 Okay, next up, we're going to review an example of the application of Gauss's law. I'm going to go through this pretty quickly. It's the one I did at the end of class on Monday, but you guys also did it in recitation yesterday, so you should be pretty familiar with this. We're using this one as our example today because we're going to contrast it with another example at the end of this class. So let's talk about a flat plane of charge. For argument's sake, this is a very, very large flat sheet of charge, so it extends out further than this diagram shows you. In that case, I've got an electric field which points straight up and straight down off the top and the bottom of the sheet of charge. I can create a Gaussian surface. Now in class, I talked about it being a cube, but you can also use a cylinder, and a lot of textbooks use that, so I wanted to show you how it's the same sort of thing. If I want to know how much charge is inside this cylinder, well I've got a surface charge density, and I have to figure out how much area is underneath inside the cylinder. And because it's a straight-sided cylinder, the area of the end cap of the cylinder is the same as the area where the cylinder intersects the sheet of charge. So I've got my surface charge density times the area of that end cap is going to give me the total charge inside. Now when I consider my Gaussian area, I don't have to consider the curved side of the cylinder. This is kind of like my four sides of my cube didn't matter, because the electric field is just going parallel along those sides. It doesn't go through those sides. The places where it goes through is at the top surface, and down here at the bottom surface. So you've got two times the end cap, two coming from the top and the bottom. When I take that cube inside and that surface of the Gaussian and plug them in to my Gaussian electric field equation, I've got sigma over the area of the end cap divided by epsilon naught two times the area of the end cap. And the area of that end cap is going to just cross out. I've got it on the top and the bottom. It cancels, and I'm left with just sigma over two epsilon naught. And notice it didn't matter whether it was a cube or a cylinder. It doesn't matter what the size of that end cap was, and it also doesn't matter how tall that cylinder or that cube is. I've got a uniform electric field pointed up above my sheet of charge and a uniform electric field pointed down away from my sheet of charge. Now that's if my surface charge is a positive charge density. If it was a negative charge density, I have the exact same formula, except for I recognize that my electric field is pointed down towards the sheet above the sheet and pointed up below the sheet, in either case pointing towards the negative charge distribution or away from a positive charge distribution. So this gives you your equation for this flat plane of charge.