This lecture introduces the idea of photons, and using the Planck constant to find a photon's
energy. A photon is a particle. Specifically, it's a particle of light. All light is made of
photons. All photons move at the speed of light, which is 3 x 10^8 meters per second. We use the
symbol lowercase c to stand in for that constant. Photons are both waves and particles. There's
no clear way to visualize this at all. We cannot really represent what a photon really looks like,
quote unquote, using pictures because there's not really a way that photons actually appear.
So even though we can't visualize them I'm going to be using this symbol to represent photons in
the lectures going forward. It's not at all what photons actually look like, just a useful visual
for a particle and a wave. Obviously it's very strange that photons are both particles and waves.
There are complicated reasons for why they behave like both. I'll go into that a little
bit in future lectures, but I've also left some additional lectures in the description of this
video explaining it more. So again, that symbol is not at all what photons actually
look like. It's just a way to symbolize the wavelength and the particle at the same time.
Photons are specifically electromagnetic waves, which is a topic that we've already covered in
the course, so if you're watching this and you haven't watched the electromagnetic waves lecture
yet I would encourage you to do that just to have a complete understanding of what kind of thing
photons fundamentally are. So that lecture goes into a lot more detail about how electromagnetic
waves work and their properties. So this lecture is assuming that you've already seen that video,
or at least know the basics of what an electromagnetic wave is. Photons carry energy.
We call the energy of light "luminous energy." The amount of energy a photon carries is based on its
wavelength. Photons with smaller wavelengths and larger frequencies carry more energy.
So students sometimes forget whether it's a larger wavelength or a smaller wavelength that carries
more energy, and I use this in the EM wave video as well, but I'm going to use this analogy again.
I'm going to imagine that I'm standing in the ocean and I try to think about if the amplitude
of the waves are equal, would I rather be hit by a wave with a long wavelength and small frequency
or a short wavelength at a high frequency? And if I think about which one I would rather be hit by,
it becomes pretty clear that the wave with the smaller wavelength
is going to be much more painful to experience because it's going to be delivering a lot more
energy in the same amount of time. So if the wavelength of the wave in water is shorter,
it's going to be delivering more energy, and so in the same way photons with smaller
wavelengths and higher frequencies deliver more energy than longer wavelength photons.
To understand exactly how much energy a photon carries we use a physical constant which relates
a single photon's frequency in Hz to its energy in Joules: the Planck constant. We symbolize the
constant with lowercase h. So this is the exact value of the Planck constant: 6.63 x 10^-34
Joule-seconds. So what this number is telling me is this is the exact amount of energy in a
photon per Hz of frequency. So if a photon's frequency is 1 Hz, that means that the amount
of energy it contains is exactly 6.63 x 10^-34 Joules, and if a photon has a frequency of 2 Hz
its energy is 2 times the Planck constant, and if it's 5 Hz it's 5 times the Planck constant.
So the Planck constant is the energy per frequency of the electron, and so an equation
for the Planck constant is the energy a photon contains over the frequency of the photon.
As a quick explanation of the unit, you'll notice that the unit is Joule-seconds.
I can see that the unit of energy over frequency would be Joules per Hz because Hz is the unit of
frequency. If I rearrange that, I get Joules times 1 over Hz, and I know that Hz is equal to 1 over
seconds, so if I rearrange that to solve for 1 over Hz. I can see that seconds on its own is
equal to 1 over Hz, so that means that the unit of energy over frequency is equal to a Joule-second,
Joules multiplied by seconds like that. So that's why the Planck constant has the unit
that it does. I can prove a few more equations by combining this identity with a few others.
First of all, if I rearrange this equation for the Planck constant I can see that energy is equal to
the constant times the frequency. So that's going to be one equation that we use a lot. I can also
rewrite this in terms of the wavelength of the wave, because I know that the velocity of a wave
is the wavelength times the frequency. Rearranging for frequency gets me velocity over wavelength,
and because we're dealing with light the velocity is going to be a constant c.
