{
  "words" : [ [ "3130", "Hi everyone. I'm Alex Hansen " ], [ "4997", "and in this video we're going to review Maxwell's equations. " ], [ "9163", "Now if you haven't studied Maxwell's equations " ], [ "10830", "before it may require a little bit more study. " ], [ "14597", "For this video we're just going to do a fairly quick review. " ], [ "19863", "Now for most of us, when we learned Maxwell's equations, " ], [ "22763", "it kind of seems like a lot of messy math thrown together. " ], [ "26397", "And so what we'll do in this video is show that there are a couple of groupings " ], [ "29863", "and symmetries that help us remember Maxwell's equations. " ], [ "33730", "We'll also work with a simplified form " ], [ "36230", "of Maxwell's equations called the magneto quasistatic approximation, " ], [ "41730", "which means that we'll allow fields to change in time, " ], [ "45963", "but slowly enough that we can ignore a couple of terms in Maxwell's equations. " ], [ "52197", "And we'll note that we're reviewing Maxwell's equations at the beginning " ], [ "56130", "of a magnetics module, because we'll need at least three of those equations " ], [ "61397", "to properly characterize a magnetic component " ], [ "64597", "in the magneto Quasistatic approximation. " ], [ "69330", "Now, the first two Maxwell's equations are both sometimes known as Gauss's law. " ], [ "77930", "There's a Gauss's law for electric fields that says that the integral " ], [ "82130", "of the D field over an area this would be a closed area, " ], [ "90663", "is equal to the total amount of charge that's inside that area. " ], [ "97863", "Now you'll notice that what I wrote here is the integral of rho dv. " ], [ "100997", "This is the mathematical way of explaining it, " ], [ "103930", "but I'm going to try as much as possible to also give the intuitive " ], [ "107497", "or a physical description of what's going on. " ], [ "110863", "So I remember that the amount of D field that goes out of a closed area. " ], [ "116363", "Now that area might look like a sphere or any other shape. " ], [ "122830", "So here's my area and I'm going to go ahead and write DCR in red " ], [ "127330", "so that you'll link those to the D field that flows out of that area. " ], [ "132330", "So let me just draw an example of a D field here, " ], [ "136363", "which applies to a small patch of area. " ], [ "139730", "And there may be D field flowing out all over the place. " ], [ "143863", "This would be the D field. " ], [ "145897", "Now that's equal to the total amount of charge in the volume. " ], [ "149463", "I'll color the volume in purple here. " ], [ "153763", "And that's everything that's contained inside this sphere. " ], [ "160097", "Let me just correct this here. " ], [ "163963", "So that's everything contained inside the sphere. " ], [ "168263", "And that's Gauss's law for electric fields. " ], [ "170230", "And you've probably used this to determine the electric field " ], [ "173863", "coming out of a sphere or a point charge, or a parallel plate capacitor. " ], [ "180897", "Gauss's law for magnetic fields is very much the same way we have an integral " ], [ "187630", "over an area. A closed area. " ], [ "192630", "This time, though, we're talking about the B field, " ], [ "196897", "and that would be equal to the integral of something contained in the volume. " ], [ "202863", "And that something would be the magnetic charge. " ], [ "207230", "The problem is, as near as physicists can tell, there is no magnetic charge. " ], [ "211797", "There are no magnetic monopoles, and therefore this term is always zero. " ], [ "218263", "I think that's kind of unfortunate, " ], [ "219763", "because there's a symmetry to these two equations that are both called Gauss's law. " ], [ "225763", "And that symmetry kind of disappears when you just write zero. " ], [ "230263", "So what I'm going to write instead is simply the integral of zero DV. " ], [ "238730", "We know that will always come out " ], [ "239963", "to zero, but it preserves the symmetry in the equations. " ], [ "245230", "Now the next equation is Ampere's law. " ], [ "251263", "And Ampere's law says that the integral around a closed loop of the h field dl, " ], [ "259030", "because this is a line integral, " ], [ "261797", "is equal to the integral over the area enclosed by that loop " ], [ "267397", "of J. This is the current density plus DDC d. " ], [ "278163", "Dot da. So this is a line integral and then a surface integral. " ], [ "285197", "Now it might be helpful to picture this in the form of a concrete scenario. " ], [ "290130", "So let me just show for example, an inductor, " ], [ "294463", "and we'll do an inductor example in much more " ], [ "296730", "detail in another video. " ], [ "298763", "But we're just going to draw it here just so we understand what all " ], [ "302363", "these integrals are. " ], [ "305563", "Now this inductor might have some current flowing through a wire okay. " ], [ "313530", "There's a current flowing through a wire. And we also expect " ], [ "316897", "there to be some magnetic field. " ], [ "322097", "So this would be a current I and a magnetic field let's say H that flows around. " ], [ "329730", "And these two fields, if you think of I as a distributed current density, " ], [ "335130", "these two fields loop around each other. " ], [ "338363", "So if I integrate HDL around a closed loop, " ], [ "342697", "it's natural to follow the loop that I've just described here, " ], [ "346797", "where this black thing that I've drawn is a magnetic material. " ], [ "355063", "And so the loop associated with the H field is this purple loop that I've described. " ], [ "363030", "And I'll go ahead and draw that integral in purple so that you can associate that. " ], [ "367630", "And that's equal to an integral over an area. " ], [ "370697", "What area are we talking about. Well we're talking about the area " ], [ "375230", "enclosed by the purple loop. So that would be this area here. " ], [ "384930", "And we're going to be asking ourselves how much current penetrates that loop. " ], [ "389830", "That's where this J comes from. " ], [ "391130", "That's the current that penetrates the green area. " ], [ "395497", "DRA. " ], [ "397430", "We'll also ask ourselves how much do flows through that area however, " ], [ "405363", "and this is where the magneto quasistatic approximation comes " ], [ "408563", "in. We're going to argue that while we allow things to vary with time, " ], [ "415230", "we're going to assume that the problem is dominated by magnetic " ], [ "419063", "fields, and things are not operating fast enough for the DD " ], [ "423763", "to be a significant contribution. " ], [ "426497", "And so we're going to let this term be zero. " ], [ "430463", "That's called the magneto " ], [ "435830", "quasi static approximation. " ], [ "441430", "Now when can you make this approximation. " ], [ "444563", "We won't derive it here. " ], [ "445797", "But you can basically make this approximation when the length scale of your problem. " ], [ "453197", "So this might be one of the dimensions of this inductor here is much " ], [ "458197", "less than the speed of light divided by the frequency that you're operating at. " ], [ "464863", "In other words, If the object is small and the frequency is fairly slow, " ], [ "471697", "then this approximation is good. " ], [ "474297", "So that's Ampere's law. And we've color coded this with a specific concrete example. " ], [ "480697", "Let's see now how Faraday's law, which is the last equation. " ], [ "486863", "Is analogous to Ampere's law, " ], [ "489197", "the same way that the two forms of Gauss's " ], [ "491097", "law are analogous to each other. " ], [ "493663", "Faraday's law says that the integral of E dot dl. " ], [ "498130", "Now I've got an electric variable instead of a magnetic variable is equal " ], [ "503397", "to the integral. Now what do you think would go here if we're replacing " ], [ "508897", "magnetic variables with electric variables and vice versa. " ], [ "511930", "I would expect this to be j magnetic. " ], [ "516563", "But of course we know there are no such things as magnetic charges " ], [ "520963", "and therefore there's no magnetic current. And therefore this term is actually zero. " ], [ "527063", "So I'll go ahead and rewrite that as zero just to maintain some of the symmetry, " ], [ "533230", "the same way I did with Gauss's laws. " ], [ "535263", "And then it turns out we're going to get minus DB. " ], [ "541630", "So we still have a derivative. " ], [ "542963", "And we've replaced the electric variable D with the magnetic variable b. " ], [ "549363", "And this is dot da. Now both Ampere's law " ], [ "552730", "and Faraday's law are true no matter what loops you choose. " ], [ "556630", "But a typical loop you might choose for Faraday's law would be the loop " ], [ "561497", "through the wire. So we would take the integral over the orange loop " ], [ "567330", "and integrate the E field that's going along that line. " ], [ "572930", "And that's going to be equal to the integral over some area. " ], [ "576530", "What area are we talking about. Well it's the area that's enclosed by this wire loop. " ], [ "584230", "And so that's going to be this dot dia here I'll color in pink. " ], [ "589163", "So it corresponds with that area I drew on the right. " ], [ "594197", "And we're interested in the B field that penetrates that loop. " ], [ "598263", "And the B field is of course a magnetic variable. " ], [ "600430", "This would be like the the H field going around corresponds to a B field as well. " ], [ "605897", "And so I'll go ahead and draw B in purple here. " ], [ "610030", "This is the magnetic field going around penetrating the pink loop. " ], [ "615597", "Now one thing to keep in mind is that in Faraday's law, just as in Ampere's law, " ], [ "621163", "the electric field integral is actually a closed integral " ], [ "625397", "which I hadn't drawn before. " ], [ "627297", "And that means when you integrate around the wire loop, " ], [ "629963", "you have to complete the loop, " ], [ "632197", "which means you have to go through whatever components make up the rest " ], [ "636430", "of the circuit. We'll see that this is important " ], [ "639497", "when we talk about leakage inductance. " ], [ "642397", "And also when we talk about layout, " ], [ "644663", "because the contribution from those extra components that are quote unquote, " ], [ "649863", "outside of the inductor can make a big difference. " ], [ "654163", "Now, strictly speaking, that's all for Maxwell's equations, " ], [ "657597", "but it still seems like there are a lot of variables. " ], [ "660197", "There's a D and an E and there's a B and an h. " ], [ "664663", "So we should just remember that there's a relationship between " ], [ "667530", "these. B is typically a function of h. " ], [ "675997", "And d is a function of e. " ], [ "682597", "And these are both material properties. " ], [ "684363", "These functions depend on what kind of material you have. " ], [ "690263", "Very frequently we run into linear materials. " ], [ "694630", "So in the case of linear materials B is proportional to H, " ], [ "701397", "and it's proportional according to some relative permeability " ], [ "705863", "times the permeability of free space times H. Not all material is obey this, " ], [ "712263", "but many materials are linear " ], [ "714363", "and will make use of linear materials in magnetics a great deal, " ], [ "719463", "but we'll also recognize where they start to become non-linear. " ], [ "722663", "That's a concept called saturation. " ], [ "725463", "Similarly, the d field is equal to some relative permittivity times the permittivity " ], [ "733063", "of free space times the electric field e. " ], [ "737597", "So those are your Maxwell's equations " ], [ "739697", "and what are sometimes called the constitutive relations " ], [ "743563", "that connect b and h and d and e. " ], [ "746530", "And we can see here how all of these integrals relate to the geometries " ], [ "752163", "that they're intended to be used on. " ], [ "756263", "That being said, this can all still seem rather abstract. " ], [ "760963", "In the following videos, " ], [ "762097", "we'll actually use Maxwell's equations to analyze inductors and transformers, " ], [ "768063", "and that will make some of these calculations " ], [ "770263", "a little bit more concrete. " ], [ "772563", "In addition, we'll see that there are ways " ], [ "775130", "to perform the same calculations without going back to Maxwell's equations " ], [ "780363", "in their raw form. We do this with electric circuits all the time. " ], [ "786463", "Maxwell's equations give us Kirchhoff's voltage law and Kirchhoff's current law. " ], [ "791530", "We will find in another video that we can do the same thing with magnetics, " ], [ "795563", "and develop magnetic circuit models that let us use a very intuitive approach " ], [ "801297", "to fields and the flow of flux quantities " ], [ "805430", "to figure out what's happening in the physical world. " ], [ "809463", "All right. So I hope this whirlwind tour through Maxwell's " ], [ "813163", "equations gives us a little bit more understanding of the geometries " ], [ "818897", "and also the symmetries in the equations, and that might help us remember them. " ], [ "823263", "We mentioned briefly that we are using the magneto quasistatic approximation, " ], [ "828063", "which makes the calculations a lot more straightforward " ], [ "831297", "because the equations are no longer coupled together in the same way. " ], [ "837163", "We saw how Ampere's law and Faraday's law will be used to analyze inductors, " ], [ "843030", "and as is commonly known, " ], [ "844963", "we use Gauss's law for magnetics to say that magnetic fields always flow " ], [ "849663", "in complete loops, which we sort of sneakily used when we drew our diagram. " ], [ "854797", "And so at least three out of these four equations play a role " ], [ "857463", "in analyzing magnetic components. " ], [ "859697", "And hopefully that underscores the importance " ], [ "862497", "of keeping Maxwell's equations close to HART. " ] ],
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}