Consider an infinite sheet of charge with uniform charge density per unit area s. What is the magnitude of the electric field a distance r from the sheet?

To apply Gauss' Law, we need to know what the field looks like.

From the symmetry of the situation you should be able to convince yourself that the field is uniform and perpendicular to the sheet, going out from the sheet if the charge is positive and in toward the sheet if the sheet is negative.

Come up with an appropriate gaussian surface. Here are some possibilities:

1. a sphere whose center lies on the sheet
2. a cylinder whose axis lies on the sheet
3. a cylinder whose axis is perpendicular to the sheet
4. a cube or rectangular box with two faces parallel to the sheet

Either choice 3 or choice 4 would be fine. Let's go with a cylinder with a cross-sectional area A.

Once again, to apply Gauss' Law, we need to answer two questions:
What is the total charge enclosed by the surface?
What is the net electric flux passing through the surface?

The total charge enclosed is q_enc = σA, the charge per unit length multiplied by the cross-sectional area of the cylinder.

To find the net flux, consider the two ends of the cylinder as well as the side. There is no flux through the side because the electric field is parallel to the side. On the other hand, the electric field through an end is E multiplied by A, the area of the end, because E is uniform. There are two ends, so:

Net flux = 2EA.

Now bring in Gauss' Law and solve for the field:

By Gauss' Law the net flux = q_enc/ε_o

2EA = σA/ε_o

The factors of A cancel, leaving:

E = σ/2ε_o