Two identical positive charges $Q$ each are fixed at a distance of $2a$ apart from each other. Another point charge $q_0$ with mass $m$ is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge $q_0$ executes SHM. The time period of oscillation of charge $q_0$ will be
Answer: (A) $\sqrt{\dfrac{4\pi^3\varepsilon_0 ma^3}{q_0Q}}$
Displace $q_0$ by a small $x$ towards one charge. With $k = \dfrac{1}{4\pi\varepsilon_0}$, the net restoring force is
$$F = kQq_0\left[\frac{1}{(a - x)^2} - \frac{1}{(a + x)^2}\right] \approx kQq_0\frac{4x}{a^3}$$
$$\omega^2 = \frac{4kQq_0}{ma^3} = \frac{Qq_0}{\pi\varepsilon_0 ma^3}$$
$$T = 2\pi\sqrt{\frac{\pi\varepsilon_0 ma^3}{q_0Q}} = \sqrt{\frac{4\pi^3\varepsilon_0 ma^3}{q_0Q}}$$
Solution by Sreeraj P, M.Sc Physics