Question

A cube of side $b$ has a charge $q$ at each of its vertices. What is the electric potential at the centre of the cube?

• A. $\frac{4q}{\sqrt{3}\pi {\in }_{0}b}$
• B. $\frac{\sqrt{3}q}{\pi {\in }_{0}b}$
• C. $\frac{2q}{\pi {\in }_{0}b}$
• D. Zero

Answer 1

The correct option is A $\frac{4q}{\sqrt{3}\pi {\in }_{0}b}$

Consider the following cube. It has $\text{8}$ charges present at each vertex and $\text{C}$ is the centre.

As the length of the body diagonal of the cube is $\sqrt{3}b$ so each point is at a distance of half of the length of the body diagonal.

Thus, the distance of a vertex from the centre of the cube of side $b$ is $r=\sqrt{3}b/2$.

Now the potential due to charge $q$ at the centre is given by

$V=\frac{q}{\left(4\pi {\in }_{0}r\right)}$,

where, $r$ is the distance between the point and the charge. In this case $r=\sqrt{3}b$

Hence the potential due to the arrangement of eight charges (each of magnitude $q$) at the centre is

$V=\frac{8q}{4\pi {\in }_{0}r}=\frac{4q}{\sqrt{3}\pi {\in }_{0}b}$

Hence, option (a) is the correct answer.