Question

A circular loop and a square loop are formed from the same wire and the same current is passed through them. Find the ratio of their dipole moments?

• A. $4\pi$
• B. $\frac{4}{\pi }$
• C. $\frac{2}{\pi }$
• D. $2\pi$

Answer 1

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

Suppose the length of the wire is $l.$

Thus the radius of the circular loop can be found out by

$l=2\pi r$

$⟹r=\frac{l}{2\pi }$

Thus the dipole moment of circular loop is $M_1=i\times \pi r^2$

$=i\times \pi \times (\frac{l}{2\pi }{)}^2$

$=\frac{i{l}^2}{4\pi }$

Same length of wire forms a square loop. The side of the loop can be found by

$l=4a$

$⟹a=\frac{l}{4}$

Magnetic dipole moment of the square loop $=M_2=i\times a^2=\frac{i{l}^2}{16}$

Thus, $\frac{M_1}{M_2}=\frac{4}{\pi }$