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

Suppose a current `I_(2)` flows through the outer circular coil. The field at the centre of the coil is
`B_(2)= (mu_(0)I_(2))/(2r_(2))`

The second co-axially placed coil has very small radius. So `B_(2)` may be considered constant over its cross-sectional area
Now, `phi_(1)= pi r_(1)^(2) B_(2)= pi r_(1)^(2) ((mu_(0)I_(2))/(2r_(2)))`
or `phi_(1)= (mu_(0)pi r_(1)^(2))/(2r_(2)) I_(2)`
Comparing with `phi= M_(12)I_(2)`, we get
`M_(12)= (mu_(0)pi r_(1)^(2))/(2r_(2))`
Also, `M_(21)= M_(12)= (mu_(0)pi r_(1)^(2))/(2r_(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 helpul. Note also that mutual inductance depends solely on the geometry.