, clearly indicate the direction of the electric field at point P
?
1
ii. How does the magnitude of the field at this point compare with the magnitude at point P
Equal
Greater
Less
Justify your answer
(c) Letting the zero of potential be at infinity, determine the potential at these points.
1
i. Point P
ii. Point
“2
.
1
(d) Determine the electric potential at an altitude of 1 km directly above point P
(e) Determine the total electric potential energy of this arrangement of charges.
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14
15
2001 AP PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
10
>
2
pm
8
6
4
V
C
(I
20
40
60
$()
IOU
I 20
140
160
180 20f)
Time (mm)
E&M2.
You have been hired to determine the internal resistance of 8.0 pF capacitors for an electronic component
manufacturer. (Ideal capacitors have an infinite internal resistance—that is, the material between their plates
is a perfect insulator. In practice, however, the material has a very small, but nonzero, conductivity.) You cannot
simply connect the capacitors to an ohmmeter, because their resistance is too large for an ohmmeter to measure.
Therefore you charge the capacitor to a potential difference of 10 V with a battery, disconnect it from the battery
and measure the potential difference across the capacitor every 20 minutes with an ideal voltmeter, obtaining the
graph shown above.
(a) Determine the internal resistance of the capacitor.
The capacitor can be approximated as a parallel-plate capacitor separated by a 0.10 mm thick dielectric
with =5.6.
(h) Determine the approximate surface area of one of the capacitor “plates.”
(c) Determine the resistivity of the dielectric.
(d) Determine the magnitude of the charge leaving the positive plate of the capacitor in the first 100 mm,
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15
16
2001 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
,_-Fixed Horizontal Wires
d_____
—Rod (mass
in.
resistance R)
Cable
E&M 3.
The circuit shown above consists of a battery of ernf 8 in series with a rod of length £, mass in, and
resistance R. The rod is suspended by vertical connecting wires of length d, and the horizontal wires that
connect to the battery are fixed. All these wires have negligible mass and resistance. The rod is a distance r
above a conducting cable. The cable is very long and is located directly below and parallel to the rod. Earth’s
gravitational pull is toward the bottom of the page. Express all algebraic answers in terms of the given quantities
and fundamental constants.
(a) What is the magnitude and direction of the current I in the rod?
(b) In which direction must there be a current in the cable to exert an upward force on the rod? Justify your
answer.
(c) With the proper current in the cable, the rod can be lifted up such that there is no tension in the connecting
wires. Determine the minimum current I in the cable that satisfies this situation.
(d) Determine the magnitude of the magnetic flux through the circuit due to the minimum current I determined
in part (c).
END OF SECTION II, ELECTRICITY AND MAGNETISM
Copyright © 2001 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
10
16
17
2002 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, ELECTRICITY AND MAGNETISM
Timc—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 the
pink booklet in the spaces provided after each part, NOT in this green insert.
,R
600,,’
‘R=O.lO m
E&M 1.
A rod of uniform linear charge density )
+1.5 x 10 C/rn is bent into an arc of radius R
is placed with its center at the origin of the axes shown above.
0.10 m. The arc
(a) Determine the total charge on the rod.
(h) Determine the magnitude and direction of the electric field at the center 0 of the arc.
(c) Determine the electric potential at point 0.
A proton is now placed at point 0 and held in place. Ignore the effects of gravity in the 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.
(e) The proton is now released, Describe in words its motion for a long time after its release.
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________IC
___
___Wv___
18
2002 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
s
R—50k2
E&M 2.
Your engineering firm has built the RC circuit shown above. The current is measured for the time t after the
switch is closed at r = 0 and the best-fit curve is represented by the equation 1(r)
milliamperes and t is in seconds.
=
5.20 e
110
where I is in
(a) Determine the value of the charging voltage V
0 predicted by the equation.
(b) Determine the value of the capacitance C predicted by the equation.
(c) The charging voltage is measured in the laboratory and found to be greater than predicted in part (a).
i. Give one possible explanation for this finding.
ii. Explain the implications that your answer to part i has for the predicted value of the capacitance.
(d) Your laboratory supervisor tells you that the charging time must be decreased. You may add resistors or
capacitors to the original components and reconnect the RC circuit. In parts i and ii below, show how
to reconnect the circuit, using either an additional resistor or a capacitor to decrease the charging time.
i. Indicate how a resistor may he added to decrease the charging time. Add the necessary resistur and
connections to the following diagram.
Wv
T
ii. Instead of a resistor, use a capacitor. Indicate how the capacitor may be added to decrease the charging
time, Add the necessary capacitor and connections to the following diagram.
T
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19
2002 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
Vertical
E&M 3.
A circular wire loop with radius 0.10 m and resistance 50 Q is suspended horizontally in a magnetic field of
magnitude B directed upward at an angle of 60° with the vertical, as shown above. The magnitude of the field
in teslas is given as a function of time t in seconds by the equation B = 4(1 C. 2t).
—
(a) Determine the magnetic Ilux
through the ioop as a function of time.
(h) Graph the magnetic flux rn as a function of time
Ô,, (T
Ofl
the axes below.
)
2
m
010
OJ)5
0
-
1
2
4
5
6
7
8
9
10
‘—i(s)
—0,05
—0.10
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20
2002 AP PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
(c) Determine the magnitude of the induced emf in the loop.
(d)
i. Determine the magnitude of the induced current in the ioop.
ii. Show the direction of the induced current on the following diagram.
Vertical
(e) Determine the energy dissipated in the ioop from
t =
0 to
t =
4 s.
END OF SECTION II, ELECTRICITY AND MAGNETISM
8
20
21
2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
PHYSICS C
Section IL ELECTRICITY AND MAGNETISM
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 the
pink booklet in the spaces provided after each part. NOT in this green insert.
