 Hi, I'm Zor. Welcome to InDesign Education. Today's lecture will be about the fourth Maxwell equation. There are only four, so this is the last one. And I do consider that you are comfortable with vector fields, scalar fields, concepts of gradient, diversion, curl, using the nabla symbol. So this is all in the past, and today I will definitely use these concepts as well. Okay, this lecture is part of the course called Physics for Teens, presented on Unisor.com. And I suggest you to watch the lecture from the website, because it's a course. So basically all lectures are logically related, they are in certain sequence, and it's very important to follow that sequence, otherwise you might not actually understand certain things. So if you found this lecture on YouTube by searching or something like this, make sure that you know that it's actually part of the course, and the course is presented on Unisor.com with menus, with problem solving, with exams, which you can take any number of times you want. The website is totally free, there are no advertisements, no strings attached. There is a prerequisite course called Math for Teens on the same website, which I do suggest you to familiarize yourself with. Definitely vector algebra and calculus are used very extensively in physics, including multivariant calculus. So it's all presented in Math for Teens course, and I do suggest you to familiarize yourself. Okay, now back to Maxwell's fourth equation. Now we all know, I mentioned it many times, that changing magnetic field is generating the electric field, that's the Faraday's law, changing electric field generates magnetic field, not necessarily changing just any electric field generates magnetic field. And that's for example how electromagnetic waves are propagating. That's how transformers are working from the primary wiring with electricity, with variable, with alternating current. It goes through the magnetic core and then to another wiring, maybe with different voltage, etc. So we all know that. Now, and it might seem that magnetic and electric field are totally symmetrical, which is not the case. And here is why. Even the direct constant electric current is capable to producing magnetic field around it. That's basically, you remember, if there is a wiring, if there is just a straight wire, if there is an electric current, even the constant, not necessarily alternating or variable, and there are magnetic field around it. So you remember this experiment, we put iron shavings on some kind of a plate perpendicular to the wiring, and if there is a current, excuse me, if there is a current in the wiring, then magnetic field generates forces between the shavings, and they are making some kind of a circle around it, right? So we know that. So we have something which is basically electric current, which is, as we know now, is a material kind of object. There are electrons which are physically moving inside the wire. There is no equivalent for the magnetic field of this concept of electron for electric field. Magnetic field is just the field. Electric can be either current with actual moving electric electrons, moving charges, moving electric charges, and electric field by itself as well exists as well, on the distance from, let's say, electric charge. So electric field is producing, is produced by electric charges, but it might actually be produced by changing magnetic field. So electric field is different from magnetic. So there are basically two different sources for generating magnetic field. One is actually physically moving electric charges, and another changing electric field. Now the non-changing, non-variable, constant electric field does not generate magnetic field. So there was an Ampère's law which basically stated how magnetic field is generated by electric, and it was related to Ampère's experiments with direct current. At the same time, there were certain ideas about generating magnetic field from changing electric field which is not covered by plain Ampère's law, and that was the Maxwell's contribution, and that's why the law is officially called Ampère Maxwell law, which basically tells about how magnetic field is generated by both real electric current with moving charges, which is Ampère's contribution, and changing electric field by itself, which is Maxwell's contribution. Because whenever we are talking about propagation of electromagnetic fields, well, initially there are some moving electrons which generate magnetic field, and moving electrons usually are not moving in a constant flow, but also magnetic field is generated as variable changing, and that it generates electric, electric generates magnetic, and they go chain connected to each other like a chain, and that's they propagate in the vacuum, and we see the light from the stars. In the vacuum there is no electrons, so there is nothing but electric field producing magnetic, electric producing magnetic, so that goes by itself, so that's why we have two different sources to generate magnetic field. One is moving collective charges, another is variable electric field, and we will talk about these two pieces separately, the Ampère's piece and the Maxwell's piece, and together they will be combined because we have a superposition of fields principle, so one source of energy generates certain magnetic field, and another generates certain magnetic field, and the field superpose, and you can just basically add the vectors which describe the intensity of the field of the one source and from another source. Okay, so let's start with this. Let's consider that we have a wire and we have electric current in it. Now, this was actually addressed in one of the previous lectures, devoted to electromagnetism, and in the notes for this lecture, by the way, and on the website every lecture has a note, I provide a direct name of the lecture which describes whatever I'm talking right now. So, basically, around the wire with I as an electric current, let's consider this is a very long wire. So, the magnetic field has circular character, the magnetic field lines are circles, and if this is, let's say, the direction of the current, then the tangential to this circle line would be vector of magnetic field intensity, and the relation is that magnetic field intensity equals, I'm talking about absolute value, mu i divided