Magnetism - Faraday's Law of Induction

where S is the surface over which the flux is evaluated.

For constant B, perpendicular to the surface, Φ_B = BA where A is the surface area of S.

The magnetic flux, Φ_B, is so important it has its own unit the Weber - 1 Weber = 1 T.m² . In the early days of electromagnetism it was common to measure the magnetic (B) field in Weber/m² .

In term of the magnetic flux Faraday's Law of Induction is given by,

The induced electromotive force (emf) in a circuit is equal to the rate of change of magnetic flux through the circuit.

An emf is not a force, rather it can be considered as the voltage induced in a closed circuit.

Faraday experimentally determined his law in the form presented above.

One of the easiest ways to change the magnetic flux through a circuit is to move a permanent (bar) magnet towards or away from the circuit as shown in the diagrams below.

(a) Magnetic flux passes through the circuit, but does not change with time, so there is no induced emf and so no induced current.

(b) The flux through the circuit increases with time causing an induced emf and current.

(c) As the magnet moves faster the rate of change of flux with time is increased causing a larger emf and current.

(d) When the magnet moves away from the circuit the flux decreases with time so the induced emf and current are reversed.

The origin of the changing magnetic flux (field) is not limited to permanent magnets. The magnetic field due to a second circuit can produce a similar effect, as described in the examples below.

In the diagram at right the current in the left circuit is constant, but the flux through the other circuit increases as the two circuits get closer.

In the situation at left both circuits are stationary. The current in the left circuit is initially zero, but rapidly increases to a constant value when the switch is closed. As the current reaches its final (constant) value the flux through the right circuit is increasing with time, thus by Faraday's Law, causing a brief pulse of induced current in the second circuit. When the switch is opened the flux in the right circuit rapidly decreases causing a short induced current pulse in the opposite direction.

Important ! In both the above examples a magnetic field (B) changing with time results in a changing magnetic flux. But it is possible to have a changing flux with a constant B if the cross sectional area - dA - can be made to change with time as in the case of the (electric) generator.

LENZ'S LAW

Mathematically the negative sign in Faraday's Law tells us about the direction of the induced (emf) current. Practically we use Lenz's Law to determine the direction in specific cases.

"The induced current will appear in such a direction that it opposes the change that produced it"

Confusing ? Yes !

The induced (flux) magnetic field (associated with the induced current) does not necessarily oppose the field which causes the change in flux, rather it opposes the CHANGE in this field.

Lenz's Law always ensures that there is a force resisting the motion of the magnet. It is the work done against this force which appears as the energy of the moving charges of the induced current.

magfaradayfig6

FARADAY'S LAW = MAXWELL EQUATION

Whenever there is an induced electric current there must also be an induced electric field, E. The work dW done by this induced field moving a charge q₀ a distance ds around a loop is given by,

where dε is the potential difference in ds.

Therefore,

So that the emf around the whole loop is

Equating this emf to that given by Faraday's Law we obtain the integral form of Faraday's Law, the third of Maxwell's equations we have encountered so far,

Note that the line integral of E must be round a closed loop (circuit).

Written in the above form the relationship between the E and B fields is clear

"A magnetic field changing with time induces an electric field"

Shortly we will see that the reverse of this statement is also true.