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 loop are coplanar and their centres coincide. The mutual inductance of the system is proportional to

Explanation

Considering the large loop to be made up of four rods 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))/(4 pi)(I)/(d)[sin alpha+sin beta]`
i.e., `B=4 xx (mu_(0))/(4phi)(I)/((L//2)) xx 2sin45`
i.e., `B_(1)=(mu_(0))/(4pi)(8sqrt2)/(L)I`

So the flux linked with smaller loop
`Phi_(2)=B_(1)S_(2)=(mu_(0))/(4pi)(8sqrt2 I)/(LO) l^(2)`
and hence, `M=(phi_(2))/(I) =2sqrt2 (mu_(0))/(pi)(l^(2))/(L) implies M prop (l^(2))/(L)`