Multiple choice questions [64 points]
Answer all of the following questions. Read each question carefully. Fill the correct
bubble on your scantron sheet. Each correct answer is worth 4 points. Each question
has exactly one correct answer.
Questions 1 through 8 all refer to the same problem.
Two positive objects with a charge +2Q are a distance a apart along the y-axis as shown in the
1
figure below. The origin is halfway between them. Let k =
= 9.0 × 10 9 N m 2 / C 2 and take
4πε 0
the zero of potential to be at infinity ( V (∞) = 0 V )
y
+2Q
a
x
+2Q
1
Which of these expressions represents the electric potential energy of these two
charges
A.
B.
C.
D.
E.
kQ 2
a
4kQ 2
k (2Q ) 2 4kQ 2
U=
=
a
a
a
2kQ
a
2kQ
a2
4kQ 2
a2
Name: ______________________________________________________________ Total Points: _______
(Last)
(First)
An object with charge –Q is brought along the x-axis from far away to rest at a distance a
from each positive charge, forming an equilateral triangle of side a (see figure below).
Point P1 is in the xy-plane, equidistant from all of the three charges, a distance
a
b=
from each one. Point P2 is halfway between P1 and the negative charge.
3
+2Q
a
b
P1
a
b
P2
–Q
b
a
+2Q
2
The work by an external agent to bring the negatively charged object from far away
to the location shown in the figure is (the object is at rest before and after being
moved).
A. positive
B. negative -Q is attracted by the two +2Q charges thus
C. zero
3
r
Felectric
r
Fagent
-Q
displacement
direction
The magnitude of the work by an external agent to bring the negatively charged
object from far away to the location shown in the figure is (the object is at rest before
and after being moved).
2kQ 2
A.
a
4kQ 2
 k 2Q

B.
Wagent = Welectric = − QV−Q = − Q
× 2
a
field
 a

2
8kQ
C.
3a
2kQ
D.
a2
4kQ 2
E.
a2
2
Name: ______________________________________________________________ Total Points: _______
(Last)
(First)
4
The electric potential energy of the three-charge configuration is given by
A.
B.
C.
D.
E.
5
4kQ 2 4kQ 2
0 Sum the answers to 1 and 3 (with the correct signs!)
−
=0
a
a
3kQ 2
a
3kQ 2
−
a
4kQ 2
−
a
4kQ
a
The electric potential at point P1 (relative to infinite distance) is
A. positive VP1 =
k 2Q k 2Q kQ 3kQ 3 3kQ
+
−
=
=
b
b
b
b
a
B. negative
C. zero
6
The magnitude of the electric potential at point P1 (relative to infinite distance) is
given by
A. 0
2 3kQ 2
B.
a
3 3kQ
C.
a
3kQ
D.
a
3kQ
E.
a2
7
The direction of the electric field at point P1 is toward
A.
B.
C.
D.
E.
8
+x Sum the electric fields acting at P1
-x
+y
-y
no direction (E=0)
r
E + 2Q
r
E −Q
r
E + 2Q
The potential difference between points P2 and P1, V21=V2-V1 is
3
Name: ______________________________________________________________ Total Points: _______
(Last)
(First)
A. positive
r
B. negative Between P1 and P2, E is along the +x
2 r
r
direction V 2−V1 = − ∫ E ⋅ ds < 0
1
C. 0
4
Name: ______________________________________________________________ Total Points: _______
(Last)
(First)
Questions 9 through 16 all refer to the same problem.
Four circuits, labeled A, B, C and D, are shown below. Each contains a battery with emf
V0=10.0V and one or more resistors. All resistors have the same resistance R=2.0 Ω.
Assume that all batteries are ideal (neglect their internal resistance).
V0
R1
R1
V0
V0
R1
R2
V0
R2
R1
R2
Circuit A
9
Circuit B
R3
Circuit C
Circuit D
Is the potential difference across resistor R1 in circuit A larger than, smaller than or
equal to the potential difference across resistor R1 in circuit B?
A.
B.
C.
D.
V1(A)>V1(B)
V1(A)<V1(B)
V1(A)=V1(B)
not enough information
10 Is the potential difference across resistor R1 in circuit A larger than, smaller than or
equal to the potential difference across resistor R1 in circuit C?
