Question 148: Electromagnetic Induction

Question

A small square loop of wire of side $$l$$ is placed inside a large square loop of wire of side $$L\left(L=l^2ight)$$. The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $$\sqrt{x} 	imes 10^{-7} \mathrm{H}$$, where $$x=$$ _________.

Accepted Answer

Flux linkage for inner loop.
$$\begin{aligned}
& \phi=\mathrm{B}_{	ext {center }} \cdot \ell^2 \
& =4 	imes \frac{\mu_0 \mathrm{i}}{4 \pi \frac{\mathrm{L}}{2}}(\sin 45+\sin 45) \ell^2 \
& \phi=2 \sqrt{2} \frac{\mu_0 \mathrm{i}}{\pi \mathrm{L}} \ell^2 \
& \mathrm{M}=\frac{\phi}{\mathrm{i}}=\frac{2 \sqrt{2} \mu_0 \ell^2}{\pi \mathrm{L}}=2 \sqrt{2} \frac{\mu_0}{\pi} \
& =2 \sqrt{2} \frac{4 \pi}{\pi} 	imes 10^{-7} \
& =8 \sqrt{2} 	imes 10^{-7} \mathrm{H} \
& =\sqrt{128} 	imes 10^{-7} \mathrm{H} \
& \mathrm{x}=128
\end{aligned}$$

Electromagnetic Induction

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

Integer / Numeric Answer Type