…current is proportional to the vector potential, necessarily, in a piece of material like this. And this means, then, that the Maxwell equations, which for a steady condition says that the Laplacian of A is proportional to the current, becomes the Laplacian…

…wants to make a theoretical physics analysis of superconductors would take this pair of equations and these, which are the Maxwell equations, where the charge and the current comes from the superconductor stuff. Of course, I should always add the charges,…

…to the cavity, but you didn't use it. In actual use, the speed of the molecules is not constant. They have a Maxwell distribution. So that, in fact, the periods of time for different molecules may vary. And it's impossible to get 100% efficiency. You…

…here, the whole glory of everything anybody does with weight, I want it to be greater over the period because of the Maxwell distribution index. do all that well they solve more complicated cases with the field bearing that you go through and so on…

…that we figured out that that's equivalent to a charge density given by the divergence of P. And if we take the original Maxwell equation, which says that the divergence of E is equal to the total charge density, we can pull out the polarization part of…

…currents that we summarize, each little current element making the M. Then, of course, it's J-submagnetism that we put in here in the Maxwell equations here to find out what fields are produced by those magnets. That's the way we do it, not by terms of…

…things at once. Let's just look at the magnetism and forget all about the polarization. So the problem is that we take in the Maxwell equations. You can leave this in or out, depending as you desire. But in this J, we'll break it into two parts, the…

…the magnetic moment in a material. So that's the first step in our analysis. Now, the next step now is to use that in the Maxwell equations and see if we can do something about this. We have c squared times the curl of b is equal to, I'll leave out…

…Every subject that we've taken up here, we've taken up in a different way. In the case of electricity, we wrote the Maxwell equations on page one and then deduced everything. You think, well, that's the way to do everything. Maybe so. But we're not…

…way of entering subjects. There are some guys who can't stand it. Where did all those equations come from, those Maxwell equations from which everything gets deduced? First explain to me about electrostatics and I'll understand the Maxwell equation. Same way…

…come from, those Maxwell equations from which everything gets deduced? First explain to me about electrostatics and I'll understand the Maxwell equation. Same way in reverse. Suppose for a guy like that, I'd start with the Schrodinger equations and all the…

…one item, which is too advanced, which is usually given in graduate school, how to deal with the reflection from surfaces by the Maxwell equations. But I can't help it. I want to finish this whole subject of electrodynamics once and for all. And so we'll…

…take the curl of something, it'll be minus i k cross the thing, and so on. So it's easy enough to substitute into the Maxwell equation. And I'll take one example. Suppose I have a plane wave and I have the Maxwell equation curl E equal minus db dt.…

…easy enough to substitute into the Maxwell equation. And I'll take one example. Suppose I have a plane wave and I have the Maxwell equation curl E equal minus db dt. And I translate that for a plane wave moving in an arbitrary direction by this trick.…

…We found out last time that the k for an isotropic material was related to the frequency necessarily in order to solve the Maxwell equation by this relationship. Now, if the thing moves in an arbitrary direction, I have to talk about the magnitude of the…

…that out, and I use, of course, the very present case here. I've written all the equations down. Those are the Maxwell equations. If you don't recognize them, you probably don't recognize them. We've been writing them in such elegant notations that you…

…means that there's charges and currents inside of the material and these charges and currents must be put into the complete Maxwell equations to find out what the fields are. We're going to solve Maxwell's equations this time in which the currents and…

…equations in the vacuum to find the waves. However, I would like to make an historical note because the original form in which Maxwell wrote his equations and in which, in fact, the equations were written for 20 years or more, and are still written by…

…but we'll just call it the other charges, charges not polarization charges. In the same manner, the currents And the second Maxwell and one of the other Maxwell equations could be separated into a polarization current and other currents. The other two…

…charges, charges not polarization charges. In the same manner, the currents And the second Maxwell and one of the other Maxwell equations could be separated into a polarization current and other currents. The other two equations of Maxwell have not changed.…

