Question

A charged particle moves with a speed v in a circular path of radius r around a long uniformly charged conductor then :

• A. $v\propto r$
• B. $v\propto \frac{1}{r}$
• C. $v\propto \frac{1}{\sqrt{2}}r$
• D. $v$ is independent of r

Answer 1

The correct option is D $v$ is independent of r

Consider a Gaussian cynlinder of radius $r$ and length $l$. Using Gauss's law the electric field at distance $r$ is $E.\left(2\pi rl\right)=\frac{\lambda l}{{ϵ}_0}\Rightarrow E=\frac{\lambda }{2\pi {ϵ}_{0}r}$

Here the electrostatic force is equal to centrifugal force. i.e,

$qE=\frac{m{v}^2}{r}$

$q\frac{\lambda }{2\pi {ϵ}_{0}r}=\frac{m{v}^2}{r}$

or$v^2=\frac{q\lambda }{2\pi {ϵ}_{0}m}$

so $v$ is independent of $r$.