Separation of variables and laplaces equation in electrostatics

The problem has to do with the potential inside a square. Separation of variables is to be used to find the general solution to the problem and then the boundary conditions need to be applied. The constant needs to be found then as well. The problem is attached. I have also attached the picture of the problem.

Electrostatics: Surface charge density and electric field

Two part question involving a spherical cavity inside a shell and calculating: a) the surface charge density b) the electric field Problem is attached in word format. Diagram is included in jpeg.

Electrostatics: Method of images for a grounded sphere

Use the method of images to find: a) the electric field at a point P. b) force on the real charge q Problem is in word format. Diagram is a jpeg.

A large conducting spherical shell has fixed surface charge density, has a small circular hole in the surface. Using integration over the surface of the sphere find the electric field in the hole at its center.

A large conducting spherical shell of radius a, and fixed surface charge density (sigma), has a small circular hole in the surface of radius b< infinity.

Electrostatics: Unequal Charge density on parallel plates

(See attached file for full problem description) --- 1. Consider two infinite parallel plane conducting plates, of finite thickness, with separation d. Suppose a charge density of sigma; is placed on one plate, while the other plate has -2sigma. A) Determine the resulting charge densities on each of the 4 surfaces B) Determine the electric field between and outside the plates Use superposition of Gaussian surfaces.