One more to go, Ampere's law. Let's turn Ampere's law into differential form. Remember, if we have a wire and it's carrying some current, we know that it makes a magnetic field and the magnetic field will follow the right-hand rule. It's going behind the wire and out in front. I'll draw it like this. The B field is going to go like that and come back around like this. With Ampere's law, we could always describe a field like this. The fact that you have these circulating B fields around wires by doing a line integral around the wire. The general statement then of Ampere's law is the integral around some closed loop of B.dl always equal to basically Mu naught times I, where B is a path you choose, and I is the current that penetrates the loop. If you wanted to be a little bit more general about it while trying to do field theory here is rather than think about just an exact I is, you could write that as an integral. You can say, well, maybe I have current distributed through space. You could write it as Mu naught and then use something called the current density that's actually a vector. You could say Mu naught times the integral of J.dA over that surface. That would be a nicer way to put it. This is the surface integral. Slightly more general, but it doesn't get us there. We aren't ready to apply Stokes theorem to this yet. The reason is something that Maxwell had to come up with. You may be wondering why they're called Maxwell's equations. We have Gauss's law, Faraday's law, Ampere's law, the no-name law. Suddenly, all Maxwell did was put them all together and he got them all named after himself. Well, he did a lot more than just put them together. We'll see at the end of this section what he did. But there is an important thing he fixed in Ampere's law and it's like this. Maxwell was a very smart guy and he knew that when you're working this out, that the area over which you do this integral, or which you would figure out J.dA, it doesn't have to be this exact surface right on this loop. You're allowed to actually make it more like a butterfly net. You can have an area out here. You can do the area around the loop, or you can have an area that sticks out. Basically, you can have any area that's bound by the loop. You can have a tube over here as long as it's bound on its edge by the loop. You do the B integral over, and then it's okay. That led Maxwell to think about something like this. What if you're charging a capacitor? Here, we have a capacitor connected to a couple of wires and we send some current in like that. Well, you know what's going to happen? Current's going to flow for awhile. You're going to get some positive charge. They're going to build up on this plate. Some negative, all the same amount of negative charge will build up on that plate and that's the story. But he said, let's apply Ampere's law because we know that current flows here. We know that if we pick a path, just like we did before, where it comes around like that, that we should be able to apply Ampere's law and get the right answer. If you do it right here, sure enough you do, you'll have the current flow integral B.dl. You will get a value for the integral of B.dl. The current flow on the other side is going right through the loop. But as Maxwell pointed out, the loop can go anywhere. It could go like here. If it goes there fine. The current can penetrate the area here, or it can penetrate back here. But then Maxwell said, what if you drew the surface where the wire doesn't actually ever go? What if we do the surface like this right down the middle of the capacitor. That mathematically is legal because that is a surface that is bound by this loop. If you do that, then you get a value for the integral of B.dl around that loop. This has a non-zero value. But then, it doesn't equal zero because here no current penetrates this entire surface. All the current is in the wire, the current stops when it gets to the plates. This is a problem and this told Maxwell that the Ampere's law is incomplete. It's missing something. Basically, what Maxwell said, I wasn't there, but he more or less, what he said is that the changing E field is an effective current. Not exactly what he said. That's what he meant. We need another term over here. We have a current, literal current charge flowing, but we need another term to account for situations like this. The thing that goes through the surface here is an E field, because an E field is created inside the capacitor. If you think about a changing E field, then you'll always have this covered. You'll always have something on the right side of this equation. You'll either have current if you're here or you'll have a changing E field if you're here, and you'll always be okay. The way then he wrote Maxwell's equations, or the way he wrote Ampere's law was the integral of B.dl equals Mu naught times the surface integral. The first term is the current that we're used to, the J. That's just the current density. We thought about the current per unit area and the direction that it flows. But then he added this other term, plus d/dt Epsilon naught E. Of course, the Epsilon naught can be outside the d/dt. Basically, Epsilon naught times the time derivative of the electric field and that's all dotted with dA. The J is added with dA, which is really just Mu naught I. But then, he has basically the changing E field flux inside the capacitor. He called this, not an effective current. He called it the displacement current. Displacement current, some people don't like the name. But basically, the idea is we don't have current, but we have charge being displaced, creating an electric field that effectively acts like a current in Ampere's law. This will make it, you can see, always be true. The full name is the Ampere Maxwell law, because Maxwell came in and fixed it.
