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 ?

Options

A. `2sqrt2 (mu_0)/(pi) (l^2)/(L)`
B. `8sqrt2 (mu_0)/(pi) (l^2)/(L)`
C. `2sqrt2 (mu_0)/(2pi) (l^2)/(L)`
D. `2sqrt2 (mu_0 L^2)/(pi l)`

Answer

A

Explanation

Let the current I be flowing in the larger loop. The larger loop is made up of four wires each of length L, the field at the centre i.e., at a distance L/2 from each wire, will be
`B = 4 xx (mu_0 I)/(4pi (L//2)) (sin 45^@ + sin 45^@) = 2sqrt2 (mu_0)/(pi)I/L`
Flux linked with smaller loop,
`phi_2 = BA_2 = 2sqrt2 (mu_0)/(pi) (l^2)/(L)`