 Hi, I'm Zor. Welcome to Unisor Education. Today's lecture will be kind of based on many previous lectures. So I would like to use material which we have discussed before and come to some kind of agreement between different formulas, basically. Now, this lecture is part of the course called Physics for Teens. It's presented on Unisor.com. So if you go to Unisor.com, you have to choose this particular course. Now, within the course there is a part which is called waves. And within this part waves you have a chapter about electromagnetic waves. So in this chapter we have, I don't remember how many, like a dozen maybe lectures. This is one of the last lectures in that group of lectures related to the chapter about electromagnetic waves. We were discussing Maxwell equations. We were discussing certain vector operations in this particular chapter. And there are a few particular facts which I would like to bring to this particular lecture and check if they agree with each other. So that's the only thing which is the purpose of this lecture. So I will use material which we were discussing before and I will try to basically check if they agree with each other. Now, the website Unisor.com contains also a prerequisite course called Math for Teens. Now, mathematics is definitely a must for studying physics. In particular, obviously all the calculus is necessary, all these differentials, integrals, etc. Vector algebra is very much used in physics. So obviously I encourage you to, before you do anything with physics, to be relatively comfortable in mathematics. Okay, so let's get back to this particular thing. So I would like to remind a few formulas which we have derived before in one of the lectures of the same chapter of the course. Now, the first formula which I would like to remind you is a total energy density of electromagnetic field. Now, energy density means amount of energy in the electromagnetic field which is in a unit of volume, like one cubic meter, for example. So it's related to two vectors, vector E, which is electric component of electromagnetic waves. There is a V, which is magnetic component. This is basically intensity of the electric field and intensity of magnetic field. Now based on these two components of electromagnetic field, we have a formula. I use the letter P with index E plus M, which means electric plus magnetic, and this is energy density. Now obviously energy density is function of time and three coordinates. As well as these vectors, they are functions of time and three coordinates, which means at every point, at every moment in time, it has some value and the value is changing. It oscillates. So electric and magnetic intensities oscillate. Now, so the formula is, and we can just put it on the board, one half epsilon comes E square plus one over mu times D square. Now when I'm saying E square, I basically means E times E, a scalar product, same thing with B. So that's why I just use square, but in theory it's this one. Now, this is the formula for the energy density. Okay, we have derived it in one of the previous lectures. Now the second formula which I would like to remind you is formula which looks like this. So this is the cross product, a vector product of these two vectors, and S is called a pointing vector. Now we basically derived with this expression for pointing vector based on properties of electromagnetic field, Maxwell equations, etc. Again, it's one of the previous lectures. I think it's not the previous one before that I guess. Now in both cases I'm using absolute and middle, which let me just remind you, epsilon is electric permittivity of the medium where electromagnetic fields are propagating, and mu is magnetic permeability, I think. It's called permittivity and permeability. Difficult words. That's the magnetic permeability of the medium where the electromagnetic field is propagating. So it's just basically constants which depend on the environment. There is a special designation for vacuum. It's epsilon zero and mu zero. That's the permittivity and permeability of empty space. So now this is energy density. Now what is this? This is basically a electromagnetic field energy flux. So what is flux? Just to remind you, that's amount of something, whatever the something is, maybe energy, maybe number of molecules, maybe air or water. So it's amount of this something which goes through unit of area in unit of time. And unit of area should be perpendicular to the propagation of that something. So in this case we're talking about energy. So the pointing vector is the energy flux, electromagnetic field energy flux. So amount of energy which goes through a unit of area, in unit of time. So my purpose right now is somehow bring these two together and see if they correspond to each other. Now they obviously should correspond because both of those things are based on Maxwell equations and they have derived them using just plain mass basically. So they must agree with each other. But I would like to basically compare them in some very simple case. So what is my simple case? My simple case is, so let me just wipe out this one. I will put it here. Okay, so what's my simple case? My simple case is flat electromagnetic oscillations. And it's supposed to be one of chromatic. So you have an electromagnetic wave basically, a flat wave which goes along a straight line. And then I can actually compare the energy using this formula in certain amount of volume. And since I have a flux, I can see how much energy is flowing into this particular volume and compare the results. Okay, so first of all, let me just simplify this formula for vacuum. So we are talking about vacuum, so which means I have to put mu0 here and mu0 here. And we are talking about flat wave. What is flat wave? Well, flat wave is E is equal to E0. That's amplitude times sine or cosine whatever you prefer, omega t minus z over c. Okay, now why did I put it this way? Well, I consider that this flat wave has electric component and magnetic component. I assume that electric component is oscillating x, y and z. So electric magnetic is oscillating only along the z, sorry, along the x-axis. Magnetic component is oscillating perpendicularly to it along y. And then the flat wave goes along z-axis. So that's my flat wave. So it just goes this way, up and up. Magnetic, magnetic this way and electric oscillating this way. That's basically kind of very, very simple, the simplest probably way of electromagnetic field. So that's why I put z here. C is the speed of light. So z over c is the time it takes from here to some point z. So it's a time delay until wave comes from the beginning to the point with coordinate z. So in the origin of coordinate it's just plain sine of omega t. But if I'm going into point z, then I have to