Question

A small square loop of wire of side \( \ell \) is placed inside a large square loop of wire of side \( L \) ( \( L = \ell^2 \) ). The loops are coplanar and their centers coincide.The value of the mutual inductance of the system is \( \sqrt{x} \times 10^{-7} \, \text{H} \), where \( x = \) _____.

Answer 1

Correct Answer: 128

Flux linkage for inner loop:

\[ \phi = B_{\text{center}} \cdot \ell^2 \] \[ = 4 \times \frac{\mu_0 i}{4\pi} \left( \sin 45^\circ + \sin 45^\circ \right) \ell^2 \] \[ \phi = \frac{2\sqrt{2} \mu_0 i \ell^2}{\pi L} \]

Mutual inductance:

\[ M = \frac{\phi}{i} = \frac{2\sqrt{2} \mu_0 \ell^2}{\pi L} = \frac{2\sqrt{2} \mu_0}{\pi} \]

Calculating:

\[ M = \frac{2\sqrt{2} \times 4\pi}{\pi} \times 10^{-7} \] \[ = 8\sqrt{2} \times 10^{-7} \, \text{H} \] \[ = \sqrt{128} \times 10^{-7} \, \text{H} \]

Thus:

\[ x = 128 \]