Consider a point charge +q at a distance d above an infinite grounded conducting plane.
(a) Determine the charge density induced on the surface, then integrate to find the total surface charge.
Compare with the result of a force calculation on the image charge.
I'll assume that you are familiar with the method of images. You can find the electric field for this problem by pretending that the conducting plate isn't there but instead a charge of -q at the position of the ''mirror image'' of the charge, i.e. at distance d on the other side of the plate.
The sum of the electric fields of the charge and the mirror charge only has a component perpendicular to the plate. This means that the correct boundary condition is satisfied for the problem in which you have the conducting plate.
Now you need to find the induced charge density on the conducting plate. Using Gauss' theorem, taking the closed surface to be a ''pillbox'' with a surface inside the conductor (where the electrical field is zero) and a surface just outside the conductor you find:
E = sigma/epsilon_0 (1)
Here E is ...
A detailed solution is given.