 Hi, I'm Zor. Welcome to InDesert Education. Today we'll talk about certain derivation of the famous Einstein formula. m is equal to 6 square, which basically specifies the energy which is concentrated in any amount of mass, and see the speed of light. So we will try to derive this formula through a slightly different avenue. We will use classic Newtonian mechanics principles and the knowledge which we have obtained by researching electromagnetic oscillations, because light is electromagnetic field oscillations. So this lecture will be very heavily related to the lectures about electromagnetic field waves, especially the lecture which is dedicated to momentum of light. So before you watch this lecture, if you did not go through the electromagnetic field waves part of the course, I do suggest you to do that because I will just use the formula from that part of the course, and this formula basically directly related to E is equal to mc square. And my personal feeling is always I want to know why that particular formula exists. So if you are satisfied with basically just taking the formula from that theory which we were presenting in the waves part of the course, well that's fine. But I think if you want to go deeper you really have to know that part of the course. And again, everything in this particular course, or course this actually, is interrelated and it's built into a sequential system of presentation of knowledge which is basically guided by the website Unisor.com. So I suggest you to basically work with the website, use all the material which is presented in there. So right now we are in a course called relativity for all. Now the electromagnetic waves, all this will be was presented in the course called physics for teens. There is also math for teens course on the website which is basically a prerequisite for for the physics. You cannot really study physics without knowing math. Calculus and vector algebra are must basically for doing this. Okay, so that's kind of a gravel and let's talk about this famous Einstein formula about the energy equals to mc2. Energy which is concentrated in any kind of a mass. Doesn't really matter which one, but the uranium or air or water. Whatever mass is, it's always carries energy basically and this is the amount of total energy which is carried by this particular mass. That was a famous achievement by physics of the beginning of the 20th century and Einstein in particular. So what am I going to use? First of all, I will use the concept of momentum from the classical physics. Momentum is mass times speed. Okay, that's basically a definition of momentum and we all know the second law of Newton which is basically relating force with acceleration which is actually, well I should use capital F of this case which is mass times first derivative of the speed by time which is actually derivative of mv by time which is derivative of momentum by time. Again, momentum of the same thing. So we all know this from classical physics. Now the thing which I would like to borrow from electromagnetic field oscillations theory and whatever the analysis we make is the formula which is related energy of light with its impulse. Now so I will just write the formula and again it's derived in the another part of the course. It's course physics for teens. Part is waves. Chapter is electromagnetic field waves and in that particular chapter there are lectures about energy of light, about momentum of light and including derivation of this formula. Actually the formula looks a little bit differently the way how I derived it was this one. If light does some kind of work or increment, well it decrements its own energy by doing certain amount of work then its impulse is in this case decremented by the same by the amount which is related to energy by this formula. Well which basically means that the entire amount of energy inside a certain amount of light is related to impulse of that amount of light by the same formula. Basically that's what it means. So amount of light. When I'm talking about the amount of light just consider you have a ray of light let's say it's a cylindrical kind of light and you just cut certain volume from it. So you have a concept of energy density. It was introduced in that course electromagnetic field part of the course. So it has certain amount of energy and this let's say one cubic meter of light if you wish. It has certain amount of energy. It's related to electric intensity and magnetic intensity of the electromagnetic oscillations. All of that was covered in that part of the course. So I'm just using a final formula which is this one. If you have certain amount of light it has certain amount of energy and it has an impulse. Great. Basically just using these two formulas I'm going to derive the energy of any mass is mass times square speed of light. Now to prove that let's imagine some kind of a thought experiment somewhere in the vacuum you have two masses equal let's call it alpha and beta. So these two masses are on the same distance from the center zero so it's in vacuum there are no external fields the external forces external fields just nothing. Now if it's nothing nobody's moving anywhere or anything everything is completely encompassed within this system. Then obviously we should have the law of conservation of momentum law of conservation of energy all that type of things. Among them by the way there is another law which I mean it's obviously the consequence of the previous two laws that the center of mass which is right now at point zero should be basically stationary it should not move. So if these are bodies alpha and beta are somehow interacting that should not really change the position of the center of mass because if we will change the position of the center of mass it means that the law of conservation of momentum is not a bait. So if all of a sudden these two alpha and beta become active like moving one relative to another then whatever they're doing the center of mass should really be at zero. Okay now after these obvious remarks let's just do the following experiment. For example let's say A has a certain switch on and off and source of light whatever battery. So it switched off and let's say we are directing the light from A from alpha to beta. So what happens? Well here is what happens. Light has energy as we know from I was already talking about this course about electromagnetic oscillations it has energy and it has momentum. Great. So whenever we send a signal a light signal from alpha to beta we are actually changing a certain amount of light which has energy and momentum. Now how do we preserve the momentum which is supposed to be conserved? Well obviously this mass should recoil so whenever we send something well you just think about this way if you just throw something like a stone from here to there obviously if we are in vacuum if you are in a spaceship you are inside spaceship so it's very inertial system as inertial as it can be if you throw if one person throws let's say a towel to another person obviously the first person recoils. Now the second person will wait until towel comes to it and when it comes to it it will also go a little bit further it will go with certain speed and two people will actually start going away from each other. So if this person throws a towel to this person first this goes recoil and this then will when it will receive the towel it will actually start moving to the same direction but their center of mass will be exactly the same as it was before so they are starting to separate from each other but the center of mass