Putting it all together 1

We are now in a position to actually evaluate a Gauss' law problem. The simplest application of Gauss' law is to derive the electric field for a point charge. Here's Gauss' law again:

$\int_{S} \vec{E} \cdot \vec{dA}=\frac{Q_{enc}}{\epsilon_0}$

where we now understand what all the pieces mean. The only ambiguity is which of the normal vectors to the area element $d\vec{A}$ we should choose. In Gauss' law, by convention we choose the outward pointing normal.

We now turn to how to evaluate Gauss' law. Place a charge of magnitude $\epsilon_0$ Coulombs at the origin. With Gauss' law you get to choose a surface $S$ for which you want to integrate over to determine $\vec{E}$ . The surface that yields the easiest algebra for solving for $\vec{E}$ using Gauss' law is