Question

Is Gauss law applicable to image charges in electrostatics?

Answer 1

In the case where the original charge is outside the sphere and the image charge is inside:

Consider a Gaussian surface just outside the conductor
In this region, the electric field due to the induced charge is exactly the the same as the electric field that would be due to the image charge
Therefore, the electric flux is equal to qin/ϵ0qin/ϵ0 where qinqin is the image charge
By Gauss's law, the charge contained inside is qinqin, and this is the induced charge.
Of course the crucial step is that you must draw the Gaussian surface outside the conductor. For a Gaussian surface inside the conductor the image fields and actual fields wouldn't match.

In the case where the original charge qq is inside the sphere, this reasoning fails, because it is impossible to draw a Gaussian surface that both lies entirely inside the sphere (for the image charge field to match) and contains the induced charges (which are on the inner surface of the sphere). But in this case the calculation is easy, we must have qin=−qqin=−q by the standard shielding argument.