Anderson Video - Relativistic Energy

Professor Anderson
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<font color="#ffffff">Okay, one thing that we talked about in class earlier was this idea that</font> <font color="#ffffff">E equals mc-squared. Okay, and that is rest mass.</font> <font color="#ffffff">So this is the energy in the rest mass particle</font> <font color="#ffffff">but when the particle is moving, we know it of course has kinetic energy,</font> <font color="#ffffff">and so there is a more general relationship for E, which is this:</font> <font color="#ffffff">E is equal to gamma MC squared, where again gamma is our</font> <font color="#ffffff">1 over square root 1 minus V squared over C squared.</font> <font color="#ffffff">Okay, but there is a much more powerful relationship between energy,</font> <font color="#ffffff">momentum,</font> <font color="#ffffff">and rest mass,</font> <font color="#ffffff">which is the following. If you take this, you can prove it to yourself but I'm</font> <font color="#ffffff">just going to tell you the answer: E squared is equal to P squared plus M squared.</font> <font color="#ffffff">Now you look at this equation and you say, wait a minute those don't</font> <font color="#ffffff">even match up in units, right? M is kilograms, how does kilogram squared</font> <font color="#ffffff">become joules squared? The what -- the reason I wrote it like this is because</font> <font color="#ffffff">it's easy to remember if you let C equal 1.</font> <font color="#ffffff">Okay, and this is how theoretical physicists think about</font> <font color="#ffffff">energy momentum and mass, how do they relate to each other?</font> <font color="#ffffff">E squared equals P squared plus M squared,</font> <font color="#ffffff">but we know if we want to get the unit's in there we got to dump in some C's,</font> <font color="#ffffff">and so in reality</font> <font color="#ffffff">we would write the following: E squared is equal to P squared C squared plus</font> <font color="#ffffff">m squared C to the fourth.</font> <font color="#ffffff">Okay? Or E squared equals P C quantity squared plus M C squared quantity squared.</font> <font color="#ffffff">What have we just written? This is a total energy of the particle.</font> <font color="#ffffff">This is the momentum of the particle and this is the rest mass energy.</font> <font color="#ffffff">Okay, and this is a very powerful relationship that tells you how all</font> <font color="#ffffff">these different terms combine and I like to just think of it as that, E squared</font> <font color="#ffffff">equals P squared plus M squared, you throw in some C's where you need to</font> <font color="#ffffff">satisfy the unit's. Now this has interesting consequences not just for</font> <font color="#ffffff">things with mass but also things without mass.</font> <font color="#ffffff">And if I look at this equation here,</font> <font color="#ffffff">particles without mass, meaning that this last term goes away.</font> <font color="#ffffff">What is a particle without mass?</font> <font color="#ffffff">Well, one of them is called the photon</font> <font color="#ffffff">and a photon is a little piece of electromagnetic wave.</font> <font color="#ffffff">Electromagnetic waves are made up of these particles called photons</font> <font color="#ffffff">and so the energy of a photon</font> <font color="#ffffff">is what?</font> <font color="#ffffff">Well, it's P squared C squared plus zero, that term goes away, and so we just get the</font> <font color="#ffffff">following relationship: energy is equal to P times C.</font> <font color="#ffffff">Okay, this is the energy of a photon.</font> <font color="#ffffff">Okay, and we know that photons have energy,</font> <font color="#ffffff">we also know that they have momentum, right?</font> <font color="#ffffff">We knew that before but this relationship tells us absolutely that if</font> <font color="#ffffff">it has energy, it has momentum. Okay, and in fact we know what that momentum is.</font> <font color="#ffffff">P is equal to energy over the speed of light, the energy of a photon is H times</font> <font color="#ffffff">the frequency of the photon. Okay, C is lambda times f, and so look what happens,</font> <font color="#ffffff">f drops out and you get h over lambda and this is something called Planck's constant.</font> <font color="#ffffff">What's the energy of the photon? Its Planck's constant divided by the</font> <font color="#ffffff">wavelength of the photon.</font> <font color="#ffffff">All right, why don't we take a five-minute break and I will see you</font> <font color="#ffffff">guys back here and a few and we'll have a little more discussion, all right.</font>
