Now, in one small table that we show we have all the fundamental laws of classical physics. We have climbed that great peak, and we're on top of the K-2, right? So [inaudible] , which mountain do you like most in the world? In the world, I think there is a really good mountain range in Colorado. Oh great, Colorado, right. Rocky Mountain for example, but here we're showing your K-2. We are nearly ready for Mount Everest which is the highest mountain in the world, which is quantum mechanics. So, we have focused on learning to understand the equations. And now that we have the whole thing put together, we're going to study what the equations mean, the meaning of equations, what new things they say that we haven't already seen. So, we will now move on to traveling field. One of the impact that I just mentioned previously about the new term is the existence, showing the existence of electromagnetic waves which is the cornerstone of the current modern telecommunication technology. So, the new consequences come from putting together all of Maxwell's equations. Let's see what would happen in a circumstance which we picked to be particularly simple. So, this is an example created by Richard Feynman, he was such a genius that he could make this kind of example to elucidate how the terms that we learned can create a traveling field, so we will see that. By assuming that all the quantities vary only in one coordinate in our case it will be x axis. So, this will be a one-dimensional problem even though you may think this is a three-dimensional problem, right? So, in order to make this one-dimensional problem, we're going to use the infinity for other two directions. We have a sheet of charge located on yz plane as you can see it's in the dashed pattern. The sheet is first at rest, which means that it's not moving, then instantaneously given a velocity u in the y-direction. All of a sudden, this dashed part will move upward. So, we have suddenly a surface current J, where surface current J is defined by the current per unit width in the z-direction, which is in this direction, and we will see what will happen. Okay? To keep the problem simple, we suppose that there is also stationary sheet of charge of opposite sign. So, if this first sheet of charge plus then there's minus sheet of charge superposed on the yz plane so that there are no electrostatic effects. Like in the current in the wire, we will have the same numbers of plus and minus charge. So, we will not have any leak field out of this, right? If we imagine that the sheet extends to infinity or as well in plus minus y and plus minus z directions, in other words, we have a situation where there is no current and then suddenly there's a uniform sheet of current with no electrostaticstatic effect. Now, what will happen? So, let me ask [inaudible] first, what's she would think would happen if all of a sudden sheet of charge starts to move. Well I think there will be corresponding magnetic and electric fields. Okay. That's a very short and brief but the correct answer. Now we will see how magnetic field will set in and then accordingly how electric field will set in as well in sequence. So, when there is a sheet of current in the plus y direction, there is some magnetic field generated which will be in the minus z-direction for positive x and the plus z-direction for negative x. As you can see here, the blue line here for the negative x, red line here for positive x. You can think this is the superposition of a lot of wires that is creating a circulation of B field, and the component that just survives is the one that is drawn here. So, we could find the magnitude of B by using the fact that the line integral of the magnetic field, will be magnetic field equal to the current over epsilon naught c-squared. So, we would get that B is equal to J over two epsilon naught c-squared. Since the current I in a strip of width w, let's say this is w, is Jw, because I told you J is the surface current where the current density is defined in terms of the current per unit width. So, you should multiply the width to get the full current, that will be Jw. The line integral of B as you can see if you do the lining club both of them, they have the same magnitude with opposite direction, but because you are doing a circular integral that will be 2Bw. So, if you equate them, you will be you will be able to see that B is equal to J over epsilon naught c-squared. This gives us the field next to the sheet for small x, but since we're imagining an infinite sheet, we would expect the same argument to give the magnetic field further out for larger values of x. So, you remember when we discussed the electric field of sheet with infinite dimension in two dimension with plus and minus charge, that the electric field will never die even though you are going further out. You will expect the same argument here, right? So, you will expect at instantaneously you will set this B field to infinite range. However, that would mean that the moment we turn on the current, the magnetic field is suddenly changed as we see, and it will produce tremendous electrical effects. So, we cannot just make that