I
E&M.
1.
A spherical cloud of charge of radius R contains a total charge
varies according to the equation
p(r)
+Q with a nonuniform volume charge density that
for r
=
R and
—
p
=
0 for r
>
R,
where r is the distance from the center of the cloud. Express all algebraic answers in terms of
fundamental constants.
Q. R. and
(a) Determine the following as a function of r for r > R.
i. The magnitude E of the electric field
ii. The electric potential V
(b) A proton is placed at point P shown above and released. Describe its motion for a long time after its release.
(c) 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 of r as it strikes the cloud.
(d) Derive an expression for Po’
(e) Determine the magnitude E of the electric field as a function of r for r
R.
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21
22
2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
E&M,
2.
In the laboratory, you connect a resistor and a capacitor with unknown values in series with a battery of emf
E = 12 V You include a switch in the circuit. When the switch is closed at time t = 0, the circuit is completed.
and you measure the current through the resistor as a function of time as plotted below.
.
i(A)
4
8
12
A data-fitting program finds that the current decays according to the equation
(s)
16
i(t)
=
E
e -rj4
(a> Using common symbols for the battery, the resistor, the capacitor, and the switch, draw the circuit that you
constructed. Show the circuit before the switch is closed and include whatever other devices you need to
measure the current through the resistor to obtain the above plot. Label each component in your diagram.
(b) Having obtained the curve shown above, determine the value of the resistor that you placed in this circuit.
(c) What capacitance did you insert in the circuit to give the result above?
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22
23
2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
You are now asked to reconnect the circuit with a new switch in such a way as to charge and discharge the capacitor.
When the switch in the circuit is in position A, the capacitor is charging; and when the switch is in position B, the
capacitor is discharring, as represented by the graph below of voltage V across the capacitor as a function of time.
‘‘c (V)
Switch at A
Charging
Switch at B
Discharging
12
6
,1
t (s)
(d) Draw a schematic diagram of the RC circuit that you constructed that would produce the graph above. Clearly
indicate switch positions A and B on your circuit diagram and include whatever other devices you need to
measure the voltage across the capacitor to obtain the above plot. Label each component in your diagram.
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23
24
2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
w
S
Vertical
•
V
E
N
Antenna
\
Top View
E&M.
\
Side View
3.
An airplane has an aluminum antenna attached to its wing that extends 15 m from wingtip to wingtip. The plane is
traveling north at 75 m/s in a region where Earth’s magnetic field has both a vertical component and a northward
component, as shown above. The net magnetic field is at an angle of 55 degrees from horizontal and has a magnitude
of 6.0 x i- T
(a) On the figure below, indicate the direction of the magnetic force on electrons in the antenna. Justify your
answer.
ni
(b) Determine the magnitude of the electric field generated in the antenna.
(c) Determine the potential difference between the ends of the antenna.
(d) On the figure below, indicate which end of the antenna is at higher potential.
-I
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24
25
2003 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
(e) The ends of the antenna are now connected by a conducting wire so that a closed circuit is formed.
i. Describe the condition(s) that would be necessary for a current to be induced in the circuit. Give a
specific example of how the condition(s) could be created.
ii. For the example you gave in i. above, indicate the direction of the current in the antenna on the
figure below.
END OF SECTION II, ELECTRICITY AND MAGNETISM
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25
26
2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, ELECTRICITY AND MAGNETISM
Tinie—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 the
booklet in the spaces provided after each part, NOT in this green insert.
a
Conductor
Cross Section
E&M. 1.
The figure above left shows a hollow, infinite, cylindrical, uncharged conducting shell of inner radius ij and
outer radius r-, An infinite line charge of linear charge density +. is parallel to its axis but off center. An
enlarged cross section of the cylindrical shell is shown above right.
.
(a) On the cross section above right,
i. sketch the electric field lines, if any, in each of regions I, II, and III and
ii. use + and
—
signs to indicate any charge induced on the conductor.
(h) In thes paces below, rank the electric potentials at points a, b. c. d. and e from highest to lowest
(1 = highest potential>. If two points are at the same potential. give them the same number.
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26
27
2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
Cross Section
(c) 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 +2, 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 r 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.
ii.
iii.
I it
r<,
i
ri
r > r,
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6
27
28
2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
14
‘l2
—.---.--
20 V
--
1
R
lskc2:
OPFT
2
0
0
10
15
Time (s)
E&M. 2.
In the circuit shown above left, the switch S is initially in the open position and the capacitor C is initially
uncharged. A voltage probe and a computer (not shown) are used to measure the potential difference across the
capacitor as a function of time after the switch is closed. The graph produced by the computer is shown above
right. The battery has an emf of 20 V and negligible internal resistance. Resistor R
1 has a resistance of 15 k2
and the capacitor C has a capacitance of 20 jiF.
(a) Determine the voltage across resistor R
2 immediately after the switch is closed.
(b) Determine the voltage across resistor 1?-, a long time after the switch is closed.
.
2
(c) Calculate the value of the resistor K
(d) Calculate the energy stored in the capacitor a long time after the switch is closed.
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28
29
2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
(e) On the axes below, graph the current in I?, as a function of time from 0 to 15 s. Label the vertical axis with
appropriate values.
LLLJ
LLL
JJ
——————-——
i—
-
L1.
L_JJ
L
LJ_JL
L_LL_
—-—-—-
Current
in R,
-•----
——--—4—4—-
___.4”__
_J
L.
—-1 ——fr-—
-•----
--H----H————
-
__4_._I_
—-————
-.-—4—.3-—I.—.
