Question

Due to the flow of current in the circular loop of radius $\text{R}$ magnetic induction produced at the centre of the loop is $B$.The magnetic moment of the loop is,

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

Answer 1

The correct option is C $\frac{2\pi B{R}^3}{{\mu }_0}$

Magnetic moment is defined as,

$\overrightarrow{M}=NI\overrightarrow{A}\Rightarrow M=NIA$

For the given loop,

$\overrightarrow{M}=I\overrightarrow{A}=I\left(\pi R^2\right)\hat{j}$ $[N=1]$

$\Rightarrow M=I\left(\pi R^2\right)...\left(2\right)$

Magnetic induction at the centre of the loop is,

$B=\frac{{\mu }_{0}I}{2R}.........\left(1\right)$

Using equations $\left(1\right)$ and $\left(2\right)$,

$M=\frac{B\times 2\pi R^3}{{\mu }_0}$

Hence, option (c) is the correct answer.