will be at the same place so that's exactly what's happening but it's happening here so light goes here so the alpha recoils here well light goes with the speed c and mass will have some kind of a d alpha speed towards this direction okay that's interesting now what if this light which carries energy and impulse does not have any mass what if mass is zero I mean we are kind of thinking about the light as having mass zero but in this particular case let's just think about it if it's zero it means this mass is going this way light doesn't have any mass this mass remains stationary during this time while light is covering this distance what is this time this time is equal to 2l divided by speed of light so during this time our center of mass would move to the left which is not supposed to happen what does it mean it means that the amount of light which we have issued this signal light itself not only has a speed but also has a mass let's use lowercase m and this guy now will have m minus m only in this particular case we can expect that the center of mass will remain where it is because we need the conservation of impulse so the impulse will be m minus m v alpha and plus m times c should be equal to zero because it was zero before it was not moving so now it's supposed to be zero afterwards because this one is always stationary doesn't change at all so this is a very interesting observation well from which we actually can conclude that the alpha is equal to minus mc divided by m minus m all right now considering the classical mechanics if something has a mass m and speed c then its momentum is mass times speed at the same time we know that this particular light carries energy e and it's related to to the momentum like this from which follows what e is equal to mc square well to tell the truth is not mathematically robust so to speak proof because we are using completely different kind of theories this is from electromagnetic oscillations this is from Newtonian mechanics which is not exactly in agreement with Maxwell equations which describe electromagnetic oscillations so i would consider this not to be like a proof but as a kind of a trick which gives correct result and what's most most important that the results of the theory of relativity were supported by numerous experiments but i think it's very interesting to see how easy it is to get to this formula if we use whatever i'm just using so again i would not really uh i i would not really emphasize the the way of this particular derivation as the proof no it's not however it's kind of interesting from my personal view that you can really derive this formula differently using something which is you see not like the whole Newtonian mechanic is wrong there are certain things in there which we are borrowing into the relativity and rightfully so and you know one of the things is probably this particular part is in correlation with electromagnetic theory with uh Maxwell's equations etc okay so basically that was my very quick derivation now i would like to talk about something which is just purely technical things related to this i would like to basically prove that our center of mass is always at the same position now how do we do this so we know that this is the law of movement of the uh alpha so the x alpha is equal to minus l that's the beginning and then i should add time times uh speed so that's negative so it's minus mc m minus m time right so that's the law of movement i should really say x alpha of t when t is equal to zero it's minus l it's this and as t increasing it goes to this way to the left with this speed okay now what's the law of movement of my light it's a signal signal of t it's equal to what minus l plus mc for t less than capital t so until it hits the beta objects right after this light disappears what does it mean it disappears well it's completely absorbed by beta object and the beta object with the energy and mass of the light will start moving to the right right the mass will be m plus m and the speed would be vb now what what would be the speed well the momentum let's just leave this alone the momentum of a light plus mass before their collision was m times c after the collision it should be the same and it should be m plus m the beta right so whatever the momentum this guy has and this one is zero right now after their after the light is absorbed the total mass would be m plus m and some speed vb would be basically from here mc divided by m plus m and what's the rule for the beta now before collision that would be zero i mean not zero l i'm sorry it would be l but after light is absorbed it would be l plus speed times time mc divided by m plus m t so these are basically the laws of motion of all the components of this system now what we have to prove is that the total momentum is always exactly the same as it was before which is zero well let's just consider two different cases for t less than capital t mc my center of momentum is not center of momentum center of mass i'm sorry center of mass how do we calculate the center of mass remember if you have certain number of uh let's say this is coordinates uh you have x alpha and you have let's say x beta this is mass m one this is mass m two what is the center of mass well you do x alpha times m one plus x beta times m two divided by m one plus m two that's the coordinate of the center of mass right and if you have three uh objects that's exactly the same thing so and this is something which you can just directly calculate what is the center of mass for example uh during this particular time well i know this is my m minus m times x alpha plus m times x sig it's of t plus m before collision my beta object has mass m uh times f it stands still on this before collision this is t less than t recall and if you will calculate this what will be well m minus m m x alpha is l minus mc m minus mc plus m times plus m times this position minus l plus mc n plus nl you will see that this is equal to zero i hope so this is m minus m l then minus mc t minus m cancels out plus m times minus l plus mc and plus ml right and it's supposed to be it's supposed to be equal to zero am i right m minus m is alpha alpha m minus m times mc mass times minus l plus no this is wrong it's something c ct this was wrong speed times time yeah i don't know why i put my mass here um so what do we do here right now um and by the way this is minus l it's another mistake which i made oh okay so many little mistakes yeah this is minus l and this is minus okay now is it zero now ml with a minus with a plus so that would reduce this one minus with plus ml plus ml and minus ml minus mc t and plus mc t yes that's zero so as we see during the t less than t we do have a zero as a center of mass now if you will do exactly the same without any small mistakes which i did for t greater than t then you will have to take into consideration this piece as well and you will have zero now every lecture has video part and textual part on the website unisor.com and on the textual part i do all this derivation much more accurately than i'm doing right now and everything is fine it goes to zero everything is okay so that's it basically i suggest you to read the notes for this lecture they are kind of more like a textbook style but after i was trying to explain something orally maybe it would be better for you to look at the at the text itself and good luck i suggest you to maybe maybe repeat the electromagnetic waves oscillation part of the course physics routines that would actually be much much more useful that's it thank you very much and good luck