JLL
0
10
15
Time (s)
Resistor R
2 is removed and replaced with another resistor of lesser resistance. Switch S remains closed for a
long time.
(f) Indicate below whether the energy stored in the capacitor is greater than, less than, or the same as it was
with resistor 1?, in the circuit.
Greater than
Less than
The same as
Explain your reasoning.
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29
30
2004 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
4e
T
3e
+
I
E&M. 3.
A rectangular loop of dimensions 3 £ and 4 £ lies in the plane of the page as shown above. A long straight wire
also in the plane of the page carries a current I.
(a) Calculate the magnetic flux through the rectangular loop in terms of 1, £, and fundamental constants.
Starting at time t
1(t)
=
=
0, the current in the long straight wire is given as a function of time t by
e, where
0
I
1 and k are constants.
(b) The current induced in the loop is in which direction?
Counterclockwise
Clockwise
Justify your answer.
The loop has a resistance R. Calculate each of the following in terms of R, J, k, £, and fundamental constants.
(c) The current in the loop as a function of time t
(d) The total energy dissipated in the loop from t
=
0 to t =
END OF SECTION II, ELECTRICITY AND MAGNETISM
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30
31
2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, ELECTRICITY AND MAGNETISM
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 the
pink booklet in the spaces provided after each part. NOT in this green insert.
(in)
(3.1
f\J
008
—ê4
(Hi)
E&M. 1.
Consider the electric field diagram above.
(a) Points A, B, and C are all located at y
0.06 rn.
i. At which of these three points is the magnitude of the electric field the greatest? Justify your answer.
ii. At which of these three points is the electric potential the greatest? Justify your answer.
(b) An electron is released from rest at point B.
i, Qualitatively describe the electron’s motion in terms of direction, speed. and acceleration.
ii. Calculate the electron’s speed after it has moved through a potential difference of 10 V.
(c) Points B and C are separated by a potential difference of 20 V. Estimate the magnitude of the electric field
midway between them and state any assumptions that you make.
(d) On the diagram, draw an equipotential line that passes through point D and intersects at least three electric
field lines.
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31
32
2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
E&M, 2.
In the circuit shown above, resistors 1 and 2 of resistance R
1 and R
2 respectively, and an inductor of
inductance L are connected to a battery of emf £ and a switch S. The switch is closed at time t = 0.
Express all algebraic answers in terms of the given quantities and fundamental constants.
,
(a) Determine the current through resistor I immediately after the switch is closed.
(b) Determine the magnitude of the initial rate of change of current, d!/dr. in the inductor.
(c) Determine the current through the battery a long time after the switch has been closed.
(d) On the axes below, sketch a graph of the current through the battery as a function of time.
Current
Time
Some time after steady state has been reached, the switch is opened.
(e) Determine the voltage across resistor 2 just after the switch has been opened.
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32
33
2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
I I.tl IIOhc
\lctcr
00
1>
(_)
c:,;
lCml
0
()
)
I
_4,
30
I
50
6)
7
I
\I)
I
‘))
1
H)
E&M. 3.
A student performs an experiment to measure the magnetic field along the axis of the long, 100-turn
solenoid PQ shown above. She connects ends P and Q of the solenoid to a variable power supply and
an ammeter as shown. End P of the solenoid is taped at the 0 cm mark of a meterstick. The solenoid can be
stretched so that the position of end Q can be varied. The student then positions a Hall probe* in the center
of the solenoid to measure the magnetic field along its axis. She measures the field for a fixed current of
3.0 A and various positions of the end Q. The data she obtains are shown below.
Trial
1
Position of End
(cm)
40
Q
Measured Magnetic Field (T)
(directed from P to Q)
n
(turns/rn)
9,70xl0
2
50
7.70xl0
3
60
6.80x10
4
80
4.90xl0
100
4.00xl0’
(a) Complete the last column of the table above by calculating the number of turns per meter.
*
A Hall Probe is a device used to measure the magnetic field at a point.
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33
34
2005 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
(b) On the axes below, plot the measured magnetic field B versus n. Draw a best-fit straight line for the data
points.
r
10.0
L.i..
J
L_LL
.L,L.11
.IjJJ ..Li
.JLL.L
;
9.0
I
.-.•_4
4.
t
.-—4--
1
T
8.0
._a
i
‘“t
70
-—
—
-t
-1—
--i———r-—r--r —
—.—-4—
4.0
r
2.0
rr’
.,LLi.
IJJJ
•i•i••
.LLI
I
0
LL_
TT1
.IJ
1
5.
I
-,
—‘r—i—i—-1——
I
I
‘__—I__
I
I
1
I
I
;
.,
,-
-
.
I
I
I
.
.
I
_a_
_J_J__L_L_
_i,J
—-1——I——--—i-—
—+—‘t—-I——I—— -—-—+‘—-,‘—-t—
I
1
I
50
100
.4...
---—r—i
.J
.‘
T11
1 A)
I
rr””
‘T’1 ‘1-
JI
—...———
,
;
F—f’—-4
—+—
—r—-
3.0
—
I:
6,u
5.0
,
———
L_L
150
LJ
—
------
200
250
300
,
I
(c) From the graph, obtain the value of ,u
, the magnetic permeability of vacuum.
0
(d) Using the theoretical value of #
value of p
0 computed in part (c).
=
4ir x lO (T.m)/A, determine the percent error in the experimental
END OF SECTION II, ELECTRICITY AND MAGNETISM
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35
2006 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 the
pink booklet in the spaces provided after each part. NOT in this green insert,
-Q
-Q
P
E&M 1.
The square of side a above contains a positive point charge +Q fixed at the lower left corner and negative point
charges —Q fixed at the other three corners of the square. Point P is located at the center of the square.
