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

Explanation:

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