Hello everybody. Welcome to Electrodynamics and Its Applications, this will be the 17th lecture. My name is Professor Seungbum Hong, and to my right side, I have my teaching assistant Melodie Glasser. So, today, we will cover the complete set of Maxwell's equations. Up to now, we have come here inch by inch, having a lot of constraints and assumptions, but now we will come back to the complete set for Maxwell equations. We have the complete and correct story for electromagnetic fields that may be changing with time in any way. So, as you can see in this table below, the complete equations are written here. So, Melodie, this was a long journey, right? Right. We're almost to the end of our full specialization lecture stairs, and this will be marked as a big victory for both of us and for our students who are listening to this lecture. So, the first equation, if you remember, is the divergence of electric field is equal to the charge density divided by the permittivity in vacuum. In other words, you can interpret that as the flux of the electric field through any closed surface is equal to the charge enclosed by the surface divided by the permittivity. The second equation would be the curl of electric field is equal to the time derivative of magnetic field, and we put a minus sign in front of it. So, the meaning would be the line integral of electric field through a closed surface is equal to the time change of flux of B field through the loop. The third is there is no magnetic monopole that is written here like divergence of B field is always zero, so the flux of B through any closed surface should always be zero. So, if there's any in-flux, there should be the same and equal amount of out-flux. The fourth equation is one of the equation that Maxwell really had a significant contribution by adding a new term to what was known that time, and this is the curl of magnetic field is equal to two terms. One is the current through the loop, current density at the point of interest divided by the permittivity in vacuum plus the change of electric field at the point of interest, and that could be interpreted as integral B on around a loop is equal to the current through the loop divided by Epsilon naught, plus the time change of flux of electric field through the loop. Okay. So, let me ask my teaching assistant among those four, what is your most favorite equation. I like Gauss's Law, which is the first one. Here is the first one we looked at, and so that looks okay. May I ask the reason. I just like it because we've used it the longest and I think maybe it's less complicated to look at in doing calculations. Exactly. In fact many of semiconductor engineer or device designers use this first law which will be changed into the Poisson's equations in one dimension, and then they will work on this charge density, which is not only coming from the electrons in the metal or holes in the metal, but also ions or defects that are included, then this equation also becomes a little bit complicated. Okay. So, now we're at the point to discuss the equations for classical physics. So, we know there are three conservations law, one of which is conservation of charge, and that could be nicely written by this differential equation in vector calculus, where you see on the left side, the divergence of current density is equal to the minus of the change of the charge density as a function of time, and this could be written as the flux of j through a closed surface is equal to the amount of charge that is inside that surface which change as a function of time and that time derivative, the negative times derivative of this charge, will be equal to that. So, this would be a very important law. We also learned about force law, where we know, if you know the position and velocity of the charge, and the electric field and magnetic field at the point of interest, then we can know the force imposed on the charge, and the law of motion states that that force should be equal to the change of momentum as a function of time, that will be the time derivative of the momentum where the momentum is defined by mv over square root one minus v squared over c squared, which is the Newton's law with Einstein's modification for the mass, and also we know the law of gravitation which resembles the Coulomb's law in electrostatics. So, let us focus on Maxwell's equations more, and we will discuss first with the Melodie's favorite equation number one and equation number three. So, let's take a look at equation number one again, which is the Delta E is equal to rho divided by Epsilon naught, and this first equation shows that the divergence of E is the charge density over Epsilon naught, and this equation is always true, not only true for statics but also dynamics and true in general. In both dynamic and static fields, Gauss' law is always valid. So, let me ask my teaching assistant Melodie, again, what Gauss' Law was. Isn't it just as it's written, that the flux of E through a closed surface is equal to the charge inside divided by the Epsilon naught. Yeah, exactly. So, that's the Gauss' law, so the name of this law is Gauss' law. So, any flux of field out of a closed surface is equal to the charge inside. So, that would be the summary of the first equation. The third