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)

$B=\frac{{\mu }_0}{4\pi }\frac{2\pi R^{2}I}{{\left(R^2+d^2\right)}^{3/2}}$
$I=\frac{q}{T}=\frac{2\pi R\lambda }{T}$
$=\frac{{\mu }_0}{4\pi }\frac{2\pi R^{2}I}{d^3}(d>>R)$
$=\frac{{\mu }_0}{2}\frac{R^2}{d^3}\times \frac{2\pi R\lambda }{T}$
$B\propto \frac{R^3}{d^3}$
$\therefore m=n=3$