 Hi, I'm Zor. Welcome to a new Zor education. I will start a new series of lectures about light. The general name is Light in Action. That's on the menu on Unizor.com. But this particular lecture is the beginning of these three or four lectures. You see, light in action means light should do something, right? And for this, light needs energy. So, today we will talk about energy carried by light. I will quantitatively evaluate it. Now, this lecture is part of the course called Physics for Teens, presented on Unizor.com. I suggest you to watch this lecture from this website, because the website has menu. And this lecture is part of the course, which means everything is interconnected. And in this lecture, for example, I'm referring to some other lectures in the same course, which contains certain material which I will definitely use. There is also Mass for Teens on the same website. This is a prerequisite course. You need mathematics to study physics. No doubts about that. Especially calculus, vector algebra. So, I recommend you to watch this lecture and all others from the website, from the Unizor.com. Also, every lecture on this website has textual material, notes, basically. Every lecture has like a piece of textbook next to it. So, you can watch the lecture and you can read the text, which is basically textbook, kind of a text. Okay, so let's start. Okay, I will definitely be based on certain information which I have already presented in some other lectures. For example, we were talking about energy carried by electromagnetic field, general electromagnetic field. It can be constant, it can be variable, doesn't really matter. And we were variating the amount of energy in, it's called density, actually. Energy density of the electromagnetic field, which is basically a local characteristic, which depends on electric field intensity, which is usually used to letter E, and the magnetic field intensity. Now, based on these two local characteristics, magnetic and electric field intensities, we have come to basically a variation of the energy density inside the unit of volume in unit of time. That's what it is. So, if you have an electromagnetic field, you can have certain energy which is in some volume, and then if you squeeze the volume down to a point, you will have the energy density, right? So, you divide energy by volume, and then you limit volume to zero. So, this energy density for electric field was at any moment time, at any point x, y, z. Electric field has this amount of energy in that particular density, energy density in that particular point, where epsilon is electric permutivity and E is electric field intensity. Magnetic component, very similarly. Again, B is the value of magnetic field, component of the electromagnetic field intensity, at moment t at point x, y, z, mu is magnetic permeability, and this is, as I was saying, magnetic field density at that particular point in time and space. Now, the sum is a total energy density of electromagnetic field. So, electromagnetic field carries this energy. All this was presented before in the course when I was talking about electromagnetic field and its properties. That's where Maxwell equations were introduced. So, it's all part of the course which has been already covered. Now, what is light? Well, we again, in that particular part, we were talking about James Maxwell and his experiments, etc. And according to his calculations, light had exactly the same speed as the speed of propagation of electromagnetic waves, which, from his opinion, was a very important argument in favor of light being basically nothing but oscillations of electromagnetic field, which happened to be confirmed by many experiments and theoretical researches. So, electromagnetic field carries this energy. So, light is electromagnetic field oscillation, so light carries energy. Okay, that's great. And now we can actually think about how to quantitatively evaluate it. Okay. Now, we will obviously start from a very simple model. Use these follows. So, our system of coordinates will be something like this, x, y, z. Now, the light will go, let's say, down, doesn't really matter. So, here we have some kind of an area where the light falls on. And my ultimate goal is to evaluate amount of light which falls onto this area, depending on the area, quantitative expression for area, and amount of time during which I'm basically measuring how much energy I'm getting. I have certain area. I have time t, and I want to know how much energy is falling onto this area during this time t. Obviously, I have to simplify my model as much as possible. So, I'm assuming that the light which goes down contains parallel rays, and they're all in sync, and they're all monochromatic, which means they all have exactly the same frequency. Frequency is omega. And the area is vacuum, so I know the speed of light, which is c. And basically, I would like to express amount of energy in terms of omega nc. Now, to simplify it even further, what I will do is I will single out one particular ray of light. And I would like to know how much energy is carried by this one particular ray of light, but not in an entire energy. An entire energy probably would be infinite. If there is an infinite source of light, then the oscillations of this ray will be basically infinite. And I would like to know only one wavelength of this ray. So, I have a ray, and it's oscillations of magnetic field and electric field, and I would like to have only one wave. The amount of energy which is carried by one