When the flux linking with a coil (or conductor) changes, an EMF is induced in the coil. This EMF can be induced by the following two ways:

- By rotating the coil in a stationary magnetic field or by rotating a magnetic field in a stationary coil (as in AC and DC generators). The induced EMF in this way is known as dynamically induced EMF. It is so called because either magnetic field or coil is moving in this case.
- The magnetic field and the coil both are stationary. But the flux linking with the coil is changed by changing the current producing this flux (as in transformers). The induced EMF in this way is called statically induced EMF. It is so called because magnetic field and coil both are stationary in this case. The statically induced EMF can be further subdivided into the following two categories:

- Self-induced EMF
- Mutually induced EMF

## Self Induced EMF

The EMF induced in a coil due to change of flux produced by it linking with its own turns is known as self-induced EMF.

Consider a coil having N turns on an iron core connected to a battery as shown in the figure. The current ** I **flows through the coil, it produces flux φ. This flux also links with its own turns. If the current flowing through the coil is changed by changing the variable resistance R, it changes the flux linking with the coil and hence an EMF is induced in the coil.

The EMF induced in this way is called self-induced EMF. The direction of induced EMF is such that it opposes the change of current i.e. the cause which produces it (Lenz’s law).

The magnitude of self-induced EMF is directly proportional to the rate of change of flux linking with the coil (Faraday’s law). But the rate of change of flux linking with the coil is directly proportional to the rate of change of current in the coil.

Therefore, self-induced EMF, e α – d

**/dt**

*I*Or e = – L(d

**/dt) ……….(i)**

*I*Where L is a constant of proportionality and is known as the coefficient of self-inductance or self-inductance or inductance of the coil.

# Self Inductance Definition | Formula

The property of the coil to oppose any change in current flowing through itself is known as self-inductance or inductance of the coil.

Self Inductance, L = (N_{2}µ_{o}µ_{r}a) ÷ l

where N = number of turns of coils,

a = cross-sectional area of magnetic circuit,

l = length of the magnetic circuit.

A coil attains this property due to self-induced EMF in the coil **when the current flowing through it is changing. **

As shown in Figure, if the current in the coil is increasing (by changing the rheostat resistance), the self-induced EMF is set up in such a direction so as to oppose the rise of current i.e. the **direction of self-induced EMF is opposite** to that of the applied voltage.

Similarly, if the current the coil is decreasing, self-induced EMF is set up in such a direction so as to oppose the fall of current i.e. the direction of self-induced EMF is in the **same direction** to that of the applied voltage.

It may be noted that self-inductance does not prevent the change of current since extra energy is supplied by the battery to overcome this opposition. However, **it delays the change of current through the coil**.

**It may be noted that the presence of inductance in the circuit is exhibited only when there is a change of current in the circuit. **

For example, if steady (direct) flows through a coil having an inductance (by virtue of its geometrical and magnetic properties) will not exhibit its presence.

However, when alternating (continuously changing) current flows through the same coil, it will exhibit its presence effectively.

The unit of inductance is Henry (H).

If in equation (i) above, e = 1 volt and d*I*/dt = 1 A/second, then L = 1 H.

Hence, a coil (or circuit) is said to have an inductance of one Henry, if an EMF of 1 volt is induced in it when current changes through it at the rate of one ampere per second.

From the above expression, it is clear that inductance is inversely proportional to the reluctance of the magnetic circuit. Therefore, a coil wound on a magnetic core (low reluctance) has more inductance than the equivalent air-cored coil.

## Mutually Induced EMF

The EMF induced in a coil due to the change of flux produced by another coil is called **mutually induced EMF.**

Consider a coil X having N_{1} turns and a coil Y having N_{2} turns are wounded on an iron bar. When these coils are connected according to the circuit, as shown Figure, the current flow through coil X and produce flux φ_{1}.

A part of this flux also links with the coil Y. If the current flowing through the coil X is changed by changing the variable resistance R, it changes the flux linking with the other coil Y and hence an EMF is induced in coil Y.

The EMF induced in this way is known as mutually induced EMF. The direction of mutually induced EMF is such that it opposes the change of current in the coil X i.e. the cause which produces it (Lenz’s law).

Mathematically, mutually induced EMF, e_{m } α – d*I*_{1}/dt

or e_{m} = – M(d*I*_{1}/dt)

Where M is constant of proportionality and is known as the coefficient of mutual inductance or mutual inductance of a coil.

# Mutual Inductance Definition | Formula

The mutual inductance between the two coils may be defined as the property of the second coil due to which it opposes the change of current in the first coil. Expressions for mutual inductance are:

M = (N_{1}N_{2}µ_{o}µ_{r1}a_{1}) ÷ l_{1}

M = (N_{1}N_{2}µ_{o}µ_{r2}a_{2}) ÷ l_{2}

where, N_{1}, N_{2}= number of turns of coils,

a_{1}, a_{2}= cross-sectional areas of magnetic circuits,

l_{1}, l_{2}= lengths of the magnetic circuits.

## Coefficient of Coupling

The fraction of the magnetic flux produced by the current in the first coil that links with the second coil is called the coefficient of coupling (k) between the two coils.

Co-efficient of coupling, k = M ÷ √(L_{1}L_{2})

where M = Mutual inductance between the two coils,

L_{1} = Self-inductance of the first coil,

L_{2} = Self-inductance of the second coil.

The value of k is unity if the flux produced by one coil links completely with the other. Due to the effect of flux leakage, the value of k is always less than unity. The value of mutual inductance depends upon the value of the coefficient of coupling (k). Its value is maximum when k = 1.

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