Faraday’s Law of Induction

AP Physics CMontwood 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.

– where Φm is the magnetic flux threading the circuit.

– Magnetic flux Φm :

dtdΦEMF m

dABΦm

• 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:

dtdΦNEMF m

dt

θ cosABd-EMF

• 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:

• 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; and4. 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:

V4.05EMFs0s0.8

Wb0Wb0.0162200dtdΦNEMF m

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:

AeBABΦ atom

aeABEMFdtdtaeABEMF

dttadeABEMFdtedABdt

dΦEMF

tao

tao

tao

ta

om

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.