Faraday’s Law of Induction
AP Physics C
Montwood High School
R. Casao
• Two simple
experiments
demonstrate that a
current can be
produced by a
changing magnetic
field.
• First: consider a loop
of wire connected to a
galvanometer as
shown.
• If a magnet is moved toward the loop, the
galvanometer needle will deflect in one direction.
• If a magnet is moved away from the loop, the
galvanometer needle will deflect in the opposite
direction.
• If the magnet is held stationary relative to the loop,
no galvanometer needle deflection is observed.
• If the magnet is held stationary and the coil is
moved toward or away from the magnet, the
galvanometer needle will also deflect.
• From these observations, you can conclude that a
current is set up in the circuit as long as there is
relative motion between the magnet and the coil.
• This current is set up in the circuit even though
there are no batteries in the circuit.
• The current is said to be an induced current,
which is produced by an induced EMF.
Faraday’s Experiment
• A coil is
connected to a
switch and a
battery.
• This is called the
primary coil and
the circuit is called
the primary circuit.
• The coil is
wrapped around
an iron ring to
intensify the
magnetic field
produced by the
current through
the coil.
Faraday’s Experiment
• A second coil, on
the right, is wrapped
around the iron ring
and is connected to
a galvanometer.
• This is secondary
coil and the circuit is
the secondary
circuit.
• There is no battery
in the secondary
circuit and the
secondary circuit is
not connected to
the primary coil.
Faraday’s Experiment
• The only purpose
of this circuit is to
detect any current
that might be
produced by a
change in the
magnetic field.
• When the switch in
the primary circuit
is closed, the
galvanometer in
the secondary
circuit deflects in
one direction and
then returns to
zero.
Faraday’s Experiment
• When the switch is
opened, the
galvanometer
deflects in the
opposite direction
and again returns
to zero.
• The galvanometer
reads zero when
there is a steady
current in the
primary circuit.
• Faraday concluded that an electric current can be
produced by a changing magnetic field.
• A current cannot be produced by a steady
magnetic field.
• The current that is produced in the secondary
circuit occurs for only an instant while the
magnetic field through the secondary coil is
changing.
• In effect, the secondary circuit behaves as though
there were a source of EMF connected to it for a
short instant.
• An induced EMF is produced in the secondary
circuit by the changing magnetic field.
• In both experiments, an EMF is induced in a
circuit when the magnetic flux through the circuit
changes with time.
• Faraday’s Law of Induction: The EMF induced in
a circuit is directly proportional to the time
rate of change of magnetic flux through the
circuit.
 dΦ
EMF 
m
dt
– where Φm is the magnetic flux threading the circuit.
– Magnetic flux Φm :
Φ m   B  dA
• The integral of the magnetic flux is taken over the
area bounded by the circuit.
• The negative sign is a consequence of Lenz’s law
and is discussed later (the induced EMF opposes
the change in the magnetic flux in the circuit).
• If the circuit is a coil consisting of N loops all of
the same area and if the flux threads all loops, the
induced EMF is:
dΦ m
EMF  N 
dt
• Suppose the magnetic field is uniform over a loop
of area A lying in a plane as shown in the figure
below.
• The flux through the loop is equal to B·A·cos ;
and the induced EMF is:
- dB  A  cos θ 
EMF 
dt
• An EMF can be induced in the circuit in
several ways:
1. The magnitude of B can vary with time;
2. The area of the circuit can change with
time;
3. The angle  between B and the normal
to the plane can change with time; and
4. Any combination of these can occur.
Application of Faraday’s Law
• A coil is wrapped with 200 turns of wire on the
perimeter of a square frame of sides 18 cm. Each
turn has the same area, equal to that of the frame,
and the total resistance of the coil is 2 . A
uniform magnetic field is turned on perpendicular
to the plane of the coil. If the field changes
linearly from 0 to 0.5 Wb/m2 in a time of 0.8 s, find
the magnitude of the induced EMF in the coil
while the field is changing.
– Loop area = (0.18 m)2 = 0.0324 m2
– At t = 0 s, the magnetic flux through the loop is 0 since
B = 0 T.
Application of Faraday’s Law
– At t = 8 s, the magnetic flux through the loop is
Φm = B·A = 0.5 Wb/m2·0.0324 m2 = 0.0162 Wb.
– The magnitude of the induced EMF is:
N  dΦm 200  0.0162 Wb  0 Wb 
EMF 

dt
0.8 s  0 s
EMF  4.05 V
Exponentially Decaying B Field
• A plane loop of wire of area A is placed in a
region where the magnetic field is perpendicular
to the plane. The magnitude of B varies in time
according to the expression B = Bo·e-a·t. That is,
at t = 0 s, the field is Bo, and for t > 0, the field
decreases exponentially in time. Find the induced
EMF in the loop as a function of time.
– At t = 0 s, B is perpendicular to the plane of the loop
and is a maximum.
– The magnetic flux through the loop at time t > 0 is:
Φ m  B  A  Bo  e
 at
A
EMF
EMF
EMF
EMF

 dΦ m
d e  a t

 Bo  A 
dt
dt
a t d  a  t 
 Bo  A  e 
dt
dt
a t
 Bo  A  e  a 
dt
 Bo  A  e a t  a

Applications of Faraday’s Law
• The ground fault interrupter (GFI) is a safety
device that protects users of electrical appliances
against electric shock by making use of Faraday’s
law.
• Wire 1 leads from the
wall outlet to the
appliance to be
protected.
• Wire 2 leads from the
appliance back to
the wall outlet.
• An iron ring surrounds the two wires, and a
sensing coil is wrapped around part of the ring.
• Because the currents in the wires are in opposite
directions, the net magnetic flux through the
sensing coil due to the currents is zero.
• If the return current in wire 2 changes, the net
magnetic flux thru
the sensing coil is no
longer zero.
• This can happen if the
appliance becomes wet,
enabling the current to leak
to the ground.
• Because household current is alternating (its
direction keeps reversing), the magnetic flux
through the sensing coil changes with time,
inducing an EMF if the coil.
• The induced EMF is used to trigger a circuit
breaker, which stops the current before it is able
to reach a harmful level.
Electric Guitar
• The coil is called a pickup coil and is placed near
the vibrating guitar string, which is made of a
metal that can be magnetized.
• A permanent magnet inside the coil temporarily
magnetizes the portion of string nearest the coil.
• When the string vibrates at some frequency, its
magnetized section produces a changing
magnetic flux thru the coil.
• The changing flux induces an EMF in the pickup
coil that is fed to an amplifier.
• The output of the amplifier is sent to the speakers,
which produce the sound we hear.
A Note on the Magnitude of an Induced
Current
• The induced current in the conducting loop
has the same magnitude at all points in the
loop.