…one of the other Maxwell equations could be separated into a polarization current and other currents. The other two equations of Maxwell have not changed. They're the same. Now, because the polarization charge density is the divergence of P, I can put this…

…some interest, I would like to discuss this equation from the point of view of relativity, because although we've put our Maxwell equations in relativistic form, It would be interesting to see what these equations look like in relativistic form, so that the…

…the classical theory of electromagnetism is an unsatisfactory theory all by itself. There are difficulties associated with the ideas of Maxwell theory which are not solved or not directly associated with the quantum mechanics. You say, but maybe it's no use…

…grand opportunity to understand something that was never understood before that came from the blue. This time it comes from Maxwell. And that is mass. You see here is a moment of proportional velocity due to electromagnetic influences. If we're conservative…

…like to discuss, though, how it might be possible to modify this. Many is the attempt to modify the electrodynamics theory of Maxwell so that the idea that the electron has a simple point charge could be maintained. And some of the theories even were…

…different possible things have been made to save the day. One way that was proposed by Born and Infeld is to change the Maxwell equations in the first place so that they're no longer linear in a complicated way, the result being that the total energy in…

…comes out finite instead of infinite, and the charge has a finite energy and momentum. But that changes the laws of Maxwell, predicts other phenomena which have never been observed, and suffers from another difficulty which I'll mention, which all of them…

…well that's impossible you can't just differentiate a lot of fields and then get a current oh yes you can if you use the Maxwell equation You see, the Maxwell equations tell us that the current is related to the field. So if we use that equation first…

…just differentiate a lot of fields and then get a current oh yes you can if you use the Maxwell equation You see, the Maxwell equations tell us that the current is related to the field. So if we use that equation first on this other side, since j…

…to t hanging out. So we almost made it, but not quite. After some mental thought, we look back at the differential equations of Maxwell, and we discover that the curl of e is fortunately db dt with a minus sign. So I'll flip it back again. See, from…

…Lecture S26, January 24, 1963, Lorentz Transformation of the Fields. Last time we began our discussion of the relativistic properties of the Maxwell field theory. We found out that the potentials phi and A together form a four vector, which we could call A…

…J mu, can be written this way. And there in one tiny space in the middle of the blackboard, there's all of the cuts, the Maxwell equations. Beautifully simple and so on. What did we do? What did we know? What did we learn from such a thing? that it's…

…other words that as a vector equation in the four space, means that it's invariant under certain transformations. Now the Maxwell equations are invariant under those transformations and therefore they can be written in a beautiful form. But before they were…

…we write only laws which are consistent with the principle of relativity. And the fact that in this particular notation the Maxwell equations are simple is not a miracle because the notation was invented with them in mind. But the interesting physical thing…

…velocity, which has been worked out. We did it two ways now. We, one, did it to the complicated way around from the Maxwell equations directly. I did that purposely to show that this is not, that this is a... It comes from Lorentz transformation, but…

…the potential so that the four vector nature of the A is obvious, is maintained. And then the equations of nature, the Maxwell equations are those. How does it come about that everything is so beautiful in the Lorentz transformation? You see, it was…

…have those nice properties would come out simple looking. So it's necessary. Anyway, here we are at the relativistic form of the Maxwell equations. Just one more point. And that I must write down because I said it implicitly. I would like to write it…

…It doesn't change anything. It's not a relativistic magnitude. No, no, no. Yeah, yeah. No, I should have emphasized the Maxwell equations were originally consistent with the principle of relativity. And they aren't changed when we go to the relativity theory.…

…of finding the new values. Because remember, if you may or may not remember, but just these we obtained from solving the Maxwell equation. So there's no more information here. than there was in the Maxwell equation. And I'd better emphasize that next time.…

…remember, but just these we obtained from solving the Maxwell equation. So there's no more information here. than there was in the Maxwell equation. And I'd better emphasize that next time. Yeah, that's why I did it before, in order to emphasize that point…

