Q 12-06-062JEE MainJEE Main 2023 (29 Jan, Shift 1)Medium
Find the mutual inductance in the arrangement, when a small circular loop of wire of radius $R$ is placed inside a large square loop of wire of side $L\ (L\gg R)$. The loops are coplanar and their centres coincide.
Answer: (C) $M=\dfrac{2\sqrt2\mu_0R^2}{L}$
Pass current $I$ in the square. Each side is at distance $L/2$ from the centre and subtends $45^\circ$ on each side:
$$B=4\times\frac{\mu_0I}{4\pi(L/2)}(\sin45^\circ+\sin45^\circ)=\frac{2\sqrt2\,\mu_0I}{\pi L}$$
Since $R\ll L$, this field is uniform over the small loop, so the flux is $\phi=B\pi R^2$ and
$$M=\frac{\phi}{I}=\frac{2\sqrt2\,\mu_0R^2}{L}$$
Solution by Sreeraj P, M.Sc Physics