by 2i r where r is radius. So, the further we are, obviously, the weaker will be the magnetic field. Now, we actually talked about this. We did not derive it exactly. It's partially experimentally driven, partially from another law when we have a very short wire, and then you can have, based on the angle, you can have the expression that that's a different law, again, half experimental, half theoretical. But this is something which has been covered in that lecture, explained in details, and I'll just take it as granted. Okay, now what is mu? It is a property, a magnetic property of the media around the wire. It's called permeability, magnetic permeability. Now, from this, we have mu i equals to b times 2 pi r, right? Now, this is the force, basically, right? And this is the length of a circle. If you remember a couple of lectures before, there is a lecture about circulation of the vector field. So, b is basically a vector field for every point in space. There is a vector, and this is actually a circulation of this constant vector with a constant magnitude directed at a tangential to a circle times the length of the circle. So, this is basically a circulation of this vector. Now, Amper's contribution was that he is saying that not only in this very simple case we have that mu times the current equals to the circulation, it's actually a universal law. So, whenever you have a distributed current, which is basically it's electrons, consider electrons which are just moving in all the different directions. Well, for simplicity, let's consider electrons are moving only in one direction. Let's say along the z-axis. And on the x-y-axis, we are trying to find out what exactly. So, this is my x-y plane. So, these are electrons flying. Maybe it's a very thick wire. So, we are talking about the thickness of the wire. So, electrons are flying. So, there is certain things which is called density of electrons. So, if you take some kind of area, there is a density of electrons here. And if area is small, you can assume that the density is the same in this area. So, now what actually follows from this that we can describe the circulation around this particular area because there is always a vector of magnetic field intensity which is perpendicular and if this is a magnetic line, consider it's a circle that would be easier. Then it would be this way. But in any case, it would be some kind of an area and the emperor's law was the circulation around this area is equal to total amount which is going through this area. Total amount of current, of electricity. Well, times magnetic field with magnetic properties of the field. So, he basically said that this is a general kind of a law whenever you have certain amount of electricity going through certain circle or whatever on the plane. In this case, if this is a very thick wire, for instance, so across the wire, that's what he is talking about. Then we will have exactly the same kind of law. The next thing is, now if this is a circulation and this is the total amount of electricity going through this particular area, the thickness of the wire, let's say in this case, right? If we will divide it by the area of this area, what we will have? Well, here we will have mu times something which is called electric flow density, electric flux density, whatever you have. It's a density of electricity per unit of area. What do we have on the right? Back to my lecture about circulation. If you remember, circulation divided by area is the curl of this field. So, we have come to this equation just dividing both sides of this initial law. That amount of electricity coming through a surface times magnetic field permeability is equal to circulation divided by area. We used the definition of curl. Obviously, there are certain issues about whether this limit exists or not. Whenever this area is squeezed to a point, you have this as a function of x, y. So, we can put it this way. When this area is very, very small around point x, y on the x, y plane, then you have basically a point-based equation because density exists, obviously, at every point. That's the limit of amount of electricity going through electric flux, if you wish, divided by area when the area is squeezed to a point. And this is the definition of curl. So, both exist. I mean, under certain smooth assumptions about the field itself, about electricity, this actually works mathematically. Limits exist. And we just call these limits this way. And basically, that's it. Right now, I will just express it. Now, as you know, curl can be expressed as nabla. Vector product with vector b at x, y is equal to mu j of x, y. That's different. Curl can be expressed as nabla vector product with vector field. Again, I did talk about this before when we talked about curl. So, that's what it is. Now, there is one more little transformation which might actually be helpful. I would like to express this electric flux density with electric field intensity. How can I do it? Well, that's actually quite straightforward, I would say. Let's think about this way. What is electric field intensity if we are talking about electric field within a wire? Well, if you remember, that's the force which is acting on the unit of electric charge. Okay, so if I will multiply by the length this charge is moving, I will have work. Right? Force times length. Okay. Now, what is the difference in potential between two points? Well, the difference in potential between two points is work again to move a unit charge from one point to another. So, basically what I can say is that e times l, let's say we are talking about some kind of material through which electrons are moving. So, e times l, l is the length of this piece would be equal to u, which is difference in potential between here and here. At the same time, u is equal to i times r, where r is the resistance and i is the current. So, this is basically Ohm's law, right? Now, what is r? What is the resistance of this piece? Well, as we know, resistance is proportional to length. The longer this piece of wire is, the more resistant it is. But it's inversely proportional to the area. The wider the wire, the less resistance it has, right? Okay. Times some kind of a resistivity of the material. Unit area, unit length has certain resistance. So, that's what it is, right? So, from this, you will see that this can be replaced with j, right? Basically, current is amount of electricity going per unit of time divided by a. That would be the density of the flow, right? So, that's our j. So, j times l times sigma. So, from this and this, you see the j. Wait a moment. That's not a sigma. That's rho. Sorry, it's a wrong letter. That's resistivity. So, from here, you will see that j is equal to e divided by rho, right? Now, sometimes the unit 1 over rho is called conductivity. So, this is rho is resistivity. 