Q. Magnetic field at the centre of a circular loop of area $A\ $is $B$. The magnetic moment of the loop will be
A) $\frac{BA^{2}}{\mu_{0}\pi}$
B) $\frac{BA^{3/2}}{\mu_{0}\pi}$
C) $\frac{BA^{3/2}}{\mu_{0}\pi^{1/2}}$
D) $\frac{2\ BA^{3/2}}{\mu_{0}\pi^{1/2}}$
Explanation
(d) $B = \frac{\mu_{0}}{4\pi}\frac{2\pi i}{r} = \frac{\mu_{0}}{4\pi}\ \ \ \ \ \ \ \ \ or\ \ \ i = \ \frac{2\ B\ r}{\mu_{0}};$
Also, $A = \ \pi r^{2}\ \ \ or\ \ r = \left( \frac{A}{\pi} \right)^{1/2}$
Magnetic moment, $M = iA = \frac{2Br}{\mu_{0}}\ A$
$= \frac{2BA}{\mu_{0}}\ \times \left( \frac{A}{\pi} \right)^{1/2} = \ \frac{2\ BA^{3/2}}{\mu_{0}\pi^{1/2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.