Calculate potential energy of a point charge - q placed along the axis due to a charge + Q uniformly distributed along a ring of radius-, R. Sketch P .F. as a function or axial distance z from the centre of the ring. Looking at graph, can you see what would happen if - q is displaced slightly from the centre of the ring (along the axis) ?

Explanation

Let us consider a ring of radius R = a having charge Q distributed uniformly over the ring.
Also a point P at a distance z on its axis passing through centre. Take r is the distance of P from charge dq on the ring.

`r= sqrt(z^(2)+a^(2))` where a=R
`:.` Electric potential at point P,
`V= int(kdq)/(r) = kint(dq)/(r)=kint(dq)/(sqrt(z^(2)+a^(2)))`
`:. V= (k)/(sqrt(z^(2)+a^(2)))intdq= (kQ)/(sqrt(z^(2)+a^(2)))[ because int dq = Q]`
If - q is at P then potential energy
U = W
` = q xxV`
` = -q xx(kQ)/(sqrt(z^(2)+a^(2)))`
`U=-(kQq)/(a[sqrt((z^(2))/(a^(2))+1)])`
Now `(kQq)/(a) =S` supposing new constant
`U=-(S)/((1+(z^(3))/(a^(2)))^(1//2))`
`z gt gt gt a ` then `(z^(2))/(a^(2)) gt 1` but `z=0 , (z^(2))/(a^(2))=0`
`:. U =-S`
The graph of `U rarr` Z is shown below ,

If charge is displaced, it would perform S.H.M.
But from the graph, we cannot predict the type of oscillations.