A.
B.
C.
D.
V1(A)>V1(C)
V1(A)<V1(C)
V1(A)=V1(C)
not enough information
11 In which circuit is the current through the battery the largest?
A.
B.
C.
D.
A
B
C (least resistance)
D
5
Name: ______________________________________________________________ Total Points: _______
(Last)
(First)
12 In which circuit is the current through the battery the smallest?
A.
B.
C.
D.
A
B (most resistance)
C
D
13 What is the equivalent resistance of circuit D (not including the battery)?
A. 3R
3
B.
R
2
2
1
1
1
C.
R R1 is in parallel with R2+R3
= +
3
Req R 2 R
5
D.
R
2
E. R
14 In circuit D, what is the current through resistor R1?
A.
B.
C.
D.
5.0 A Use V=IR with V=10V and R=2Ω
0.4 A
10 A
2.5 A
15 In circuit D, what is the current through resistor R3?
A.
B.
C.
D.
5.0 A
0.4 A
10 A
2.5 A The potential difference across R3 is 5V(V2+V3=10V and V2=V3) and R3=2Ω
16 In circuit D, what is the power dissipated in resistor R3?
A.
B.
C.
D.
5W
10 W
12.5 W P = VI = 5 × 2.5
20 W
6
Name: ______________________________________________________________ Total Points: _______
(Last)
(First)
PROBLEM [36 points]
Consider the circuit below containing a 10.0 Volt battery and three parallel-plate
capacitors with capacitance C1= 2.0 µF, C2= 1.0 µF, and C3= 1.0 µF.
Switch S
C1=2.0 µF
V0 = 10.0 V
C2=1.0 µF
C3=1.0 µF
Questions 1 through 3: the switch S has been closed for a long time.
1). [6 pts] Find the charge Q1 on capacitor C1.
C2 and C3 are in parallel. We can replace them with an equivalent capacitor Ceq=2µF
which is equal to C1.
Then Veq=V1 by symmetry
Also V1+Veq =Vbat
Thus V1=5V
Q1 = V1C1 = 5 × 2 × 10 −6 = 10 −5 C
2). [7 pts] Find the potential difference V2 across capacitor C2
V2 + V1 = Vbat = 10V
We know V1=5V from 1)
Thus V2=5V
3). [7 pts]Find the potential energy stored in capacitor C3.
V3=V2=5V
U=
1
1
C 3V32 = × 10 −6 × 25 = 12.5 × 10 −6 J
2
2
7
Name: ______________________________________________________________ Total Points: _______
(Last)
(First)
Questions 4 and 5: the switch S is opened. A dielectric with relative dielectric constant
k=3.0 is then inserted into capacitor C2, completely filling the space between the plates.
The circuit reaches a new equilibrium.
4). [7 pts] Does the charge stored on capacitor C1 increase, decrease, or stay the same
relative to your answer to question 1 (with no dielectric in C2)? Explain your
reasoning.
C1=2.0 µF +Q1
-Q1
V0 = 10.0 V
+Q’2
C’2=3.0 µF -Q’2
This plate is insulated. Its
charge can’t change. Q1 on
C1 remains the same.
+Q’3
C3=1.0 µF
-Q’3
5). [7pts] Does the potential energy stored in capacitor C3 increase, decrease or stay the
same relative to your answer to question 3. Explain your reasoning.
The capacitance of C2 becomes C 2 ' = κC 2 = 3µF
The charges on C’2 and C3 become Q’2 and Q’3.
The potential differences across C’2 and C3 become V’2 and V’3.
The total charge on the bottom plate of C1 and top plates of C’2 and C3 is still 0. It can’t
change since this part is not in contact with any other part.
Q2 '+Q3 ' = Q1
Since C’2 and C3 are in parallel, V’2 = V’3. Thus,
Q2 ' Q3 '
=
C 2 ' C3
It follows that
Q1
Q '3 =
C
1+κ 2
C3
V '3 =
Q' 3
10 −5
= −6
= 2.5V
C 3 10 × (1 + 3)
1
V’3 is less than 5V. The potential energy (= C 3V32 ) stored in C3 decreases (precisely by a
2
factor of 4).
8