Electromagnetic Induction Question 8
Question 8 - 2024 (31 Jan Shift 1)
A small square loop of wire of side $\ell$ is placed inside a large square loop of wire of side $L\left(L=\ell^{2}\right)$. The loops are coplanar and their centers coinside. The value of the mutual inductance of the system is $\sqrt{x} \times 10^{-7} H$, where $x=$
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Answer: (128)
Solution:
Flux linkage for inner loop.
$$ \begin{aligned} & \phi=B _{\text {center }} \cdot \ell^{2} \\ & =4 \times \frac{\mu _0 i}{4 \pi \frac{L}{2}}(\sin 45+\sin 45) \ell^{2} \end{aligned} $$
$\phi=2 \sqrt{2} \frac{\mu _0 i}{\pi L} \ell^{2}$
$M=\frac{\phi}{i}=\frac{2 \sqrt{2} \mu _0 \ell^{2}}{\pi L}=2 \sqrt{2} \frac{\mu _0}{\pi}$
$=2 \sqrt{2} \frac{4 \pi}{\pi} \times 10^{-7}$
$=8 \sqrt{2} \times 10^{-7} H$
$=\sqrt{128} \times 10^{-7} H$
$x=128$
