A slab of insulating material has a uniform positive charge density ? , as shown

Question

A slab of insulating material has a uniform positive charge density ?, as shown in the figure below. The slab is infinite in the y and z directions.

Derive expressions for the field for the following regions. (Use the following as necessary: ?0, ?, d, and x as necessary.)

d 2 A slab of insulating material has a uniform positive charge density ?, as shown in the figure below. The slab is infinite in the y and z directions. Derive expressions for the field for the following regions. (Use the following as necessary: ?0, ?, d, and x as necessary.)

Explanation / Answer

Note that the only electric flux will be going through the flat faces of the cylinder, this is because the electric field in the + and - y and z directions cancels out (because the area is assumed to be "essentially infinite").

Therefore, the electric field from the slab goes in the + and - x directions.

Gauss's law states that the electric flux through a surface (electric field times area through which it acts) is proportional to the charge enclosed by the surface. In this case the constant of proportionality is 1/e (e being the permittivitty of free space). The electric flux going through each flat face of the cylinder is going to be EA, so the total flux through the cylinder is 2EA. The charge enclosed is going to be 2dA*p. Therefore the equation becomes: 2EA = 2dA*p/e.

This implies that E=d*p/e.

Now for finding the electric field inside of the insulator, we use a cylinder with a dimensions: r and h.

Once again, the electric flux is only through the flat faces of the cylinder. The surface area of each face is pi*r^2. Thus the electric flux through the cylinder is E*2*pi*r^2. The charge enclosed by the cylinder will be V*p (V=pi*r^2*h). Therefore the equation is: E*2*pi*r^2 = (pi*r^2*h*p)/e.

This gives us the equation: E = hp/2e. (h being the distance away from the x axis). *note that if it had mentioned otherwise, you would have used some other constant of proportionality for the first part of this questions than 1/e, because the area inside the insulator would not be considered "free space".
I hope this helps.

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