Question

Identical charges q each are placed at the eight corners of a cube of side b. Find the electrostatic potential energy of a charge +q placed at the center of the cube

Answer 1

Length of the cube diagonal is $\sqrt{3}b$. Distance of center of the cube from its each corner is

$r=\frac{\sqrt{3}}{2}b$

Total PE of charge $q$ at the center is

$\frac{-8q\left(q\right)}{4\pi {\epsilon }_{0}r}=\frac{-8q^2}{4\pi {\epsilon }_0\sqrt{3}\left(b/2\right)}=\frac{-4q^2}{\pi {\epsilon }_0\sqrt{3}b}$