Question

A particle of mass $m$ and carrying charge $-q_1$ is moving, around a charge $+q_2$ along a circular path of radius $r$. The period of revolution of the charge $-q_1$ about $+q_2$ is

• A. $4\sqrt{\frac{{\pi }^3{\epsilon }_{0}m{r}^3}{q_1{q}_2}}$
• B. $\sqrt{\frac{{\pi }^3{\epsilon }_{0}m{r}^3}{q_1{q}_2}}$
• C. $2\sqrt{\frac{{\pi }^3{\epsilon }_{0}m{r}^3}{q_1{q}_2}}$
• D. $3\sqrt{\frac{{\pi }^3{\epsilon }_{0}m{r}^3}{q_1{q}_2}}$

Answer 1

Answer: (A)

Suppose that the charge $-q_1$ moves around the charge $q_2$ along a circular path of radius $r$ with speed $v$. The necessary centripetal force is provided by the electrostatic force of attraction between the two charges, i.e.,
$\frac{1}{4\pi {\epsilon }_0}\cdot \frac{q_1\times q_2}{r^2}=\frac{m{v}^2}{r}$
or $v={\left(\frac{1}{4\pi {\epsilon }_0}\cdot \frac{q_1\times q_2}{mr}\right)}^{1/2}$
If $T$ is the period of revolution of the charge $-q_1$ about $q_2$, then
$T=\frac{2\pi r}{v}$
Substituting for $v$, we get $T=\sqrt{\frac{16{\pi }^3{\epsilon }_{0}m{r}^3}{q_1{q}_2}}$