Question

A charged particle moves with a speed $v$ in a circular path of radius $R$ around a long uniformly charged conductor.

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

Answer 1

Answer: (D)

$E_{\text{rod }}=\frac{\lambda }{2\pi {\epsilon }_{0}R}$
If charge goes around rod, it means the electric force provides the centripetal force, then

$\frac{m{v}^2}{R}=qE$
$\frac{m{v}^2}{R}=q\left(\frac{\lambda }{2\pi {\epsilon }_{0}R}\right)\Rightarrow v={\left(\frac{q\lambda }{2\pi {\epsilon }_{0}m}\right)}^{\frac{1}{2}}$
$\Rightarrow v$ is independent of $R$