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 centres coincide. Find the mutual inductance of the system.

Explanation

Let a current I pass through the large square loop of side L. Magnetic field at the centre O of the loop.
B = 4 x magnetic field due to each side
`therefore B=4xx(mu_0I)/(4pi(L/2)) [ sin 45^@+ sin 45^@]`
`therefore B=4xx(mu_0I)/(4pixxL/2)[1/sqrt2+1/sqrt2]`
`=(2mu_0I)/(piL)(1/sqrt2+1/sqrt2)`
`therefore B=(2sqrt2mu_0I)/(piL)`....(1)
Since l is very small compared to L, value of B can be considered uniform over the area `A =pil^2` of the inner loop.
`therefore` magnetic flux linked with small square loop
`phi=AB`
`therefore phi=l^2 xx (2sqrt2mu_0I)/(piL)`...(2) [`because` From equation (1)]
Mutual inductance of the system of two loops
`M=phi/I`
`therefore M=(2sqrt2mu_0l^2)/(piL)` [`because` From equation (2)]
`therefore M prop l^2/L`