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. $4 \sqrt{\frac{\pi^{3} \varepsilon_{0} m r^{3}}{q_{1} q_{2}}}$
• B. $\sqrt{\frac{\pi^{3} \varepsilon_{0} m r^{3}}{q_{1} q_{2}}}$
• C. $2 \sqrt{\frac{\pi^{3} \varepsilon_{0} m r^{3}}{q_{1} q_{2}}}$
• D. $3 \sqrt{\frac{\pi^{3} \varepsilon_{0} m r^{3}}{q_{1} q_{2}}}$

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 \varepsilon_{0}} \cdot \frac{q_{1} \times q_{2}}{r^{2}}=\frac{m v^{2}}{r} $
or $ v=\left(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{1} \times q_{2}}{m r}\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 $T=\sqrt{\frac{16 \pi^{3} \varepsilon_{0} m r^{3}}{q_{1} q_{2}}}$