1 over rho is conductivity. So, that would be sigma times e. Now, if you know this, then you can say that this is equal to mu, oops, mu times sigma times e of xy. And this is basically the final form of the first half of the Maxwell's equation describing how electric field is acting on the magnetic field. It's not the full equation. That's half of the equation. This is the contribution of direct electric current, which goes through some point xy. I will make even one more slight modification. I will put time as well here, because if this is variable, then obviously magnetic field will be correspondingly variable. At any moment of time, this is valid. So, if this is changing with the time, these are constants, obviously, then this will be changing. So, the curl of magnetic field generated by direct electricity is field, and the field again can be multiple source. This is field generated by electric current with direct moving of electric charges of electrons, basically. Now, let's talk about a different component, a second component which is supposed to go to fourth Maxwell's equation. The component when electric field just by itself changing electric field without any moving charges, without any electrons present, somewhere in the vacuum, let's say. We know that if we somehow produce magnetic field, which is changing in time, it will produce electric field. That's the Faraday's law, even in vacuum. So, and then moving, changing electric field would in turn produce magnetic field without this component. So, we need a second component to this. Okay, now to do this, I will consider capacitor. Capacitor is a perfect example of electric field between the plates without moving electrons in between. So, if you have some kind of a source of variable electricity, let's say alternating current, then as we know from the theory of capacitors, which again were presented in this course in one of the previous lecture, I have a link in the notes for this lecture, as we know the electric current would circulate. Why? For a very simple reason. Let's say first we generate plus and minus. Minus is where electrons are actually going, right? So, electrons will be moving here and this generator will basically take electrons from the left and put it to the right. Something like this, or from the top to the bottom in this case. And electrons will go this way. Now, the absence of electrons will be on this side and on this side as well. So, this will be charge plus, this will be charge minus. So, not only this direction, not only electrons will move to this direction using this generator, also they will actually attract to each other. So, that's why when we change from plus to minus, it's not immediately go, because this plus will still hold it for a while until this particular potential will grow to a certain extent. So, that's why we have the current shifted its phase by pi over 2 over u. It's always kind of behind a little bit. And again, all the calculations were presented there. So, that's not what's important right now. What is important is the following. That amount of electricity which is accumulated on the plates of this capacitor is proportional to difference of potential on this particular generator and between these plates as well. The difference in potential between these plates is proportional to amount of electricity accumulated on this place, which is reasonable. And this is the coefficient of proportionality which depends, in turn, it depends on area of the capacitors. It's inversely proportional to D. And then you have some kind of electric permeability of the space between permeability. There are two terms, permittivity for magnetic and permeability for electric field. I always mix them. But anyway, it's usually called epsilon. Epsilon is permeability of the space between the plates. A is area of the plates. D is distance. So, the wider they are, the more electricity they can hold, obviously. And the closer to each other they are, the more they attract to each other. So, the more accumulated electricity will be there. Okay, that's great. So, we know this formula. Now, and that was very, very important. Something which is basically a good guess by Maxwell. You see, we know what circulation of the electrons in the wiring is. Now, but if you will change it from plus to minus from minus to plus, this will circulate. Now, no electrons will go from plate to plate here, directly. Everything will be going through this way or that way. So, the circuit would actually operate. It would work. However, it looks like there is no movement of electrons here. And that seems to be kind of strange. What James Maxwell suggested was, let's just conditionally think that there is some kind of a movement of electricity between these two plates as well. It was actually easy for him to consider because it was before electrons were discovered. They didn't really know the real nature of the electricity related to electrons. So, for them, even electricity in the wiring was also kind of the same thing. They didn't know what's the carrier. They were considering there is something like ether. It's called ether, which is just everywhere. And electricity is basically some kind of waves of the ether as a media. Like, for instance, sound is waves of oscillations of air molecules. So, they were considering the ether to be the most important carrier of electricity. So, there is a ether here and there is a ether there. So, he just suggested, okay, let's just close this loop and consider that we have some kind of current within this loop. It's called displacement current, which is basically equal to the current in the circuit itself, right? So, if we know this, and this is equal to C times, not C times, it's equal to E times B, right? So, the difference in potential, that's work of moving one