(Solved): The figure shows a uniformly charged ring with radius R and total positive charge Q>0. The ring' ...

The figure shows a uniformly charged ring with radius $R$ and total positive charge $Q>0$. The ring's central axis coincides with the $x$-axis and its centre is at $x=0$. In the lectures we found that the electric field generated by the ring at points on the $x$-axis is given by $\overrightarrow{\mathbf{E}}(x)=\frac{1}{4?{?}_0}\frac{Qx}{{\left(x^2+R^2\right)}^{3/2}}\hat{\mathbf{i}}.$ Suppose we release a negative point charge with charge $?q<0$ and mass $m$ from rest at a point $x_0$ close to the ring's centre on the $x$-axis. By "close to the ring's centre" we mean that $?x_0??R$. Show that the charge will undergo approximate simple harmonic motion around the centre of the ring, and determine the frequency with which it will oscillate.