…No waves traveling down. So let's try getting some guess as to what the mathematical form is to see if it'll satisfy the Maxwell equations, which I remind you are equivalent inside the guide where there are no... where there are no currents in an empty…

…field goes like this. And how does it maintain itself? The moving electric field maintains the magnetic field by the equation of Maxwell. And how can the electric field stay here? Because the electric field here is 0. But this is possible because the…

…box. So the wavelength fits into the box. Well, I'm putting together two equations which are equivalent to the wave, to the full Maxwell equations, and therefore the wave equation. In fact, you can try to derive it from those two if you want. That's what…

…the following items that remain. First, on a theoretical level, we want to discuss the question of relativity and the Maxwell equations, what happens when you look at the Maxwell equations and moving coordinate systems. Also a question of the conservation…

…level, we want to discuss the question of relativity and the Maxwell equations, what happens when you look at the Maxwell equations and moving coordinate systems. Also a question of the conservation of energy and the electric and magnetic fields. Those…

…we needed for the theory of light. Now, it behooves us, it's necessary for us to connect these two things together. We have the Maxwell equations, which are right up here, and we have this stuff which we had before. So, you have a perfect right to ask…

…circumstance, we can find the potentials directly from these integrals and then differentiate and get the fields. So we've finished with the Maxwell theory. And this permits us to close the ring of our theory with light and so on. Because to get the…

…So for historical purposes, in order to appreciate where things have come from, I would like to show that the Maxwell equation contains the information about the Lorentz transformation. That's how Lorentz got it. And therefore, I would like to calculate…

…I would like to calculate the potentials of a charge moving at a uniform velocity once without relativity directly from the Maxwell equation. We get that then from the equations underneath, or therefore from this. Now I have pointed out how to make…

…discuss the relativity of the electrodynamics later. It was only the point to demonstrate that the Lorentz transformation comes out of the Maxwell equations and is not necessary to be put in. Thank you. That's the old thing that you knew a long time ago.…

…and so we're just repeating the subject. So now I begin the main lecture today. Two lectures ago, we were talking about the Maxwell equations when we had them in complete form, and I've written them at the top of the Blackboard. That's all you have to…

…instead of working with the potential, just to work more directly, let's try another way of looking at it. Let's start with the Maxwell equation, a region where the currents and charges are 0, in which case the equations are that the divergence of E is…

…Well, I claim to have already done it for B by arguing about the potentials. But let's just for fun look again at the Maxwell equations and now not restrict ourselves to any direction and see if we can find out what equation E satisfies. By taking the…

…that the entropy rate is a maximum, but the distribution of the velocities of each of the electrons. See, the Maxwell distribution is the distribution of equilibrium. But when there's a current present, the electrons have a different distribution. In fact,…

Well, what have we decided? Is this lecture S18 on December 3rd? Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd,…

…lecture S18 on December 3rd? Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. To begin…

…3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. To begin today's lecture, I would like to tag something…

…S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. Lecture S18, December 3rd, 1962, The Maxwell Equations. To begin today's lecture, I would like to tag something on the end of the last lecture that I…

…was before was an approximation or a special case under certain circumstances only, such as fixed current. Now, here are the Maxwell equations. I've written them down as equations. And for those who can't read equations yet, I've written them in words. The…

…has the vector and scalar potentials as the fundamental laws and the equations for those are the equations that replace the Maxwell equations because the E and the B slowly disappear as the physics becomes more modern and they're replaced by A and…

…argument by Ampere. But we're going to cut corners on history and simply say that we're going to say we know what the Maxwell equations are, which are the result of sublimation of a large number of experiments, and that they are here. The equations that…

…steady currents, currents which don't change with time. And nothing otherwise changing with time, we can take the other term in the Maxwell equation and erase it. So in the same way as we did with electrostatics for a while, we'll discuss the equations…