(a) On the diagram, indicate with an arrow the direction of the net electric field at point P.
(b) Derive expressions for each of the following in terms of the given quantities and fundamental constants.
i.The magnitude of the electric field at point P
ii. The electric potential at point P
(c) A positive charge is placed at point P. It is then moved from point P to point R, which is at the midpoint of
the bottom side of the square. As the charge is moved, is the work done on it by the electric field positive,
negative, or zero?
Positive
Negative
Zero
Explain your reasoning.
(d)
i. Describe one way to replace a single charge in this configuration that would make the electric field at
the center of the square equal to zero. Justify your answer.
ii. Describe one way to replace a single charge in this configuration such that the electric potential at the
center of the square is zero but the electric field is not zero. Justify your answer.
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35
36
2006 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
K.
E&M 2.
The circuit above contains a capacitor of capacitance C, a power supply of emf E, two resistors of resistances R
1
and R
2 and two switches. S
1 and S
. Initially, the capacitor is uncharged and both switches are open. Switch S
2
1
then gets closed at time t = 0.
.
(a) Write a differential equation that can be solved to obtain the charge on the capacitor as a function of time
(b) Solve the differential equation in part (a) to determine the charge on the capacitor as a function of time
t.
t.
Numerical values for the components are given as follows:
E = 12 V
C=0.060F
1 = R-, = 4700 2
R
(c) Determine the time at which the capacitor has a voltage 4.0 V across it.
After switch S has been closed for a long time, switch 5, gets closed at a new time r
=
0.
(d) On the axes below, sketch graphs of the current I in R
1 versus time and of the current 12 in K. versus time,
beginning when switch 5, is closed at new time t = 0. Clearly label which graph is I and which is 12
Current
rine
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2006 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
ix
)<
X
)<
X
w —H
E&M 3.
A loop of wire of width w and height/i contains a switch and a battery and is connected to a spring of force
constant k, as shown above. The loop carries a current tin a clockwise direction, and its bottom is in a constant,
uniform magnetic field directed into the plane of the page.
(a) On the diagram of the loop below, indicate the directions of the magnetic forces, if any, that act on each side of
the loop.
(b) The switch S is opened, and the loop eventually comes to rest at a new equilibrium position that is a distance x
from its former position. Derive an expression for the magnitude B
0 of the uniform magnetic field in terms of
the given quantities and fundamental constants.
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37
______
______
______
38
2006 AP® PHYSICS C: ELECTRICITY AND MAGNETISM
FREE-RESPONSE QUESTIONS
The spring and loop are replaced with a loop of the same dimensions and resistance R but without the battery and
switch. The new loop is pulled upward, out of the magnetic field, at constant speed v
0 Express algebraic answers to
the following questions in terms of B
. v
0
, R. and the dimensions of the loop.
0
(c)
i.
On the diagram of the new loop below, indicate the direction of the induced current in the loop as the loop
moves upward.
f
0
LI
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
ii. Derive an expression for the magnitude of this current.
(d) Derive an expression for the power dissipated in the loop as the loop is pulled at constant speed
Out of
the field.
(e) Suppose the magnitude of the magnetic field is increased. Does the external force required to pull the loop at
speed v
0 increase, decrease, or remain the same?
Increases
Justify your
Decreases
Remains the same
answer,
END OF EXAM
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38
39
2007 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
PHYSICS C: ELECTRICITY AND MAGNETISM
SECTION II
Tirne—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 the
pink booklet in the spaces provided after each part, NOT in this green insert.
Switch
-
__C=4000,uF
E&M
I.
A student sets up the circuit above in the lab. The values of the resistance and capacitance are as shown, but the
constant voltage E delivered by the ideal battery is unknown. At time t = 0, the capacitor is uncharged and the
student closes the switch. The current as a function of time is measured using a computer system. and the following
graph is obtained.
2.5
2.0
0.5
1
1-I
-iI
1
----
1.5
1.0
T
I
—-——
‘—-----1
---
•t+•••+••+..64,4_
—
o
2
4
6
lime (s)
8
,
10
(a) Using the data above, calculate the battery voltage S.
(b) Calculate the voltage across the capacitor at time t = 4.0 s.
(c) Calculate the charge on the capacitor at t
4,0 s.
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-539
______Greater
______Less
________________________
______The
40
2007 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(d) On the axes below, sketch a graph of the charge on the capacitor as a function of time.
o
—
10
Time (s)
(e) Calculate the power being dissipated as heat in the resistor at t
=
4.0 s.
(f) The capacitor is now discharged, its dielectric of constant = 1 is replaced by a dielectric of constant = 3.
and the procedure is repeated. Is the amount of charge on one plate of the capacitor at t = 4.0 s now greater than.
less than, or the same as before’? Justify your answer.
than
than
same
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-640
41
2007 AP® PHYSICS C ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
E&M
2.
In the figure above, a nonconducting solid sphere of radius a with charge +Q uniformly distributed throughout
its volume is concentric with a nonconducting spherical shell of inner radius 2a and outer radius 3a that has a
charge —Q uniformly distributed throughout its volume. Express all answers in terms of the given quantities and
fundamental constants.
(a) Using Gauss’s law, derive expressions for the magnitude of the electric field as a function of radius r in the
following regions.
i. Within the solid sphere (r <a)
ii. Between the solid sphere and the spherical shell (a <r
iii. Within the spherical shell (2a <r
iv. Outside the spherical shell (r
>
<
<
2a)
3a)
3a)
(b) What is the electric potential at the outer surface of the spherical shell (r
(c) Derive an expression for the electric potential difference V
—
=
3a )? Explain your reasoning.
