A small circular wire loop of radius r is placed inside a large square loop ABCD of side a (where a >> r ). The loops lie in the x-y plane with their centres at the origin O. The mutual inductance of the system is proportional to:

Refer to Fig. Let I be the current in the square loop.

The magnetic field at centre O due to current I in the square loop of side a is:

$B=\frac{2\sqrt{2}{\mu }_{0}I}{\pi a}$

Since r << a, the magnetic field can be assumed to be constant throughout the inner circular loop. Therefore, magnetic flux through the circular loop is

$\varphi =BA=B\times \pi r^2$

$\varphi =\frac{2\sqrt{2}{\mu }_{0}I}{\pi a}\times \pi r^2=\frac{2\sqrt{2}{\mu }_{0}I{r}^2}{a}$

$\text{ By definition }\varphi =MI\text{. Hence, }$

$M=\frac{2\sqrt{2}{\mu }_0{r}^2}{a}$

$\mathrm{Thus}, \mathrm{M}\propto \frac{{\mathrm{r}}^2}{\mathrm{a}}$