Question

Magnetic field at the centre of a circular loop of area A is B. The magnetic moment of the loop will be

• A. $\frac{B{A}^2}{{\mu }_0\pi }$
• B. $\frac{B{A}^{3/2}}{{\mu }_0\pi }$
• C. $\frac{B{A}^{3/2}}{{\mu }_0{\pi }^{1/2}}$
• D. $\frac{2B{A}^{3/2}}{{\mu }_0{\pi }^{1/2}}$

Answer 1

Answer: (D)

$B=\frac{{\mu }_0}{4\pi }\frac{2\pi 1}{r}=\frac{{\mu }_{0}I}{2r}$
Or $I=\frac{2Br}{{\mu }_0}$
Also, $A=\pi r^2$
Or $r={\left(\frac{A}{\pi }\right)}^{1/2}$
Magnetic moment, $M=IA=2\frac{Br}{{\mu }_0}A$
$=2\frac{BA}{{\mu }_0}{\left(\frac{A}{\pi }\right)}^{1/2}=2\frac{B{A}^{3/2}}{{\mu }_0{\pi }^{1/2}}$