Question

A Charge of $Q$ coulombs is uniformly distributed throughout the volume of a solid hemisphere of radius $R$ metres. Find the potential at centre $\text{O}$ of the hemisphere (in volts):

• A. $\frac{1}{4\pi {ϵ}_0}\frac{3Q}{2R}$
• B. $\frac{1}{4\pi {ϵ}_0}\frac{3Q}{4R}$
• C. $\frac{1}{4\pi {ϵ}_0}\frac{Q}{4R}$
• D. $\frac{1}{4\pi {ϵ}_0}\frac{Q}{8R}$

Answer 1

The correct option is A $\frac{1}{4\pi {ϵ}_0}\frac{3Q}{2R}$

Let $q$ be the total charge distributed in a solid sphere.

The potential at the centre of solid sphere in which $q$ charge is uniformly distributed throughout the volume is given by

$V=\frac{1}{4\pi {ϵ}_0}\frac{3q}{2R}$

By symmetry the potential at the centre of the hemisphere will be half of that of the complete sphere i.e.,

$V_O=\frac{V}{2}$

$\Rightarrow V_O=\frac{1}{4\pi {ϵ}_0}\frac{3\left(\frac{q}{2}\right)}{2R}$

From the data given in the question, we can deduce that, $\frac{q}{2}=Q$

$\therefore V_O=\frac{1}{4\pi {ϵ}_0}\frac{3Q}{2R}$

Hence option (a) is the correct answer.