Question

Suppose the x − y plane is electrically charged with a constant charge density σ.
Find the electric field at a hight h from the x − y plane. Using gauss’s law.

Answer 1

According to the Gauss law,

$\displaystyle \oint \bar{E} \cdot \bar{dS} = \frac{q}{\epsilon_0}$

where q is the charge inside region S.

Let's take cylinder with base area S and heigh H. Cylinder will intersect charged plane at H/2, so from the opposite sides of the plane there are equal parts of cylinder. Because of the planar symmetry, the flux of electric field through the lateral surface of the cylinder is zero.

The total flux trough the cylinder is

$\displaystyle 2 S E = \frac{q}{\epsilon_0}$

But the amount of charge inside the cylinder is equal to $q = \sigma S$.

So

$\displaystyle 2 S E = \frac{\sigma S}{\epsilon_0}$

$\displaystyle E = \frac{\sigma}{2\epsilon_0}$