Question

Using Gauss' law in electrostatics, prove
that the charge resides only on the surface
of a conductor.

Answer 1

Electrostatic field inside a fixed conductor equal to zero. Why? Because if the electrostatic field inside a conductor is not equal to zero, then, by the definition of a conductor, a current flows in it, then the charges move, and the conditions of electrostatics are not fulfilled.

According to the law of Gauss:

$$\oiint \vec{E}\vec{dS}=4\pi Q_{inside}$$

But $\vec{E}=0$, then $Q_{inside}=0$ and all charge resides only on the surface of a conductor.