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

The coefficient of mutual induction found between two coils depends on the following factors:

Size and Shape of the coils, number of turns and permeability of material on which the coils are wound

proximity of the coils

S_{1} and S_{2} are two long solenoids each of length l. The solenoid S_{2} is wound closely over solenoid S_{1} as shown in figure.

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}.

Lets find the formula that denotes the **mutual inductance** of two solenoids.

The **magnetic field** B_{1} produced at any point inside the solenoid S_{1} due to the current I_{1} is

B_{1} = `mu_0N_1/lI_1` ---------------------------> (1)

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

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

`phi_2` = B_{2}AN_{2}

Substituting B_{1} in (1)

`phi_2 = ( mu_0 N_1/l I_1) A N_2`

`phi_2 = (mu_0N_1N_2AI_1)/l` ------------------------> (2)

But `phi_2` = MI_{1} ---------------- (3)

where M is the coefficient of mutual induction for the two solenoids S_{1} and S_{2}.

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From equation (2) and (3)

MI_{1} = `(mu_0N_1N_2AI_1)/l`

M = `(mu_0N_1N_2A)/l`

If the core is filled with a magnetic material of permeability `mu`

M = `(muN_1N_2A)/l`