V between points X and Y shown in the figure.
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-741
______Clockwise
______Counterclockwise
42
2007 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
P
x
v
B
L
x
x
x
x
x
x
x
x
x
x
x
x
Q
x
x
x
x
x
x
x
x
x
x
x
x
Top View
E&M
3.
In the diagram above, a nichrorne wire of resistance per unit length 2 is bent at points P and Q to form horizontal
conducting rails that are a distance L apart. The wire is placed within a uniform magnetic field of magnitude B
pointing into the page. A conducting rod of negligible resistance, which was aligned with end PQ at time t = 0. slides
to the right with constant speed v and negligible friction. Express all algebraic answers in terms of the given
quantities and fundamental constants.
(a) Indicate the direction of the current induced in the circuit.
Justify your answer.
(b) Derive an expression for the magnitude of the induced current as a function of time t.
(C) Derive an expression for the magnitude of the magnetic force on the rod as a function of time.
(d) On the axes below, sketch a graph of the external force f, as a function of time that must be applied to the rod
to keep it moving at constant speed while in the field. Label the values of any intercepts.
Time
o
(e) The force pulling the rod is now removed. Indicate whether the speed of the rod increases, decreases, or remains
the same.
Increases
Decreases
Remains the same
Justify your answ er.
END OF EXAM
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-842
43
2008 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 the
pink booklet in the spaces provided after each part. NOT in this green insert.
E&M. 1.
A metal sphere of radius a contains a charge +Q and is surrounded by an uncharged, concentric, metallic shell
of inner radius b and outer radius c, as shown above. Express all algebraic answers in terms of the given
quantities and fundamental constants.
(a) Determine the induced charge on each of the following and explain your reasoning in each case.
i.
The inner surface of the metallic shell
ii. The outer surface of the metallic shell
(b) Determine expressions for the magnitude of the electric field E as a function of r, the distance from the
center of the inner sphere, in each of the following regions.
i.
r<a
ii.
a<r<b
iii.
b<r<c
iv.
c<r
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44
2008 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(C)
On the axes below, sketch a graph of E as a function of r.
t
I
a
I)
r
(d) An electron of mass ,n carrying a charge —e is released from rest at a very large distance from the spheres.
Derive an expression for the speed of the particle at a distance lOr from the center of the spheres.
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_____________________
45
2008 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
R
=
1
200Q
VV’v
85()()V
R=3(X2
R=lOO2
IA
Ill
E&M, 2.
In the circuit shown above, A and B are terminals to which different circuit components can be connected.
(a) Calculate the potential difference across R
2 immediately after the switch S is closed in each of the
following cases.
i. A 50 2 resistor connects A and B.
ii.
iii.
A 40 mH inductor connects A and B.
An initially uncharged 0.80 ltF capacitor connects A and B.
(b) The switch gets closed at time t = 0. On the axes below, sketch the graphs of the current in the 100 Q
resistor R
3 versus time t for the three cases. Label the graphs R for the resistor, L for the inductor, and
C for the capacitor.
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46
2008 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
R2
Figure 1
E&M. 3.
The circular loop of wire in Figure 1 above has a radius of R and carries a current I. Point P is a distance of R/2
above the center of the loop. Express algebraic answers to parts (a) and (b) in terms of R, I, and fundamental
constants.
(a)
i. State the direction of the magnetic field B
1 at point P due to the current in the loop.
ii. Calculate the magnitude of the magnetic field B
1 at point P.
Axis
?
R12
Figure 2
A second identical loop also carrying a
Figure 2 above.
Figure 3
current
I is added at a distance of R above the first loop, as shown in
tb) Determine the magnitude of the net magnetic field B,
1 at point P.
A small square loop of wire in which each side has a length s is now placed at point P with its plane parallel to
the plane of each loop, as shown in Figure 3 above. For parts (c) and (d), assume that the magnetic field between
the two circular loops is uniform in the region of the square loop and has magnitude B,,
1
(c) In terms of
and s, determine the magnetic flux through the
square
loop.
(d) The square ioop is now rotated about an axis in its plane at an angular speed o. In terms of Bnet, s, and ai,
calculate the induced emf in the loop as a function of time t, assuming that the loop is horizontal at t
0.
END OF EXAM
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-846
47
2009 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, NOT in the green insert.
E&M. 1.
A spherically symmetric charge distribution has net positive charge Q
0 distributed within a radius of R.
Its electric potefltial V as a function of the distance r from the center of the sphere is given by the following.
=
1rEJRL
4
V(r)=
r
0
4lrE
—2 + 3(i-)
R
for r
forr>R
Express all algebraic answers in terms of the given quantities and fundamental constants.
(a) For the following regions, indicate the direction of the electric field E(r) and derive an expression for its
magnitude.
i. r < R
Radially inward
Radially outward
ii. r> R
Radially inward
Radially outward
(b) For the following regions, derive an expression for the enclosed charge that generates the electric field in
that region, expressed as a function of r,
i. r < R
ii. r> R
(c) Is there any charge on the surface of the sphere (r
Yes
No
If there is, determine the charge. In either case, explain your reasoning.
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48
2009 AP’’ PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(d) On the axes below, sketch a graph of the force that would act on a positive test charge in the regions r
and r > R. Assume that a force directed radially outward is positive.
<
R
F
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-648
49
2009 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
jg6
5.0 x
9.OV
E&M. 2.
A 9.0 V battery is connected to a rectangular bar of length 0.080 m, uniform cross-sectional area 5.0 x l0 rn
.
2
and resistivity 4.5 x l0 2.m, as shown above. Electrons are the sole charge carriers in the bar. The wires have
negligible resistance. The switch in the circuit is closed at time I
=
0.
