Many of you who watch this channel actually know quite a bit about physics, which makes sense
of course, because, well, you’re smart and that whole “physics is everything” thing. And
you ask questions about things that don’t make sense to you. A super common question is “how
can a photon, which has no mass, have momentum?” That’s a good question, with a lot of complexity
in the answer. So, let’s dive into it. (intro music) Before I begin, I guess I should apologize.
This video has more math in it than most. That’s because the reason that people ask
the question of how photons can have momentum is that they learned about momentum
in a physics class and they learned the equation for momentum. So I have to build the
explanation around that equation. This equation says that momentum, which is written as p,
is equal to an object’s mass, written as m, times its velocity, which is written as
v. “p equals m v” that’s the equation. And you immediately see the problem. While a
photon of light travels at the speed of light, which is written as c by the way, its mass is zero. And zero times a number is zero. So
the momentum of a photon should be zero. Right? I mean, it’s a really good question. Yet
physicists are certain that photons have momentum. So let me start with explaining this in terms
of equations, then go to a more intuitive explanation, although even that really isn’t
so intuitive. It’s going to be pretty cool, but you should expect that your brain cells
are going to get a real serious workout here. First, let me remind you of equations that you
would have encountered in any introductory physics class. I already reminded you of the momentum one.
There’s also another equation that connects mass, velocity and kinetic energy. This is the kinetic
energy, which is written as KE, equals one half times the mass m times the velocity v squared.
So we say that KE equals one half m v squared. You probably remember all that. Now let me show
you how kinetic energy and momentum are related. Let’s start with the momentum
equation, p equals m v. Now square both sides. You get p squared
equals m squared, v squared. So far, so good? Now divide both sides by
m. On the right hand side, the bottom m cancels one of the top m’s. And you
get p squared divided by m equals m v squared, which should remind you of something,
specifically the kinetic energy equation. You can replace the m times v squared
in the kinetic energy equation with this p squared over m thing and
what you find is that the kinetic energy equals the momentum squared divided by twice
the mass. KE equals p squared divided by 2 m. Okay, the first conclusion to draw here is that
in classical physics – and I remind you that these equations were devised back in Newton’s time when
they didn't worry about things moving near the speed of light – but even back then we can see
that kinetic energy and momentum are related. We have an equation with energy on
one side and momentum on the other. And this kind of means that anything with
kinetic energy must also have momentum. Of course, we still have that m kicking
around, which is zero for photons, so I haven’t really answered the question.
And I didn’t intend to at this point. I just wanted to show you that kinetic energy
and momentum are related, even classically. Okay- now to get into something you might
not have known. Let’s move into this new material. And I want to tell you something and
that’s these equations aren’t completely true. That’s right. You’ve been lied to! Wake
up sheeple! The man is lying to you! Man, I have always wanted to
say that in a video. Why should others have all the fun? Of course, like a lot of
times when people toss around the word sheeple, what I just said was a little misleading
and the real story is a bit complicated. The real truth is that those equations are
special cases that only work for objects with mass that is moving at slow speeds. And,
of course, I should be up front that when they say “the man,” the man is someone like me,
so I guess you have to decide whether to believe me or not. But you should. ‘Cause,
you know- I know stuff. And it’s way cool. Now comes the next surprise and that
surprise involves Einstein. Of course, Einstein was all about moving at super fast
speeds, speeds near the speed of light, so you know that he’s going to pop up in any discussion
involving photons and momentum. And, when you think about Einstein, you almost always think
of his famous equation, E equals m c squared. And, guess what? That equation’s wrong too! Come on, sheeple! Wake up! How
long are you going to stay fooled? Okay- I promise that’s the last rant. But the rant
has a super-tiny kernel of truth. E equals m c squared is also an approximate equation and
one that only works for objects with mass that are stationary. It’s not a general equation
and it doesn’t work for massless objects, nor objects that are moving. In fact, I made a
whole video just digging into the limitations of this equation. The link is down there in the
description. Watch it later, if you’re interested. So now we’re ready to get to the truth. I
