Two wires of same length are made to from a circle and a square. They carry same current. The ratio of the…
Two wires of same length are made to from a circle and a square. They carry same current. The ratio of the magnetic moment of circle to that of square is
$4: \pi$
$2: \pi$
$\pi: 4$
$\pi: 2$
Solution
Let length of wire be $l$. For a square loop, side of square $\frac{l}{4}$.
Hence its magnetic moment
$M_{\mathrm{sq}}=i A=i\left[\frac{l}{4}\right]^2=\frac{i l^2}{16}$
Radius of a circular loop $r=\frac{l}{2 \pi}$
Hence, its magnetic moment
$\begin{aligned} & M_{\mathrm{cir}}=i A=i \pi r^2 \\ & M_{\text {cir }}=i \pi\left[\frac{l^2}{4 \pi r^2}\right]=\frac{i l^2}{4 \pi} \\ & \Rightarrow \frac{M_{\text {cir }}}{M_{\mathrm{sq}}}=\frac{4}{\pi}\end{aligned}$