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A cube of side b has a charge $q$ at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube.

Given:

Sides of the cube $=b$

Charge at the vertices $=q$

Diagonal of one of the sides of the cube

$d^{2}=\sqrt{b^{2}+b^{2}}=\sqrt{2 b^{2}}$

$\mathrm{d}=\mathrm{b} \sqrt{2}$

Length of the diagonal of the cube

$l^{2}=\sqrt{d^{2}+b^{2}}=\sqrt{2 b^{2}+b^{2}}=\sqrt{3 b^{2}}$

$1=b \sqrt{3}$

The distance between one of the vertices and the centre of the cube is

$r=1 / 2=(b \sqrt{3 / 2})$

The electric potential $(V)$ in the cube’s centre is caused by the eight charges at the cube’s vertices, which add up to one volt.

$V=8 q / 4 \pi \varepsilon_{0}$ $V=\frac{8 q}{4 \pi \epsilon_{0}\left(b \frac{\sqrt{3}}{2}\right)}=\frac{4 q}{\sqrt{3} \pi \epsilon_{0} b}$

Therefore, potential at the centre of the cube is $\frac{4 q}{\sqrt{3} \pi \epsilon_{0} b}$

In the centre of the cube, where the eight charges are concentrated, the electric field strength is zero. The charges are dispersed in a symmetrical manner with respect to the cube’s centre of mass. As a result, they are no longer valid.