Mutual Inductance

We know that if a current builds up or varies in a coil, the flux change leads to induced e.m.f in the same coil. This can happen event mutually between two interacting coils are close together, and if current is passed in one of them, it sets up a magnetic flux surrounding itself. When the second coil is near the first coil, the changes in the magnetic flux of the first coil produces similar changes in the second. Thereby, producing induced e.m.f in the second coil. To distinguish it from self-induction, it is called as mutual induction.

It is the property of two circuits (or coils) by virtue of which each oppose any change in the magnitude of the current flowing through the other circuit by producing an induced EMF in it.

Consider two coils P and S placed near coil P connected to a battery and key and is called the primary coil. Coil S is connected to a sensitive galvanometer and is called the secondary coil.

When the key K is closed, the flux linked with the coil in the primary circuit changes. This induces an EMF in the secondary coil indicated by the deflection in the galvanometer.

When the key K is opened, an EMF is induced in the secondary coil, but in a direction opposite to that induced during the make, i.e., current in S always oppose any change in current in P.

Sub Topics

Coefficient of Mutual Induction
Note:
Mutual Inductance of Two Long Solenoid
Note:

Coefficient of Mutual Induction

If f₂ be the flux linked with the secondary or neighboring coil at a given time and t and I₁ be the current flowing in the primary coil at time t,

It is found that

f₂ a I₁

f₂ = M₁₂I₁

where M₁₂ is constant and called the coefficient of mutual induction or mutual inductance of two coils.

Also induced EMF,

The -ve sign shows that e₂ and dI₁/dt are opposite signs, but M is positive.

Similarly one can show that, the induced EMF on the first coil e₁ due to the current flowing in the second coil I₂ is

If I = 1 ampere

f=M then the coefficient of mutual induction of two coils is numerically equal to the amount of magnetic flux linked with one coil when a unit current flows through the neighboring coil.

Note:

The mutual inductance of two coils depends on the geometry of the two coils, distance between the coils and orientation of the two coils.

The following diagrams indicate the maximum coupling between the two coils.

(i) Coupling between the coils is maximum.

(ii) Coupling is less than in case (a)

(iii) Coupling is minimum

Mutual inductance of two long solenoids

Mutual Inductance of Two Long Solenoid

Consider a solenoid P within the core having N₁ turns. Another solenoid S having N₂ turns is wound over the solenoid P. Let 'l' be the length of each solenoid and let them have nearly the same area of cross-sections A.

The magnetic field B₁ at any point inside P due to current I₁ is

The flux f linked with each turn of S

= B₁ x area of each turn

= B₁ A

Total magnetic flux linked with S

f₂

B₁A x N₂

Now f₂= M₁₂ I₁

On comparing,

Note:

If the area of cross section was different from the area of cross section of the inner solenoid, the smaller one is to be considered.