A small square loop of wire of side `l` is placed inside a large square loop of wire of side `L (L gt gt l)`. The loops are coplanar and their centre coincide. What is the mutual inductance of the system ?

Explanation

Considering the large loop to be made up of four rod each of length `L`, the field at the centre, i.e., at a distance `(L//2)` from each rod, will be
`B = 4 xx (mu_(0))/(4pi) (I)/(d)[sin alpha+sin beta]`
i.e., `B = 4 xx (mu_(0))/(4pi)(I)/((L//2))xx2 sin 45`

i.e., `B_(1) = (mu_(0))/(4pi)(8sqrt(2))/(L)I`
So the flux linked with smaller loop
`phi_(2) = B_(1)S_(2) = (mu_(0))/(4pi)(8sqrt(2))/(L)l^(2)`
and hence, `M = (phi_(2))/(I) = 2sqrt(2)(mu_(0))/(pi)(l^(2))/(L) rArr M prop (l^(2))/(L)`