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.E. as a function of a axial distance z from the center of the ring, Looking at graph, can you see what happen if `-q` is displaced slightly from the centre of the ring (along the axis) ?

Explanation

let us take point P to be a t a distance x from the centre of the ring, as shown in figure. The charge element dq is a t a distance x from point P. therefore, V can be written as
`V=K_(e)int(dq)/(r)=k_(e)int(dq)/(sqrt(z^(2)+a^(2)))`
ltBrgt Considering -q charge at P, the potential energy is given by
`U=W=qxx`potential difference
`U=(K_(e)Q(-q))/(sqrt(z^(2)+a^(2)))`
or `U=(1)/(4piepsi_(0))(-Qq)/(sqrt(z^(2)+a^2))`
`=(1)/(4piepsi_(0)a)(-Qq)/(sqrt(1+((z)/(a))^(2)))`

This is the required expression.
The variation of potential energy with z is shown in the figure. the charge -q displaced would perrform oscillations.
Nothing can be concluded just by looking at the graph.