The magnetic field at the centre of a current carrying circular coil of area ' $A$ ' is ' $B$ '. The…

The magnetic field at the centre of a current carrying circular coil of area ' $A$ ' is ' $B$ '. The magnetic moment of the coil is ( $\mu_0=$ permeability of free space)
  1. $\frac{2 \mu_0 \pi^{1 / 2}}{\mathrm{BA}^{3 / 2}}$
  2. $\frac{\mathrm{BA}^{3 / 2}}{\mu_0 \pi}$
  3. $\frac{2 B \mathrm{~A}^{3 / 2}}{\mu_0 \pi^{1 / 2}}$
  4. $\frac{\mathrm{BA}^2}{\mu_0 \pi}$

Solution

$\begin{array}{ll} & A=\pi r^2 \Rightarrow r=\sqrt{\frac{A}{\pi}} \\ \therefore \quad & B=\frac{\mu_0 I}{2 r}=\frac{\mu_0 I}{2 \sqrt{\frac{A}{\pi}}} \\ \therefore \quad & I=\frac{2 B}{\mu_0} \sqrt{\frac{A}{\pi}} \end{array}$
Magnetic moment $\mathrm{m}=\mathrm{IA}$ $\mathrm{IA}=\frac{2 \mathrm{~B}}{\mu_0} \sqrt{\frac{\mathrm{~A}}{\pi}} \times \mathrm{A}=\frac{2 \mathrm{BA}^{3 / 2}}{\mu_0 \pi^{1 / 2}}$ :

Asked in: MHT CET 2024 (02 May Shift 1)

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