infinite, right? So, because we moved the sheet of charge, we make the change in magnetic field, and therefore, electric field must be generated. So there is another effect that we have to consider, right? If there are electric fields generated, they had to start from 0 and change to something else. There will be some [inaudible] t that will make a contribution as well. So, now this is like chicken and egg kind of discussion, right? Together with the current J to the production of the magnetic field. Through the various equations there is a big intermixing and we have to try to solve all the fields at once. That's a challenging task. So, by looking at the Maxwell's equations alone is not easy to see directly how to get the solution. So, here we will first show you what the answer is. We will change the approach instead of going step one to the final answer, we're going to give you the final answer and then we will go back to this step one. That's kind of a reverse engineering of our solution to this specific problem. Now, the answer is the following, the field B which we just calculated J over epsilon naught C squared that we computed, is generated right next to the current sheet which is good for small x. But the field B out farther for larger x, is at first zero. It stays zero for a while and then suddenly turns on. So, now we have a region where we will maintain our B field and region beyond that, where we will have no magnetic field at all. So in short, we turn on the current and the magnetic field immediately next to it turns on to a constant value B. Then the turning on of a B spreads out from the source region. So, now we have spreading out of magnetic field. So, that's like spreading out of a good news or spreading out of any influence from the source. After a certain time, there is a uniform magnetic field everywhere out to some value x, and then zero beyond. Because of the symmetry, it spreads in both the plus and minus x directions, okay? So, that reminds me of the origin of marathon. The soldier who wanted to bring the news of the victory to his own country. He ran almost more than 40 kilometers, 42 point something, right? Do you happen to know the lengths of the marathon? Oh no, sorry. I guess among our students there should be those who do run marathon and know by heart what the length is, but it's around 42 kilometers if I'm correct. So, that reminds me of that. So this will be where this guy, the soldier is moving, but here, we're moving at the speed of light. So, the E-field does the same thing before t equals zero. When we turn on the current the field is zero everywhere, that we know, that's the initial condition. Then after the time t both E and B are uniform out to the distance x equal vt and zero beyond. The fields make their way forward like a tidal wave with a front moving at a uniform velocity which turns out to be c which is the speed of light, but for a while we will just call it v which is the general term for velocity. The region between x equals plus minus vt, we just draw a plus vt, but imagine you also have minus vt. This is the region filled with the fields, but they haven't yet reached beyond. We emphasize that we are assuming that the current sheet and therefore the fields E and B extend infinitely far both y and z directions. So, maybe that's why we can see the light of a star that are so far away from us because the light emitting from the star conveys the information of the star when it was emitting it to us at the speed of light. To analyze quantitatively what is happening we want to look at two cross sectional views, a top view, so our eyes are on top of this box and that's looking down along the y-axis and the side view where our eyes is located here, looking back along the z-axis. So, we have the top view and side view. We will start with the side view. Now, suppose we start with the side view, okay? We see the charge sheet moving up. So here, we see charge sheet moving up here. The magnetic fields points into the page, here into the page and out of the page, into the page for positive x, out of the page for negative x. So, this is for them. The electric field is downward everywhere. You see the blue arrow here, downward everywhere to x equals plus minus vt. Now, you already see that magnetic field and electric field, they're perpendicular. Let's see if these fields are consistent with Maxwell's equations. You notice that one side of the rectangle gamma, two which is the red box here is in the region where there are fields, but one side is in the region the fields have still not reached. So, this is the boundary. You see, this is the boundary and boundaries moving here to the dashed line. So, here on the left side you have fields, on the right side you don't have any fields. So, that's like the boundary that we have in South Korea as well between South and North Korea. Now, there are some magnetic flux through the loop gamma two, if it is changing there should be an EMF around it. So, if the magnetic flux here changed you should develop an EMF, electromotive force. You see the magnetic field itself doesn't change, only the area where you have magnetic field