Derive the expression for the mutual inductance of two long coaxial solenoids of same length l having radii r_{1} and r_{2} (r_{2}>r_{1} and l>>r_{2}). is it the derivation for mutual inductance due to two solenoids?

Suppose a current i is passed through the inner solenoid

S_{1}. A magnetic field

is produce inside S_{1}

whereas field outside it is zero.

is permeability of vaccum.

Let n_{1} and n_{2} be the no of turns it each solenoid of equal length l.

Therefore, flux through each turn of solenoid S_{2} is

The total flux through all the turns in a length l of S_{2} is

And we know that

where M is the mutual inductance.

Therefore mutual inductance of two long coaxial solenoids of same lenth is