Last week we learned about Ampere's Law for finding magnetic fields. This week we will look into Gauss's Law in order to be better able to find Electric Fields.

Back at the beginning of the semester, we talked about how to find the electric field from point charges, lines of charge, and distributions of charge (i.e., cylinders, spheres, or planes of charge). The method of building a field from a point charge can also be used for a 2D sheet of charge or a 3D volume of charge and will always work; however, the mathematical calculation needed to determine the field can become much more complicated (you have to use either a double or triple integral). In some of these cases, Gauss's Law works as a shortcut for finding the electric field from complicated charge distributions. However, there are trade offs for using this shortcut – it is alway true, but only useful in highly symmetric situations. In the end, both of these methods are built around the same idea, even though the mathematics we will use is different: charges create electric fields. These notes will introduce the general concept of Gauss's law before we talk about the mathematics and the advantages and disadvantages to using it.

First, let's go back to our example of the point charge. The electric field points away from a positive charge. If we imagine a spherical bubble (like a very thin shell) around the point charge, we could think about the strength of the electric field that is on the surface of our imagined bubble (shown in the figure to the right).

Now if we imagine our bubble (typically called the “Gaussian surface”) to be a little bit bigger than it was before, two things happen:

1. The strength of the electric field decreases. The decrease is proportional to $1/r^2$ because $\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$ for a point charge.
2. The area of the imagined bubble or imagined Gaussian surface increases. The increase in area is proportional to $r^2$ because the surface area of a sphere is $A=4\pi r^2$.

This suggests that the electric field at the surface of the imaginary sphere multiplied by the surface area of the imaginary sphere is a constant; this product is called electric flux.

If we are considering a closed surface, like a bubble that encapsulates a charge, we are dealing with the total electric flux. If you draw the Gaussian surface with a larger radius, the electric field will be smaller, but the total electric flux is constant. If you draw the Gaussian surface with a smaller radius, the electric field will be greater, but the total electric flux is constant.

Gauss's Law is built around this idea that the total electric flux is constant through the imagined bubble (no matter where we draw the surface of the bubble) and that electric flux is related to the amount of charge inside the bubble (the point charge in this case). It turns out this relationship is always true - for any shape of charge (line, sphere, blob, etc.) and for any closed Gaussian surface, the electric flux on that surface is directly related to the charge inside the surface.

Note that the Gaussian surface is completely imaginary - we are not physically placing some surface around the charge. As we will talk about later, the choice of where to draw the surface and what shape to draw it is up to you; however, certain choices will lead to math that is virtually impossible, so we will want to be careful in our choice of surface. We will spend the next few pages of notes talking about the mathematical formalism for Gauss's Law, but this is the general conceptual idea behind the math.