A small square loop of side $a$ and one turn is placed inside a larger square loop of side $b$ and one turn $(b \gg a)$. The two loops are coplanar with their centres coinciding. If a current $I$ is passed in the square loop of side $b$, then the coefficient of mutual inductance between the two loops is:
Answer: (A) $\dfrac{\mu_0}{4\pi}8\sqrt2\dfrac{a^2}{b}$
Field at the centre of a square loop of side $b$: each side is at distance $\dfrac b2$ and subtends $45^\circ$ on both sides.
$$B = 4\times\frac{\mu_0I}{4\pi(b/2)}(\sin45^\circ + \sin45^\circ) = \frac{\mu_0}{4\pi}\cdot\frac{8\sqrt2I}{b}$$
Since $b \gg a$, this field is uniform over the small loop: $\phi = Ba^2$.
$$M = \frac{\phi}{I} = \frac{\mu_0}{4\pi}\cdot\frac{8\sqrt2a^2}{b}$$
Solution by Sreeraj P, M.Sc Physics