A long solenoid of length l, cross-sectional area A and having N_{1} turns (primary coil) has a small coil of N_{2} turns (secondary coil) wound about its centre. Determine the Mutual inductance (M) of the two coils.

B = µ_{0}nI_{s} ……………….. (1)

Then, the magnetic flux through each turn of the coil due to the current in the solenoid is

Φ_{CS} = BA = (µ_{0}nI_{s})(πR^{2}) ……………… (2)

Thus, their mutual inductance is

M = \(\frac{N \Phi_{\mathrm{CS}}}{I_{\mathrm{S}}}\) = µ_{0}πR^{2}nN ……………. (3)

Equation (2) is true as long as the magnetic field of the solenoid is entirely contained within the cross section of the coil C. Hence, M does not depend on the shape, size, or possible lack of close packing of the coil.

replacing n with N_{1}/l and N with N_{2}. M = µ_{0}A = \(\frac{N_{1} N_{2}}{l}\)