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

Explanation

Suppose a time varying current `I_(2)` is made to flow through the outer circular coil, Fig.
`:.` Magnetic field at the centre of the coil is `B_(2) = (mu_(0) I_(2))/(2 r_(2))`
As the inner coil placed co axially has very small radius, therefore, `B_(2)` may be taken as constant over its cross sectional area. Hence, flux associated with inner coil is
`phi_(1) = pi r_(1)^(2) B_(2) = pi r_(1)^(2) (mu_(0) I_(2))/(2 r_(2)) , phi_(1) = ((mu_(0) pi r_(1)^(2))/(2 r_(2))) I_(2) = M_(12) I_(2)`
`:. M_(12) = (mu_(0) pi r_(2)^(2))/(3 r_(2))`
Here, `M_(21) = M_(12) = (mu_(0) pi r_(1)^(2))/(2 r_(2))`