 Now that we've discussed what a conductor is, we can move on to discuss Gauss's law on conductors. So remember from our previous discussion that all of the charge is on the surface of that conductor. For right now I'm going to stick with a nice sphere, but we'll see it doesn't actually have to be a sphere. So let's use Gauss's law on the inside. So I've got a little Gaussian surface here I've placed on the inside. Well remember Gauss's law depends on how much charge is inside that. And I'm never going to have a net charge inside it. All the unbalanced charges are on the outside edge of the conductor. So in here in the middle I've got no net charge. And that means I've got no electric field inside. Now anywhere I draw this sphere I still have no net charge in it. Even all the way up to almost on the surface it's always going to have no net charge inside of it. Even if instead of using a circle I use some sort of weird wiggly amoeba shape in here I still have no net charge inside that surface. And if this is going to be true for any Gaussian surface I put inside I cannot ever have an electric field inside. This actually kind of makes sense if there was a free electron inside and there was an electric field inside well that electric field would move the electron. So it would tend to move it where? Out to the surface. And as soon as it's moved it out to the surface I no longer have any free charges on the inside. So it turns out that for a conductor when you're inside the conducting material there's no electric field. Now we need to be careful this was a solid conducting sphere before we put this little Gaussian surface in here. I don't want you to think that this is a hollow sphere where the metal is just in a shell around the edge. If it was then the electric field would have to be zero inside the metal but not necessarily inside the open hollow area. So now we can talk about what if I have a conductor but instead of putting charge on that conductor and seeing what kind of electric field happens I actually put it in a region of field where there's an electric field. So I've got some outside electric field and this outside electric field is going through a conductor, a piece of metal. Well I've got free electrons in here and those free electrons are going to feel a force and in particular electrons always move opposite of the electric field and so they're going to be pulled over to this surface. When the electrons are pulled over to this surface it's going to leave an unbalanced proton over here on this surface. So I'm going to end up having positive charges on one side, negative charges on the other side. But once I've built up that charge well that's going to cause an electric field from my positive charges to the electrons and that electric field is exactly opposite the outside electric field and it's going to end up cancelling it all out. So I end up really only having electric field coming up to the surface, a charge distribution on the surface, no electric field inside because the two have cancelled each other out, positive charge distribution over on this side and then electric field lines emanating out on the other side. So our uniform electric field doesn't go all the way through the block, it's actually a region inside here where I'm shielded from the electric field. Now we can use this understanding that there's never an electric field inside a conductor but you can have an electric field outside the conductor and you've got some sort of surface charge density on the surface of the conductor to help us figure out what kind of electric field and charge density is going to build up. Now you'll notice I'm using my same diagram as what I had before in my earlier example where I had a flat plane of charge. In that case I had an electric field on both sides but now I don't have any electric field on the bottom inside surface. I'm assuming this is the upper surface of a conductor and the conductor expands on down into this region of space. I still have that electric field on the top side though. So let's take this and apply it to our Gauss's law. Well, my charge inside my little Gaussian cylinder which is partly sticking up out of my conductor and partly extending down into my conductor. Remember this whole region is filled with conducting material. I still have the same amount of surface charge distribution being the sigma surface charge density over the area where the cylinder intersects that surface. But now my Gaussian area, well before I had both the top and the bottom had electric flux going through it now only the top does. So where before I had two times the area of the end cap now I only have one times the area of the end cap. When I plug that into my Gauss's law again my area of the end cap disappears but I end up with sigma over epsilon naught. If this wasn't a conductor and it was just a surface which was charged and I had empty space on both sides it would have been sigma over two epsilon naught. But because I've got a conductor the electric field on the top side is actually more, it's sigma over epsilon naught instead of sigma over two epsilon naught. So now you've got your equation here for figuring out the electric field right near the surface of a conductor. Make sure you've got that written down and now you're going to apply that to do some actual problem solving.