 Hi, I'm Zor. Welcome to new Zor education. Today we will talk about momentum of light. Well, first of all, let's just simplify our work. We will consider only the right light, which means it's flat wave front. It's electromagnetic oscillations, which are completely synchronous, which means all in phase, all these oscillations, sinusoidal oscillations. And they are monochromatic, which means the same frequency. So it's basically like a flat wave front of electromagnetic oscillations. And they are propagating with the speed of light in vacuum. So as simple as possible. Can't be anything simpler than that, right? And we will examine what happens when these electromagnetic oscillations are falling on some kind of a surface. Well, we'll definitely consider flat surface, which is perpendicular to the direction of light. So that's what we're going to do in this lecture. Now, this lecture is part of the course called Physics for Teens, presented in Unisor.com. I definitely suggest you to take a look at the whole course, because I will definitely use certain things which I have already covered in previous lectures in this one. And that includes formulas. Now, every lecture on the Unisor.com has a text shell part, which is basically like a textbook, where I'm basically referring you to exactly the previous lecture where this or that particular formula or statement was used from. So I do suggest you to read that part of Unisor.com. And whenever I'm referring to the previous lecture, make sure that you know the contents of that lecture. So all the lectures are logically related, and they are combined into a course with specific menu, of course. So it's all connected to each other. So if you found this lecture, for instance, somewhere on YouTube, and just by itself, basically without even the text shell notes which follow this lecture on Unisor.com, I suggest you to go to Unisor.com. The website is completely free, there are no advertisements, so fall for your studies. Okay, so we are considering this flat wave now. What's next? Next is, again, I refer you to one of the previous lectures where we were talking about energy, which electromagnetic oscillations are carrying with that. So the energy which basically is the energy which is carried by two different forces, the electric component and magnetic component of the electromagnetic oscillations. So these are basically two forces, if you wish. Well, let's assume that the electrical component goes this way. The magnetic component is perpendicular to it, and then the direction of propagation of electromagnetic waves is perpendicular to both. So let's say this is x, this is y, and this is z. So my electric component goes up to y, my magnetic component z and propagation along the x-axis. So the electric component is oscillating this way, the magnetic component is oscillating this way, and the waves are going this way. Now, in terms of E and B, in terms of electrical and magnetic components, my total energy, electrical plus magnetic, is equal to, and I'm using the formula which I have already derived in some other lecture before it, epsilon zero E square plus one over mu zero B square. Now, epsilon is electric permittivity, mu is magnetic permeability of the vacuum. It's all about vacuum, that's why the index zero is. And the square of electrical and magnetic components means basically, since it's a vector, it's a vector. So E square basically means E times E as a scalar product. And obviously, all of those guys are dependent on the time because they're oscillating. And that's why the energy is also oscillating. Now, what is this energy? This is a density of the energy, which means amount of energy in the unit of volume, basically. So this formula was, again, proven, presented in the previous lecture. The reference to the previous lecture is in the textual part for this particular course on Unison.com. That's number one. Number two. Okay, so let's assume that there is some kind of flat surface here. So all these oscillations are falling on this flat surface. And we will assume that the area of this surface is A. And we also have to understand that there are electrons in it. And all the electrons are basically affected by electric component because it's the force. The force is acting on certain charges. Now, the nucleus of the atoms are heavy. But electrons are very light. So obviously, the most important action which is performed by electric force is against the electron. So as force is oscillating up and down, the electrons in the surface of this particular object, where the light actually is falling, they are also oscillating. Well, when the force goes up, the electron goes down because the electron is negatively charged. And the force is always defined against the positive. So this goes up and down, the electrons go down and up. Okay, now the total charge of the surface electrons, let it be Q, which is basically if you have some kind of a density of the surface density of the electrons times area. It's the same thing happens. I will use Q primarily. So there is a charge. These are all the electrons which are concentrated on the surface on which the light actually falls. So what happens with these electrons? Well, they are moving. And let's talk about the forces. The electric force acting on the electron is basically one force. And then, now the E is the intensity of the field, which means that the force which is acting on the charge Q is the product of Q times intensity. That's basically definition of intensity. But there is another force which also acting on the electrons. Electrons are moving because of this force, because of electric force. It's moving down and up. Okay, now it's perpendicular to the magnetic force. And there is a Lorentz force. You know that if there is a magnetic force and these are lines of magnetic force. And then there is some charge which is going perpendicular. There is a force which is perpendicular to both. In this particular case, if this is B, if this is the speed of the electrons, then the charge is perpendicular to both of them. So it's perpendicular to the surface of the board. So in this particular case, since this is the direction of the B. This is direction of the electrons, the speed of the electrons, because of electric force. Then that would be direction of the movement of these electrons. So