Question

A thin charged rod is bent into the shape of a small circle of radius $R$ 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 $\frac{R^{m}}{d^{n}}$. The values of $m$ and $n$ respectively are

• A. $m=2, n=2$
• B. $m=2, n=3$
• C. $m=3, n=2$
• D. $m=3, n=3$

Answer 1

Answer: (D)

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