equation is corresponding equation for magnetic field or magnetic charge, and as you can see on the right side is zero, so there is no magnetic charge, meaning, since there is no magnetic charge, the flux of B through any closed surface should always be zero. So, let me then see, if you remember, that if you have no divergence, if any vector field has zero divergence, do you remember how we can change this in other forms? I think you can also say that there's something else that's equal to zero. Yeah. So, you can change this to a curl of another vector field. So, any curl of a vector field has zero divergence, therefore you will see this magnetic field can be expressed in terms of magnetic potential, the curl of magnetic potential which is del cross A. Okay. Now, we will jump to Maxwell's equations two and four, which are relatively more complicated when you compare with equations number one and three. So, the second equation tells us that the curl of electric field is equal to the minus of rho B over rho t, which is interpreted as the line integral of E through a closed surface is equal to the minus of the time derivative of flux of B through the loop. So, in fact, the second equation that the curl of E is minus rho B over rho t is called Faraday's Law, and it's generally true. So, if you imagine you have a ring, metallic ring, and you're approaching it with a magnet, and you change the distance between them, then you are changing the magnetic flux through the ring, and then you will have alternating current around the ring, and that cannot be explained by gradient of electrostatic potential, because if you have gradient, is like a stepwise, and you cannot say which one is higher than the other one, because it's closed loop. So, that's only explained by the time derivative of magnetic potential, which we will cover later. Now, the fourth equation is c square del cross B, which is equal to j over Epsilon naught plus rho E over rho t, and the last equation has something new, which is the second term, and until Maxwell's work, the equation for the magnetic field of steady currents was known only as del cross B is equal to j over Epsilon naught c squared. So, if you have a current, you know there is a circulation of magnetic field around it, but people didn't know, even without current, if you have displacement of electric field, then you will also have the same effect, which is the case for capacitor. So, Maxwell began by considering these known laws and expressing them as differential equations, and notice that there was something strange about del cross B equals j over Epsilon naught c squared. So, in the history of science, Melodie, as you can see, it is important to explore and do experiments and gather data, but it's also equally important to survey the data, investigate that data which was already published, and extract some useful knowledge or find a missing term, like Maxwell did. So, let's focus on Maxwell's equation four. So, if one takes the divergence of delta cross B is equal to J over epsilon naught C squared, the left-hand side will be zero, because as we just discussed, the curl of any vector field has zero divergence, which means the divergence of the current density should also be zero. Now, if the divergence of current J, density is zero, then the total flux of current out of any closed surface is also zero. Now, the flux of current from a closed surface from the conservation of charge is equal to the decrease of the charge inside a surface, which certainly cannot in general be zero. Sometimes, there are cases where you are taking out charge from one place and moving it to the other. In such an instance, the diversion of current density is non-zero. So, if you have to impose that this should be zero, you are excluding many cases in natural phenomena. So, the equation above has been our definition of our J, which expresses the very fundamental law that electric charge is conserved, any flow of charge must come from some supply. Because of this inconsistency, you can now also see from Maxwell perspective that this equation should be modified. So, Maxwell appreciated this difficulty and proposed that it could be avoided by adding the term round E over round t to the right-hand side of delta cross B is equal to J over original naught C squared. Now, we know that we should add the term round E over round t. Next, we'll discuss his ideas in terms of a model where the vacuum was like an elastic solid. So Melodie, at that time, they thought vacuum is an elastic solid, but now we know this is not true. So sometimes, the results or output that we derive from our experiments might be wrong. But nonetheless, some of the equations or some of the findings may stay true, which was the case with Maxwell. So, do you have such an experience in your research career? Yeah. I think that happens a lot. When we're making something, we have a whole bunch of different options for why the problem might happen, and sometimes, actually, fixing the problem let's us know what happened wrong in the first place. Exactly. So, research is a never ending process of correcting what was being published or being known in the past, but nonetheless, it is being accumulated as a function of time, so it will be time-tested knowledge. Now, we