A small square loop of wire of side l is placed inside a large square loop of wire L L ≫ l . Both…

Let a current, $I$ pass through the square loop of side, $L$. The magnetic field at the centre $O$,

$B=4\times \mathrm{Magnetic} \mathrm{field} \mathrm{due} \mathrm{to} \mathrm{each} side$

$\Rightarrow B=4\times \frac{{\mu }_0}{4\pi }\cdot \frac{I}{L/2}\left(\sin {45}^{\circ }+\sin {45}^{\circ }\right)$

$\Rightarrow B=\frac{2{\mu }_{0}I}{\pi L}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\right)=\frac{2\sqrt{2}{\mu }_{0}I}{\pi L}$

Magnetic flux linked with the small square loop,

$\varphi =BA=B{l}^2=\frac{2\sqrt{2}{\mu }_{0}I{l}^2}{\pi L}$

Mutual inductance of the coil, 

$M=\frac{\varphi }{I}=\frac{2\sqrt{2}{\mu }_0{l}^2}{\pi L}$.