Question

Consider a ring of radius a containing q amount of negative charges. Derive an expression for the electric field strength at a point at a distance x on the axis of this charged ring. Hence prove that at a great enough distances, the charged ring behaves like a point charge.

Answer 1

dQ is portion of Q along a segment ds of the ring
E = (1/4pieo) dQ/(x^2+a^2)
the component of this in the axial direction is
dEx = dE cos angle to x axis = dEx/sqrt(x^2+a^2)
(component in direction perpendicular to axis is adds to zero by symmetry)
so
dEx = (1/4pieo) x dQ/(x^2+a^2)^1.5
at a given constant x our integral is only over d Q and is Q
so
Ex = (1/4pieo)Q x/(x^2+a^2)^1.5

if x is >> a
then
Ex = (1/4pieo)Q x/(x^2 + 0)^1.5
=(1/4pieo)Q x/x^3
= (1/4pieo)Q/x^2 which is point charge