Two concentric circular coils, one of small radius `r_1` and the other of large radius `r_2`, such that `r_1 lt lt 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_2` 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,
`M_(12)= (miu_0 pi r_1^2)/(2r_2)`
From Eq. (6.14)
`M_(12) = M_(21) = (mu_0 pi r_1^2)/(2r_2)`
Note that we calculated M12 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`.