wavelength of one single ray of light. Then, knowing that, I will obviously divide this length by how many wavelengths fit into this length, multiply by this and multiply by area, and this is how I will get the total amount of energy I need. So, right now we are talking only about one wavelength of one single ray of light. Okay. Now, based on expressions which I have already presented, and magnetic, the same thing. Now, when we are talking about one particular ray, and obviously we assume that these are synchronous oscillations of electromagnetic field, and we are always presuming, again, for simplicity, and that's probably true, that these oscillations are harmonic. Which means, our e of t x y z is equal to... So, this is electric field intensity. They are oscillating, and they are harmonic oscillations, which means it's e0 times cosine of omega t minus z over c. And let me explain you why I put it this way. If you have some kind of a source of light, and then the light goes down, let's say, and we were actually talking about the axis that we go down z, and this is x, and this is y. So, it goes down. All rays of lights are parallel, so one single ray doesn't really matter what x and y characteristics it has. It does not depend. It goes down, and the oscillations are, electrical oscillations are perpendicular to z, and in the direction of the x axis. So, these will be electrical oscillations, and these will be magnetic oscillations, which are towards the y axis, but the whole waves are going down. So, one wavelength from here to here contains one wave of electric and one wave of magnetic oscillations, and both are sinusoidal. Now, why did I put it this way? Well, let's assume that in the beginning, where the source of light is, the electrical will be cosine omega t. Now, that's something which is familiar. This is the regular oscillations, harmonic oscillations. That's the source of light. Now, by the time it goes the distance z, the oscillations at this particular point are exactly the same as oscillations at this point, but with a delay. And what is delay? Delay is the time it takes to cover the distance for the wave. So, if c is the speed of propagation, and z is the distance, then z divided by c would be the time delay, and that's why I subtracted from t this time delay. So, the oscillations at point z would look exactly like oscillations at point zero, z equal to zero, let's say it's here, but with this particular time delay. So, this is a general formula again. We were addressing this formula many times before when talking about electromagnetic oscillations. And rope oscillations, actually. I didn't want to start from electromagnetic. I started with rope, so just moving one end of rope up and down, the waves are going all the way. So, all these formulas were addressed before. So, I assume this is my electric component, so it does not depend on x and y, only on z. E zero is amplitude, and in this particular case, I will use absolute zero and mu zero. These are electric permittivity of vacuum, and this is magnetic permeability of vacuum. So, we're assuming that we are in a vacuum. So, we are simplifying our problems as much as possible. So, it's a vacuum, it's a single ray of light, and that's how its electric component looks like, and this is magnetic component. By the way, magnetic and electric are always connected, and they are always in sync with each other. Let me change my marker. So, knowing this, what I have to do is, how do I calculate the amount of energy? Now, this is energy in one, and this is energy in another component of electromagnetic field, but these are densities at point in time and in space. Now, let's forget about time, let's fix the point in time, and just think about space. So, my wave goes like this, and I have to calculate, R, E, and B. They are basically the same, with different amplitude. Now, this is the length, wavelength, lambda. Let's say from zero to zero, or from peak to peak, doesn't really matter. Now, this wavelength, to find the amount of energy, I have to basically integrate all the energy here. I know the density, energy density at each point. So, this is my z-axis, right? So, it propagates along the z-axis. So, I know what's the electric and magnetic components at every point. So, how can I find out the total amount of energy, which is in one particular wavelength? Well, I have to integrate, basically, from zero to lambda, one-half epsilon zero E squared, which is E zero squared, cosine squared of omega t minus z over c d z. That's my electrical component. Energy, electrical energy. And magnetic energy would be similar, but instead of epsilon zero, I will have one over mu zero, and instead of E zero, I will have B zero. So, I have to really take these two integrals, and their sum would be amount of energy, electric and magnetic energy, concentrated in one particular wavelength of one single ray of light, okay? Now, let's talk about lambda. What is lambda? Lambda is a wavelength. So, we have many different characteristics of waves. We have lambda, which is the wavelength. We have c, which is the speed of propagation. We have tau, which is a period. Now, period is the time which is needed for a wave to cover the distance equal to wavelength, right? That's the period, which means that lambda is equal to c times tau, okay? Now, what else do we have? One thing is period, another is how many oscillations per second, so how many periods per second fits. That's frequency. Frequency is equal to one over tau. So, if tau is amount of