Define the term 'mutual inductance' between the two coils. Obtain the expression for mutual inductance of a pair of long coaxial solenoids each of length l and radii r1 and r2 (r2 >> r1). Total number of turns in the two solenoids are N1 and N2 respectively.

A coil B kept near another coil A has magnetic flux passing through it when kept near coil A. The ratio of magnetic flux through the coil B to the current in the coil A is called as mutual inductance of coil A and coil B.
M12 = Φ2/i
Let S1 carries a current i. Magnetic field inside S1 will be
B = μoN1i/l
Flux through each turn of S2 is
Φ = B × πr2²
Φ = μoN1πr2²i/l
Flux through all turns of S2 is
ΦB = ΦN2
ΦB = μoN1πr2²iN2/l
M = ΦB/i = μoN1N2πr2²/l