Whenever there is a change in the magnetic flux linked with a coil, there is also a change of flux linked with the neighbouring coil, producing an induced emf in the second coil. This phenomenon of producing an induced emf in a coil due to the change in current in the other coil is known as mutual induction.

### The coefficient of mutual induction:

At any instant, Magnetic flux linked with the secondary coil α current in the primary coil

where M is mutual inductance or coefficient of mutual induction of the two coils.

**Case1:** If I = 1 , then ϕ = M

Thus the mutual inductance of two coil is numerically equal to the magnetic flux linked with one coil when a unit current passes through the other coil.

**Case 2:** If dI/dt = 1, then e = -M

Thus the mutual inductance of a coil may be defined as the induced emf set up in one coil when the current in the neighboring coil changes at the unit rate.

**Unit of mutual inductance: 1henry (H)**

### Mutual inductance of two long solenoids:

S_{1} and S_{2} are two long solenoids each of length *l*. The solenoid S_{2} is wound closely over the solenoid S_{1} (Fig 4.8).

N_{1} and N_{2} are the number of turns in the solenoids S_{1} and S_{2} respectively. Both the solenoids are considered to have the same area of cross section A as they are closely wound together. I_{1} is the current flowing through the solenoid S_{1}. The magnetic field B_{1} produced at any point inside the solenoid S_{1} due to the current I_{1} is

The magnetic flux linked with each turn of S_{2} is equal to B_{1} A.

Total magnetic flux linked with solenoid S_{2} having N_{2} turns is

Substituting for **B _{1} **from equation (1)

where M is the coefficient of mutual induction between S_{1} and S_{2}.

From equations (2) and (3)

If the core is filled with a magnetic material of permeability μ.