Two conducting plates A and B are placed parallel to each other. A is given a charge `Q_1` and B a charge `Q_2`. Prove that the charges on the inner surfaces are of equal magnitude and opposite sign.

Explanation

Consider a Gaussian surface as shown in figure.
Two faces of this closed surface lie completely inside the conductor
where the elctric field is zero. The flux through these faces
is, therefore, zero. The other parts of the closed surface, which
are outside the conductor, are parallel to the electric field, and
hence the flux on these parts is also zero. The total fulx of the
electric field through the closed surface is, therefore, zero.

From Gauss's law, the total charge inside this closed surface
should be zero. Thus , `x + y = 0 or x = -y`
Hence, the charge on the inner surface of A should be equal
and opposite to that on the inner surface of B.