# Can you Prove E=MC˛ ?

Jun 14, 2021hello everyone welcome to my channel first of all i want to thank everyone who has subscribed to my channel because we have finally reached 5000 subscribers and even though 5000 is not that big a number in youtube terms it is still something , Yes and me. I have been working on this channel for some time now and it makes me very happy to see that 5000 of you have subscribed to this particular channel and for that thank you very much to all of you who have subscribed and for the sake of the celebration I wanted to do. something very interesting today, so let's talk about what is arguably the most famous equation in physics. e is equal to MC squared.

This is the mass energy equivalence principle given by Albert Einstein himself, which is probably the most famous equation in physics. They all know her. Physics students know. Even normal people who have nothing to do with physics know about this particular equation, but if someone asks you as a physics student, can you derive this equation? you

#### prove

this equation or you can get to it through some argument how many of you can actually do that i think very few of you so let's spend some time today talking about the interesting very interesting thought experiment with which we can#### prove

this equation, we can actually say that e equals MC squared so this is a thought experiment that was done by Albert Einstein himself almost 100+ years ago and it is a very simple thought experiment involving very simple math however a very conceptual and fundamental understanding So the thought experiment goes something like this.Imagine that there is a container or a box in deep space free of any kind of force. It does not experience friction. It also does not experience gravitational force. So, there is a box or a container floating. in deep space in complete isolation ok and inside this box is a light bulb or light emitting source and on the other hand there is an absorber that can t absorb the light falling on it and this container or this system is initially at rest with respect to some kind of inertial observer so let's say there is a person or a scientist who is looking at this container and is in some kind of frame of reference he is an inertial observer and with respect to him this container is at rest now what happens in this thought experiment is to imagine that the bulb emits a small beam of light at a particular instant, so when this bulb emits a small beam of light, this light travels from the left side to the left side right yes very simple now what happens when this light is emitted is the container recoils because light carries momentum this is an idea before Einstein this is an idea that can be derived from electr Maxwell's magnetism is proper that whenever light is absorbed by a surface or light is emitted by a surface or light travels from point A to point B, then there is a momentum transfer that is associated with that motion of the light beam. and you don't have to think of the beam of light in terms of photons or in terms of electromagnetic waves, that's not the point, but the point is that whenever there is electromagnetic radiation going from point A to point B, then there is a impulse associated with it and using electromagnetism we can come up with an expression for the impulse associated with a particular beam of light which is given by P equals I over C so what is I here?

I is the intensity of that light radiation and C is the speed of light what is intensity intensity is a way of measuring the energy of that light radiation of that light beam which is basically energy per unit time per unit of area, so the intensity here can be written simply as the energy carried by this particular light beam as it travels from left to right now that the light beam is emitted by the bulb and is traveling in the direction of the right hand side what happens is the container experiences recoil And well you can think of this as if a gun fires a bullet so if you have a gun that fires a bullet the bullet goes from left to right the gun experiences our recoil because we have to conserve linear momentum if light carries its own momentum. that the same amount of momentum will be carried by the container as it moves to the left hand side this has to happen otherwise we will violate the conservation of linear momentum and that is one of the fundamental laws of consideration of our universe the conservation of momentum linear now what happens after a while the light as it travels will eventually reach the other end of the container it gets absorbed at the other end of the container so when the light reaches the end wall of the container the light is correct and brings some momentum to the container. it carries some momentum, they cancel each other and the system comes to a stop again, so initially the container was at rest, but when a light beam is emitted inside the container, I experience recoil and the container will start moving towards the direction from the left side when the light reaches the other end they will both stop again, so this happens due to conservation of momentum.

However, there is a problem here. Can you identify the problem? The problem is very simple, I already told you that from the perspective of this inertial observer, this container was initially at rest and this container is an isolated system free from any kind of external force, so how can an isolated system move without external forces act on it? violates Newton's first law Newton's first law tells us that objects at rest will remain at rest and objects in motion will continue in uniform motion along a straight line until acted upon by some external force and no amount of change Internal forces or internal forces or internal interactions can change this particular motion until and unless some external force acts on it, this is a very necessary law to explain the workings of our universe why because otherwise we could not explain why motion occurs objects would start moving randomly over objects would randomly start to stop there would be total chaos Newton's first law is needed to make sense of why objects move in the first place, if I want to conserve momentum, so somehow this first law seems to be being violated, no what if I imagine the container is always in rest, in that case I'll have to get rid of the conservation of linear momentum now conservation of linear momentum and Newton's first law are two of the most fundamental laws of mechanics and they somehow contradict each other we can't just get rid of these two things because otherwise the universe would be completely chaotic, objects would start to accelerate. start slowing down automatically without any explanation objects should start moving and stop automatically in a random chaotic way without us being able to predict or explain what is happening which is not feasible at all so what is the solution To this particular problem, the solution was proposed by Einstein in a very simple way, but before I tell you what the solution is, let's do some calculations, shall we? so i already told you initially that the momentum carried by the light is equal to the energy of the line given by the velocity, now the recoil of the container is the conservation of momentum that must be conserved, we will also have some momentum associated with it, so Suppose initially the container was here, it started moving and finally it got here and again it stopped.