So I can replace v with c in this equation. So the frequency of a photon is equal to c (the speed of
light) over the photon's wavelength, and I know that c is equal to 3 x 10^8 meters per second,
so I can replace f in my equation with c over lambda, and when I do that this is what I get.
And if I rearrange this to find the wavelength I get this equation here by multiplying both sides
by lambda and dividing both sides by e, so these two facts here are constants that you're given on
every test. You would never have to memorize the constant or the speed of light. Those are given
in your data booklet, and these two equations are also given in your data booklet. So the
equation on the top tells you how to find the energy of a photon if you have its frequency
and the equation on the bottom tells you how to find the wavelength of a photon if
you have its energy, because you always know the speed of light and the Planck constant,
so you just need to worry about the energy to find its wavelength. So we now have a way of
connecting the frequency the wavelength and the energy of any photon that we're dealing with.
So I'm going to keep these equations up here and do a few examples with them in just a moment.
Before we go into that I just need to note when we're dealing with photons energy is
often reported in "electron-volts" or eV. To convert from electron-volts to Joules remember
that 1 electron-volt is equal to the charge of one electron multiplied by one volt, so multiplying
that out I find that 1 electron-volt is equal to 1.60 x 10^-19 Joules. If you've forgotten how
electron-volts work I've left a link to my lecture on electron-volts in the video description.
There will also be times when you need to identify what type of EM wave a photon is based
on its energy. To do this you would solve for its frequency or wavelength and use that to identify
where the photon is on the EM spectrum. So again I talk about this spectrum a lot more in my
electromagnetic waves lecture which I've linked in the description. Okay, let's do some examples with
these new equations. What is the wavelength of a photon which has an energy of 38 electron-volts?
So we're trying to find the wavelength, we know the Planck constant (that's always given) and we
also know the speed of light, and we also have the energy as 38 electron-volts. Because the Planck
constant works with Joules and not electron-volts, I'll have to convert to Joules before solving the
problem. So I'll use the factor-label method for that, where I know that for every 1 electron-volt
there is 1.6x10^-19 Joules, so multiplying that out gets me an energy of 6.08x10^-18
Joules. So I now have enough information to solve the problem, because looking at my equation
I only have 1 missing variable, and if I only have one missing variable I can plug in my numbers,
so when I plug in my numbers here I find that the wavelength is equal to 3.27x10^-8
meters. So that's kind of cool. If you know exactly how much energy a single photon contains,
you also automatically know the exact wavelength of the photon.
Example 2 says: what is the frequency of a photon with an energy of 112 electron-volts?
So now we're looking for the frequency. As usual we're given the Planck constant, and we know the
energy is 112 electron-volts. Converting that to Joules gets me 1.79x10^-17 Joules. So the
energy is the Planck constant times the frequency. Isolating the frequency there gets me the energy
over the Planck constant. So I plug that in, and you'll notice the units also work here where
my final unit will be 1 over seconds, because the Joules cancel out and that's equal to Hz,
so the frequency of this photon is equal to 2.70x10^16 Hz. So again if you're given
the energy you can always automatically calculate the frequency of the photon.
Example number 3 says: what is the energy in electron-volts of a photon with a wavelength
of 6.5x10^-7 meters? So now we're looking for the energy, and we know the Planck constant,
and we know the wavelength, and we'll also need the speed of light here because that's going to
show up in the equation. So I'll plug this in and start by isolating energy in the equation.
When I do that this is the equation that I get for energy, and then I plug in my numbers
and I get an answer of 3.06x10^-19 Joules, but I want that answer in electron-volts,
not Joules, because that's what the problem is asking for. So to convert from Joules to
electron-volts I use the factor-label method and I find that this is equal to 1.91 electron-volts,
so that's exactly how much energy a photon would have if it had that exact wavelength. Finally,
example number 4 says: what is the energy in Joules of a photon with a frequency of 2x10^6 Hz?
So I'm looking for the energy. I have the Planck constant and I have the frequency, so this is
a pretty simple equation. I just plug it into E equals Planck constant times frequency, and when
I do that I find a final answer of 1.33x10^-27 Joules. So that's everything that you need to
know about the connection between the frequency, wavelength, and energy that photons contain.