(a) Calculate the power delivered to the circuit by the battery.
(b) On the diagram below, indicate the direction of the electric field in the bar.
-______
Side View
Explain your answer.
(c) Calculate the strength of the electric field in the bar.
A uniform magnetic field of magnitude 0.25 T perpendicular to the bar is added to the region around the bar, as
shown below.
0.080 in
m
6
10
x
x
x
ax
><
x
x
x
x
x
x
X
x
x
x
x
)<
X
X
X
X
X
X
xx
x
x
x
x
x
x
x
x
Side View
(d) Calculate the magnetic force on the bar.
(e) The electrons moving through the bar are initially deflected by the external magnetic field. On the diagram
below, indicate the direction of the additional electric field that is created in the bar by the deflected
electrons.
Side View
(f The electrons eventually experience no deflection and move through the bar at an average speed of
3.5x l0 rn/s. Calculate the strength of the additional electric field indicated in parue.
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-749
_______
50
2009 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
L
jx
x
J.
x
X
(J)
X
1
X
X
X
x
X
B
X
X
X
X
E&M, 3.
A square conducting loop of side L contains two identical lightbulbs, 1 and 2, as shown above. There is a
magnetic field directed into the page in the region inside the loop with magnitude as a function of time t given by
B(t) = at + b, where a and b are positive constants. The lightbulbs each have constant resistance R
0 Express
.
all answers in terms of the given quantities and fundamental constants.
(a) Derive an expression for the magnitude of the emf generated in the loop.
(b)
i. Determine an expression for the current through bulb 2.
ii. Indicate on the diagram above the direction of the current through bulb 2.
(c) Derive an expression for the power dissipated in bulb 1.
Another identical bulb 3 is now connected in parallel with bulb 2, but it is entirely outside the magnetic field, as
shown below.
L
(1)
X
X
x
x
TX
X
IX
X
XX
x
B
2
X
X
(d) How does the brightness of bulb 1 compare to what it was in the previous circuit?
Brighter
—
Dimmer
The same
Justify your answer.
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51
2009 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
Now the portion of the circuit containing bulb 3 is removed, and a wire is added to connect the midpoints of the
top and bottom of the original ioop, as shown below.
L
x
x
x
x
B
x
x
x
x
xx
x
x
x
x
B
x
2
)
x
(e) How does the brightness of bulb I compare to what it was in the first circuit?
Brighter
Dimmer
The same
Justify your answer.
END OF EXAM
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-951
52
2010 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 the
pink booklet in the spaces provided after each part, NOT in this green insert.
y
A
Figure I
E&M. 1.
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 1
being greatest. If two points have the same potential, give them the same ranking.
Justify your rankings.
Point P is at the origin, as shown below, and is the center of curvature of the charge distribution.
y
A
+Q1
j
/
Figure II
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53
2010 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(hi Determine an expression for the electric potential at point P due to the charge
Q.
(c) A positive point charge q with mass in 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,
(d) On the dot representing point P below, indicate the direction of the electric field at point P due to the
charge
Q.
V
4
(e) Derive an expression for the magnitude of the electric field at point P.
30 V
E&M. 2.
In the circuit illustrated above, switch S is initially open and the battery has been connected for a long time.
(a) What is the steady-state current through the ammeter?
(b) Calculate the charge on the 10 jiF capacitor.
(C)
Calculate the energy stored in the 5.0 tiF capacitor.
The switch is now closed, and the circuit comes to a new steady state.
(d) Calculate the stead-state current through the battery.
(e) Calculate the final charge on the 5,0 iF capacitor.
(f Calculate the energy dissipated as heat in the 40 12 resistor in one minute once the circuit has reached
steady state.
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54
2010 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
I
Lighthulb
(resistance R)
+
I
(I
E&M. 3.
The long straight wire illustrated above carries a current I to the right. The current varies with time t according
to the equation I = I Kt, where I and K are positive constants and I remains positive throughout the time
period of interest. The bottom of a rectangular loop of wire of width b and height a is located a distance d
above the long wire, with the long wire in the plane of the loop as shown. A lightbulb with resistance R is
connected in the loop. Express all algebraic answers in terms of the given quantities and fundamental constants.
—
(a> Indicate the direction of the current in the ioop.
Clockwise
Counterclockwise
Justify your answer.
(b) Indicate whether the lightbulb gets brighter, gets dimmer, or stays the same brightness over the time period of
interest.
Gets brighter
.,__Gets dimmer
Remains the same
Justify your answer.
(c) Determine the magnetic field at t
long wire.
=
0 due to the current in the long wire at distance r from the
(d) Derive an expression for the magnetic flux through the loop as a function of time.
(e) Derive an expression for the power dissipated by the lightbulb.
END OF EXAM
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-754
55
2011 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 the
pink booklet in the spaces provided after each part. NOT in this green insert.
E&M.
1.
A nonconducting, thin, spherical shell has a uniform surface charge density o on its outside surface and no charge
anywhere else inside.
(a) Use Gauss’s law to prove that the electric field inside the shell is zero everywhere. Describe the Gaussian
surface that you use.
(b) The charges are now redistributed so that the surface charge density is no longer uniform. Is the electric field
still zero everywhere inside the shell?
Yes
No
It cannot be determined from the information given.
Justify your answer.
Now consider a small conducting sphere with charge
below.
+Q whose center is at corner A of a cubical surface, as shown
B
G
I
I
L
I)
E
(c) For which faces of the surface, if any, is the electric flux through that face equal to zero?
DEF
ABED
ABGH
EFGH
BCFG
ADEH
Explain your reasoning.
(d) At which corner(s) of the surface does the electric field have the least magnitude?
