# Work to induce charge on grounded conductors

I'm currently studying Method of Images in Griffiths book and in section 3.2 he introduces the method of images for a point charge at a distance $$d$$ from a grounded conducting plane at potential $$V = 0$$. In subsection 3.2.3, Griffiths compute the energy of the real system and the image charge system and obtain results differing by a factor of 2, The explanation of griffiths is that in the image problem, we do work on both charges bringing them from infinity to a distance $$2d$$ apart from each other. However in the real problem, griffiths says that we do work only on the point charge $$q$$, since the induced charge on the conductor moves along an equipotential. My problem is: if the potential inside the grounded conductor still $$V=0$$ even with the external electric field due to the point charge, how charges can be induced, since no force acts on them? I've tried to re-read conductors section in griffiths chapter 2, but as long as I remember, he makes no mention to this problem.

Also sorry for possible typos, I'm still learning english

Imagine moving the point charge $$q$$ a tiny bit closer to the grounded plate (a displacement $$dx$$, say). Before this displacement, the charges in the plate are in equilibrium and the plate is at $$V=0$$. After the displacement, then instantaneously, the charges in the grounded plate will feel a force (and $$V$$ will not be uniformly zero), and very quickly rearrange themselves into a new equilibrium position.
However, since the displacement of the charge $$q$$ was infinitesimal, the charges in the plate will only move an infinitesimal distance $$dr$$ due an infinitesimal force $$dF$$ in order to reach their new equilibrium, so that the total work done on the charges in moving from one equilibrium to the other is of the order $$(dr)(dF)$$, which is a product of differentials so can be ignored, i.e., the work done is zero!