Download H-ElectromagneticInduction-Solutions

Solutions to Electromagnetic
Induction
1. Suppose this page is perpendicular to a uniform magnetic field and the magnetic
flux through it is 5 Wb. If the page is turned by 30 around an edge the flux through it
will be:
A) 2.5 Wb
B) 4.3 Wb
C) 5 Wb
D) 5.8 Wb
E) 10 Wb
Solutions:
5 Wb  cos30  4.3 Wb
Ans: B
2. One hundred turns of insulated copper wire are wrapped around an iron core of
cross-sectional area 0.100 m2. The circuit is completed by connecting the coil to a
10- resistor. As the magnetic field along the coil axis changes from 1.00 T in one
direction to 1.00 T in the other direction, the total charge that flows through the
resistor is:
A) 10–2 C
B)
C)
D)
E)
2  10–2 C
1C
2C
0.20 C
Solutions:


I
Q
E  L
 I R  
R
t
t
t
 100   0.1 m    2 T 
Q 

2C
R
0.1 
2

Ans: D
3. A car travels northward at 75 km/h along a straight road in a region where Earth's
magnetic field has a vertical component of 0.50  10–4 T. The emf induced between
the left and right side, separated by 1.7 m, is:
A)
B)
C)
D)
E)
0
1.8 mV
3.6 mV
6.4 mV
13 mV
Solutions:
3
4
vB  75  10 / 3600 m / s  0.5  10 T 
E 

 1.77 mV
L
1.7 m
Ans: B
4. A rectangular loop of wire has area A. It is placed perpendicular to a uniform
magnetic field B and then spun around one of its sides at frequency f. The maximum
induced emf is:
A) BAf
B) BAf
C) 2BAf
D) 2BAf
E) 4BAf
Solutions:
E 
d  B A cos  
d
d

 B A sin 
 B A sin   2 f 
dt
dt
dt
Max at   /2.
Ans: D
5. The circuit shown is in a uniform magnetic field that is into the page. The current
in the circuit is 0.20 A. At what rate is the magnitude of the magnetic field changing:
Is it increasing or decreasing?
A) zero
B) 140 T/s, decreasing
C) 140 T/s, increasing
D) 420 T/s, decreasing
E) 420 T/s, increasing
Solutions:
Let path C follows the current:
E 
dB
A
dt
E  EBattery  IR  0
dB
2
  0.12 m   4V   0.20 A  10    0
dt

dB
 139 T / s
dt
Ans: B
6. A changing magnetic field pierces the interior of a circuit containing three
identical resistors. Two voltmeters are connected to the same points, as shown. V1
reads 1 mV. V2 reads:
A)
B)
C)
D)
E)
0
1/3 mV
1/2 mV
1 mV
2 mV
Solutions:
I
V1
R
V2  I  2R   2V1  2mV
Ans: E
7. A rectangular loop of wire is placed perpendicular to a uniform magnetic field
and then spun around one of its sides at frequency f. The induced emf is a maximum
when:
A) the flux is zero
B) the flux is a maximum
C) the flux is half its maximum value
D) the derivative of the flux with respect to time is zero
E) none of the above
Solutions:
E 
d
d B A cos 
d

 B A sin 
 B A sin   2 f 
dt
dt
dt
 emf is a maximum when /2, i.e., when   0 .
Ans: A
8. A merry-go-round has an area of 300 m2 and spins at 2 rpm about a vertical axis
at a place where the Earth's magnetic field is vertical and has a magnitude of 5  10–5
T. The emf around the rim is:
A) 0
B)
C)
D)
E)
0.5 mV
3.1 mV
15 mV
30 mV
Solutions:
E 
d  BA
d

0
dt
dt
Ans: A
9. As a loop of wire with a resistance of 10  moves in a constant non-uniform
magnetic field, it loses kinetic energy at a uniform rate of 5 mJ/s. The induced current
in the loop:
A) is 0
B) is 2 mA
C) is 2.8 mA
D) is 20 mA
E) cannot be calculated from the given data
Solutions:
P  IE  I 2 R
I
P
5  103 J / s