time needed for one wave, so how many waves are per second? You basically divide one by tau, right? And we have omega, which is not number of oscillations per second, but it's number, this is an angular frequency, which means I have to multiply number of oscillations by 2 pi, right? So, that's an angular, which is 2 pi f. Now, I would like to express my lambda in terms of omega and c. So, instead of tau, I will put c over f, and instead of f, I will put omega divided by 2p 2 pi. So, it would be c omega and 2pi, okay? So, lambda is equal to 2pi c over omega. That's the same thing as we see, right? Now, what do we have? We have one-half epsilon zero e zero squared cosine omega t minus zc dz, right? So, let me open these parentheses. I will put separately omega t minus omega z. What's interesting here is that if I will replace omega z over c as u, so I will have here u. I will have here, if z is from zero to 2pi c over omega, then omega divided by c would be from zero to 2pi, right? So, integral for zero to 2pi for u. So, that would be one-half epsilon zero e zero squared cosine square of omega t minus u. And instead of dz, I have to put du c over omega. So, c over omega du. Okay, this is much simpler integral, obviously. It's very easy actually to calculate it. But there is a kind of a more mathematical way. You replace cosine square with, what's the formula? Cosine square is equal to 1 plus 2 cosine of two angles. Yeah, let me check it. Cosine square minus sine square, this is cosine of double angle, right? So, sine square is 1 minus cosine square. So, it's 2 times square minus 1. Right, so it's 1 plus cosine of 2pi divided by 2. And that will get rid of the square here. And then, you know, take whatever the integral from a cosine. That's very simple. All these calculations I put in the notes for this lecture. However, there is kind of a trick here which might actually help. You see, cosine square and sine square are really not much different if we are talking about one period. It's just shifted. So, the integral of 1 should be equal to integral to another. But if we will add them together, you will have 1. So, that actually gives you that integral of cosine is equal to, cosine square is equal to integral basically of, without cosine square, equal to 1 divided by 2. And I can give you right now the answer. Again, either you do it yourself or you will take a look at the notes. These integrals are really kind of trivial and I don't want to spend much time on this. I will give you the answer as I have already calculated. Epsilon 0 pi c over omega. And obviously, e0 square. And here, more or less similar, pi c omega b square. So, their sum, which is equal to 1 half, pi over c over omega, epsilon 0 e0 square plus 1 over mu 0 b square. That's energy of one particular wavelength of the ring. Again, that's energy of one lambda, wavelengths. Okay, now it's simpler. So, if you have certain area, again let me go back to my picture. This is area and this is the time. So, this cylinder contains all the energy which falls on to this area during this time. What's the height of this cylinder? If t is time, then t times speed of light, speed of propagation of waves is the height of the cylinder. Now, we know one ray of one wavelength. So, what should we do now? We should multiply this, this. We should multiply it by what? How many waves fit in the height, which is c times t divided by lambda, and multiply by area. That's my total amount of energy falling on to area A during the time t. Okay? Now, what do they say about lambda? Remember, lambda is c times tau, which is c divided by f, and this is c divided by omega times 2 pi, right? So, this is A times t times c divided by lambda, which means divided by 2 pi c, and multiply by omega. Nw at, right? A times t times omega 2 pi, and this is my, so it's one-half pi c divided by omega epsilon zero z zero square plus mu zero b zero square. This is total amount of energy which is falling. Now, this is not really very convenient way of presenting this. It's much better to say what's the density of the energy. And density of the energy is basically the same thing divided by A and divided by t. So, how much energy falls on to unit of area during unit of time? Now, if you have these, then whenever you have any area and any amount of time, you just multiply it by area by time and you will get the result. So, the density of energy called p at, how much energy per unit of time? Now, I shouldn't really have these indices. Per unit of time per unit of area, that's equal to one-quarter epsilon zero e zero square plus mu one over zero mu zero b zero square. So, this is the result. This is something which can be used for any kind of calculations. If you know the amplitude of electric and magnetic fields, that's basically, I think I missed something. Did I miss something? I think I missed c. Let me check. Yeah, I missed c, so it should be c. I forgot, c did not omega and pi actually cancel down, but c remained. Yeah, and I think I call it letter i for intensity of the light, intensity of the light, amount of energy falling onto unit of area perpendicularly to the light during unit of time. Okay, that's basically what it is. Yes, I forgot c. Alright, now, I would like to simplify it a little bit further. I need e and b. I need electrical component and magnetic component. Actually, for the harmonic oscillations, we can express, let's say, b over e in terms of e. And here is how. Again, I'm referring you to the lectures about electromagnetic field in general. That's