 Hi, I'm Zor. Welcome to Unisor Education. It will be a lot of different things today. First of all, this lecture is part of the course. It's called Physics for Teens. It's presented on Unisor.com. I suggest you to watch this lecture from the website because it has notes immediately next to the video. There are some notes, and it's very detailed notes like a textbook. And sometimes I can miss something during my lecture, but notes are much more complete usually. Or some calculations I might just skip here not to waste any time, but I'm putting all the details into the notes. So that's number one. Number two, the website is totally free. There are no advertisements. It is not really necessary unless you do some functionality related to supervising something for somebody, etc. Now, there are other courses on the same website. First of all, there is a prerequisite mathematics course called Math for Teens. Certain things like calculus or vector algebra are mandatory to know, to study physics. So I suggest you to familiarize yourself with all these concepts in mathematics from whatever source you want, including Unisor.com. Okay, now what we are talking today about is electromagnetic energy continuity. And basically as a kind of continuation, it's a conservation law, but a little bit more strong conservation law than something which we used to think about conservation laws. Okay, so first of all, I would like to refer you to another lecture within the same topic. So the topic is actually within the Physics 14 there is a part called Waves, and in this part there is a topic called electromagnetic field waves. Now in that topic, I have a lot of different things including Maxwell equations, which everything is prerequisite basically to whatever the lecture I'm doing. This lecture is one of the last ones within the topics. But I start from certain vector algebra concepts, and one of them is diversion. So there is a gradient, there is a diversion, there is a curl. All these are presented in the corresponding lectures of the same topic. Now I'm talking about electrical diversion. And I spent a lot of time in that lecture basically using example of airflow what is basically a diversion of vector field. So first of all we are talking about vector field in the space, which means at every point there is a vector defined. So the vector is a function of a point and sometimes of time as well. And the wind actually gives you a perfect example. At every point there is a direction and the magnitude of the wind in that point in space. And obviously it's changing with time as well. So we have at any point in space a vector which depends on coordinates of that point and time. So if you have such a vector which depends on position and time, it's basically a combination of three components. Every vector in three-dimensional space has three components. And we can call it Vx of x, y, z, t, Vy that's a y component of the same coordinates and Vz of the same components. So this vector has three components. Okay, now the divergence of this particular vector is basically defined. It's a definition. It's g per dx Vx plus d of Vy per dy plus d of Vz per dz. Now each of them is a function of this, all three parameters. So this is called the version and it's usually symbolized as this. This is nubla vector which is not really a vector. It's just a triplet, d by dz. It's a triplet. But if it's a triplet, you can basically deal with this as a vector. And this is a scalar dot product of two vectors which means the first component of this multiplied by the first component of that but in this case it's not really a multiplication. It's application of this operator of differentiation to the Vx which is this plus this applied to this and plus this applied to this and that's what we will get. So this is the version. Okay, we don't need it anymore. This is just the definition, just a repetition basically. And you're supposed to know about this and again from the same lecture on diversions I have come up with an equation which is nubla times V it's equal to minus d by dt of rho of x, y, z, t. Now, what is rho? Rho is a density of some subject in our particular example in the divergence lecture rho was a density of the air. So if you have some material, some substance which basically is distributed in space with this characterizes basically what kind of quantities of this material are coming into the point. And this is, as a result, a density of that point. But it's actually not a density, it's a rate of change of density. Rho is a density and the derivative by time is the rate of change. So if you have certain amount of substance coming in a certain amount coming out so the difference between them constitutes the net change in the density of that material which we're talking about and we're talking about air over there. Now, so right now we can generalize this example about air to basically any substance. It can be water, it can be oil in the pump, in the pipe. By the way, oil in the pipe is the simplest example because obviously if you have a pipe and you have a constant flow of oil it's definitely whatever comes in to a certain area comes out and the density remains constant. Now, whenever it comes in and comes out which means that this thing is equal to zero. This is a net between coming in and going out and if it's zero this derivative is zero and if derivative is zero then the Rho itself is constant. So this is just a trivial example. With air wind it's much more complicated example. Now, another example which I would like to talk about is electric charge. Now, electric charge is also something which doesn't disappear in the thin air and doesn't move somewhere else with infinite speed which means it just disappears from here and appears there. Everything goes smoothly. Now, let's just talk about electric charge a little bit. So let's consider that in some area of space you have certain distribution of electric charge. So let's again call it Rho of X, Y, Z, T. So somewhere there are certain amount of electric charges. Okay, great. And they are continuously distributed. We are talking about continuous distribution. Now, as such, we basically can talk about what if there is some kind of electric current. Now, what is electric current? Electric current is amount of electric charge which is moving through certain place with certain speed. So basically it's a derivative of charge, rate of change of charge per unit of time. That's what electric current is. Now, if it goes along certain conductor, the conductor has certain