Hi everyone. I'm Alex Hansen and in this video we're going to review Maxwell's equations. Now if you haven't studied Maxwell's equations before it may require a little bit more study. For this video we're just going to do a fairly quick review. \r\nNow for most of us, when we learned Maxwell's equations, it kind of seems like a lot of messy math thrown together. And so what we'll do in this video is show that there are a couple of groupings and symmetries that help us remember Maxwell's equations. We'll also work with a simplified form of Maxwell's equations called the magneto quasistatic approximation, which means that we'll allow fields to change in time, but slowly enough that we can ignore a couple of terms in Maxwell's equations. And we'll note that we're reviewing Maxwell's equations at the beginning of a magnetics module, because we'll need at least three of those equations to properly characterize a magnetic component in the magneto Quasistatic approximation. \r\nNow, the first two Maxwell's equations are both sometimes known as Gauss's law. There's a Gauss's law for electric fields that says that the integral of the D field over an area this would be a closed area, is equal to the total amount of charge that's inside that area. Now you'll notice that what I wrote here is the integral of rho dv. This is the mathematical way of explaining it, but I'm going to try as much as possible to also give the intuitive or a physical description of what's going on. \r\nSo I remember that the amount of D field that goes out of a closed area. Now that area might look like a sphere or any other shape. So here's my area and I'm going to go ahead and write DCR in red so that you'll link those to the D field that flows out of that area. So let me just draw an example of a D field here, which applies to a small patch of area. \r\nAnd there may be D field flowing out all over the place. This would be the D field. Now that's equal to the total amount of charge in the volume. I'll color the volume in purple here. \r\nAnd that's everything that's contained inside this sphere. Let me just correct this here. So that's everything contained inside the sphere. And that's Gauss's law for electric fields. \r\nAnd you've probably used this to determine the electric field coming out of a sphere or a point charge, or a parallel plate capacitor. Gauss's law for magnetic fields is very much the same way we have an integral over an area. A closed area. This time, though, we're talking about the B field, and that would be equal to the integral of something contained in the volume. And that something would be the magnetic charge. \r\nThe problem is, as near as physicists can tell, there is no magnetic charge. There are no magnetic monopoles, and therefore this term is always zero. I think that's kind of unfortunate, because there's a symmetry to these two equations that are both called Gauss's law. And that symmetry kind of disappears when you just write zero. \r\nSo what I'm going to write instead is simply the integral of zero DV. We know that will always come out to zero, but it preserves the symmetry in the equations. Now the next equation is Ampere's law. And Ampere's law says that the integral around a closed loop of the h field dl, because this is a line integral, is equal to the integral over the area enclosed by that loop of J. This is the current density plus DDC d. \r\nDot da. So this is a line integral and then a surface integral. Now it might be helpful to picture this in the form of a concrete scenario. So let me just show for example, an inductor, and we'll do an inductor example in much more detail in another video. \r\nBut we're just going to draw it here just so we understand what all these integrals are. Now this inductor might have some current flowing through a wire okay. There's a current flowing through a wire. And we also expect there to be some magnetic field. \r\nSo this would be a current I and a magnetic field let's say H that flows around. And these two fields, if you think of I as a distributed current density, these two fields loop around each other. So if I integrate HDL around a closed loop, it's natural to follow the loop that I've just described here, where this black thing that I've drawn is a magnetic material. And so the loop associated with the H field is this purple loop that I've described. \r\nAnd I'll go ahead and draw that integral in purple so that you can associate that. And that's equal to an integral over an area. What area are we talking about. Well we're talking about the area enclosed by the purple loop. So that would be this area here. And we're going to be asking ourselves how much current penetrates that loop. That's where this J comes from. That's the current that penetrates the green area. DRA. We'll also ask ourselves how much do flows through that area however, and this is where the magneto quasistatic approximation comes in. We're going to argue that while we allow things to vary with time, we're going to assume that the problem is dominated by magnetic fields, and things are not operating fast enough for the DD to be a significant contribution. And so we're going to let this term be zero. That's called the magneto quasi static approximation. Now when can you make this approximation. We won't derive it here. But you can basically make this approximation when the length scale of your problem. So this might be one of