 2.2  102 A
R
10 
Ans: D
10.
A rod lies across frictionless rails in a uniform magnetic field B, as shown. The
rod moves to the right with speed v. In order for the emf around the circuit to be zero,
the magnitude of the magnetic field should:
A)
B)
C)
D)
not change
increase linearly with time
decrease linearly with time
increase quadratically with time
E) decrease quadratically with time
E 
Solutions:
E 

d
d
   B vt l 
dt
dt
where l is the length of the rod.
dB
vtlBvl 0
dt
dB
dt

B
t
ln B t  C
B
C
t
Ans: C
11. When the switch S in the circuit shown is closed, the time constant for the
growth of current in R2 is:
A) L/R1
B)
C)
D)
E)
L/R2
L/(R1 + R2)
L(R1 + R2)/(R1R2)
(L/R1 + L/R2)/2
Solutions:
Ans: B
When S is closed, emf of battery acts directly on the series of L & R2 .
12.
The diagrams show three circuits with identical batteries, identical inductors,
and identical resistors. Rank them according to the current through the battery just
after the switch is closed, from least to greatest.
A) 3, 2, 1
B) 2 and 3 ties, then 1
C) 1, 3, 2
D) 1, 2, 3
E) 3, 1, 2
Solutions:
L is open circuit.
I1  0
I2 
Rank least to greatest:
Ans: C
13.
E
R
I3 
E
2R
1,3,2
A 3.5 mH inductor and a 4.5 mH inductor are connected in parallel. When the
total emf of the combination is 16 V, the rate of change of the current in the larger
inductor is:
A)
B)
C)
D)
E)
2.0  103 A/s
3.6  103 A/s
4.6  103 A/s
7.0  103 A/s
8.1  103 A/s
Parallel configuration 
Solutions:
E L
dI
dt
same E for both L.
dI 4.5
16 V

 3.6  103 A / s
dt
4.5  103 H
Ans: B
14. A current of 10 A in a certain inductor results in a stored energy of 40 J. When
the current is changed to 5 A in the opposite direction, the stored energy changes by:
A) 20 J
B) 30 J
C) 40 J
D) 50 J
E) 60 J
U
Solutions:
U 
1
L I2
2
L
2U 2   40 J 

 0.8H
2
I2
10 A
1
2
2
  0.8H   5 A  10 A   30J


2
Ans: B
15. An inductance L and a resistance R are connected in series to an ideal battery. A
switch in the circuit is closed at time 0, at which time the current is zero. The rate of
increase of the energy stored in the inductor is a maximum:
A)
B)
C)
D)
E)
just after the switch is closed
at the time t = L/R after the switch is closed
at the time t = 2L/R after the switch is closed
at the time t = (L/R)ln 2 after the switch is closed
a long time after the switch is closed
Solutions:
U
1
L I2
2

dU
dI
LI
dt
dt
 d 2 I  d I 2 
d2 U
Rate extrema at:
 L I

 0
2
d t2
 d t
 d t  
dI
0
dt
Circuit:
E I RL

dI E I R

dt
L
R E  I R 
d 2I
R dI


2
dt
L dt
L2
I R E  I R  E  I R 


0
L2
L2
E
I
or
2R
dI
dU
0 
 0 so it is a min.
dt
dt
2

Extrema at:
I
i.e.
At
I
E
R
E
,
R
Max is at I 
E
:
2R
I
E
1  e  Rt / L 

R

1
 1  e  Rt / L
2
t
L
ln 2
R
Ans: D
16. A 6.0-mH inductor and a 3.0- resistor are wired in series to a 12-V ideal
battery. A switch in the circuit is closed at time 0, at which time the current is zero.
2.0 ms later the energy stored in the inductor is:
A) 0
B) 2.5  10–2 J
C) 1.9  10–2 J
D) 3.8  10–2 J
E) 9.6  10–3 J
Solutions:
I
U
Ans: C
E
12 V
1  e  Rt / L  

R
3

  3     2 ms   
1
1  exp  
   4 1  e  A  2.53 A
6
mH



1
1
2
L I 2   6 mH    2.53 A  19.2 mJ
2
2