 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about amplitudes of electric and magnetic components of electromagnetic field. Now, this lecture follows the one where we were talking about speed of light, and we have come up with a formula which I will put on the board. So, it's basically a continuation of that lecture, and I will use basically all the results which have been derived in that lecture. So, I do suggest you to refresh it if you don't remember what's in there. It's a previous item on the menu just prior to this one. It's called speed of light. So, you go to coursephysics14 on Unisor.com and choose the waves part, and in that wave, waves part, you will have this particular lecture on electromagnetic field amplitude. And prior to this is speed of light. Now, the website unisor.com contains not only this particular course, but also the prerequisite which is mass for teens. Also, right now I'm working on another part which is called relativity for all, which is basically a little bit higher, more advanced level after the physics for teens. The site is totally free. There are no advertisements. Sign-in is optional only if you would like to use some functionality related to educational philosophy. If you wish, if you have a teacher and he would like to supervise you, so you need sign-in on both sides to basically establish the relationship. Otherwise, if you just do it by yourself, you don't need any sign-in. What else? Every lecture including this one obviously contains on the website right next to it the very detailed explanation of the same material in a textual format. It's actually a textbook format. So, I mean, if you will combine all the different notes for all the different lectures in proper order, that would be a textbook. So, I basically kind of divided it into lecture and notes, lecture and notes, et cetera. And it's all related through the menu. You have the top-level menu and then sub-menus, et cetera, which will give you a proper sequence and logical dependency between the lectures. Okay, so now back to business. We are talking right now about the simplest kind of electromagnetic field, so-called monochromatic flat waves or plane waves, if you wish. Now, what is this? Imagine system of coordinates. Let's say this is x, this is y, and this is z. The magnetic field has electric and magnetic component. These are intensities of the field, the forces basically. This is the force per unit of charge and this is also per unit of something. So, magnetic and, now, they are perpendicular to each other. And let's say this is direction where vector of electric component is directed and this is directed along the y-axis. Now, so the electric component goes along the x-axis. The magnetic component goes along the y-axis around zero. And obviously the direction, this is the speed of light, this is the propagation of the waves. So this is a monochromatic plane waves. And obviously I assume that, and again that was in the previous lecture, the result of Maxwell equation in this plane simplest, I would say, case can be expressed basically analytically. Something like vector E has, in theory, it has three components. E x, E y, and E z. But these two are equal to zero. This one not equal to zero. Same thing with vector B. It has three components. B x, B y, and B z. And B x and B y and B z are equal to zero. And only B y not equal to zero. And the analytical expression for correspondingly E x and B y is E x. It depends on time and location along the z-axis. So it basically goes this way and propagates that way. So it depends on the time and position of the z. So it's equal to some kind of amplitude. And then I believe I'm using sine. Yes, I'm using sine. It doesn't really matter sine or cosine because it's all dependent on the phase. So it's omega c minus z divided by c. Now, in text book you might actually see slightly different expression. You might see expression like omega t minus or plus kz, where k is equal to omega divided by c. Now, this expression is exactly the same as this one. However, I personally prefer this one because it has physical meaning. And here is the physical meaning. If regular oscillations are usually expressed as this one, for example, means t equals to zero, we are in a neutral position on the zero, right? And then we're oscillating with frequency omega, angular frequency omega. If you are further from the source, now this is basically dependency on the source only. So at source, these are oscillations. At zero, it goes this way all the time, okay? Back and forth. Now, if I would like to find out what it is on certain distance, well, there is a time delay. So the same kind of oscillations will be here as well. But with certain time delay, what is the time delay? Well, time delay is z divided by speed of light, right? So you divide distance by speed and you have the time delay. And that's why I prefer to have this time delay here. So in my personal opinion, it looks better than kz where k doesn't really, you know... I mean, they put some kind of name to this constant, but I think I prefer this one. So I will use this one. It doesn't really matter, quite frankly. So the b, according to... Okay, now this formula was actually derived in that previous lecture as a solution to Maxwell equation in this simplest