Question

Calculate potential on the axis of a ring due to charge Q uniformly distributed along the ring of radius R?

Answer 1

Given,

Consider a small differential element
$dl=Rd\theta$ on the ring.

The element carries a charge

$dq=\lambda dl=\lambda Rd\theta$

The electric potential at $P$ due to this small differential element,

$dV=\frac{Kdq}{r}$

Electric potential at $P$ due to entire ring,
$V=\int \frac{Kdq}{r}$

$V=K\int \frac{\lambda Rd\theta }{r}$

Where $r=\sqrt{R^2+z^2}$

$V=\frac{K\lambda R}{\sqrt{R^2+z^2}}\int d\theta =\left(\frac{1}{4\pi {\epsilon }_o}\right)\frac{2\pi R\lambda }{\sqrt{R^2+z^2}}$

$V=\left(\frac{1}{4\pi {\epsilon }_o}\right)\frac{Q}{\sqrt{R^2+z^2}}$

Final Answer: $V=\left(\frac{1}{4\pi {\epsilon }_o}\right)\frac{Q}{\sqrt{R^2+z^2}}$