(e) Determine the electric field strength at the positions you have indicated in part (d) in terms of
fundamental constants, as appropriate.
Q, L, and
(f) Given that one-eighth of the sphere at point A is inside the surface, calculate the electric tiux through
face CDEE.
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-555
56
2011 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE”RESPONSE QUESTIONS
500Q
1
S
25 mF
9,0 V
E&M.
S
5.0 H
2.
The circuit represented above contains a 9.0 V battery, a 25 mF capacitor, a 5.0 H inductor, a 500 Q resistor.
and a switch with two positions, S
2 Initially the capacitor is uncharged and the switch is open.
1 and S
.
(a) in experiment I the switch is closed to position S
1 at time r1 and left there for a long time.
i. Calculate the value of the charge on the bottom plate of the capacitor a long time after the switch
is closed.
ii. On the axes below, sketch a graph of the magnitude of the charge on the bottom plate of the capacitor
as a function of time. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with
numerical values or algebraic expressions, as appropriate.
11
‘Time
iii. On the axes below, sketch a graph of the current through the resistor as a function of time. On the axes,
explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic
expressions, as appropriate.
Time
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57
2011 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(h) In experiment 2 the capacitor is again uncharged when the switch is closed to position 1
S at time r1 The switch
is then moved to position S
2 at time t
2 when the magnitude of the charge on the capacitor plate is 105 inC.
allowing electromagnetic oscillations in the LC circuit.
i, Calculate the energy stored in the capacitor at time 12
ii. Calculate the maximum current that will be present during the oscillations,
iii. Calculate the time rate of change of the current when the charge on the capacitor plate is 50 mC.
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58
2011 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
Cross-sectional View
(current into page)
E&M.
3.
A section of a long conducting cylinder with inner radius a and outer radius b carries a current
! that has
a uniform current density, as shown in the figure above.
(a) Using Ampere’s law, derive an expression for the magnitude of the magnetic field in the following regions
as a function of the distance r from the central axis.
i.
ii.
iii.
1’<(l
a <r<b
r=2b
(b) On the cross-sectional view in the diagram above, indicate the direction of the field at point P, which is at
a distance r = 2b from the axis of the cylinder.
(c) An electron is at rest at point P. Describe any electromagnetic forces acting on the electron. Justi your
answer.
\ic th
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-858
59
2011 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
Now consider a long, ijd conducting cylinder of radius b carrying a current I. The magnitude of the magnetic
field inside this cylinder as a function of r is given by B = poJor/2,rb2 An experiment is conducted using
a particular solid cylinder of radius 0.010 m carrying a current of 25 A. The magnetic field inside the cylinder is
measured as a function of r, and the data is tabulated below.
.
Distance r (m)
Magnetic Field B (T)
0.002
0,004
1.2 x l0
2.7 x i0
I 0.006
I 0.008
3.6 x l0
4.7 x i0
0.010
6.4 x l0
(d)
i. On the graph below, plot the data points for the magnetic field B as a function of the distance r, and label
the scale on both axes. Draw a straight line that best represents the data.
E
z
zzzzz
—
.
..Ezr:zEzEzz
.
ii. Use the slope of your line to estimate a value of the permeability
,
0
t
END OF EXAM
\‘hit
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-959
60
2012 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 IS points each. The parts within a question may not have equal weight. Shoss all your work in this booklet in the
spaces provided after each part.
V
E&M,
1,
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 —100 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
1 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
r. and fundamental constants. as appropriate.
answers in terms of Q
.
1
.
(a) Using Gauss’s Law, derive an algebraic expression for the electric field E(r for 0.10 m
ht Determine an algebraic expression for the electric field E(r) for r
>
r
<
0.20 m
0.2() m
ci Determine an algebraic expression for the electric potential Vr) for r
(d
<
>
0.2() m
Using the numerical information given, calculate the value of the total charge Q on the two spherical shells
—
÷
(2,,)
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-560
61
2012 AP’ PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(e) On the axes below, sketch the electric field E as a function of r Let the positive direction he radially outward.
(J.
r(rn)
10
(0 On the axes below, sketch the electric potential V as a function of
r.
V
r(1fl)
0 h
U
20
U 0
Ih (oHe BoarcL
Boi n
Web:
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-661
62
2012 APa PHYSICS C: ELECTRICITY AND MAGNETISM FREERESPONSE QUESTIONS
Length
in
ENl.
2.
ii)
m
.\ plivic %tudent i\he to rneaure the resistlvit ot slightly conductic paper that ha a thickncs of 1 .()
The student cut’. a sheet of the conductive paper into strips of idth 00)2 in and varx ing lengths, making live rcsistor
labeled RI to R5. Using an ohmmeter, the student measures the resistance of each strip, as shown aho e. The data are
recorded below.
/
RI
R5
Length ()
0.020
0100
Resistance ()
b0,000
440,000
Resistor
t
a) Use the grid below to plot a linear graph of the data points from which the resistivity of the paper can be determined.
Include labels and scales for both axes, Draw the straight line that best represents the data.
Ib) Using the graph. calculate the resistivity of the paper.
‘.i’’h(
ih (i Br
1 ,,. Brd•n :h \\h:
.
rd’r,.
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-762
63
2012 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
Capas.dtor
Battery
rue student uses resistors R4 and R5 to build a circuit using wire, a 1.5 V batter, an uncharged 10 iaF capacitor. and an
open switch, as shown above,
(C)
Calculate the time constant of the circuit,
(d) At time t
=
0, the student closes the switch. On the axes below, sketch the magnitude of the voltage V across the
capacitor and the magnitudes of the voltages VR4 and VR5 across each resistor as functions of time r. Clearly label
each curve according to the circuit element it represents. On the axes, explicitly label any intercepts, asymptotes,
maxima, or minima with values or expressions, as appropriate.