changed. So, you're changing the magnetic flux by changing the area where we have the magnetic field. If the wave front is moving we will have a changing magnetic flux because the area where B exists is progressively increasing at the velocity v as you can see. So, the flux inside gamma two is B times a part of the air inside gamma two which has a magnetic field. Now, the rate of change of the flux since the magnitude of v is constant is, the magnitude times the rate of change of the area, as we just discussed. So, if the width of the rectangle gamma two is L, you can see the width is L, this one, the area where B exists changed by L times v delta t which is the area here in the time delta t. The rate of change then of flux is then BLv. So, according to Faraday's law this should equal the line integral of E around gamma two which is just EL, why? Because on the right side you don't have any electric field, on the left side you have electric field that is parallel to one side of the rectangle. So, the other side will not have any contribution. So, this one side have lengths of L, so that's why you will have EL. So, if BLv is equal to EL then we see E is equal to vB. This is very important. So, electric field is velocity times B. That is very interesting because if you think about the Lorentz force which says the force is equal to q times parenthesis E plus v cross B, then you already know that E should have a relationship with v times B. So, now you can intuitively know that this should be true. Now, we have our equation relating E and B, so now you see the fourth equation which is c squared del cross B is equal to j over H0 naught plus round E or round t, and we have seen that this equation will give us the value of B next to the current sheet as Melody just mentioned before. Now, we're going to see, also for any loop drawn outside the sheet but behind the wavefront, there is no curl of B nor any j or changing E, you see here, nothing. So, the equation below is correct here as well. You see this is zero, and this is zero. So, that's correct for this condition. Let's look at what happens for the curve Gamma one that intersects the wavefronts. So now, we're moving from side view to top view. So now, we're moving our eyes from here to here. Then, we are creating another loop, which is different from this one, so don't be confused. Here, you had magnetic field that is going in or out of the page, but here you have magnetic field that is going up or down on the page. Okay. So, here there are no currents as well. So, we don't have to worry about the current term, we just have to worry about the electric field terms. So then, if we change the form with the knowledge that there is no current, then, the line integral around the loop of Gamma one of B field will result in the area integral of electric field, which will result in the flux of electric field. In front of that, you will have the time derivative. So, the line integral of B as you see is just B times L, because here you see you have B field up, going up, and the contribution from the line integral will only come from the section that is parallel to this B field. The other part will be zero, and on the right side, you have zero B field. So, that will be BL. The area inside Gamma_1, where E is not zero is increasing at the rate of vL again, the same. So, therefore, you will see the rate of change of the flux of electric field through this surface area will be EvL. So, the right hand side of the equation is EvL or vLE. The left-hand side we know that is c squared B times L. Am I right? Right. So, we have a solution where you have a constant B and constant E behind and front posts at right angles to direction where the front is moving and at right angles to each other. Now, Maxwell's equations specify the ratio of E to B based on the discussions we had from the side view and top view. Now, is the time to solve these two equations. So, there is only one velocity, v, for which both of these equations can hold, that is, the velocity should be equal to speed of light. The wavefront must travel with the velocity, c, which is very fast as you know. We have an example where the electrical is influenced from a current propagates at a certain finite velocity, c. So, this would be a very good example to show that the electrical influence can propagates at the speed of light. Now, let's ask what happens if the motion of charge sheet suddenly stopped after it was on for short time, capital T. Now, we can see what will happen by the principle of superposition. So, we had a current that was zero and then were suddenly turned on. We know that case's solution. Now, we're going to add another set of fields. We take another charged sheet, and suddenly start moving in the opposite direction with the same speed, only at time, capital T, after starting the first current. In order to do that, you probably have to make another two sheets to remove the electrical effects, but alternatively, what can we do? Melodie. Instead of adding. Adding two more sheets to this problem. We had two sheets, which was minus charge and plus charge and plus charge, suddenly moved up. But we want to superpose