case. Now, the Maxwell equations in vacuum looks very much similar to each other, and the solution would be exactly the same. So dy of t and z. So this is the magnetic component. It would look exactly the same way. And what's important actually is the same omega and the same shift in phase. Because electric and magnetic components, we did discuss this, they are always in phase with each other. When magnetic component is increasing, the electric component is increasing as well. When electric component is decreasing and magnetic component is decreasing as well. So they are always like in sync with each other. And that's basically how they are going all the time. And then it's propagating with the same speed of light further along the z-axis. Okay, so this is something which was partially discussed, derived actually in the previous lecture. And I just added the magnetic component which is exactly the same thing. Now, my purpose in this particular lecture, I would like to know the relationship between E0 and B0, between amplitudes, how big they are, relative to each other, of course. Okay, so that's the main purpose of this lecture. Now, how can we do it? Okay, now there is a so-called third Maxwell equation, which is Faraday's law, which looks like this nabla. Vector product with E equals minus dB by dt, where nabla is a pseudo-vector. I mean, I can put this as well. It's a pseudo-vector which is actually differentiation by covenants. Now, these are all topics which were covered in previous lecture. I was talking in details about this pseudo-vector. I mean, it's not a vector. It's just kind of a syntactical vector, if you wish, in previous lectures. So it's all explained over there. So right now we're discussing only this one. However, let's just consider that we have only a very, very simple case. This is the general form of third Maxwell's equation, the Faraday's law. In our case, things are much, much simpler. So if you will actually open up this vector product by covenants, you will have only one thing, you will have only this one. It's actually dEx of tz by dz times j. j is the unit vector along the y-axis. And minus db by dt, again, considering only dy component is equal, that's actually minus ddy of tz by dt and also the same unit vector. This is a vector, so that's why I have to put vectors. Now, both of them are along the y-axis. So if they are equal to each other, my equation would be very simple in this way. I can drop the vector part because it's unit vector along the same direction. And now allow me just to drop this index x and y because it's the only one. So it's easier to put it this way. dE by dz of tz equals to minus db tz by dt. So that's my equation. This is the simplest form of third Maxwell's equation in vacuum for monochromatic plane waves described by this. Okay, fine. So I have this equation. Now let's use it since I have explicit expression for my e and v. So I just decided to drop index just for brevity. So what is it? Well, let's just have this derivative taken and we'll see what happens. So derivative by z. Okay, so dE of tz by dz is equal to, okay, e0 is a constant. Derivative of sine is a cosine of this function times derivative of inner function. What's inner function under sine of z? We're talking about derivative by z. So it's just a constant minus omega nc. So I have to multiply by minus omega times c. Okay, great. So derivative by time of the magnetic component db of tz by dt by time is equal to, similarly, b0 is a constant. Derivative of sine is equal to sine of the same thing times derivative of inner function. Now we have a derivative by t, so the only coefficient is omega. Let's compare it. This, which is this, is equal to minus this. Okay, so this is equal to minus this. This is the same, minus here, and we are talking about minus here, right? So minus would be minus here. Omega is the same. So what remains? The only thing remains is e0 divided by c is equal to b0. Now this is basically the result which I wanted to get today. So as you see, the amplitude of electric component is by c, which is a very, very big number, speed of light, greater than component than the magnetic component. That's basically my result for this particular lecture. We might use it in some particular cases, but right now let's just stop here. And, well, basically that's it for today. That's the only thing which I wanted to derive today. I do suggest you to read the notebook notes for this lecture because there are maybe a little bit more details. I skipped, for example, from NABLA, I skipped a simple expression. In the notes it's a little bit more detailed, but in any case it's always useful to read the same material if you understand everything here. And don't forget that this lecture is not by itself. It depends on the previous lecture, the previous lecture depends on even more early one, etc. So you always have to consider the whole thing as a course. And that's what I recommend you to take the whole course because it's all logically related things, not just individual facts. It's the whole picture which actually makes you comfortable in your knowledge. Just if you just consider separate facts that might not actually be very useful, you will forget them anyway. All right, that's it for today. Thank you very much and good luck.