the dimensions of this inductor here is much less than the speed of light divided by the frequency that you're operating at. In other words, If the object is small and the frequency is fairly slow, then this approximation is good. So that's Ampere's law. And we've color coded this with a specific concrete example. Let's see now how Faraday's law, which is the last equation. Is analogous to Ampere's law, the same way that the two forms of Gauss's law are analogous to each other. Faraday's law says that the integral of E dot dl. Now I've got an electric variable instead of a magnetic variable is equal to the integral. Now what do you think would go here if we're replacing magnetic variables with electric variables and vice versa. I would expect this to be j magnetic. But of course we know there are no such things as magnetic charges and therefore there's no magnetic current. And therefore this term is actually zero. So I'll go ahead and rewrite that as zero just to maintain some of the symmetry, the same way I did with Gauss's laws. And then it turns out we're going to get minus DB. So we still have a derivative. And we've replaced the electric variable D with the magnetic variable b. And this is dot da. Now both Ampere's law and Faraday's law are true no matter what loops you choose. But a typical loop you might choose for Faraday's law would be the loop through the wire. So we would take the integral over the orange loop and integrate the E field that's going along that line. And that's going to be equal to the integral over some area. What area are we talking about. Well it's the area that's enclosed by this wire loop. And so that's going to be this dot dia here I'll color in pink. So it corresponds with that area I drew on the right. And we're interested in the B field that penetrates that loop. And the B field is of course a magnetic variable. This would be like the the H field going around corresponds to a B field as well. And so I'll go ahead and draw B in purple here. This is the magnetic field going around penetrating the pink loop. Now one thing to keep in mind is that in Faraday's law, just as in Ampere's law, the electric field integral is actually a closed integral which I hadn't drawn before. And that means when you integrate around the wire loop, you have to complete the loop, which means you have to go through whatever components make up the rest of the circuit. We'll see that this is important when we talk about leakage inductance. And also when we talk about layout, because the contribution from those extra components that are quote unquote, outside of the inductor can make a big difference. Now, strictly speaking, that's all for Maxwell's equations, but it still seems like there are a lot of variables. There's a D and an E and there's a B and an h. So we should just remember that there's a relationship between these. B is typically a function of h. And d is a function of e. And these are both material properties. These functions depend on what kind of material you have. Very frequently we run into linear materials. So in the case of linear materials B is proportional to H, and it's proportional according to some relative permeability times the permeability of free space times H. Not all material is obey this, but many materials are linear and will make use of linear materials in magnetics a great deal, but we'll also recognize where they start to become non-linear. That's a concept called saturation. Similarly, the d field is equal to some relative permittivity times the permittivity of free space times the electric field e. So those are your Maxwell's equations and what are sometimes called the constitutive relations that connect b and h and d and e. And we can see here how all of these integrals relate to the geometries that they're intended to be used on. That being said, this can all still seem rather abstract. In the following videos, we'll actually use Maxwell's equations to analyze inductors and transformers, and that will make some of these calculations a little bit more concrete. In addition, we'll see that there are ways to perform the same calculations without going back to Maxwell's equations in their raw form. We do this with electric circuits all the time. Maxwell's equations give us Kirchhoff's voltage law and Kirchhoff's current law. We will find in another video that we can do the same thing with magnetics, and develop magnetic circuit models that let us use a very intuitive approach to fields and the flow of flux quantities to figure out what's happening in the physical world. All right. So I hope this whirlwind tour through Maxwell's equations gives us a little bit more understanding of the geometries and also the symmetries in the equations, and that might help us remember them. We mentioned briefly that we are using the magneto quasistatic approximation, which makes the calculations a lot more straightforward because the equations are no longer coupled together in the same way. We saw how Ampere's law and Faraday's law will be used to analyze inductors, and as is commonly known, we use Gauss's law for magnetics to say that magnetic fields always flow in complete loops, which we sort of sneakily used when we drew our diagram. And so at least three out of these four equations play a role in analyzing magnetic components. And hopefully that underscores the importance of keeping Maxwell's equations close to HART.