A solenoid has length L, radius R, and number of turns N. A second smaller solenoid of length L, radius R and number of turns N is placed at the center of the first solenoid, such that their axes coincide. What is the mutual inductance of the pair of solenoids

Mutual inductance is FLUX induced in second coil due to unit current passed in first coil.

Let i be the current in the bigger coil.

magnetic field at its center

B = μ₀ n i, n is no of turns per unit length

= μ₀ (N / L) i

Magnetic flux associated with small coil placed near its axis

= B X πR² X N

=μ₀ (N / L) i X πR² X N

FLUX = μ₀ (N² / L) i X πR²

FLUX induced by unit current

M = μ₀ (N² / L) X πR²