A small square loop of wire of side l is placed inside a large square loop of side L (L >> l). If the loops are coplanar and their centres coincide, what is the mutual inductance directly proportional to? A) L/l B) l/L C) L^2/l D) l^2/L

What we are comparing when $L \gg l$

Mutual inductance is $$M=\frac{\Phi}{I},$$ where $\Phi$ is the magnetic flux through the small loop due to current $I$ in the large loop. Since $L \gg l$, the field from the large loop does not change much across the area of the small loop, so we can treat it as roughly constant over that small area.

Magnetic field of the large square near its center

At the center of a square loop of side $L$, the distance to each side is $r=L/2$. The field from one finite straight side scales like $$B_{\text{side}} \propto \frac{\mu_0 I}{r} \propto \frac{\mu_0 I}{L}.$$ All four sides contribute with the same scaling, so the total field at the center still scales as $$B_{\text{center}} \propto \frac{\mu_0 I}{L}.$$ (Only the numerical constant changes when you add the four sides.)

Flux through the small loop and the scaling of $M$

The small loop area is $A=l^2$. With nearly uniform $B$ over the small loop, $$\Phi \approx B_{\text{center}}\, A \propto \left(\frac{I}{L}\right) l^2.$$ Therefore, $$M=\frac{\Phi}{I} \propto \frac{l^2}{L}.$$

Correct option

So the mutual inductance is directly proportional to $\dfrac{l^2}{L}$, which is option D.