1. A N turns circular coil of wire has diameter D. It is placed with its axis along the
direction of the Earth's magnetic field of B and then in t is flipped 180°. An average emf
of what magnitude is generated in the coil?
Solution:
180o
B
2
1
D
A      D 2
4
2
1
( B ) i  NAB  NBD 2
4
1
( B ) f   NAB   NBD 2
4
1
1
 B  ( B ) f  ( B ) i   NBD 2  NBD 2
4
4
1
  NBD 2
2
2
 B NBD

  
t
2t
2. The flexible loop in the figure below has a radius of R and is in a magnetic field of
magnitude B. The loop is grasped at points A and B and stretched until its area is nearly
zero. If it takes t to close the loop, what is the magnitude of the average induced emf in
it during this time interval?
Solution:
Initial :
A  R 2
( B ) i  AB  R 2 B
Final :
A0
( B ) f  0
 B  ( B ) f  ( B ) i  0  R 2 B
 R 2 B
 B R 2 B

  
t
t
3. A circular loop of wire of radius R is placed in a magnetic field directed perpendicular
to the plane of the loop as shown in the figure below. If the field decreases at the rate of r
(in T/s) in some time interval, find the magnitude of the emf induced in the loop during
this interval.
Solution:
A  R 2
 B  AB  R 2 B
d B
dt
d
  (R 2 B)
dt
d
  R 2 B
dt
  
  R 2 (r)
 R 2 r
(
d
B  - r)
dt
4. A N-turn circular coil of radius a and resistance R is placed in a magnetic field
directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies
in time according to the expression B = At + Ct2, where B is in teslas and t is in seconds.
Calculate the induced emf in the coil at t.
Solution:
A  a 2
 B  NAB  Na 2 B
 Na 2 (At  Ct 2 )
d B
dt
d
  Na 2 (At  Ct 2 )
dt
d
  Na 2 (At  Ct 2 )
dt
2
  Na (A  2Ct)
  
|  | Na 2 (A  2Ct)
5. An aluminum ring of radius r1 and a resistance of R is placed around one end of a long
air-core solenoid with n turns per meter and radius r2 as shown in the figure below.
Assume the axial component of the field produced by the solenoid is one-half as strong
over the area of the end of the solenoid as at the center of the solenoid. Also assume the
solenoid produces negligible field outside its cross-sectional area. The current in the
solenoid is increasing at a rate of r (in A/s). (a) What is the induced current in the ring? (b)
At the center of the ring, what is the magnitude of the magnetic field produced by the
induced current in the ring? (c) At the center of the ring, what is the direction of the
magnetic field produced by the induced current in the ring?
Solution:
(a)
B at center of solenoid   0 nI
1
 0 nI
2
1
1
2
2
  B   0 nI   r2   0 n r2 I
2
2
d
d 1
1
1
2
2 d
2
|  |   B  (  0 n r2 I)   0 n r2
I   0 n r2 r
dt
dt 2
2
dt
2
| |
1
I

 0 n r2 2 r
R
2R
B at the aluminum ring 
(b)
At the center of the ring,
 I 
1
1
B 0  0 
 0 n r2 2 r 
 0 2 n r2 2 r
2r1 2r1 2R
4Rr1
(c) Lenz’s Law:
Induced current
Magnetic field
due to induced
current
Increasing flux
6. A square, single-turn wire loop length on a side is placed inside a solenoid that has a
circular cross section of radius r , as shown in the end view of the figure below. The
solenoid is L long and wound with N turns of wire.
(a) If the current in the solenoid is I, what is the magnetic flux through the square loop?
(b) If the current in the solenoid is reduced to zero in t, what is the magnitude of the
average induced emf in the square loop?
(a)  B  BA
B  0 n I  0
N
I
L
A  2
  B  0
(b) |  | 
N 2
I
L
d
d
N
N
d
B 
(  0 I 2 )   0  2
I
dt
dt
L
L dt
N
I
 0  2
L
t
2
N
 0I
Lt
7. Use Lenz's law to answer the following questions concerning the direction of induced
currents. Express your answers in terms of the letter labels a and b in each part of the
figure below.
(a) What is the direction of the induced current in the resistor R in Figure a when the bar
magnet is moved to the left? (b) What is the direction of the current induced in the
resistor R after the switch S in Figure b is closed? (c) What is the direction of the induced
current in the resistor R when the current I in Figure c decreases rapidly to zero?
(a)
Decreasing
flux
Induced current flows from a to b to compensate the decreasing flux.
(b)
Increasing
flux
Induced current flows from b to a to reduce the increasing flux.
(c)
Decreasing
flux
Induced current flows from a to b to compensate the decreasing flux.
8. The figure below shows a top view of a bar that can slide on two frictionless rails. The
resistor is R, and a magnetic field B is directed perpendicularly downward, into the page.
(a) Calculate the applied force required to move the bar to the right at a constant speed of
v. (b) At what rate is energy delivered to the resistor?
Solution:
I
Fapp
x
(a)  B  BA  Bx
d
d
d
 B  (Bx)  B x  B v
dt
dt
dt
|  | B v

I
R
R

B v
 B
| FB | IBsin90 o  IB 
R
B2 2 v

R


For constant velocity, Fapp   FB
| |


B2 2 v
 | Fapp || FB |
R
(b) P  I 2 R  (
B v 2
B2 2 v 2
) R
R
R