# Physics > Electromagnetic Induction

• The magnetic flux through a surface of area A placed in a uniform magnetic field B is defined as,  where θ is the angle between B and A.
• Faraday's laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of flux through it,

Here  is the flux linked with one turn of the coil. If the circuit is closed, a current  is set up in it, where R is the resistance of the circuit

• Lenz's law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it. The negative sign in the expression for Faraday's law indicates this fact.
• When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is

• Changing magnetic fields can set up current loops in nearby metal (any conductor) bodies. They dissipate electrical energy as heat. Such currents are called eddy currents.
• Inductance is the ratio of the flux-linkage to current. It is equal to
• A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1). This relation is given by,

The quantity  is called mutual inductance of coil 1 with respect to coil 2. One can similarly define . There exists a general equality,

M12 = M21

• When a current in a coil changes, it induces a back emf in the same coil. The self-induced emf is given by,

L is the self-inductance of the coil. It is a measure of the inertia of the coil against the change of current through it.

• The self-inductance of a long solenoid, the core of which consists of a magnetic material of permeability , is given by

Al where A is the area of cross-section of the solenoid, l its length and n the number of turns per unit length.

• In an ac generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction. If coil of N turn and area A is rotated at ν revolutions per second in a uniform magnetic field B, then the motional emf produced is

where we have assumed that at time t = 0 s, the coil is perpendicular to the field.

### Sample Examples

#### Question

A wheel with 10 metallic spokes each 0.5 m long is rotated with a speed of 120 rev/min in a plane normal to the horizontal component of earth's magnetic field HE at a place. If HE = 0.4 G at the place, what is the induced emf between the axle and the rim of the wheel?

Note that

#### Solution

Induced

The number of spokes is immaterial because the emf's across the spokes are in parallel.

#### Question

A circular coil of radius 10 cm, 500 turns and resistance 2 Ω is placed with its plane perpendicular to the horizontal component of the earth's magnetic field. It is rotated about its vertical diameter through 180°in 0.25 s. Estimate the magnitudes of the emf and current induced in the coil. Horizontal component of the earth's magnetic field at the place is 3.0 × 10–5 T.

#### Solution

Initial flux through the coil,

Final flux after the rotation,

Therefore, estimated value of the induced emf is,

Note that the magnitudes of ε and I are the estimated values. Their instantaneous values are different and depend upon the speed of rotation at the particular instant.