…complicated circumstances. Imagine. that the electric field is like elastic deflections of things that are stiff, then the equations of Maxwell may be understood, perhaps, as being the equations of an elastic material, the ether. This has turned out not to…

…charged as hard as you can, and you detect with as sensitive an instrument as you can, and you find out there's no effect. Maxwell did this and showed there was no effect to a part in 10,000, but If you want a more recent measurement, the more recent…

…Starting with this lecture, we begin our study of electromagnetic theory in detail. And as it all begins with the Maxwell equations, but the situations that are described there are fairly complicated, and we'll break them down first into simpler situations.…

…fields stay constant, then we need the case in which nothing depends upon the time. Under those circumstances, the terms in the Maxwell equations which depend on the derivatives of the fields don't change. This case is called a static case, and the Maxwell…

…the Maxwell equations which depend on the derivatives of the fields don't change. This case is called a static case, and the Maxwell equations become this pair. Now, you'll notice an interesting thing. I mean, this set, this set of four, that you break it…

…curl of E, we proved, therefore, that the curl of the electric field is zero. And that, of course, is the one of the Maxwell equations. Everything's all right. You say we proved it. Well, in a way, when did we prove it? The first moment that we proved…

…a feel as to what should happen under different circumstances in electromagnetic situations. It is not possible to solve the Maxwell equations in all the circumstances exactly. On the other hand, in presenting the subject of electricity and magnetism, none of…

…very mathematical mind are often led astray in studying physics because they don't study the physics. They say, look, the Maxwell equations are all there is to electrodynamics. It's emitted by the physicist. Nothing else is contained. So since the equations,…

…the abstract mathematical end of the theory of electricity and magnetism. The ultimate idea is to explain the meaning of the Maxwell equations. And in doing so, we find that we're written in a very peculiar notation with upside-down triangles and so on,…

…peculiar notation with upside-down triangles and so on, and I have to explain what that is. And so I disregard completely the Maxwell equations and start to discuss the mathematics of vector fields. This will be of very great importance, not only for the…

…equations and start to discuss the mathematics of vector fields. This will be of very great importance, not only for the Maxwell equations, but for all kinds of physical circumstances. You'll find the same things, the same writing, the same notation again…

…Now to one technical point about the other piece of this law, it's a small technical point that was discovered by Maxwell and of great importance. The law without that would be mathematically impossible because there are such things as open circuits. For…

…the fact that the laws of mechanics are reversible. Incidentally, for historical interest, I'd like to remark on a device invented by Maxwell, who first worked out the dynamical theory of gases. He says, suppose that we have a little hole here in the gas…

…the fast ones through that way. Well, pretty soon, the heat will develop in here and remove over here. And so Maxwell might argue first that the ideas of thermodynamics are impossible. You can't get thermal equilibrium fundamentally because you could have…

…direct words, but I have a summary by somebody else. A criticism, I mean, a remark by Clausius On the calculation of Maxwell, Maxwell was the first man to calculate the heat conductivity. And Clausius wrote a paper on it, admiring it greatly. And Maxwell…

…words, but I have a summary by somebody else. A criticism, I mean, a remark by Clausius On the calculation of Maxwell, Maxwell was the first man to calculate the heat conductivity. And Clausius wrote a paper on it, admiring it greatly. And Maxwell…

…Maxwell was the first man to calculate the heat conductivity. And Clausius wrote a paper on it, admiring it greatly. And Maxwell compared his formulas with the data of Rankine, who had measured the heat conductivity in some gases. And Clausius says this,…

…measure and have still to be multiplied by 0.4356, the ratio of the English pound to the kilogram. That's the criticism of Maxwell. The numbers have, furthermore, been calculated with one hour as the unit of time, whereas Maxwell has used them as if a…

…to the kilogram. That's the criticism of Maxwell. The numbers have, furthermore, been calculated with one hour as the unit of time, whereas Maxwell has used them as if a second had been the unit. So you see that even the great ones do exactly the same…