Suppose the change in distance from this particular point to is . a time period of let's assume of T well then what is going to be the momentum of the container the momentum of the continua can be defined as the mass of the container multiplied by its velocity now let's assume the mass of the container is M ok and what is the velocity V is the distance traveled per unit time, so the velocity is nothing more than the distance traveled by the car in I'm Del T, so the momentum of this container is nothing more than M del X about T now the law of conservation of momentum tells us that the momentum carried by light is going to be exactly equal to the momentum of the container itself, which means that the momentum of the container is the mass of the container multiplied by Del X over del T is equal to the momentum of light which is e over if ok now what is Del T here so del T is the time it takes for the container to go from this position to this position yes , is the same time period that light goes from one container to the other end of the container, so when the light is emitted at point a, it starts moving in the direction of the right hand side and finally reaches point B, which is now B double dash which is the time period of T so when light travels from one end of the container to the other end of the container, it doesn't travel this particular distance because the container itself also travels in the opposite direction, basically travels from a to this particular point so this is a distance that light basically travels before it hits the opposite end of the container so light travels from a to B Double Dash and the time period taken as Del T now let's assume this distance this length that light has to travel as it is emitted at point a and is how it is absorbed at point B Double Dash is L so note that L here is not the length of the container but rather what is a di distance between the initial point where the light is emitted and the final point where the light is absorbed then this is a distance L then if the time period taken is Del T in that case del T is no more than the distance traveled by the light divided by the speed of light yes then del T is the distance traveled ed by light divided by the speed of light C now we know from the second postulate of special relativity that the speed of light is always the same no matter what what kind of fire frame of reference you're looking at it from, whether you're at rest or moving doesn't matter, the speed of light will always equal C, so from this person's perspective the time taken by light is basically this distance divided by the speed of light and we can use this here in this particular. equation so let's use this equation here in that case M of X over T is nothing more than L over C equals e over C or M of X equals EI over C squared times L now let's take note of this particular equation because this is going to be useful to us, let's assume this is equation number one, right now, going back to what Einstein proposed to make sure that Newton's first law and the law of conservation of momentum didn't violate each other, would How can we resolve this particular contradiction?

Fiction, as I already told you, conservation of momentum tells us that a container moves to the left, however, Newton's first law tells us that isolated systems must be at rest and must not move until external forces act. on it so the way Einstein figured that out is something like this now according to electromagnetism light radiation or wave has an associated momentum but no mass so Einstein proposed that along with the associated momentum with light there is also a mass associated with the light, so when light moves from A to B, then there is also an amount of mass associated with this energy of light moving from A to B, so with this simple explanation we can avoid the violation of the law of conservation of momentum and Newton's first law because if there is a mass associated with the movement of light and that means that although from the outside this container has moved the center of mass of this can and this container will remain the same e which is what Newton's first law is in a system of particles, the center of mass will not change until, unless acted on by external forces, even though the particles within the system can change their location based on internal forces, so what is the center of mass of these two systems, so the center of mass initially for the container before the light was emitted is you can write it simply as, let's say the mass of the container itself is Capital m, but the mass associated with the movement of light energy is, let's say, small M, ok so this was a center of mass initially, but what happens to the center of mass after the container appears to have moved a certain amount?

So later or eventually the center of mass is nothing more than capital M times x1 minus the X Y minus Galax that's because the container has shifted an amount of from the X toward the negative x axis, so if this is the positive x-axis, the container from the outside appears to have been displaced an amount Del X, so the final center of ma ss for the container is M X 1 minus X plus the mass of the light and the final position of the light that is initially the light was that X 2 correct user a but once the light has moved to the other end the distance the light has traveled is a distance of L so this can be written as M X 2 plus L well divided by upper case M plus lower case M, as I said, the whole system consisting of thecontainer and the photon of light has a center of mass that will not change because this configuration is in an isolated environment, free from any kind of external force, so the center of mass of these two initial and final configurations must not change, which means this is equal to this so if we equate these two equations this becomes x1 plus the small M x2 divided by the capital M plus the small M equals the capital M plus the small M equals to the uppercase M x1 minus the X plus the small M x2 plus the L L divided by the uppercase M plus the small M so the denominators cancel out and you're left with the uppercase M x1 plus the small M x2 equals uppercase M x1 minus Capital m of the smallest X M x2 smallest M capital L M x1 is canceled small M X 2 is also canceled and you are left with the capital M of X is equal to small M capital and suppose that this is the equation number 2 that we have obtained, now let's use equation 2 into equation number 1 and see Now what we get if we use equation 2 in equation number 1, we can say that M of X is equal to m small capital n, which is equal to e over C squared m, so this can be written as small capital m L equals e over C squared L capital L cancels out and ends up getting E equals M C squared so this is the principle of mass-energy equivalence or e equals mc-squared where e is the energy of the photon of light and M is the mass of light photon and C is the speed of light so this is how you can prove that the equation e equals MC squared using a very simple idea in this particular thought experiment now what Einstein is proposing is that not only light energy, any kind of energy, sound energy, chemical energy, wherever there is energy, that energy is always associated with a certain amount of mass where the mass is given by M equals e over C squared in this case the movement of the energy of the l Light from point A to point B is also associated with the transfer of the amount of mass M equals e over C squared now, of course this is a very specific example for the case of the photon of light, but Einstein proposed that this is a general statement that can be made for both energy and mass if there is an energy E then that energy is associated with it an amount of mass M and if there is an amount of mass M that mass has associated with it a certain amount of energy E where the quantization of these two values â€‹â€‹is given by E equals MC squared now how to convert energy to mass or how to convert mass to energy is a completely different kind of challenge there is an experimental challenge that is not of the same kind.

What Einstein is talking about, but he's talking about the quantization of an energy and a mass that are associated with each other, so that's an interesting way that you can derive this particular equation or prove this one. particular equation, the most famous equation in physics, so now I think if someone tells you how to prove this, you know how to do it, so I hope you learned something from it. I also enjoyed this particular interesting problem, so thank you very much for watching the video and thank you once again for subscribing to my channel. I hope to see you in the next one.

If you have any copyright issue, please Contact