a
When the negative charge is shifted at a distance \(x\) from the centre of the ring along its axis, then force acting on the point charge due to the ring.

\(F=q E\) (towards centre)

\(=q \cdot \frac{k Q x}{\left(R^{2}+x^{2}\right)^{3 / 2}}\)

If \(R \gg x,\) then \(R^{2}+x^{2} \approx R^{2}\)

and \(F=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q q x}{R^{3}}\) (towards centre)

\(\Rightarrow a=\frac{F}{m}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{Q q x}{m R^{3}}\)

since, restoring force \(F_{E} \propto x,\) therefore motion of charge particle will be \(SHM.\)

Time period of \(SHM,\)

\(T=\frac{2 \pi}{\omega}=\left[\frac{16 \pi^{3} \varepsilon_{0} R^{3} m}{Q q}\right]^{1 / 2}\left[\because \omega^{2}=\frac{Q q}{4 \pi \varepsilon_{0} R^{3}}\right]\)