Gauss law and the method of images

Consider a grounded spherical shell conductor (radius $$R$$) with a point charge $$q$$ inside a distance $$a$$ from the center. In order to calculate the electric field inside we can place an image charge with charge $$q'=-q\frac{R}{a}$$ and position $$d=\frac{R^2}{a}$$ but if we apply Gauss law around the sphere, the electric field outside is zero $$\rightarrow$$ flux is zero $$\rightarrow$$ net charge inside is zero. That means the total charge induced on the sphere is $$-q$$. But how does that make sense, shouldn't the charge on the shell be the same as the point image charge we used to calculate the potential inside? So we got $$q'=-q$$ instead of $$q'=-q\frac{R}{a}$$. Where is the contradiction?

• @Darkenin No, the grounding just fixes the total charge of the point charge and sphere to zero. You could also do the same problem but say that the sphere has a total charge $Q$ on it. In that case, the extra uniform charge density would be whatever is necessary to get a total charge of $Q$ on the sphere. May 9, 2020 at 19:17