Prove that "a perfect conductor cannot contain an electrostatic field within it".

The proof is below

Explanation:

The net electric charge of a conductor resides entirely on its surface. (The mutual repulsion of like charges from Coulomb's Law demands that the charges be as far apart as possible, hence on the surface of the conductor.) The electric field inside the conductor is zero.

When two parallel plates one having the positive charge and the other having the negative charge, then the electric field will generate from positive to the negative direction.

When a conductor is placed between the two parallel plates:

The conductor has large number of electrons and holes are also produced by the electrons movement.

If the electric field is applied, the positive charge plate will attracts the electrons (negatively charged) from the conductor on the 1 side and the negative charged plate will attract holes.

On opposite side, the conductor will induce the positive and the negative charge and it will create the electric field inside it and the magnitude of this field is equal and the opposite to the electric field applied.