Magnetic field at the centre of a circular loop of area ' $A$ ' is ' $B$ '. The magnetic moment of the loop…

Magnetic field at the centre of a circular loop of area ' $A$ ' is ' $B$ '. The magnetic moment of the loop will be
  1. $\frac{\mathrm{BA}^2}{\mu_0 \pi}$
  2. $\frac{2 \mathrm{BA}^{3 / 2}}{\mu_0 \pi}$
  3. $\frac{\mathrm{BA}^{1 / 2}}{\mu_0 \pi^{1 / 2}}$
  4. $\frac{2 \mathrm{BA}^{3 / 2}}{\mu_0 \pi^{1 / 2}}$

Solution

Magnetic field at centre of circular loop is given by, $B=\frac{\mu_0 I}{2 r} \Rightarrow I=\frac{2 B r}{\mu_0}...(i)$
For circular loop, $\mathrm{A}=\pi \mathrm{r}^2 \mathrm{r}=\sqrt{\frac{\mathrm{A}}{\pi}}...(ii)$
Magnetic moment of the loop will be, $\begin{aligned} & \mathrm{M}=\mathrm{IA} \\ & \mathrm{M}=\frac{2 \mathrm{Br}}{\mu_0} \times \mathrm{A}...[From(i)] \\ & \mathrm{M}=\frac{2 \mathrm{BA}}{\mu_0} \times \sqrt{\frac{\mathrm{A}}{\pi}}...[From(ii)] \\ & \mathrm{M}=\frac{2 \mathrm{BA}^{\frac{3}{2}}}{\mu_0(\pi)^{\frac{1}{2}}} \end{aligned}$

Asked in: MHT CET 2024 (11 May Shift 2)

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