<font color="#ffffff">Okay, one thing that we talked about in class earlier was this idea that</font> <font color="#ffffff">E equals mc-squared. Okay, and that is rest mass.</font> <font color="#ffffff">So this is the energy in the rest mass particle</font> <font color="#ffffff">but when the particle is moving, we know it of course has kinetic energy,</font> <font color="#ffffff">and so there is a more general relationship for E, which is this:</font> <font color="#ffffff">E is equal to gamma MC squared, where again gamma is our</font> <font color="#ffffff">1 over square root 1 minus V squared over C squared.</font> <font color="#ffffff">Okay, but there is a much more powerful relationship between energy,</font> <font color="#ffffff">momentum,</font> <font color="#ffffff">and rest mass,</font> <font color="#ffffff">which is the following. If you take this, you can prove it to yourself but I'm</font> <font color="#ffffff">just going to tell you the answer: E squared is equal to P squared plus M squared.</font> <font color="#ffffff">Now you look at this equation and you say, wait a minute those don't</font> <font color="#ffffff">even match up in units, right? M is kilograms, how does kilogram squared</font> <font color="#ffffff">become joules squared? The what -- the reason I wrote it like this is because</font> <font color="#ffffff">it's easy to remember if you let C equal 1.</font> <font color="#ffffff">Okay, and this is how theoretical physicists think about</font> <font color="#ffffff">energy momentum and mass, how do they relate to each other?</font> <font color="#ffffff">E squared equals P squared plus M squared,</font> <font color="#ffffff">but we know if we want to get the unit's in there we got to dump in some C's,</font> <font color="#ffffff">and so in reality</font> <font color="#ffffff">we would write the following: E squared is equal to P squared C squared plus</font> <font color="#ffffff">m squared C to the fourth.</font> <font color="#ffffff">Okay? Or E squared equals P C quantity squared plus M C squared quantity squared.</font> <font color="#ffffff">What have we just written? This is a total energy of the particle.</font> <font color="#ffffff">This is the momentum of the particle and this is the rest mass energy.</font> <font color="#ffffff">Okay, and this is a very powerful relationship that tells you how all</font> <font color="#ffffff">these different terms combine and I like to just think of it as that, E squared</font> <font color="#ffffff">equals P squared plus M squared, you throw in some C's where you need to</font> <font color="#ffffff">satisfy the unit's. Now this has interesting consequences not just for</font> <font color="#ffffff">things with mass but also things without mass.</font> <font color="#ffffff">And if I look at this equation here,</font> <font color="#ffffff">particles without mass, meaning that this last term goes away.</font> <font color="#ffffff">What is a particle without mass?</font> <font color="#ffffff">Well, one of them is called the photon</font> <font color="#ffffff">and a photon is a little piece of electromagnetic wave.</font> <font color="#ffffff">Electromagnetic waves are made up of these particles called photons</font> <font color="#ffffff">and so the energy of a photon</font> <font color="#ffffff">is what?</font> <font color="#ffffff">Well, it's P squared C squared plus zero, that term goes away, and so we just get the</font> <font color="#ffffff">following relationship: energy is equal to P times C.</font> <font color="#ffffff">Okay, this is the energy of a photon.</font> <font color="#ffffff">Okay, and we know that photons have energy,</font> <font color="#ffffff">we also know that they have momentum, right?</font> <font color="#ffffff">We knew that before but this relationship tells us absolutely that if</font> <font color="#ffffff">it has energy, it has momentum. Okay, and in fact we know what that momentum is.</font> <font color="#ffffff">P is equal to energy over the speed of light, the energy of a photon is H times</font> <font color="#ffffff">the frequency of the photon. Okay, C is lambda times f, and so look what happens,</font> <font color="#ffffff">f drops out and you get h over lambda and this is something called Planck's constant.</font> <font color="#ffffff">What's the energy of the photon? Its Planck's constant divided by the</font> <font color="#ffffff">wavelength of the photon.</font> <font color="#ffffff">All right, why don't we take a five-minute break and I will see you</font> <font color="#ffffff">guys back here and a few and we'll have a little more discussion, all right.</font>