Two concentric circular coils, one of small radius `r_(1)` and the other of large radius `r_(2)`, such that `r_(1) gtgt r_(2)`, are placed co-axially with centres coinciding. Obtain the mutual inductance of the arrangement.

Explanation

Let a current `I_2 ` flow through the outer circular coil The field at the centre of the coil is `B_2 = mu_0 I_2// 2r_2` . Since the other co-axially placed coil has a very small radius, B may be considered constant over its cross-sectional area, Hence.
`phi_1 = pi r_1^2 B_2 = (mu_0 pi r_1^2)/(2r_2) I_2 = M_12 I_2 ` thus

mutual inductance of solenoid `S_1` with respect to `S_2`
`M_12 = (mu_0 pi r_1^2)/(2r_2) ......(i) " but " M_12 = M_21 = (mu_0 pi r_1^2)/(2r_2)`
Note that we calculated `M_12` from an approximate value of `phi_1` , assuming the magnetic field `B_2` to be uniform over the area `pi r_1^2` . However we can accept this value because `r_1 lt lt r_2` ?
It would have been difficult to calculate the flux through the bigger coil of the non-uniform field due to the current in the smaller coil and hence the mutual inductance `M_12` The equality `M_12 = M_21` is helpful. Note also that mutual inductance depends solely on the geometry.