Question

Find the mutual inductance of the system of a small square loop of wire of side $l$ placed inside a large square loop of wire of side $L(>>l)$ in the same plane as shown in the figure.

• A. $2\sqrt{2}\frac{{\mu }_0}{\pi }\frac{l^2}{L}$
• B. $8\sqrt{2}\frac{{\mu }_0}{\pi }\frac{l^2}{L}$
• C. $2\sqrt{2}\frac{{\mu }_0}{2\pi }\frac{l^2}{L}$
• D. $2\sqrt{2}\frac{{\mu }_0{L}^2}{\pi l}$

Answer 1

Answer: (A)

Let the current $I$ be flowing in the larger loop.
The larger loop is made up of four wires each of length $L$ , the field at the center $i.e.,$ at a distance $\frac{L}{2}$ from each wire, will be
$B=4\times \frac{{\left(\mu \right)}_{0}I}{4\pi \left(\frac{L}{2}\right)}(sin{\left(45\right)}^o+sin{\left(45\right)}^o)$
$=4\times \frac{{\mu }_0}{4\pi }\frac{2I}{L}\frac{2}{\sqrt{2}}=2\sqrt{2}\frac{{\mu }_0}{\pi }\frac{I}{L}$
Flux linked with smaller loop
${\varphi }_2=B{A}_2=2\sqrt{2}\frac{{\mu }_0}{\pi }\frac{I}{L}\times l^2$
Hence, $M=\frac{{\varphi }_0}{I}\Rightarrow M=2\sqrt{2}\frac{{\mu }_0}{\pi }\frac{l^2}{L}$