0
Time
I
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64
2012 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
Crossbar
B
x
x
H
E&.M.
x
x
L
i
x.
H
3.
A closed loop is made of a U-shaped metal wire of negligible resistance and a movable metal crossbar of
resistance R, The crossbar has mass in and length L. It is initially located a distance h
) from the other end of the
1
loop. The loop is placed vertically in a uniform horizontal magnetic field of magnitude B( in the direction
shown in the figure above. Express all algebraic answers to the questions below in terms of B L, in. h,. 1?. and
fundamental constants, as appropriate.
.
(a) Determine the magnitude of the magnetic flux through the loop when the crossbar is in the position shown.
The crossbar is released from rest and slides with negligible friction down the U-shaped wire without losing
electrical contact.
cb) On the figure below, indicate the direction of the current in the crossbar as it falls.
Justify your answer.
(c) Calculate the magnitude of the current in the crossbar as it falls as a function of the crossbar’s speed v.
(d) Derive, but do NOT solve, the differential equation that could be used to determine the speed
crossbar as a function of time t.
(e) Determine the terminal speed
f
UT
U
of the
of the crossbar.
If the resistance R of the crossbar is increased, does the terminal speed increase, decrease, or remain the
same
J)ecreases
Increases
Remains the same
Give a physical pistitication for your answer in terms of the forces on the crossbar.
STOP
END OF EXAM
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2013 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 ot 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 L
A very long, solid, nonconducting cylinder of radius R has a positive charge of uniform volume density p.
A section of the cylinder far from its ends is shown in the diagram above, Let r represent the radial distance from
the axis of the cylinder. Express all answers in terms of r. R, p, and fundamental constants, as appropriate.
(a) Using Gauss’s law, derive an expression for the magnitude of the electric field at a radius r
appropriate Gaussian surface on the diagram.
<
R. Draw an
(b) Using Gauss’s law, derive an expression for the magnitude of the electric field at a radius r> R.
(C)
On the axes below, sketch the graph of electric field E as a function of radial distance r for r = 0 to r = 2R.
Explicitly label any intercepts, asymptotes. maxima, or minima with numerical values or algebraic
expressions. as appropriate.
1:
2!?
1?
(d)
i. Derive an expression for the magnitude of the potential difference between r
ii. Is the potential higher at r
r
=
0
=
0 or r
=
0 and r
=
R.
R ?
r=R
2013
The Collcee Hoard.
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2013 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE GUESTIONS
e) The nonconducting cylinder is replaced with a conducting cylinder of the same shape and same lin charge
density. On the axes below, sketch the electric field F as a function of r for r 0 to r = 2R. Explicitly label
an intercepts, asymptotes. maxima, or minima with numerical alues or algebraic expressions, as
appropriate.
I
R
,()i
\
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2013 AP PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
E&M 2,
In a lab, you set up a circuit that contains a capacitor c, a resistor R, a switch S. and a power supply, as shown in
the diagram above, The capacitor is initially uncharged. The switch, which is initially open, can be moved to
positions A or B.
(a)
i. Indicate the position to which the switch should be moved to charge the capacitor.
A
B
ii. On the diagram, draw a voltmeter that is properly connected to the circuit in a manner that will allow
the voltage to be measured across the capacitor.
After a long time you move the switch to discharge the capacitor, and your lab partner starts a stopwatch. You
collect the following measurements of the voltage across the capacitor at various times.
t(s)
V(V)
6
18
30
42
54
252
74
33
10
6
You wish to determine the time constant v of the circuit from the slope of a linear graph.
i. Indicate two quantities you would plot to obtain a linear graph.
ii. Use the remaining rows in the table above, as needed, to record any quantities that you indicated that
are not given. Label each row you use and include units.
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2013 Al) PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(c) On the axes below, graph the data from the table that will produce a linear relationship. Clearly scale and
label all axes including units, if appropriate. Draw a straight line that best fits your data points.
L
I
I
I
e
I
r
I
I
———
1
I
1
I
I
I
1
1
I
I
———
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
(d) From your line in part (c), obtain the alue of the time constant r of the circuit.
(e)
.iF. Calculate an experimental value for
i. In the experiment, the capacitor C had a capacitance of 1.50 1
the resistance R.
ii. On the axes in part (c), use a dashed line to sketch a possible graph if the capacitance was greater than
1.50 pP but the resistance R as the same. Justify your answer.
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69
2013 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
xxxxxxx
X
X
B
xxx xxx
x
xxx xx
x
x
XXX
)(
)(
xX
xxxxxxx
E&M 3,
The figure above shows a circular loop of area 0.25 m
2 and resistance 12 ] that lies in the plane of the page.
A magnetic field of magnitude B directed into the page exists in the area of the loop. The field varies with time t.
as shown in the graph below.
B (T)
B
0
4
8
=
1 ,8e©
12
16
18
(a)
i. Derive an expression for the magnitude of the induced emf in the loop as a function of time for the
interval t = 0 s to r = 8 s.
ii. Calculate the magnitude of the induced current I in the loop at time t
=
4
5,
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2013 AP® PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS
(b)
i. Sketch a graph of the induced current I in the ioop as a function of time i from r
axes below. assumine that a counterclockwise (CCW) current is positive.
=
0s
to
r
=
18 s
on
the
IA)
0
4
8
12
16
18
Cw
ii. For the time interval 12 s to 16 s, justify the direction of the current you have indicated in your graph.
(C)
Calculate the total energy dissipated in the loop during the first 8 s shown.
STOP
END OF EXAM
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