put the three equations I’ve mentioned so far here. E equals m c squared, p equals m v and KE
equals p squared divided by 2 m. And all of these equations are just fine and correct – but they’re
all equations that only describe objects with mass that move slowly. Those equations fail in other
situations. It turns out that there is a more general equation that works for all situations and
it looks kind of like Einstein’s famous equation. This equation is energy squared equals
the momentum times the speed of light squared plus the mass of the object times
the speed of light squared, all squared. E squared equals p c squared
plus m c squared, all squared. Now this equation works for everything – massive
objects, massless objects, things moving near the speed of light, things moving at the speed of
light, and things that are stationary. And we can look at the special cases. For instance, let’s
see how it works in the special case of a photon. Since the photon has zero mass, this second
term on the right hand side goes to zero, and we have that energy equals
momentum times the speed of light. So here we see that energy and momentum
are equal except for a constant. And, if an object isn’t moving, then its
momentum is zero, which means that p is zero, which gets you back to Einstein’s
equation E equals m c squared. So, I guess the first lesson is
that when you learn an equation, it’s super important to understand
if it’s a special case or not, and, if it’s a special case, when
it applies and when it doesn’t. The bottom line is that the familiar equation
p equals m v just doesn’t apply for massless photons and the question of photon
momentum, while completely sensible, arises from using an equation in a
situation where it was never intended to be. I could spend a long time showing you how
you can manipulate this general equation that equates energy, momentum, and mass, but I will
leave that to you to play around if you want. And I put a link in the description to
a couple websites that do a lot of the heavy lifting for you. By the way, let me
tantalize you a bit – that one half m v squared thing is itself only an approximation for
kinetic energy that works at very low velocities. There are more precise approximations.
There are endless levels of complexity. So that’s the equation-y way of thinking about
this. The equations very clearly show that energy and momentum are related and an object that has
energy also has momentum, even if it has no mass. But maybe you’re looking for something deeper –
something more intuitive. You think that perhaps mass is somehow special. Well
let me disappoint you there too. To first approximation, modern physics has proven
that mass is an illusion. There is no mass. Now I made an entire video on this topic
and I put a link in the description for you to watch. It’s one of my favorite videos
and definitely one of the most mind-blowing, but let me sketch it here for you. Just expect
that everything you think you know about mass is about to be upended. And watch
that video to get the whole story. You are made of mass, which is true
of anything that is made of atoms. The mass of objects made of atoms is found in
the atom’s protons and neutrons. The electrons pretty much don’t matter. But protons and neutrons
are made of smaller objects which have very little mass – nearly none. So where does the mass of
protons and neutrons come from? Those objects are orbiting around each other at crazy fast
speeds. You can think of protons and neutrons as tiny subatomic tornadoes – vortices of motion and
energy. And that’s all they are. If you look at them, there is nearly no mass in the way that we
intuitively think about mass. Mass is just energy. So, assuming that you’re willing to accept
that – and you really should watch that video, because it shows this idea in far more detail
– then moving mass is just moving energy. And, of course, a photon is moving energy.
So, when you get right down to it, a moving photon or a moving proton are not
so different. Both are nothing more than moving energy. So, if a moving proton
has momentum, so does a moving photon. And that’s really all there is to it. On the other
hand, I predict that this whole topic has made you think very differently about mass and energy and
how they aren’t really all that different. The more you learn about the subatomic world, the more
you learn that it’s some crazy stuff down there. Okay- so this concludes our very quick
tour through a very complicated topic. If you learned something from the video, you
know the drill – like, subscribe, and share. While the first message is that photons certainly
do carry momentum, I think the most important message is to always remember that equations
only apply in certain situations and if you try to use them incorrectly, you’re going to get
confused. And this leads me to my final message, which is that as you study physics at a
deeper and deeper level, there’s always more to appreciate and, the more you appreciate,
the more you realize that physics is everything. (outro music)
