Question

Magnetic induction at the centre of a circular loop carrying a current is $B$ . If $A$ is the area of the coil, the magnetic dipole moment of the loop is

• A. $\frac{B A^{2}}{\mu _{0} \pi }$
• B. $\frac{B A \sqrt{A}}{\mu _{0}}$
• C. $\frac{B A \sqrt{A}}{\mu _{0} \pi }$
• D. $\frac{2 B A}{\mu _{0}} \, \sqrt{\frac{A}{\pi }}$

Answer 1

Answer: (D)

$ B =\frac{\mu_0 i}{2 r}, A=\pi r^2, r=\sqrt{\frac{A}{\pi}} $
$m =i A=\frac{B \times 2 r}{\mu_o} \times \pi r^2$
$=\frac{2 B}{\mu_o} \pi r^3 $
$=\frac{2 B \pi}{\mu_o}\left(\frac{A}{\pi}\right)^{\frac{3}{2}}$
$=\frac{2 B A}{\mu_o} \sqrt{\frac{A}{\pi}} $