When flux linking with a conductor (or coil) changes, an emf is induced in it. This change in flux linkages can be obtained in the following two ways:

By either moving the conductor and keeping the magnetic field system stationary or moving the magnetic field system and keeping the conductor stationary, in such a way that conductor cuts across the magnetic field (as in case of DC and AC generators). The emf induced in this way is called dynamically induced emf

By changing the flux linking with the coil (or conductor) without moving either coil or field system. However, the change of flux produced by the field system linking with the coil is obtained by changing the current in the field system (solenoid), as in transformers. The emf induced in this way is called statically induced emf

By either moving the conductor keeping the magnetic field system stationary or moving the field system keeping the conductor stationary so that flux is cut by the conductor, the emf thus induced in the conductor is called dynamically induced emf

\[\begin{aligned} \text{Area swept by the conductor}, A &=l \times d x \\ \text{Flux cut by the conductor}, \phi&=B \times A=B l d x \\ e&=\frac{\text { flux cut }}{\text { time }}=\frac{\phi}{d t}=\frac{B l d x}{d t}=B l \text { v } \end{aligned}\]

If conductor is moved at an angle \(\theta\) with the direction of magnetic field at a velocity \(v\) metre/second .

A small distance covered by the conductor in that direction is \(d x\) in time \(d t\) second.

Then the component of distance perpendicular to the magnetic field, which produces emf, is \(d x \sin \theta\)

\[\begin{aligned} \text{Area swept by the conductor}, A=l \times d x \sin \theta \\ \text{Flux cut by the conductor}, \phi=B \times A=B l d x \sin \theta \\ \text { Induced emf } e=\frac{B l d x \sin \theta}{d t}=B l v \sin \theta \end{aligned}\]

When the coil and magnetic field system both are stationary but the magnetic field linking with the coil changes (by changing the current producing the field), the emf thus induced in the coil is called statically induced emf

The statically induced emf may be:

Self induced emf

Mutually induced emf

Emf induced in a coil due to the change of flux produced by it linking with its own turns

Direction of this induced emf is such that it opposes the cause which produces it (Lenz’s law) i.e., change of \(I\) in the coil.

Since the rate of change of flux linking with the coil depends upon the rate of change of current in the coil. Therefore, the magnitude of self induced emf is directly proportional to the rate of change of current in the coil.

Magnitude of self induced emf is directly proportional to the rate of change of current in the coil, i.e., \[e \propto \frac{d I}{d t} \quad \text { or } \quad e=L \frac{d I}{d t}\] where L is a constant of proportionality and is called self inductance or co-efficient of self inductance or inductance of the coil

Emf induced in a coil due to the change of flux produced by another (neighbouring) coil, linking with it

Since the rate of change of flux linking with coil ‘B’ depends upon the rate of change of current in coil ‘A’. Therefore, the magnitude of mutually induced emf is directly proportional to the rate of change of current in coil ‘A’, i.e., \[e_{m} \propto \frac{d I_{1}}{d t} \text { or } e_{m}=M \frac{d I_{1}}{d t}\] where M is a constant of proportionality and is called mutual inductance or co-efficient of mutual inductance