Solutions to Alternating Current
Circuits
1. A capacitor in an LC oscillator has a maximum potential difference of 15 V and a
maximum energy of 360 J. At a certain instant the energy in the capacitor is 40 J.
At that instant what is the potential difference across the capacitor?
A) zero
B) 5 V
C) 10 V
D) 15 V
E) 20 V
Solution:
U
2U
C  2max
Vmax
1
CV2
2
2
40 J
2U
2UVmax
U
 15V  5V
V


Vmax 
360 J
C
2U max
U max
Ans: B
2. In the circuit shown, switch S is first pushed up to charge the capacitor. When S
is then pushed down, the current in the circuit will oscillate at a frequency of:
A) 318 Hz
B) 0.01 Hz
C) 12.500 Hz
D) 2000 Hz
E) depends on V0
Solution:

1

LC
50  10
1
3
H  5  10 6 F 
 2000 rad / s
f 

 318Hz
2
Ans: A
3. An LC series circuit with an inductance L and a capacitance C has an oscillation
frequency f. Two inductors, each with inductance L, and two capacitors, each with
capacitance C, are all wired in series and the circuit is completed. The oscillation
frequency is:
A) f/4
B) f/2
C) f
D) 2f
E) 4f
Solution:
Leq  2 L
eq 
1

LeqCeq
1
Ceq  C
2
1

LC
Ans: C
4. An RLC circuit has a resistance of 200  and an inductance of 15 mH. Its
oscillation frequency is 7000 Hz. At time t = 0 the current is 25 mA and there is no
charge on the capacitor. After five complete cycles the current is:
A) 0
B) 1.8  10–6 A
C) 2.1  10–4 A
D) 2.3  10–3 A
E) 2.5  10–2 A
Solution:
I  t   I 0 e Rt /2 L cos  t
I  5 cycle   I 0 e R /2 L 5/ f 

200
5 

4
  25  103 A  exp  


  2.13  10 A
3
7000
Hz

 2  15  10 H  
Ans: C
5. A circuit contains an inductor, a capacitor, and a light bulb connected as shown.
In which frequency limit is the light bulb the brightest?
A) the low frequency limit
B) the high frequency limit
C) both
Solution:
2
Pbulb  I bulb
Rbulb  I C2 Rbulb
IC 

Ans:
V
R
1
iC
IC 
V
 1 
R 

 C 
2
2
IC & hence Pbulb are larger in the high frequency limit.
B
6. An RLC series circuit has L = 100 mH and C = 1 F. It is connected to a 1000-Hz
source emf is found to lead the current by 75. The value of R is:
A) 12.6 
B) 126 
C) 175 
D) 1750 
E) 1810 
Z  R  i L 
Solution:
V  I Z  I Z ei 
1
i C
with
tan  
Im Z

Re Z
1
C
R
L 
1
C
tan 
L 
R

 2  3.14  1000 Hz  100  103 H  
R
1
 2  3.14  1000 Hz  106 C 
tan 75
 125.7 
Ans: B
7. In the diagram, the function y(t) = ymsin(t) is plotted as a solid curve. The
other three curves have the form y(t) = ymsin(t + ), where  is between –/2 and
+/2. Rank the curves according to the value of , from the most negative to the
most positive.
A) 1, 2, 3
B) 2, 3, 1
C) 3, 2, 1
D) 1, 3, 2
E) 2, 1, 3
1:  < 0
2:  > 0
3:  > 0, less than 2
Rank (- to +):
1,3,2
Solution:
Ans: D
8. An RLC series circuit is driven by a sinusoidal emf with angular frequency d.
If d is increased without changing the amplitude of the emf the current amplitude
increases. If the L is inductance, C is the capacitance, and R is the resistance, this
means that:
A) d L > 1/d C
B) d L < 1/d C
C) d L = 1/d C
D) d L > R
E)
d L < R
Z  R  id L 
Solution:
I 
V
Z
1
id C


1 
Z  R   d L 
d C 

|Z| must decrease when d increases.
2
2
dZ 1 1

1 
1 
   2  d L 
 L  2   0
d d 2 Z
d C 
d C 


d L 
1
0
d C
Ans: B
9. A series circuit consists of a 15- resistor, a 25-mH inductor, and a 35-F
capacitor. If the frequency is 100 Hz the power factor is:
A) 0
B)
C)
D)
E)
0.20
0.45
0.05
1.0
Solution:
P  I V cos 
V IZ

  Z
Z  R  i L 
cos Z 

1
i C
R

1 
R   L 
 C 

2
2

15 


1
15     2  3.14  100 Hz   25  103 H  

 2  3.14  100 Hz   35  106 F  

2
2
 0.45
Ans: C
10.
is:
A)
B)
C)
D)
A coil has a resistance of 60 and an impedance of 100. Its reactance, in ohms,
40
60
80
117
E) 160
Solution:
Z  R  i  L  R  i XL
Z  R 2  X L2
2
Ans: C
XL 
Z  R2 
2
100
2
  60  80
2