Question

Two wires of same lengths are shaped into a square and a circle. If they carry same current, then the ratio of their magnetic moments is :

• A. $2:\pi$
• B. $\pi :2$
• C. $\pi :4$
• D. $4:\pi$

Answer 1

The correct option is C $\pi :4$

Magnetic moment of a loop is given by $p_m=IA$

where, $I=$ current through the loop

$A=$ area of loop

Perimeter of square should be equal to the circumference of circle, since wires have same length.

Therefore, $4a=2\pi r\Rightarrow a=\frac{\pi r}{2}$ .....(i)

In figure (i), magnetic moment of square loop is given by

$p_{m_1}=I\left(A_1\right)\Rightarrow$

$\Rightarrow p_{m_1}=\frac{I{\pi }^2{r}^2}{4}$ [from equation (i)] ....(ii)

In Fig. (ii), magnetic moment of circular loop is given by

$p_{m_2}=I\left(A_2\right)$

$p_{m_2}=I\left(\pi r^2\right)$ ....(iii)

From equations (ii) and (iii), we get

$\Rightarrow \frac{p_{m_1}}{p_{m_2}}=\frac{\pi }{4}$