Question

Find an expression of mutual inductance for shown concentric co-planner circular and regular hexagonal loops $(a>>r):$

Answer 1

Let us flow a current $I$ in the hexagonal loop.

Now, field at $O$ due to one side of hexagon is $B_1=\frac{{\mu }_{o}I}{2\pi d}\cos {60}^o=\frac{{\mu }_{o}I}{4\pi \frac{\sqrt{3}a}{2}}$

Hence, total field at O is $B=6B_1=\frac{\sqrt{3}{\mu }_{o}I}{\pi a}$

Hence, flux through the circular loop is $\varphi =B\times \pi r^2$

But, $\varphi =MI=\frac{\sqrt{3}{\mu }_o{r}^2}{a}I$

Hence, $M=\frac{\sqrt{3}{\mu }_o{r}^2}{a}$