A small square loop of wire of side $l$ is placed inside a large square loop of wire of side $L (L>>l)$. The loops are co- planar and their centres coincide. The mutual inductance of the system is proportional to

$\frac{l^2}{L}$

The correct answer is Option (2) → $\frac{l^2}{L}$

Given:

Small loop of side $l$ inside large loop of side $L$ ($L \gg l$), co-planar, centres coincide.

Mutual inductance $M$ is given by:

$M = \frac{\text{Flux through small loop due to current in large loop}}{\text{Current in large loop}}$

Magnetic field at the centre of a large square loop carrying current $I$:

$B \propto \frac{\mu_0 I}{L}$

Flux through small loop:

$\Phi = B \cdot \text{area of small loop} \propto \frac{\mu_0 I}{L} \cdot l^2$

Mutual inductance:

$M = \frac{\Phi}{I} \propto \frac{\mu_0 l^2}{L}$

Answer: Mutual inductance $M \propto \frac{l^2}{L}$