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A thin charged rod is bent into the shape of a small circle of radius $\mathrm{R}$ the charge per unit length of the rod being $\lambda$. The circle is rotated about its axis with a time period $T$ and it is found that the magnetic field at a distance ' $d$ ' away $(d>>R)$ from the center and on the axis, varies as $\frac{\mathrm{R}^{\mathrm{m}}}{\mathrm{d}^{\mathrm{n}}} .$ The values of $\mathrm{m}$ and $\mathrm{n}$ respectively are
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The correct answer is:
$m=3, n=3$
$\mathrm{B}=\frac{\mu_{0}}{4 \pi} \frac{2 \pi \mathrm{R}^{2} \mathrm{I}}{\left(\mathrm{R}^{2}+\mathrm{d}^{2}\right)^{3 / 2}} \quad \mathrm{I}=\frac{\mathrm{q}}{\mathrm{T}}$
$=\frac{2 \pi \mathrm{R} \lambda}{\mathrm{T}}$
$=\frac{\mu_{0}}{4 \pi} \frac{2 \pi \mathrm{R}^{2} \mathrm{I}}{\mathrm{d}^{3}}(\mathrm{~d}>>\mathrm{R})$
$\frac{\mu_{0}}{2} \frac{\mathrm{R}^{2}}{\mathrm{~d}^{3}} \times \frac{2 \pi \mathrm{R} \lambda}{\mathrm{T}}$
$\mathrm{B} \propto \frac{\mathrm{R}^{3}}{\mathrm{~d}^{3}}$
$\therefore \mathrm{m}=\mathrm{n}=3$
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