where I covered Maxwell equations. And one of the equations, general equations for any electromagnetic field was minus d by d z by d equals epsilon 0 mu 0 x. What is this? This is a simplified version of Maxwell equation for vacuum. Epsilon 0 and mu 0 are permittivity and permeability of the vacuum. Now, b is magnetic component, e is electric component. Now, these indices were used in that presentation just to show that the direction of propagation of the waves is z. Direction of electric component is oscillations over the x-axis. And magnetic components, again, let me just draw this picture. x, y, z. So the wave goes this way. Now electric component is parallel to x. And magnetic component is parallel to y. And they are perpendicular to each other. So that's basically the picture. And that's what kind of an equation we have. This is the fourth Maxwell equation. And in our case, I can actually draw these indexes because we don't really need them. We know that x, in our concrete example, we have postulated that x is direction of changing electric component and y is magnetic component. So these are the same. Now, let's use this for our particular case where the e of tz is equal to e0 times cosine of omega t minus z over c. And b, correspondingly, is exactly the same. b0 times cosine, same thing. All right. So let's differentiate b by z. That would be what? So minus db of tz by dz. So this is my d. So what is it? Minus. OK. Now the constant goes out, b0. Derivative of cosine is sine with a minus sign. So I put plus here and sine of omega t minus z over c times derivative of inner function. Inner function we are calculating by z. So it's omega over c. That's my derivative. So here we have epsilon 0 mu 0 dE of tz by dt equals, OK, epsilon 0 mu 0 remains. e0 is constant. I think in this case it's not just w over c. I think it's minus w over c. Sorry. Minus. Now here I have sine but with a minus sign. So put minus in front. Minus sine of the same thing times inner function which is again minus omega c over omega over c. OK. Minus and minus. So that's negative. And here we have minus sine. I'm sorry. I'm not multiplying by derivative by z. We are multiplying derivative by t. It's by t. So it's omega. I see something is wrong. OK. And they are equal to each other. OK. Great. That's how we would derive. They are equal to each other. And that's how we derive dependence between b0 and e0. Right? So this is minus. And this is minus. So what we can have is b0 times omega over c is equal to epsilon 0 minus 0 omega. Again in the same lecture we were deriving the speed of light as 1 over square root of epsilon 0 minus 0. See I'm always referring you to the previous. That's why you really have to take the course and not just listen to this particular lecture from which c square is equal to 1 over epsilon 0. So here I can actually change this to 1 over c square. Now this goes out. And this goes out. And what do I have? I have e0 equals cb0. Such a simple dependency. Between them if we are talking about harmonic oscillations and we are talking about monochromatic light without an in vacuum etc. So all is clean. Which basically allows us to change slightly our equation. So again, so we can do instead of b0 square I can put i is equal to 0 over 4 epsilon e0 square plus 1 over 0. b square which is c square. No, b is e divided. c square would be here. Now this can be simplified even further. Since c square is equal to 1 over epsilon 0 mu 0 mu 0 is equal to what? mu 0 is equal to 1 over epsilon 0 c square, right? So mu 0 times c square is equal to 1 over epsilon 0. And 1 over mu c square is equal to epsilon 0. So instead of this I can write this epsilon 0 e0 square plus epsilon 0 e0 plus which is equal to... This is the same thing, right? So it's c over 2 epsilon 0 e0 square. And this is my total formula for density of light amount of energy light carries of this amplitude when it falls on some flat surface per unit of area, per unit of time. That's what density actually is. Very simple formula. Well, as everything counts in real life, real life is much more complex. Light is never monochromatic unless it's a laser light. And well, in any case, whatever it is we are talking about, as always we are talking about model and our model has allowed us to relatively simply express the amount of energy which light carries. In this case it's monochromatic light which has the same amplitude. It's harmonic oscillation. C is the speed of light. Epsilon 0 is electric permittivity of the vacuum. So that's what it is. I do suggest you to read the notes for this lecture again. And whenever I'm referring to prior material in the course, especially when I was talking about general energy of the electromagnetic field. So I refer to this particular chapter of the course in the notes and I suggest you to read that to refresh your memory. It would be great if you go through all the four Maxwell equations. They are explained in detail. And I think it's presented basically in a rather rigorous way. So there should be no questions basically remained unanswered. I don't really put something and tell you, okay, that's what it is, just believe me. I'm trying to put everything with a proof. I do suggest you to review that particular chapter of the course. And this is just a direct consequence of that. So that's the amount of light which can be used, this one, when we are investigating basically what the light can do. And that will be in the next lectures. So that's it for today. Thank you very much and good luck.