Question

An electron is in a circular orbit of certain radius $r$ around an infinite line of charge with linear charge density $\lambda c{m}^{-1}$. lf the plane of the orbit is perpendicular to the line, then, point out correct statement(s) from the following:

• A. the orbital period of the electron is independent of the radius
• B. the total energy of the electron is constant and equal to $e\lambda /4\pi {\epsilon }_0$
• C. the speed of the electron is inversely proportional to $\sqrt{r}$
• D. the kinetic energy of the electron is independent of the radius of the orbit and equal to $e\lambda /4\pi {\epsilon }_0$

Answer 1

The correct option is D the kinetic energy of the electron is independent of the radius of the orbit and equal to $e\lambda /4\pi {\epsilon }_0$

Let $m$ be the mass , $e$ be magnitude of electric charge, $r$ be the radius and $\lambda$ be the linear charge density of the line .
Electric field at distance r from the line charge, $E=\frac{\lambda }{2\pi {\epsilon }_0}.\frac{1}{r}$
Due to circular motion, the centrifugal force is equal to the electric force on the electron.
Let $v$ be the speed of the electron,
$\frac{m{v}^2}{r}=\frac{\lambda e}{2\pi {\epsilon }_0}.\frac{1}{r}$
$\therefore \frac{1}{2}m{v}^2=\frac{e\lambda }{4\pi {\epsilon }_0}$
$v=\sqrt{\frac{e\lambda }{2\pi {\epsilon }_{0}m}}$
so, speed as well as Kinetic energy is independent of $r$.
Time period,$T=\frac{2\pi r}{v}=2\pi r\sqrt{\frac{2\pi {\epsilon }_{0}m}{e\lambda }}$
$\therefore T\propto r$