Question

Find the expression for electric potential at a point $P$ located on the central perpendicular axis of a uniformly charged ring of radius $a$ and total charge $Q$, at a distance of $x$ from the centre.

• A. $\frac{Q}{4\pi {ϵ}_{0}x}$
• B. $\frac{Q}{4\pi {ϵ}_0\sqrt{(x^2-a^2)}}$
• C. $\frac{Q}{4\pi {ϵ}_{0}a}$
• D. $\frac{Q}{4\pi {ϵ}_0\sqrt{(x^2+a^2)}}$

Answer 1

The correct option is D $\frac{Q}{4\pi {ϵ}_0\sqrt{(x^2+a^2)}}$

Let us consider an element of charge $dq$.

Potential at P due to this charge element is given by-

$dV=\frac{dq}{4\pi {ϵ}_{o}r}$

Every point on ring is at distance $r$ from point P and $r=\sqrt{x^2+a^2}$

$⟹dV=\frac{dq}{4\pi {ϵ}_o\sqrt{a^2+x^2}}$

$⟹V=\int dq=\frac{1}{4\pi {ϵ}_o\sqrt{a^2+x^2}}\int dq$

$⟹V=\frac{Q}{4\pi {ϵ}_o\sqrt{a^2+x^2}}$

Answer-(D)