will see that he explained the meaning of his new equation in terms of mechanical model, which means he used cogs and wheels to understand that, that was the fashion that time. But now with the advanced knowledge and technology, we know that we don't need to use those tools that Maxwell used before. There was much reluctance to accept his theory. But first, because of the model and second because there was at first no experimental justification. So, sometimes, in the history of science, you will see an old idea can be revisited with the new technology and be verified, and in that way, we can advance the knowledge as well. Today, we understand better that what counts are the equations themselves, and not the model used to get them. So, we may only question whether equations are true or false, and untold numbers of experiments after Maxwell have confirmed his equations. So, this is also beauty of mathematics and equations. Equations are so abstract and they are universal. So, we can use it for a longer time and test our equations through future experiments. So, Maxwell brought together all of the laws of electricity and magnetism and made one complete and beautiful theory, which we are looking now in the table. So, let's go further with the extra term, round E over round t. Let us show that the extra term is just what is required to straighten out the difficulty Maxwell discovered. So, this will be another perspective, not only from the perspective of the conservation of charge, which we just discussed, but also from the other inconsistency when we add some components in the circuitry. So, taking the divergence of the equation, we must have the divergence of the right-hand side is zero, as you can see. So, you can see the divergence of the current density plus divergence of the change of electric fields should be zero. In the second term, the order of the derivatives with respect to coordinates and time can be reversed. So, we can reverse space and time, the sequence of the derivative. So, the equation can be rewritten as follows: the delta j plus epsilon naught round, round T of delta E, and delta E is, again, Melodie's favorite equation, which is the Gauss law. This is where you can see the charge density where epsilon naught can be replaced by this one. If that's the case, again, we get this beautiful equation, which we just told you is the conservation of charge. So conversely, if we accept Maxwell equation and this new term, we must conclude that charge is always conserved. Now, what happens if a charge is suddenly created? So, let's ask Melodie. What happens if the conservation of charge is suddenly not observed? Yes. We need some new equation to describe that. Exactly. So, we need a new equations than the equations that we have dealt with. So, no answer can be given because our equation say it doesn't happen. So, it doesn't give you anything about beginning or ending of the charge. So, if it were to happen, we would need new laws, but we cannot say what they would be for the moment. We haven't had a chance to observe how a world without a charge conservation behaves. According to our equations, if you suddenly place a charge at some point, you have to carry it there from somewhere else. In which case, we can say what would happen, and that's where the second term do an important job. So, when we added a new term to the equation for the curl of electric field, we found that a whole new class of phenomenon was described. We shall see that Maxwell's little addition to the equation for delta cross B also has far reaching consequences. So Melodie, can you guess one of the far-reaching consequences of adding this term in the force equation? Maybe it helps with nuclear science where there are charges being created. That's one thing. The other thing is the electromagnetic waves; that is the ingredient of our smartphones; to enable us to talk to each other through smartphones or even with the videos and Internet. Good. So, let's take a look at the new terms in other's perspective, and Melodie just mentioned about nuclear science. So we're going to use an example from nuclear science. So, we consider what happens with a spherically symmetric radial distribution of current. So, how do we create that? Suppose we imagine a little sphere with radioactive material on it, which is squirting out some charged particles. So, let's assume there's a radioactive materials where you are having charge coming out of this sphere in radially symmetric way. In this case, we would have a current that is everywhere radially outward. So Melodie, what does it mean radially outward? So, we talked about that before, radially means symmetric. It's at the same and every single distance from the center. Exactly. So, it doesn't matter which angle it is going out from. It only matters the distance from the origin, or the center of the sphere. So, we will assume that it has the same magnitude in all directions if you have the same distance. Okay. So, let the total charge inside any radius r are be Q of r. If the radial current density at the same radius is J of r, then Q decreases at the rate of round Q of r over round T is equal to minus four pi r squared j of r. This is because J of r is current density, which is current per unit area, and the