Question

A straight wire of length $l$ is bent into a $\left(i\right)$ a circular loop $\left(ii\right)$ a square loop, and each having single turn. ${\text{M}}_C$ and ${\text{M}}_S$ indicate their dipole moments respectively. If the loops carry the same current, the value of $\frac{{\text{M}}_C}{{\text{M}}_S}$ is:

• A. $\frac{\pi }{4}$
• B. $\frac{4}{\pi }$
• C. $\pi$
• D. $\frac{1}{\pi }$

Answer 1

The correct option is B $\frac{4}{\pi }$

For a circular loop,

$l=2\pi r$ (where, $r$ is the radius of loop)

$\therefore r=\frac{l}{2\pi }$

For a square loop,

$l=4a$ (where, $a$ is the side of square)

$\therefore a=\frac{l}{4}$

Now, $M=NIA$

And, ​​​​as $I$ and $N$ is same for both the loops

$\therefore \frac{M_C}{M_S}=\frac{A_C}{A_S}$

$\frac{{\text{M}}_C}{{\text{M}}_S}=\frac{\pi .\frac{l^2}{4{\pi }^2}}{\frac{l^2}{16}}=\frac{4}{\pi }$

Hence, option $\left(B\right)$ is the correct answer.