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(D) The magnetic field $B$ at the centre of a circular loop of radius $R$ carrying current $i$ is given by:$B = \frac{\mu_0 i}{2 R}$ ...$(i)$The magne

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$\frac{2 B A \sqrt{A}}{\mu_0 \sqrt{\pi}}$

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(D) The magnetic field $B$ at the centre of a circular loop of radius $R$ carrying current $i$ is given by:$B = \frac{\mu_0 i}{2 R}$ ...$(i)$The magnetic moment $M$ of the loop is given by:$M = i A$ ...(ii)From equation $(i)$,we can express the current $i$ as:$i = \frac{2 B R}{\mu_0}$Substitute this into equation (ii):$M = \left( \frac{2 B R}{\mu_0} \right) A$Since the area of the loop is $A = \pi R^2$,we have $R = \sqrt{\frac{A}{\pi}}$.Substituting $R$ into the expression for $M$:$M = \frac{2 B A}{\mu_0} \sqrt{\frac{A}{\pi}}$$M = \frac{2 B A \sqrt{A}}{\mu_0 \sqrt{\pi}}$Therefore,the correct option is $D$.

Magnetic field at the centre of a circular loop of area $A$ is $B$. Then the magnetic moment of the loop is ($\mu_0$ is the permeability of free space).

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