area of the sphere on the surface is four pi r squared, and they are uniform. Therefore, this will be the equation converted from the charge conservation. So now, we ask about the magnetic field produced by the currents in this situation. So, suppose we draw some loop gamma on a sphere of radius here. You can see here, next to Melodie. Then, we might expect to find a magnetic field circulating in the direction shown. But here, we can do some thought experiment. So, you remember if you have a current through an open surface, then the rotation of magnetic field will be only the magnetic field around the outer loop, which is the boundary, because inner loop will always cancel out, because you will have a component that is in opposite direction. Now, if you expand that idea, and you increase the area of the open surface, and then you certainly remove the boundary, then what happens? Do you have any space to have circulation of B? No. No. So now, with this thought experiment, you can already guess that for this special situation, you may not have any circulation of B, but let's see if that is really the case. So, how can B have any particular direction on the sphere? That is the question. A different choice of gamma would allow us to conclude that it's direction exactly opposite to that shown here. So, how can there be any circulation B around the currents? Now, we are saved by Maxwell's equations in addition to intuitive inference. So, the circulation of B depends not only on the tool current through gamma, but also on the rate of change with time the electric flux through it. So, it must be that these two parts just cancel. So then, we have consistent idea about the magnetic circulation. So, let's take a look at the electric field at radius r. That should be Q of r over 4 Pi Epsilon_naught r squared, from the Gauss's law, which is again, Melodie's favorite equation, so long as the charge is symmetrically distributed as we assume. Now, it's radial and its rate of change is then round E over round t should be just round this one or round t, which will be 1 over 4 Pi Epsilon_naught r squared times round Q over round t. Now, let's compare that with the conservation law that we just discussed. Round Q over round t is equal to minus 4 Pi r square j of r. Then if we replace this one here, then you will see that it ends up being round E over round t is equal to minus j over Epsilon_naught, and if we put it into our fourth equation, you see it is equal to 0. So now you can see the curl of magnetic field of this radially squirting charge, the curl is always 0. It's like the magnetic static's case. So, let's take a look at another example. A capacitor, parallel-plate capacitor and see the magnetic field of a wire used to charge this capacitor. So, we consider the magnetic field on the wire used to charge a pair of plate condenser. Condenser, capacitor , they can be interchangeable words. So, if the charge Q on the plates is slowly changing with time, the current in the wires is equal to dQ over dt. We would expect that this current will produce a magnetic field that encircles the wire. You can see this is the 3D perspective view. And this is the cross-sectional view. You can see you're supplying charge to the capacitor. As you can see, because it's disconnected with an insulator, they cannot go through this unless it's a tunneling device. So, if it is now being prevented to move, then you will see there will be a limitation to provide the charge to the capacitor, and that's when the current will cease to flow, but until then, you will have a change in current to charge the capacitor. Suppose we take a loop Gamma_1, which is a circle with radius r. You see this fictitious circle. Sometimes we use the circle around our head to show when we're dizzy or confused, right? The line integral of the magnetic field should be equal to the current divided by Epsilon_naught C square. We have learned that. So, let's take a look at this again. So, you will see on the left side, the curl of magnetic field, and if you do the aerial integral is equal to the line into a B field around a closed loop that encircles the surface that we did the surface integral. So, what's the name of this law? Do you remember? This is Stokes' theorem. Is Stokes' theorem. Yes, this is Stoke' theorem. As you can see, if I do the area integral of the current density, that will be the current, and therefore, we know it's 2 Pi rB because we have a symmetry here. So, with the same distance, you will have the same magnetic fields that's why it's 2 Pi rB, which is the circumference of the circle times the B field, and is equal to I over Epsilon_naught C square. So, from the previous slide, we just derive this equation in the red box. So, this is what we will get for a steady state current. But it is also correct with Maxwell's addition because if we consider the plane surface, S, inside the circle, there are no electric fields on it. Nothing is changing and, therefore, the surface integral of round E or round t will be 0. Now, suppose that we slowly move the curve Gamma downward. We get always the same result until we draw even with the place of the condenser where the current I goes to 0, which is the center part of our capacitor. So, magically, does the magnetic field