(See attached file for full problem description)
A plastic sheet of thickness t has a uniform free charge density, +, embedded inside, and also one surface has a surface charge of -. Find the electric field and the potential as functions of distance from one surface. [Please neglect the issues of dielectric polarizability, the plastic simply functions to fix the charges in place].
You can solve this problem by using Gauss' Law:
The surface integral of E dot dO is Q/epsilon_0,
where Q is the charge enclosed by the surface you integrate over, dO is an area element that has the direction of the outward normal.
In this problem you assume that the sheet is infinite, so the electric field is constant parallel to the sheet. Also the electric field only has a component orthogonal to the sheet. Let's denote the coordinate orthogonal to the plates as x. Put the plate with zero surface charge density at x = 0 and the plate with charge density sigma at x = t.
Let's first calculate the electric field outside the plastic medium. Apply Gauss's law to a box ...
A detailed solution is given.