…ever known, ever discovered, is this one. It's the first. You see, first there was something known. There was something wrong. Maxwell knew there was something wrong, but the problem is what was right instead of the kt. Now, here is the quantitative…

…all together, still more, still more, still more. It's ridiculous. It's wrong. And in the first paper on the dynamical theory of gases by Maxwell, He had a summary at the end after he did all the wonderful things with it. And he showed how good it was.…

…if the particle corresponds to non-relativistic energies, the Dirac equation, if it's an electron at relativistic or other energies, The Maxwell equations, if it's a photon that you're describing, and so on. So I'm sorry I started too early, apparently. So…

…the subject of electromagnetic radiation. The most dramatic moment during the 19th century in the development of physics occurred to JC Maxwell one day in the 1870s. The dramatic moments of physics are those moments in which great syntheses take place. That…

…writing about Genesis to arrange for it a special creation. It was appreciated to be a mysterious and wonderful thing. Yet what Maxwell could say when he was finished with his discovery was, let there be electricity and magnetism, and there was light. What…

…when he was finished with his discovery was, let there be electricity and magnetism, and there was light. What Maxwell discovered was this. The gradually developed properties, the gradually discovered properties of electricity and magnetism, and electric forces…

…As a consequence of that, at sufficiently great distances, there's very little influence of one electrical system or charges on another. Maxwell noted that the equations or the laws that had been discovered up to his time when he tried to put them…

…because they do not satisfy the inverse square, but go down in only inversely as a distance, which is what Maxwell discovered. All these phenomena we summarize under the word radiation, or more specifically these days, electromagnetic radiation, there being…

…problems. Second, it turns out that, and this is more important, that the fundamental laws of physics are often linear. The Maxwell equations, for example, the laws of electricity are linear. The great laws of quantum mechanics turn out, as far as we…

…Maxwell's equations, which describes electricity, magnetism, and light in one uniform system. Now, the thing that's interesting is that the Maxwell equations did not seem to obey the principle of relativity. That is to say, if I took Maxwell's equations and…

…was wrong with the equations of physics. Now, the first thing to think of, of course, is that the new equations, the Maxwell equations of electrodynamics, since they're mostly new and they were only 20 years old, were obviously the ones that were wrong.…

…quite good. In the meantime, Lorentz noticed a very remarkable curiosity. which was that if he made the following substitutions in the Maxwell equation, x prime equals x minus ut over the square root of 1 minus u squared over c squared, y prime equal…

…of 1 minus u squared over c squared, which is different than this. If he made that substitution in the Maxwell equations, this transformation called the Lorentz transformation, then the Maxwell equation would remain the same in form. Now, what Einstein's,…

…than this. If he made that substitution in the Maxwell equations, this transformation called the Lorentz transformation, then the Maxwell equation would remain the same in form. Now, what Einstein's, or actually it was Poincare's idea originally, was to…

…they travel with velocities in a certain region. And I've plotted over here on this blackboard the curve which was developed by Maxwell on basis of common sense and the help of mathematics, the curve which gives the probability that I will find a…

…to talk about that but won't. So I'll just summarize by again two quotations. Probability is marvelous for quotations. James Clark Maxwell, who was the one who developed this curve and was one of the important theoretical physicists who developed the theory…

…and magnetism and the electromagnetic effects, which were finally worked out, the full equations for everything was worked out by Maxwell in 1873, is probably the most fundamental transformation of the most remarkable thing in history, the biggest change in…

…it turns out that as time went on, new laws were discovered after Newton, and those were the laws of electricity by Maxwell. And one of the consequences of the laws of electricity were that there should be waves, electromagnetic waves, light, in fact, is…

…the first but now it doesn't work so good. Now the next guy who did something another man who did something great was Maxwell who obtained the laws of electricity and magnetism. But what he did was this: he put together all the laws of electricity, due…