unit of electricity between the plates. This is the force for a unit of electricity between the plates. Times the distance between the plates, that's also work, right? So, that's why we did exactly the same in the previous part. So, the difference in potential is equal to intensity of electric field times distance between the points we are measuring the potential, right? So, that we know on one hand. On another hand, if I know the amount of electricity accumulated between the plates, what is the displacement current as I will put in index G here? What is displacement current as Max will basically suggest it? Well, we know that current is amount of electricity per unit of time. So, it's derivative basically. That's what it is, right? If we will differentiate left and right, we will have, on the left we will have this. On the right we will have this, D and D E of T for DT, right? Okay, which is equal to, instead of C we will substitute this. D will go away, right? So, it's A E A epsilon divided by D multiplied by D times D E of T for DT. From here we go straight to density because if I will divide by A, I will have I D by T divided by A, that's density. Also, this displacement density of the density of the flow of electricity because electricity is going here across the old plates, right? So, I will divide it by area, I will have density. So, density is equal to epsilon times D E of T over DT. Now, what if I am talking about electric field in general? Well, this is a general equation, not necessarily between these two plates of the capacitor, but in any case, whenever I have an electric field, this is something which is completely independent of anything outside because I can actually reduce this to a point. If I have a very, very small, a tiny capacitor around any point where electric field is located, I can just use the same logic basically, consider that the field is created by electricity, by some kind of a tiny capacitor, and all my logical steps would be exactly the same. This and this are characteristics of a point because I can actually put J of T, X and Y is equal to epsilon times, I will use now partial derivative only by time. So, for any changing electric field, and I need changing, right? Because this is the derivative by time, I will have this particular type of the law. Now, if I have this particular type of electric density at point, which is equal to this, now, from the previous part of this lecture in the beginning, we actually derived the formula that nabla multiplied, vector multiplied by B is equal to mu J, mu J, yes, f xy, and this is also at xy. This was the first part of this lecture when my J was related to real current with electrons moving. Now, again, what Maxwell suggested is that whenever there are no electrons but there is still a movement of some kind of electric field is changing, we can talk about this displacement current, which basically plays exactly the same role as in this case, which means that the same type of equation, nabla times B of xy, should exist with JD of xy, with displacement current in between here. So, if we equate displacement current which exists this, and which is expressed by this density of electricity at every point, if we equate the properties of this displacement current and the real electric current, material, I mean, this is a field basically concept, and this is a real material concept in our general understanding of what material actually is. Then, if we equate these properties, then we should really say that this particular electric current, the displacement electric current should also create the magnetic field of the same type as this one. So, these two equations kind of exist simultaneously. The direct electric current generates this part of the field, magnetic field. Changing electric field generates another piece, and the sum of them is supposed to be the real generated field, because we have a superposition of different kind of fields. So, the real equation would be a sum of these two, nabla times B of xy is equal to j, j of xy plus mu JD of xy. So, this is piece from direct moving charges, and this is piece from changing electric field not related to those charges. And we can obviously express it in terms of electric field intensity as nabla B. Okay, the first part I have already done, right? So, the second part would be plus. So, instead of E, I will put, instead of j, I would put sigma E. So, it would be mu sigma, no, epsilon, sorry, epsilon, mu epsilon, mu epsilon, dE of txy, where dt. So, this is a fourth Maxwell's equation, which is a result of contribution of direct current, maybe changing in time, but it's still a real material movement of charges, and changing electric field, which might be independent of these charges. Now, that's why it's important, it's changing because it's derivative by time. If it's not, then this part, if there is no changing electric field, this part doesn't exist. It's zero in this case, if E is not dependent on t. Well, basically that's it. Okay, one more thing, last one. There are different units of measurements of fields, electric field, magnetic field. That's just kind of historical kind of development. I will put a couple of more things. h is B divided by mu, d is epsilon times E. Well, obviously j is sigma times E. So, using these, we can have different form of this fourth Maxwell's equation, but I don't want to basically spend time on this. My purpose was more towards understanding that the fourth Maxwell's equation represents generating of magnetic field by both moving electric charges and changing electric field as well. And it's this part, the second component of this sum, is actually what makes electromagnetic waves propagate in vacuum where there are no electrons, basically. So, that's why both pieces are very, very important. Because of this part, we see the light, which is electromagnetic field, oscillations. That's it. I suggest you to read the notes for this lecture. So, you go to the same website I mentioned, Physics 14 is the course. Part of this course is called Waves, and then there is a topic called Field Waves. And in this topic, I have all these Maxwell's equations and the properties of vector fields, etc. So, I suggest you to read, and there are some exams actually. Do whatever you want. Thank you. Thank you very much, and good luck.