Induced e.m.f can be either dynamically induced emf or statically induced emf. in this first case, usually the field is stationary and conductors cut across it (as in d.c. generator). But in the second case, usually the conductor or the coil remains stationary and flux linked with it is changed by simply increasing or decreasing the current producing this flux (as in transformers).

Let the flux linking with the coil of turns N be changed by an amount in short time dt.

EMF induced, *e* = Rate of change of flush linkage

= Number of turns rate of change of flux

=

A minus sign is required to be placed before the right hand side quantity of above expression just to indicate the phenomenon explained by Lenz’s law, therefore, expression for induced emf may be written as

……(1)

## Dynamically Induced EMF

We have learnt that when the flux linking with the coil or circuit changes, an emf is induced in the coil or circuit.

EMF can be induced by changing the flux linking in two ways:

- By increasing or decreasing the magnitude of the current producing the linking flux. In this case, there is no motion of the conductor or of coil relative to the field and, therefore, emf induced in this way is known as statically induced
- By moving a conductor in a uniform magnetic field and emf produced in this way is known as
*dynamically induced emf*

Consider a conductor of length *l* meters placed in a uniform magnetic field of density , as shown in Fig. 1(a).

Let this conductor be moved with velocity v m/s in the direction of the field, as shown in Fig. 1(b). In this case no flux is cut by the conductor, therefore, no emf is induced in it.

Now if this conductor is moved with Velocity *v* m/s in a direction perpendicular to its own length and perpendicular to the direction of the magnetic field, as shown in Fig. 1(c) flux is cut by the conductor, therefore, an emf is induced in the conductor.

Area swept per second by the conductor = m^{2}/s

Flux cut per second = Flux density area swept per second = Bl*v*

Rate of change of flux, = Flux cut per second = B*l*v Wb/s

Induced emf, e = Bl*v* volts

If the conductor is moved with velocity *v* meters per second in a direction perpendicular to its own length and at an angle to the direction of magnetic field, as shown in Fig. 1(d).

The magnitude of emf induced. is proportional to the component of the velocity in a direction perpendicular to the direction of the magnetic field and induced emf is given by

The direction of this induced emf is given by Fleming’s right hand rule.

If the thumb, forefinger and middle finger of right hand are held mutually perpendicular to each other, forefinger pointing into the direction of the field and thumb in the direction of motion of conductor then the middle finger will point in the direction of the induced emf as shown in Fig. 3.

Fig. 4 illustrates another way of determination of induced emf, known as right hand flat palm rule. This law states that if right hand is so placed in a magnetic field along the conductor that the magnetic lines of force emerging from the north pole enter the palm and the thumb points in the direction of motion of conductor, the other four fingers will give the direction of induced emf or current.

## STATICALLY INDUCED EMF

Statically induced emf may be (a) self-induced emf or (b) mutually induced emf

**Self-Induced EMF**

When the current flowing through the coil is changed, the flux linking with its own winding changes and due to the change in linking flux with the coil, an emf, known as self-induced emf, is induced.

Since according to Lenz’s law, an induced emf acts to oppose the change that produces it, a self-induced emf is always in such a direction as to oppose the change of current in the coil or circuit in which it is induced. This property of the coil or circuit due to which it opposes any change of the current in the coil or circuit, is known as self-inductance.

Consider a Solenoid of N turns, length *l* meters, area of X-section a square meters and of relative permeability . When the solenoid carries a current of *i* amperes, a magnetic field of flux webers is set up around the solenoid and links with it.

If the current flowing through the solenoid is changed, the flux produced by it will change and, therefore, an emf will be induced.

self-induced emf,

……..(2)

The quantity is a constant for any given coil or circuit and is called coefficient of self-inductance. It is represented by symbol L and is measured in henries.

Hence self-induced emf,

Where

**Coefficient of Self Induction**

The coefficient of self-induction (L) can be determined from any one of the following three relations.

**First Method.** In case the dimensions of the solenoid are given, the coefficient of self-induction may be determined from the relation

……(3)

**Second Method.** In case the magnitude of induced emf in a coil for a given rate of change of current in the coil is known, self-inductance of the coil may be determined from the following relation.

…….(4)

**Third Method.** In case the number of turns of the coil and flux produced per ampere of current in the coil is known, the self-inductance of the coil may be determined from the following relation

…..(5)

The above relation can be derived as follows:

Magnetic flux produced in a coil of N turns, length *l* meters, area of x-section *a* meters^{2} and relative permeability when carrying a current of I amperes is given by

and self-inductance of the coil

From the above relation, it is obvious that the self-inductance of a coil or circuit is equal to weber-turns per ampere in the coil or circuit.

In the above relation if =1Wb-turn and i = 1 A then L = 1 H.

*Hence a coil is said to have a self-inductance of one henry if a current of 1 A, when flowing through it, produces flux linkage of I Wb-turn in it.*

### Mutually Induced EMF

Consider two coils A and B placed closed together so that the flux created by one coil completely links with the other coil. Let coil A have a battery and switch S and coil B be connected to the galvanometer G.

When switch SW1 is opened, no current flows through coil A, so no flux is created in coil A, i.e. no flux links with coil B, therefore, no emf is induced across coil B, the fact is indicated by galvanometer zero deflection. Now when the switch S is closed current in coil. A starts rising from zero value to a finite value, the flux is produced during this period and increases with the increase in current of coil A, therefore, flux linking with the coil B increases and an emf, known as mutually induced emf is produced in coil B, the fact is indicated by galvanometer deflection. As soon as the current in coil A reaches its finite value, the flux produced or flux linking with coil B becomes constant, so no emf is induced in coil B, and galvanometer pointer returns back to zero position. Now if the switch S is opened, current will start decreasing, resulting in decrease influx linking with coil B, an emf will be again induced but in direction opposite to previous one, this fact will be shown by the galvanometer deflection in opposite direction.

*Hence whenever the current in coil A changes, the flux linking with coil B changes and an emf, known as mutually induced emf is induced in coil B.*

Consider coil A of turns N_{1} wound on a core of length *l* meters, area of cross-section *a* square meters and relative permeability . When the current of i_{1} amperes flows through it, a flux of is set up around the coil A.

Mutually induced emf, e_{m} = -Rate of change of flux linkage of coil B

= -N_{2} rate of change of flux in coil A

…..(6)

The quantity is called the coefficient of mutual induction of coil B with respect to coil A. It is represented by symbol M and is measured in henrys.

Hence mutually induced emf,

Where

**Coefficient of Mutual Induction**

Mutual inductance may be defined as the ability of one coil or circuit to induce an emf in a nearby coil by induction when the current flowing in the first coil is changed. The action is also reciprocal i.e. the change in current flowing through second coil will also induce an emf in the first coil. The ability of reciprocal induction is measured in terms of the coefficient of mutual induction M.

The coefficient of mutual induction (M) can be determined from any one of the following three relations.

**First Method.** In case the dimensions of the coils are given, the coefficient of mutual induction may be determined from the relation

…..(7)

**Second Method.** In case the magnitude of induced emf in the second coil for a given rate of change of current in the first coil is known, mutual inductance between the coil may be determined from the following relation

Or …..(8)

**Third Method.** In case the number of turns of the coil and flux linking with this coil per ampere of current in another coil is known, the mutual inductance of the coil may be determined from the following relation

……(9)