there is another component which is Q, the same thing, times speed vector product with B. So this is basically how we will arrange this mutually perpendicular, proportional to B and proportional to B. So this is the speed caused by the electric force, this one. And this is the magnetic component. So this is the whole force which is acting on the all electrons, all charges here and here. Alright, so what do we do with this force? Well, again it has two components. This component is acting in this direction, this component acting in this direction. Since electrons are moving in this direction, it presents some kind of pressure, it's called radiation pressure. So just because the light falls on the surface, there is certain amount of pressure on the surface itself. Now, but we know that the direction of these vectors is changing all the time. It's oscillating, right? So it goes up and down, up and down. But what happens with this force? Well, what's interesting thing is, whenever the vector E goes to the opposite direction, vector B goes also to the opposite direction. And the result of this thing is again towards the increasing of the X component. So that's what's very important. It's oscillating, but the force is always going this way. Well, obviously it's not constant force because since these are oscillating, the force is pulsating, but it's always in the same direction. That's what's important. And that's what makes the pressure, this radiation pressure on this surface. Okay, so what is exactly the pressure itself? The pressure itself is only the magnetic component of this. Because this part goes perpendicularly to the surface, I mean parallel to the surface. This part goes perpendicular to the surface. So we have to really talk about only this piece. So Fx direction of the X, this vector is actually Q times V times V. And obviously this is all functions of time because they're all pulsating. Okay? All right. So we have the force, total force which is acting on the whole surface. Now, before proceeding any further, I'd like to have certain deviation back to Newtonian mechanics. Here is Newtonian mechanics. Force is equal to mass times acceleration. Remember the second law of Newton, which is mass times, what is acceleration? It's the first derivative of speed by time. Again, that's all covered in previous lectures related to mechanics. Now, M is considered to be constant in Newtonian mechanics, which means I can actually write it down like this way, put it under differential. It's a constant anyway. And what is this? Well, this is the momentum, if you remember, momentum of movement. So we're talking about dp by dt. And obviously it's all dependent on time. So force at any moment of time is equal to derivative or a rate of change, if you wish. Not mathematically, it's what is derivative. It's a rate of change. Force is equal to rate of change of momentum. Okay, that's very important. And that's exactly what we're talking about. So now we can talk about this momentum, because our purpose is to find momentum of light. So let's talk about momentum. So increment of momentum is equal to force times dt from this. dp is equal to f times dt. Now this thing is called impulse of the force. So basically, increment of momentum is equal to impulse of the force. So I'm using this for force, and I can say that increment of momentum of the object on which light is pushing, because this is the force. So I'll multiply it by dt, and I will have q times dbt times dt, right? So I multiply force by differential of time, by increment of time, both sides. So that's why I have this, and force times increment of time is increment of impulse, increment of momentum. Okay, fine. Now let's talk about this again in a simpler format. What is vector product of v times b? Since they are perpendicular to each other, v is directed along the y-axis, along the component, electric component e. B is against along the z-axis, right? So they are perpendicular to each other, which means the vector product can be replaced by plain product. Because there is no, because the sign of the angle between them is equal to one. So that's basically q quantitatively. v of t, v of t, v of t. Now, again in the previous lecture we were talking about relationship between the magnetic component and electric component. And we have basically come up with the very important formula for all these simple flat wave oscillations. We had the formula b is equal to quantitatively to the electric divided by the speed of light. Again, if you would like, there is a reference in the text for this lecture to the previous lecture. You can examine the previous lecture where this is derived. So using this thing, I'll just replace this with e over c. Now I always put c as a parameter because it's all oscillating, it's all moving up and down sinusoidally. Okay, so we have this and that's very important, that's basically almost everything I wanted to show you. Everything else is pure manipulation in formulas. So what is this? Okay, now this, we're right this way. Now what is this? This is the force of electric, a component of the field. This is intensity times charge. Intensity times charge is the whole force which is acting by the field. So field actually is performing certain work and the force is this. Now what is this? Speed times time is the distance covered which is equal to, now the distance is along this force. That's basically the y of t divided by, I mean d y of t divided by c. That's the increment of movement along the y axis. So e is the force acting along the y axis. Remember, this is y, this is z, this is x, e, b and this is how y is. So this is the force acting along in the direction of y axis. This is differential, so this is 1 over c times differential of work. Performed by who, who performs the work? Electromagnetic field. Primarily it's electric component. So this is a very important formula. Increment of the momentum of the object is equal to increment of the work which is done by electromagnetic field divided by speed of light. Well, now we just have to talk. Now, there is a law of conservation of energy and the law of conservation of momentum. If my object gets this particular increment of momentum, where does it take it? Well, it took it from the speed, from the light. So somehow when light falls onto the surface and is absorbed