have this time delay. And same thing I can say about b. b would be exactly the same thing. Now they are always in sync with each other and in phase. So that's why this part is exactly the same. So whenever x goes to a maximum, y goes to the maximum. Electric and magnetic components are synchronized in the same phase. Okay, so these are two equations which describe my oscillations at any point on the z-axis. Okay, and it's a flat wave which means the whole area x, y actually is going in sync with one particular ray along the x and y is equal to zero. So that's what it means flat wave. Alright, now in this particular case let me remind you another formula which we have derived before. We were talking about amplitudes of electric and magnetic components and we came up with the formula b0 is equal to e0 divided by c. Now that was again one of the previous lectures and based on this I can replace this with e0 divided by c. Now using these expressions for electric and magnetic components I'll just simplify my equation for the power, I mean for energy density. One-half epsilon zero e square plus one over mu zero and instead of b square I will put, instead of b square I will put e square and I will divide it by c. Square, c square, right? So since my e and b are related this way b square is just equal to e square divided by c square. Okay, now let me just remind you another formula which was derived again. It's all in the lectures of the same chapter about electromagnetic waves. Speed of light, square field of light is equal to, that was a consequence from Maxwell equations. Okay, which means that one over c square is equal to epsilon mu. So mu and mu, so instead of this, instead of c square in the denominator I can put epsilon zero mu zero, mu is going down and the whole thing is equal to basically epsilon zero e square. Very simple. So this is the energy density of electromagnetic waves. In case we have this type of a flat electromagnetic field oscillations. Simple, right? It depends only basically on the electric component because magnetic component is very much related to electric component. So that's a simple thing. So I know the energy density. Okay, so let's assume that I have a space, a piece of space and electromagnetic waves are, light is propagating this way. It's in vacuum, so it's light, I'm talking about light which is electromagnetic waves, so it propagates this way. I want to know how much energy is flowing into this area during the time of one second considering this area is equal to one square meter. So why am I doing this? Well, this thing is the flux. What is the flux? It's amount of substance which goes through the unit of area which is one square meter during unit of time, one second. So if this is the light and the light is moving into this parallel, if you wish. So this side is one square meter. Now in one second considering the light is propagating with the speed C I will have obviously C meters, right? In one second. So this is one square meter, this is C meters. That's the length of this parallel pipette. So the volume is equal to C, length times the area. Okay, if the volume is C, if I will multiply it by density I will have amount of energy flowing into this parallel pipette during one second through the area of one square meter. Now I'm supposed to basically have the same thing if this pointing vector is expressing basically the same thing, a flux which means how much energy in this particular case is flowing perpendicularly to one square meter in one second. So let me just check if I will have this thing corresponding to this thing. Well, that's very easy to check. Since we are talking about two vectors which are propagating this, is oscillating along the x-axis and this oscillating along the y-axis perpendicular to each other, then I can say that absolute module if you wish of this particular vector is equal to one over mu square times E times B, right? Now B as we have already discussed is equal to E over C. So what's the result? Mu zero is equal to mu zero is equal to one over C square, am I right? Yes. So one over mu is equal to epsilon zero C square. Now C and C square goes out and what do I have? Epsilon C and E square. So I'm not surprised that they coincide. They're supposed to basically be equal to each other because these are things which we have derived from the same Maxwell equations. But nevertheless it's always nice to basically come to the same thing from two different roots and the same thing is supposed to be equal if the roots are correct. So for this particular case when you have flat waves, the simplest case, it actually gives you a nice kind of feeling that, okay, we are still in logically correct way of researching this particular problem. So basically that's all I wanted to present to you. So what's important is this expression for pointing vector which is basically the vector which gives you the flux of electromagnetic energy which means amount of energy which is flowing through the unit of area perpendicularly to it during the unit of time. That's basically what I have described as this parallel pipette. And it corresponds basically to my main formula about energy density of electromagnetic field which again we have derived before that differently. Okay, now I do recommend you to read notes for this and every other lecture on Unisor.com. And actually if you found this lecture somewhere else, not on the Unisor.com, still it's much better if you go to this website because every lecture has textual description right next to it. Plus lecture is part of the course. As you see here I'm referring to previous lectures where I have already derived certain formulas and if you would like to be comfortable with this you have to really go to these lectures as well. So it goes without saying that to take the course is much more important than if you accidentally hit the lecture because you will not really understand where the formulas are coming from. Unless again you go back and research etc. etc. So the course gives you basically a complete picture of all these things. Okay, and also I have to mention that the website is totally free. There are no advertisements, no strings attached. You don't even have to sign in if you're doing just basically self-educating yourself. You do need to sign in if you would like to work under the supervision of somebody else like your parents or teachers. And then the teachers can basically assign you a particular piece of the course maybe to study. Also the website in many places, not in all places but in many places it contains exercises, problem solving and exams which I do recommend you to take. There are no restrictions. You can take it as many times as you want. Okay, that's it for today. Thank you very much and good luck.