area and we can talk about density of electric charge. So if you will divide by the area of the conductor, that would be density of the electric charge. So basically, let's call it J. So we are talking about movement of electric charge in time. And then if it moves along certain conductor, which has certain cross-section area, we are talking about per unit of area. So J is electric current density, which is amount of electricity coming through this infinitesimal area during the infinitesimal time period. That's what J is. Now, how can we just generalize? Is there any difference between this, let's say, and the airflow? Well, basically there is no principal difference. It's exactly the same process. So there is a term for this in physics. It's called flux. So flux is basically a flow of something. Maybe it's a flow of air, a flow of oil in the pipe, or a flow of electricity, electric charge. So it's a flow. And whenever we are talking about something like this, it's the flow density. So flow density is amount of something, in this case it's charge or air or oil or whatever. So it's amount of something which is flowing through the unit of area during the unit of time. So that's what it is. So in this case this is a flux density. And it's exactly the same logic as I was using in the divergence lecture about the air to derive the formula about this. We have exactly the same thing. So the formula is nabla times j is equal to minus d rho by dt. Exactly the same equality. This is actually called continuity equation. Well, continuity, why is it continuity? Well obviously because right now we are talking about something which is coming in and going out. So this is basically a net result of the coming in and going out. And this is the resulting rate of change of the density. The greater the difference between coming in and going out, the greater change of the density. And if this is by definition, whenever we are getting more out than in, the density is changing and that's why there is a minus sign here. So it's exactly the same continuity equation with electric charge which basically is a little bit stronger law of conservation than the traditional law of conservation. Because traditionally we are talking about if we have some kind of closed system, then electric charge or mass or something doesn't really disappear from it. However, this doesn't really exclude the situation when something like electric charge or energy or momentum, whatever law of conservation we are talking about, whenever something instantaneously disappears from one place the system and appears in another. No, that is not possible because the law of continuity actually tells that it's not really momentarily movement from one place to another. It's a flow. So there is a flux, there is a flow and there is the density of this flux and everything goes through this basic equation. Okay, so this is electric charge. Now we were talking about electromagnetic energy which is basically the purpose of this lecture. So I would like to talk about energy, electromagnetic energy in exactly the same terms as I was just talking about charge for example and come up with the same equation. Okay, so first of all, do we have a flux when we are talking about the electromagnetic field? Well yes, because let's say you have a light. You know if you direct the light onto something, that's something, if light is really intense that something might actually be heated up by light if it's a strong light. But if it's not so strong light it's still heated up but not to a noticeable extent. And there is a photo effect whenever you bombard with light some metal plate it can actually kick out the electrons from the metal plate, right? That's photo effect, we were talking about it before. So there are really different observations which basically come up with very simple result. Light carries energy. Electromagnetic waves carry energy. Oh, by the way, radio, whenever electromagnetic waves, radio waves are heating some kind of antenna there is movement of electrons in it, right? We can obviously amplify them but no matter what it is it's still moving. If they are not moving we will not be able to amplify anything. So electromagnetic field carries energy. So there is a flux and there is an energy density. There is all this type of things. Now speaking about energy density we already came up with the formula. Again that's not in this particular topic of the course. It's the previous one where the topic is called energy waves and within that topic we were talking about electromagnetic field and its electric component and its magnetic component and we came up with the formula which I'm going just to use as is. If you just don't remember I refer you to this previous lecture. So we have a formula which basically tells that electric and magnetic energy which is energy density is equal to 1 half epsilon p square plus 1 over mu b square. Now e is electric component intensity of the field b is magnetic component intensity of the field epsilon is permittivity of the medium where the whole thing is going on if it's vacuum it's epsilon zero and mu is magnetic permeability of the space where the whole thing is going on. So e and b are very important characteristics they are actually defining electromagnetic field and again whatever we were talking before about Maxwell's equation etc. e and b were involved. e and b are basically defining the field and I would like to express the flux and flux density of electromagnetic field in terms of e and b so I know there is some flow of energy but what is it? Well obviously it depends on the flow of the electromagnetic field characteristics now these are characteristics of magnetic field functions of x, y, z and t the same as this one and the same as energy so that's why if I will be able to express in terms of these two the flow of energy more importantly flux density of the energy flow then that would be basically the answer to my question about what exactly this energy continuity is. Now we are talking about energy continuity and yes we will come up with a very similar formula of continuity as with electric charge or with air. Alright, so let's do it. Now, so here is the plan. Now if I know the energy density I know basically the left part of this equation so let me just repeat the equation. So this is equation of continuity. Now in case of a charge this was electric current density and that was charge rate of change of the density of charge rate of change. Now I would like to actually come up with something similar. Now