# 1.4 Gauss' Law Peter Apps

### What the Flux?!?

Flux is a very useful concept to help describe a wide variety of physics concepts. We're going to apply it here for electric fields, and then use it to help describe magnetic fields. Basically flux describes how much of something goes through a given area.
We're going to imagine an area on the surface of a charged object. It doesn't matter what the object is. The electric flux is then described by how many electric field lines pass through the area. Generally, we define the area to be parallel to the electric field, since this simplifies the math. However, if we can't do that, we take the dot product between the area vector and the electric field to determine the flux. Let's look at the total flux in this image below. Image from opentextbookbc.ca

When field lines enter a closed surface, the flux is negative. When they exit the surface, the flux is positive.
In this case, we see that there are 11 field lines entering the area ABCD and 11 field lines exiting FGHK. The flux from these two areas cancel out, and, therefore, the total flux is 0. A simple way to think about this is that if there isn't a source of charge or a sink (something that absorbs charge) **inside the area, the net flux is 0.

### Gauss' Law

Gauss's Law relates the flux on a closed surface to the amount of charge enclosed by the surface. Let's break this formula down a bit and see where it comes from.   Let's try to find the flux. Remember that E is constant across the entirety of the surface. Adding up all the partial areas of the sphere gives us the surface area. (A= 4*pi*r^2). In short, Gauss's Law states that sum of the charge sources within a closed surface is equal to the total electric flux through the surface.

### Practice Problems

1. Image from AP Classroom 2. Image from AP Classroom

We know there is +Q enclosed in the inner cylinder. We can see 4 lines of flux leaving that cylinder. With the larger cylinder, there are 8 lines entering the cylinder. This means that the net charge enclosed by the larger cylinder is -2Q. So the outer cylinder must have a charge of -3Q.
3. Image from AP Classroom 4. Image from AP Classroom

The sphere is conducting, so any fields outside it can't penetrate inside and don't influence the internal field. The only charge enclosed is +Q, so choice A is the correct answer.

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