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

• A

  $m=2, n=2$
• B

  $m=2, n=3$
• C

  $m=3, n=2$
• Correct answer
  D

  $m=3, n=3$

Solution

$\mathrm{B}=\frac{\mu_{0}}{4 \pi} \frac{2 \pi \mathrm{R}^{2} \mathrm{I}}{\left(\mathrm{R}^{2}+\mathrm{d}^{2}\right)^{3 / 2}} \quad \mathrm{I}=\frac{\mathrm{q}}{\mathrm{T}}$ $=\frac{2 \pi \mathrm{R} \lambda}{\mathrm{T}}$ $=\frac{\mu_{0}}{4 \pi} \frac{2 \pi \mathrm{R}^{2} \mathrm{I}}{\mathrm{d}^{3}}(\mathrm{~d}>>\mathrm{R})$ $\frac{\mu_{0}}{2} \frac{\mathrm{R}^{2}}{\mathrm{~d}^{3}} \times \frac{2 \pi \mathrm{R} \lambda}{\mathrm{T}}$ $\mathrm{B} \propto \frac{\mathrm{R}^{3}}{\mathrm{~d}^{3}}$ $\therefore \mathrm{m}=\mathrm{n}=3$

Hence, the correct answer is option: 'D'.