A solenoid `S_1` is placed inside another solenoid `S_2` as shown in the fig. The radii of the inner and the outer solenoids are `r_1` and `r_2` respectively and the numbers of turns per unit length are `n_1` and `n_2` respectively. Consider a length I of each solenoid. Calculate the mutual inductance between them.

Explanation

Suppose a current I is passed through the inner solenoid `S_1` . A magnetic field `B= mu_0 n_(1)i` is produced inside `S_(1)` whereas the feild outside it is zero. The flux through each turn of `S_(2)` is
`Bipir_1^2 = mu_(0)n_(1)i pi r_(1)^(2)`
The total flux through all the turns in a length l of `S_2` is `Phi = (mu_0 n_1 ipir_1 ^2)n_2l =(mu_0 n_1 n_2 pir_1 ^2l)i`
Thus, `M =mu_0 n_1 n_2 pir_1 ^2l`.