Question

The magnetic field at the centre of 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

The correct option is D $\frac{2B{A}^{3/2}}{{\mu }_0{\pi }^{1/2}}$

The magnetic field at the centre of the circular loop is given by,

$B=\frac{{\mu }_{0}I}{2r}$

$\Rightarrow I=\frac{2Br}{{\mu }_0}$

For a circular loop, $A=\pi r^2$

$r={\left(\frac{A}{\pi }\right)}^{1/2}$

Now the magnetic moment of the loop will be,

$M=IA$

$M=\frac{2BrA}{{\mu }_0}$

$M=\frac{2BA}{{\mu }_0}\times {\left(\frac{A}{\pi }\right)}^{1/2}$

$\therefore M=\frac{2B(A{)}^{3/2}}{{\mu }_0(\pi {)}^{1/2}}$

Hence, option $\left(D\right)$ is the correct answer.