this with minus charge and plus charge to suddenly it goes down. Yes. So, they should be canceling each other out, right? So, we don't need to add another few sheets. Yeah. So, in order to do that, let me try to draw it. So, let's see you have minus charges here and plus charges here, and this suddenly moves up and in order to make, so this is when it moves at T equals zero. But then, you want to make it stop. So, in order to make that happen, you can add two more sheets where these sheets start to move in the opposite direction. So, you will have a delayed superposition. But there are too many sheet of charge here. So, what you can do instead of doing that, you can do like this, plus. So, this moves up at T equals zero and at T equals T. Instead of adding one more, the equivalent motion plus going down is equivalent motion of minus going up. So then, you will start following the suit of plus, then they will move together. In that way, you can create the superposition effect that we just discussed. Okay. So, we take another charge sheet and suddenly started moving in the opposite direction with the same speed only at time, capital T, after starting the first current. So, this is where we have two more sheets. But as I told you, you can do it with our current sheet. The total current of the two added together is first zero. Then, on for time, capital T, then off again because the two currents cancel. We have a square pulse of current. As you can see after time, T, you will have the same influence with negative sign. So, if we add them up, we will have this one left, which is the square pulse of current. So, you can understand that if a star is created and emitting light, and then it suddenly die, then, you will know that you will have a packet of that lifetime. So, if we receive that packet of a lifetime, we can predict the creation and destruction of a star, right? I think there's a certain percentage of stars that are already dead, but we can still see that for it, which is cool. Exactly. So, this is the soul of the star. So, even though it's no more there, it has left a fingerprint or some footprint in the universe. All right. So, the new negative current produces the same fields as the positive one, with all signs reversed and delayed in time by capital T. A wavefront again travels out at the velocity of c as we discussed. At the time, small t, it has reached a distance x equals plus or minus parenthesis small t minus capital T. So, we have two blocks of field matching out at speed c. Now, the combined fields are zero for x is larger than c times small t, they are constant between x equals c times parenthesis t minus capital T and x equals ct, and again zero for x is smaller than c times t minus capital T. In short, we have a little piece of field, a block of thickness, c times capital T, which has left the current sheet and it's traveling through space all by itself. The fields have taken off. They are propagating freely through space, no longer connected in any with the source. So, Melodie, if this is true, is there any case that this wave packet will disappear? Will it go forever? Theoretically, yes, right? No? Can you think of any case that some entity can absorb this, for example? No, sorry. Okay. So, think about photovoltaic cell. So, photovoltaic cell, the solar cell, is receiving the photons and changing it to electricity. When they change it to electricity, there is no more light. They were absorbed. So, the materials in the world like Earth, Sun, or planets, can absorb the light and stop the motion of the light propagation. That will be another topic of material science. So, how can this bundle of electric and magnetic field maintain itself? The answer is by the combined effects of the Faraday law, Del cross E is equal to minus round B over round t, and the new term of Maxwell, c squared Del cross B is equal to round E over round t. So, that reminds me of a leadership. Leaders by themselves cannot lead the group. They should have push and pull from the members that they are leading, right? The same is true for example of pigeons or other birds that are flying. They rotate the leadership position to make the travel safe and long enough that they can move from one place to more safer place. Now, suppose the magnetic field were to disappear. Right. Think about that. There would be a changing magnetic field which would produce an electric field. If this electric field tries to go away, the changing electric field would create a magnetic field back again. So, this is like a virtuous cycle. Where if someone is needing help, the other can help him or her. If the other is needing help, vice-versa can happen. So by a perpetual interplay, by switching back and forth from one field to the other, they must go on forever. They maintain themselves in a kind of a dense when making the other, the second making the first propagating onward through space. So this is beautiful, isn't it? All right. So, let's talk about speed of light. We have a wave which leaves the material source and goes outward at velocity c, which is the speed of light. From the point of view of electricity and magnetism, we just start