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

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:

$x = 128$