Question

A straight wire current $I$ is turned into a circular loop. If the magnitude of magnetic moment associated with the $MKS$ unit is $M$, the length of wire will be

• A. $4\pi IM$
• B. $\sqrt{\frac{4\pi M}{I}}$
• C. $\sqrt{\frac{4\pi I}{M}}$
• D. $\frac{M\pi }{4I}$

Answer 1

Answer: (B)

$\sqrt{\frac{4\pi M}{I}}$

Magnetic moment of straight current carrying wire $M=IA$............(i)
$M=I\left(\pi r^2\right)$
where r=radius of loop.
When wire of length $L$ is turned into loop of radius $r$, then
$L=2\pi r$
$r=\frac{L}{2\pi }$
Substituting the value of $r$ Eq.(i)
$M=I\pi (\frac{L}{2\pi }{)}^2$
$L^2=\frac{4\pi M}{I}$
$\Rightarrow$ $L=\sqrt{\frac{4\pi M}{I}}$