I know this one because rho is a density of the substance we were talking about and I know the energy density so if I will differentiate it by time I will basically have the right side. Now if I will be able to express the right side of this equation which is derivative of this thing by time in terms of divergence of some vector that vector would be the flux density, right? Okay, so let's do this. So first I will differentiate this part by time and then I will try to convert it into something purely mathematically into divergence of something. Okay, so let's try to do it. So first we have to differentiate it by time. Now this is not exactly the good way to do it because these are actually vectors. Now when we're talking about E squared B squared considering E and B are vectors well E squared is actually E vector scalar product with E vector, right? That's what it is. So now we can deal with real vectors. Same thing with B. So if I will differentiate that same what I will have well first of all we'll have one half then I will have epsilon. Now derivative of this, derivative of product is the first component times derivative of the second plus the second component times derivative of the first. Now they're both the same so it would be 2E times GE, this is scalar product, right? Plus 1 over mu 2B times dB by dG. Okay, obviously 2 will cancel out and what I will have is... So I will have epsilon E times GE by dG plus, now this is scalar product of 2 vectors plus 1 over mu B times dB by dG. Okay, great. Now we will use the... Now this doesn't look like the divergence of a fantasy, right? Remember divergence is basically d by dx of the first component plus d by dy of the second component plus d by dz of the third component of the vector. Now this doesn't look like. However, let's just be more inventive. Now these definitely kind of disturb my... the way to convert this into divergence of some vector but I do have Maxwell equations. Okay, now what do we do now? We know the Faraday law which is the third Maxwell equation in my course. So this is delta... vector product of E is equal to minus dB by dG. So this thing gives me this. Now there is an ampere Maxwell law which is the fourth Maxwell equation which says nabla vector product with B is equal to epsilon mu dE by dt. So both equations are Maxwell equations which are presented in previous lectures within the same topic. So I have the whole lecture dedicated to this and the whole lecture dedicated to this. So again, I'm just using these formulas. Faraday's law and Maxwell law. Now this is Maxwell, this is actually in vacuum and there is no other sources of electricity. So electromagnetic field just goes by itself. There is no electric current which feeds it with something else, etc. Let's just talk about very, very simple case electromagnetic field in vacuum. So that's the equation. From this equation I have a very simple expression for both derivatives which I will substitute here. So what it will be? So instead of dB I will put minus nabla E. So that would be minus 1 over mu B times nabla E. Okay, Faraday is no longer needed and for E I will use this. So I will have epsilon E times 1 over epsilon mu nabla vector product B. Right? So epsilon mu on this side I will transfer to this. That's why it's 1 over mu nabla vector product B. Okay, this goes out. And what's the result? The result is 1 over mu. 1 over mu and 1 over mu. And what do we have here? We have epsilon E, E scalar product nabla vector product B minus B scalar product nabla E. Now these are all vectors. Now again in the same topic I have a couple of vector identity. Now these are, this is just pure mass, just vector algebra. And from the vector algebra I will just use this formula. The proof of which again in the topic which is dedicated to different vector identities where I prove it actually. So I'll use it as it is. And it's equal to nabla times B. Again, just take my word that this is correct for any kind of vector expression. This is scalar product, this is vector product, this is scalar vector. So this is identity which is proven and I have proved it in the lecture which is dedicated to different vector identities in the same topic of the course. So you go to, again, Physics 14, waves, electromagnetic field waves and among them there are many different lectures. This is one of the lectures about identities of the vector algebra. And now, what do I need? Well, first of all, this is a dp, actually we can use rho now. Rho is energy density, d rho to dt. Now we need it with an opposite sign, right? Remember we have this continuity equation with a minus. So if I would like to invert the sign of it I will have that now. By the way, if you change the order of vector product it changes the sign. So 1 over nu nabla e times d vector product is equal to minus d rho of x, y, z, t by dt which is basically a continuity equation if we say that the vector s which is e vector product v well, actually with a mu. This is a vector, right? e is a vector, b is a vector and we were talking about expressing in terms of this vector, these two vectors. So this is some vector which we can actually treat as the flux density vector and using this I can say nabla s well, obviously it's x, y, z, t equals minus d rho by dt also x, y. And this is a continuity equation of the electromagnetic field. And by the way, this vector e times b, e vector product e divided by mu is called pointing vector. Pointing is a physicist and mathematician pointing. And it basically tells this continuity equation which exists for electromagnetic field energy. Energy is conserved, not only just conserved it's also continuously changing. Well, basically that's the most important part of this which I wanted you to come up with from this lecture that energy of electromagnetic field adheres to equation of continuity and the vector s which is pointing vector characterizes the magnitude and direction of an energy flow of electromagnetic field and what is this direction? Well, since it's e vector product b it's perpendicular to both. Remember, the vector product is perpendicular to both components and these are two perpendicular to themselves among themselves. So we basically have three vectors e, b and s they're all mutually perpendicular to each other and that's what happens. That's why we have transversal oscillation of electric field, magnetic field and the propagation goes perpendicular to both of them. That's it. Do read the notes for this lecture. Other than that, that's it. Thank you very much and good luck.