by the surface, the object which has the surface gets the increment of momentum. And the light obviously is losing all its momentum because there is no more light, light is absorbed completely. Which means that exactly the same amount was in the light. So the same increment which object gains, the light loses when it basically absorbed by the object. So that's the very important consideration, which means that the light has momentum, the light performs work, and momentum of the light is related to the amount of energy which light is losing. Again, because this is when the light is consumed by the object during that infinitesimal time period, dt. So this is the amount of work which light is supposed to do because this is the amount of energy consumed by the object. So there is a law of conservation. So this is the amount of energy which light is losing and this is the amount of momentum which light is losing. So not only light has an energy which we were talking about before, but light also has a momentum. And the momentum is related to energy of light in this particular fashion. So whenever certain amount, if you wish, of energy is concentrated in the field, in the electromagnetic field, there is certain amount of momentum which is also in that exact field because it's oscillating, because it's doing something, etc. So it's a source of energy and obviously we know that the light is a source of energy because, for example, sun is a source of energy which is coming onto the earth. Everything, all energy which we have here is the result basically of the sun's activity. So all these rays of light which are coming onto earth from the sun, they all carry energy and that's what the energy we are using, ultimately. I mean it's converted into certain other forms, etc., etc., chemical energy. So that's a very important formula. Sometimes this formula is written shorter. So instead of differential, sometimes people use something like this. Without providing that this is function of T, without using only incremental, etc., but that's basically the same thing. Because you can always integrate this and this and you will have certain amount of energy which is concentrated in the volume and therefore certain amount of momentum which is concentrated in the same value. So that's very important and that's the ultimate function, ultimate thing which I wanted to talk about. Now, this is energy which is flowing towards the object in the infinitesimal amount of time, dT. Now, remember we were talking about energy flux density. Now, energy flux density is amount of energy which is going through a unit area during the unit of time. So if we will divide this by the area, now this object, it has certain area A. So if I will divide by area, I will have the dW of T divided by area. Now, this is per unit of time because this is derivative. So if derivative is basically by time, so it's how much energy flow per unit of time. If I divide by A, that's how much energy is flowing through a unit of surface in the unit of time, which is flux density of energy flux density. Now, energy flux density, if you remember, was related again in a previous lecture, so-called pointer vector. Now, pointer vector which is actually 1 over mu E times B. Again, the same vector product which is going towards propagation of light, that's how S-pointing vector is directed. So this vector basically gives you the flow of energy per unit of time per unit of area, through unit of area. So basically this can be expressed as the pointer vector. So, well, basically that's exactly what I wanted to show, that there is a relationship between the flow of energy and the pointing vector. So you can always say that, again, increment of the impulse, increment of impulse is equal to 1 over C, increment of work over T, 1 over C, A times pointing vector, right, from here. Pointing vector is per unit of time per unit of area, we multiply by area, will be per unit of time, which is rate of change, which is the rate of change of momentum. And the last thing which I wanted to talk about is, everything we were talking about right now was about complete absorption of the light by this surface. What if it's not absorption, what if it's a complete reflection? Well, if it's a complete reflection, then look at it this way. In absorption case, light has certain momentum in the beginning, and then it's completely absorbed and the light has no momentum at all. So all this momentum goes to the object. In case of reflection, light has momentum P, and then it has momentum minus P. Because the speed will be in a different direction, but it's exactly, we're talking about absolute ideal reflection. So the light goes this way, after reflection goes this way, which means momentum changes the sign. Now the difference is, in this case, difference is P, in this case, difference is 2P. So light changed momentum by the divide. They actually reduced the amount of momentum by 2P. P minus 2P is equal to minus P, right? So this 2P is reduction of momentum of light. Well, which means that the object should be pushed with the momentum 2P, which means stronger than if it was absorbed, a complete absorption. So complete reflection gives more push towards the object. The mirror-like object will be pushed more by the light than the dark object. And basically there were certain very important experiments about this. And experiments were like this. So if you take some kind of a reservoir without air vacuum, completely vacuum, and they had something like a little propeller with squares of some material. So one side of the square was mirrored, and another was black. So these are blacks, and this is mirrors. So whenever light goes this way, now, and this is like a needle. So if light goes this way, since this is reflecting surface, it will have more push than this one. So it will start turning. But the back of this dark thing is again mirror, so it will continue rotating. So that's a very important and famous actual experiment just to prove that the whole thing actually is working as we predict through the calculations. So basically this is again the result of the law of conservation of energy. Okay, that's it. I do recommend you to read the notes for this lecture. They are basically the same as this one, but there are some references, etc. And other than that, that's it. Good luck.