Question

Total charge $-Q$ is uniformly spread along length of a ring of radius $R$. A small test charge $+q$ of mass $m$ is kept at the centre of the ring and is given a gentle push along the axis of the ring.
a- Show that the particle executes a simple harmonic oscillations
b- Obtain its time period.

Answer 1

(a) Let the charge q is displaced towards the axis of ring.

Electric field at the axis of the ring at a distance z fro the axis of ring is

$E=\frac{1}{4\pi {\epsilon }_0}\frac{Qz}{{(R^2+Z^2)}^{\frac{3}{2}}}$

$F_{net}=qE$

$F_{net}=\frac{1}{4\pi {\epsilon }_0}\frac{qQz}{{(R^2+Z^2)}^{\frac{3}{2}}}$

$\because z\ll R$

$F_{net}=\frac{1}{4\pi {\epsilon }_0}\frac{qQz}{R^3}$

$Let,K=\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{R^3}$

$so,F_{net}=Kz$

force is directly proportional to the z and directed towards O. so, net force can be written as:

$F_{net}=-Kz$

This shows that the particles executes SHM.

(b) $\omega =\sqrt{\frac{K}{m}}$

$T=\frac{2\pi }{\omega }$

$T=2\pi \sqrt{\frac{m}{K}}$

$T=2\pi \sqrt{\frac{m}{\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{R^3}}}$

$T=2\pi \sqrt{\frac{4\pi {\epsilon }_{0}m{R}^3}{Qq}}$