out with two constants, Epsilon naught and c squared. That appear in the equations of electrostatics and magnetostatics. As you can see, this is the melodie's favorite equation, where you have epsilon naught in the denominator, and Del cross B equals j over Epsilon naught c squared. This is for magnetostatics without the Maxwell's additional term. Again, you see Epsilon naught c squared, which is the constant appearing there. If we take any arbitrary definition of a unit charge, we can determine experimentally the constant Epsilon naught, say by measuring the force between two unit charges at rest using Coulomb's law. This can be also measured by atomic force microscopy. Because you can measure the force acting on the cantilever by using the position sensitive for the dial. You can define the charge by applying voltage, well-defined voltage between the tip and the sample, or the bottom electrode, and measure the force. If you measure the force, you will be able to get the Epsilon naught. So, this is one way you can experimentally prove or measure the permittivity of vacuum. So, we must also determine experimentally constant Epsilon naught c squared, which we can also do by measuring the force between two unit currents. So another thing you can do is to have current flow of your wire. Again, afm can be one way to do that. You have a narrow wire where you flow current I here, and you flow current here, which could be displacement current. Then again, measure the force between them. Then you can get the constant for Epsilon naught c squared. If you do that, then without really measuring the speed of the light, you will be able to get the speed of light. So that's amazing, isn't it right? So, the ratio of these two experimental constant will be c squared, just another electromagnetic constant. So notice that this constant c squared is the same no matter what we choose for our unit of charge. If we put twice as much charge, say twice as many proton charge in our unit of charge, Epsilon naught would need to be one-fourth as large. When we pass two of these unit currents through two wires, there will be twice as much charge per second in each wire, so the force between two wires is four times larger. The constant Epsilon naught c squared must be reduced by one-fourth. But the ratio between those two will be unchanged. So just by experiments with charges and currents, we find a number c square which turns out to be the square of the velocity of propagation of electromagnetic influences. From static measurements, this is important, from static measurement, we found c is equal to 3 times 10 to the 8 meters per second, which is exactly the number measured independently or separately using the light directly. When Maxwell first made his calculation with his equations, he said that the bundles of electric and magnetic fields should be propagated at this speed. He also remarked on the mysterious coincidence that this was the same as the speed of light. So he said, "We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of the electric and magnetic phenomena". So, Maxwell had made one of the great unifications of physics. Before his time, there was light and there was electricity and magnetism. The latter two had been unified by the experimental work of Faraday, Oersted and Ampere. You see they are all appearing the units in the electrical engineering. Then all of a sudden, light was no longer something else, but was only electricity and magnetism in this new form, little piece of electric and magnetic fields propagate through space on their own. Now also we know magnetism is just special relativistic phenomenon of electric effect. So, it could be combined within the frame of electricity. We have called your attention to some characteristic of this special solution, which turn out to be true for any electromagnetic wave. So, three things I want you to keep in mind. First, the B and E fields are transverse. Meaning, they are perpendicular to the direction of propagation. The two vectors E and B are perpendicular to each other. So, you can see three of them are orthogonal. And E is equal to cB. Now, we want to write Maxwell equations in a simpler form. So, we're going to go through a conversion of the four equations to two equations using electrostatic potential and magnetic vector potential. So you may consider that we are complicating them. But if you will be patient, they will suddenly come out simpler. Although by this time you are thoroughly used to each of the Maxwell equations. I guess Melodie is also used to the four equations we just displayed. There are many pieces that must be put together. Okay. So, we will begin with the third equation, Del dot B is equal to zero. The simplest of the equations. And we know if any vector field has no divergence, then that vector field can be expressed in terms of a curl of another vector field, right? So, we can say B is equal to Del cross A. And we have already solved one of Maxwell's equations. Incidentally, you appreciate that it remains true that another vector A prime would be just as good If A prime is equal to A plus Del Phi, where Phi is any scalar field because the