A hypothetical magnetic field existing in a region is given by $\vec{B} = B_{0} {\vec{e}}_{r}$ where ${\vec{e}}_{r}$ denotes the unit vector along the radial direction. A circular loop of radius $a$ , carrying a current $i$ , is placed with its plane parallel to the $x y$ - plane and the center at $(0, 0, d) .$ Find the magnitude of the magnetic force acting on the loop.

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$\frac{B i π a^{2}}{\sqrt{a^{2} + d^{2}}}$

$\frac{4 B i π a^{2}}{\sqrt{a^{2} + d^{2}}}$

$\frac{6 B i π a^{2}}{\sqrt{a^{2} + d^{2}}}$

$\frac{2 B i π a^{2}}{\sqrt{a^{2} + d^{2}}}$

answer is D.

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Detailed Solution

The situation is shown in the fig

$\sin θ = \frac{a}{\sqrt{a^{2} + d^{2}}}$

The given field can be resolved into two components: $B_{0} \sin θ,$ which acts all round the periphery of loop and lie in the plane of the loop and other component $B_{0} \cos θ,$ acts all round the periphery and perpendicular to the plane of the loop.

Force on the loop due to $B_{0} \cos θ,$ is zero. The force is only due to $B_{0} \sin θ,$ which is

$F = (B_{0} \sin θ) \times 2 π a = \frac{2 B i π a^{2}}{\sqrt{a^{2} + d^{2}}}$