disappear? I don't think it should, so we'll see. Yeah, we'll see. So in physics or science, most of the phenomena, they are continuous. We can think of phase transition where you have discontinuity in ideal case, but still in that case also, the phase transitions goes continuously. So, let's see what Maxwell's equations says for the curve Gamma 2, which is the red curve here, which is a circular radius r whose plane passes between the condenser plates. The line integral of B around Gamma_2 is 2 Pi rB again because of the symmetry, which must equal the time derivative of the flux of E through the plane circular surface S2, because as you know, there is no current through it. So, C square 2 Pi rB will be equal d over dt of Q over Epsilon_naught, and that's from again, Melodie's favorite Gauss's law. You will see that indeed, the dQ over dt is I, and then Epsilon_naught is a constant, so it comes out and you will have the same results. Then you will wonder, look, there were no direct flow of current were charged species through insulator, how can I have current? That is called displacement current, and you will learn that for capacitive charging or discharging. So, integrating or changing electric field gives the same magnetic field as integrating over the current in the wire. So, now our theory is complete. It is easy to see that this must always be so by applying our same arguments to the two surfaces, S1 as S1 prime that are bounded by the same circle Gamma_1. This discussion was already done in the first lecture if you remember. Through S1, there is a current I but no electric flux, whereas through S1 prime, there is no current but an electric flux changing at the rate of I over Epsilon_naught. So, this is how you can design your thought experiment to put separate terms in action and then see how you can be consistent in explaining the same phenomenon. Now, from our discussion so far of Maxwell's new term, you may have the impression that it doesn't add much, that it just fixes up the equations to agree with what we already expected. It is true that if we just consider the force equation by itself, nothing particularly new comes out. The words "by itself" are however all important. So, let's revisit this equation again. So, we have C squared Del plus B is equal to j over Epsilon_naught plus round E over round t. Maxwell's small change in this equation when combined with other equations does indeed produce much that is new and important. So, let's make a summary here for the classical physics. So, at the beginning of the lecture, we showed that all that was known of fundamental classical physics is the physics that was known by 1905. The year 1905, what do you think has happened then Melodie? Do you have any idea this year? Historically, I'm not entirely sure besides what you just said. So, 1905 was the year when Albert Einstein wrote a very important paper entitled "On the Electrodynamics of Moving Charge," where he published the special relativity, where he understood the speed of light is constant no matter what frame of reference you're observing from. Due to that fact, he was able to show that space and time should change depending on the frame of reference. That was already predicted from the Maxwell's equations that the speed of light should be constant no matter which frame of reference you're observing the phenomenon from. With these equations, we can understand the complete realm of classical physics. First, we have the Maxwell's equations, as we just discussed, written in both the expanded form and the short mathematical form. So, the conservation of charge in this box you see Del.j is equal to minus round j over round t, and this is the moment we have the complete Maxwell equations. We can deduce from the conservation of charge as well that we just did before. The force law is that having all the electric and magnetic fields doesn't tell us anything until we know what they do to the charge. So, the destiny of charge that we are interested in is where they are from, where they are now, and where they are going to. So knowing E and B, however, we can find the force on an object with a charge Q moving with velocity v, so we can tell deterministically the destiny of the charge. So, we revisit the law of motion where the momentum change as a function of time is equal to the force, where the momentum is defined by mv over square root of 1 minus v squared over C squared. So, finally, having the force doesn't tell us anything until we know that what happens when a force pushes on something. We need the law of motion which is that the force is equal to the rate of change of the momentum. This also includes the law of inertia as also in the human life, like when we have a habit, we don't change that habit until there's external or internal force to change it. I was told that dream, love, and pain are the three forces that are very efficient in changing one's habit, and in the case of the charge, I think that would be magnetic field and electric field. Finally, we will have a law for gravitation which we'll not deal with in too much detail, but if we really want to be complete, we should add one more law, that is Newton's law of gravitation where you can see this is very similar to Coulomb's law in electrostatics.