curl of Del Phi is zero and B is still the same. So, this would be like constant when you do integration, where it doesn't matter if you do differentiation. All right. So, now we're going to take the next, the Faraday law, where the Del cross E is equal to minus round B over round t because it doesn't involve any currents or charges. If you write B as Del cross A and differentiate with respect to t, we can write Faraday's law in the form here where you can see Del cross E is equal to minus round over around t, Dell cross A. So, we just replace B by Dell cross A. Now since we can differentiate either with respect to time or to space first, so we can change the sequence of the time, the derivative, or differentiation. We can also write this equation as Del cross parenthesis E plus round A over round t is equal to zero, okay? Now Melodie? If some vector field has no curl. The curl of the vector field is zero. Then we don't have any circulation, right? Right. If we don't have any circulation, what is the characteristic? What can we do with this? Like for divergence free field, we could express that in terms of the curl of another field. Now for for curl free vector field, what can you do? Oh sorry, I forgot. Okay. So, if you also forgot as Melodie did, then think about this. If there is no circulation, you can't have gradient, okay? You can't have divergence. So, this could be a divergence of something else. This could be a gradient of something else. This could be a gradient over something else. So like in the mountain hiking, the gradient, if you have a gradient, you cannot circulate them, okay? As you can think of a stair. If you have a gradient there is no circulation, is one way, start and end. So likewise, if there is no circulation, you can express this in terms of a gradient of a scalar field. So, we see that E plus round A over round T is a vector whose curl is equal to zero. Therefore, that vector is the gradient of something. When we worked on electrostatics, we had the curl of electric field is zero; meaning there is no circulation of electric field. Then, we decided that E itself was the gradient of something and we took it to be the gradient of minus Phi. The minus for technical convenience because, the E field, the driving force is from high to low, right? So, we took it to the gradient of minus phi. We do the same thing for E plus round A over round T. We said E plus round A over round T is equal to minus Del.Phi. Now, if there is no change of magnetic potential as a function of time, then you see it will go to the old form. So, we use the same symbol Phi so that in the latter static case where nothing changes with time and the round A over round T term disappears, E will be our old Del Phi. So, we can use all the equations we learned under the constraint of electrostatic situation. So Faraday's equation can be put in the form E is equal to minus Del Phi minus round A over round T. So, now in addition to the gradient of the electrostatic potential, we have the time derivative of magnetic potential which will create electric fields. So, there are two sources to create electric field. So we have solved two of Maxwell's equations already and we have found that to describe the electromagnetic fields E and B we need for potential functions, a scalar potential Phi and a vector potential A. How many components do we have in vector potential A? Three. Yes, three. So, one plus three is four. Now that A determines part of E as well as B, what happens when we change A to A prime equals A plus Del Phi. In general, E would change if we didn't take some special precautions. We can still allow A to be changed in this way without affecting the fields E and B, that is to say without change the physics. If we always change A and Phi together by the rules written here. So A prime is equal to A plus Del Phi and Phi prime is equal to Phi plus Del Phi. Then, neither B nor E obtained from the equation above is changed. So previously, we chose to make Del dot A is equal to zero and we call that gauge transformation. This was to make the equations for magnetic vector potentials be in similar form with the Poisson's equations. Now, here we are going to make a different choice, but we'll wait a bit before saying what the choice is because, later it will be clear why the choice is made. Now, it returned to the two remaining Maxwell equations which will give us relations between the potentials and the sources Rho and j. So, once we can determine A and Phi from the currents and charge, we can always get E and B from the equations below. So, we will have another form of Maxwell's equations. So, again, magnetic field is Del cross A and electric field is minus Del Phi minus round A over round T, and now we are moving to the sources. So, we begin by the following substitution. Again, we return to marry this favorite, that that E is equal to Rho over epsilon naught and now we insert what we just derived; E is equal to minus Del Phi minus round A over round T. If you put it in, then we have this form. If we make this a little bit neater. It is; minus Laplacian of Phi minus round over round T Del dot A. Again, we can change the