Question

Magnetic induction produced at the centre of a circular loop carrying current is $B$. The magnetic moment of the loop of radius $R$ is
(${\mu }_0=$ permeability of free space)

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

Answer 1

Answer: (B)

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

Magnetic moment

$M=I\times A$ .....(i)

Current $\times$ Area of enclosed by loop

Magnetic induction at the centre of circular loop

$B=\frac{{\mu }_{0}I}{2R}\Rightarrow I=\frac{2BR}{{\mu }_0}$ ......(ii)

Here, $A=\pi R^2$ ......(iii)

Substituting equations (ii) and (iii) in equation (i), we get

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