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. Skecth 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 slighlty from the centre of the ring (along the axis) ?

Explanation

In Fig, we have shown a charge `+Q` distributed uniformly along a ring of radius R, with center at O,
Let `lambda` be the linear charge density of the ring. Therefore,
`lambda = Q//2 pi R` …(i)
Consider a small element of length dl of the ring with center at A.
`:.` Charge of the element, `dq = lambda dl`
At any point P on the axis of ring.
where `OP = z`, potential due to the charge element,
`dV = (dq)/(4pi in_(0) (AP)) = (lambda dl)/(4pi in_(0) sqrt(R^(2) + z^(2)))`
`:.` Potential at P due to entire ring.
`V = oint (lambda dl)/(4pi in_(0) sqrt(R^(2) + z^(2)))`
`= (lambda (2 pi R))/(4pi in_(0) sqrt(R^(2) + z^(2))) = (Q)/(4pi in_(0) sqrt(R^(2) + z^(2)))`
Potential energy of charge `-q` held at P
`U = -qV = (-qQ)/(4pi in_(0) sqrt(R^(2) + z^(2)))`
The varition of potential energy with axial distance z from the center of the ring is shown in Fig. IF charge `-q` si displaced slightly from the center of the ring along the axis of the ring, and left the charge would perform oscillations. However, by just looking at the graph, we cannot come to a firm conclusiion about the nature of these oscilations.