sequence of the differentiation with respect to space and time and we will have this one. So this is one equation that is merged. Our final equation will be the most complicated. We start by rewriting the fourth Maxwell equation by sending one of the term on the right side to the left and using, again, the two equations that we just mentioned before, B is equal to.Del cross A and E is equal to minus Del Phi minus round A over round T. If we put it in, then we will have this Del cross Del cross A minus round over round T parenthesis minus Del Phi minus round A over round T, equals j over epsilon naught. So this is the most complicated one. Now, the first term can be rewritten using the algebraic identity. We learned that Backup rule where you have A cross B cross, A cross B cross C, where you have Backup but then you can replace some of the factors to operators. This is what we learned in the earlier chapters. If we do that replacement, then we come up with this one where you have minus C squared Laplacian of A plus C squared Del of Del dot A plus round over round T Del Phi plus second derivative of A with respect to time is equal to j over epsilon naught. Now, that's very complicated. So, I'm just reading out the equations but for you guys you probably want to write it down and see what each term really means. So, fortunately, we can now make use our freedom to choose arbitrarily divergence of A. So, that's the case transformation. What we are going to do is to use our choice to fix things so that the equation for A and for Phi are separated but have the same form. So, we can do this by taking the Lorentz case. So Del dot A is equal to minus one over C squared times round Phi over round T. Choosing the Del dot A is called choosing a gauge, as I mentioned, and changing A by adding the Phi is called a gauge transformation. So, if I do this and put it in here, you can find it will change this into the minus from this, so these two will be cancelled out and you will have only these two terms on the left side which is Laplacian of A minus one over C squared times the second derivative of A with respect to T is equal to minus j over epsilon naught C squared. So, if you take a look at this, you have now a space derivative, time derivative and the source of those magnetic vector potential. So, in our equation for Phi takes in the same form because you have the Laplacian of Phi minus round over round T Del dot A is equal to rho over epsilon naught and we know the gauge. Now we choose Lorentz gauge, Del dot A is equal to minus one over C squared round Phi over round T. So, if we replace that here, now you have another similar form where you have space derivative, time derivative and the source of that this electrostatic potential which is the charge density. So, putting it side-by-side, we now have two equations instead of four and that's beautiful, simpler. Now, instead of having electric field and magnetic field, we have magnetic vector potential and electric scalar potential and we have charged density here and the current density here and we have epsilon naught and C square which are the ingredients for our Maxwell equations. So, there are beautiful because they are nicely separated. Now, we separate them out. There's no intermixing of magnetic field and electric field. With the charge density goes Phi, with the current goes A. So now, it's easier to solve them as well. So, when we unfold the equation, it has a nice symmetry in x, y, z and t. The one over C squared is necessary because time and space are different. Time and space, they are related by velocity. So, that you can see. They have different units, of course. So, for the first equation, if I do it in cartesian coordinates x, y, z, then you can spell them out like this. If you look at this form, they have the same kind of format. So Maxwell equation has led us to a new kind of equation for the potential Phi and A but to the same mathematical form for all four functions; phi, A sub x, A sub y and A sub z. So, Melody, do you think these equations are beautiful? Yes, I think they are very nice. Yes. I hope our students will appreciate the beauty of these equations as well. So, once we learn how to solve these equations, we can get B and E from the operation here. B is equal to Del cross A and E is equal to minus Del Phi minus round A over round T. So, we have another form of electromagnetic laws exactly equivalent to Maxwell's equations. In many situations, they are much simpler to handle. So the equation below is the corresponding wave equation for three dimensions. If you look at this, this is a form of wave equations. So, there are solutions in which there is some set of Phi and A which are changing in time but always moving out at the speed C. The field travels onward, through free space and with Maxwell's new term in the force equation. We have been able to write the field equations in terms of A and Phi in a form that is simple and that makes immediately apparent that there are electromagnetic wave. So, let there be light, right? Let there be light. With that, I'd like to thank you for your attention and see you in the next lecture. Bye-bye. Bye.