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Introduction to Mutual Inductance of two Solenoids

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

S1 and S2 are two long solenoids each of length l. The solenoid S2 is wound closely over solenoid S1 as shown in figure.


N1 and N2 are the number of turns in the solenoids S1 and S2 respectively. Both the solenoids are considered to have the same area of cross section A as they are closely wound together. I1 is the current flowing through the solenoid S1.

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

Derivation of Mutual Inductance of Two Solenoids :

The magnetic field B1 produced at any point inside the solenoid S1 due to the current I1 is

B1 = `mu_0N_1/lI_1`                                      ---------------------------> (1)

The magnetic flux linked with each turn of S2 is B1A

Total magnetic flux linked with solenoid S2 having N2 turns is

           `phi_2` = B2AN2

Substituting B1 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` = MI1 ---------------- (3)

 where M is the coefficient of mutual induction for the two solenoids S1 and S2

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Mutual Inductance of Two Solenoids is Given by M:

From equation (2) and (3)

   MI1 =   `(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`