Question

A charge $Q$ is spread non-uniformly on the surface of a hollow sphere of radius $R$, such that the charge density is given by $\sigma=\sigma_{0}\left(1 \sin \theta\right)$ where $\theta$ is the usual polar angle. The potential at the centre of the sphere is

• A. $\frac{Q}{2\pi\varepsilon_{o}R}$
• B. $\frac{Q}{\pi\varepsilon_{o}R}$
• C. $\frac{Q}{8\pi\varepsilon_{o}R}$
• D. $\frac{Q}{4\pi\varepsilon_{o}R}$

Answer 1

Answer: (D)

Potential due to an elemental charge $dQ$ at centre of sphere is
$dV=\frac{kdQ}{R}$
Potential due to complete charge distribution at centre is
$V=\oint dV=\oint\frac{kdQ}{R} $
$=\frac{k}{R}\oint dQ=\frac{kQ}{R} $
or $V=\frac{kQ}{R}=\frac{Q}{4\pi\varepsilon_{o}R}$