Question

A non conducting charged cube has a constant charge density $\rho$. The electric potential at infinity is taken to be zero. i.e. $V\left(\infty \right)=0$. The electric potential at the center of the cube is $V_0$ and at the corner of the cube is $V_1$. If $n^{-1}\left(\frac{V_0}{v_1}\right)=1$, then the value of $n$ is :

Answer 1

Electric potential at the center of the cube is $V_0$.

Let the length of each edge be $l$.

Let the total charge of the cube be $Q$.

We construct 7 more such cubes and keep them such that one of the corners of first cube is that the center of the new cube formed of edge of length $2l$.

Thus the corner is at the center of cube of charge density $\rho$ and edge length $2l$

The total charge of this cube is $8Q$.
Dimensionally , potential at center of such cubes is $\left(constant\right)\times \frac{Q}{l}$

Thus $\frac{V_0}{V_1}=\frac{\frac{Q}{l}}{\frac{1}{8}\frac{8Q}{2l}}$.

$\frac{1}{8}$ comes from the fact that potential at center of cube of length $2l$ is 8 times the potential at that point due to one of the cubes, which is that potential at corner of that cube.

Therefore $n=2$