14.2 Photoelectric Effect

Okay, today's lesson builds on what we talked about yesterday with Planck and Einstein, so now we're gonna look at a topic called the photoelectric effect. So, how it started was Heinrich Hertz did some experiments where he was trying to detect electromagnetic radiation on a coil of wire, and he would see sparks coming off when he stimulated it. But he noticed that there were electrons leaving the coil. Sort of by accident, he tried to see the spark better, so he put it inside a dark box to eliminate any outside influences. He hoped that by doing this, he would see the spark better. However, when he looked through the glass window on the box, it seemed that the spark got worse instead of better. You would think that inside the dark box, the spark would get better, but in fact, it actually got worse. Then, he thought maybe it had something to do with the glass window. So, he changed the glass to a piece of quartz, and the spark got a lot bigger than it was with the glass. This led him to realize that it had something to do with the glass. He thought that because quartz doesn't block ultraviolet light like regular glass does, then maybe the ultraviolet light shining on the metal had something to do with it. He never got any further in proving or trying to figure out why, but he just left it at that. Einstein then came along and said that, well, what's happening is we have some sort of ultraviolet light coming in, and it's hitting a metal surface. Because of the frequency, whatever the magic frequency is of that light source, that's going to cause the electrons to leave the surface. So, Einstein basically said that we're still dealing with Planck's formula, which equals HF, and the energy that the UV light has, each photon has that energy and gets transferred to the electron. And if it's big enough, if it has enough energy, then the electron will leave. So, he said that basically the key to this thing is having the right frequency, which he called the threshold frequency, symbolized as Fo. So, depending on what metal it is, it might have a different threshold frequency than some other metal. So, basically, the idea is that the energy that's going to release the electrons can't be created magically. So, there has to be some sort of work or energy that has to be used to get that electron off the surface. Einstein said that we can modify Planck's formula again to be equal to HF, but some of that energy is going to be transferred to work to get the electron off, and that frequency needed is the threshold frequency. We call this the work function. It's how much work the light gives off in order to get the electron off the metal. So, the threshold frequency is the minimum frequency needed, or if we were given it in wavelength, we can convert it using wavelength and it would be the maximum, because remember, frequency and wavelength are inversely related. So, in the formula, it would be HC/lambda. So, because the wavelength is on the bottom, it now would be a maximum, but it's the same logic no matter which one we use. Let's do a calculation. It's asking what the threshold frequency is. So, we just have to go word equals HFO, and since they're giving us it in electron volts, let's use our H as 4.14 electron volts, and solve it for our threshold frequency. That gives us 2.4 x 10^-15. In the next example, it's asking what the work function is if the maximum wavelength is 1.1. So, we'd use HC/lambda instead of HF. We do the same thing this time. It's in joules, so let's use our H as joules times the speed of light and divide by 1.1 times 10 to the negative 7. That'll give us our answer, which is 1.8 x 10^-18. So, pretty easy formulas in terms of dealing with work function, threshold frequency, threshold wavelength, and so on. Later on, Milliken set up a different experiment to kind of take this one step further. He said, "If we have ultraviolet light hitting zinc like Hertz predicted, and we have electrons flying off, then we should be able to create electricity or create a current." So what he did was, he set up a tube with no outside influence, shone the UV light on the zinc plates, and got electrons to go across to another plate and kept it going. The idea was that we'd get current going around the circuit. So, the current - electrons would leave the metal plate, go around, build back up on the zinc plate, and the UV light would shoot them out, and we would just keep on repeating around and around. He used an ammeter to detect the current. So, if he had stronger UV or whatever, then maybe the current would get higher. If there's less, then it would show less current. Or if it was the wrong light source, then nothing would happen. So, that was just a way to detect how much current was there. A couple of key things: our electrons are leaving, so that would leave that plate positive. The electrons would pile up on the other plate, making it negative, and our current flow would go towards the positive plate. What he also added to the experiment was trying to figure out how much energy those electrons actually had. So, let me erase this. We have our electrons going across, and we get our current flowing around the circuit like that. The idea was, then, how much energy do those electrons have? We know they're moving, so they're going to have some sort of kinetic energy. He tried to calculate that kinetic energy by putting a battery in there and reversing the battery. In this case, the electrons inside the battery are going to go the other way. So, our current is going to sort of cancel each other out. If our battery is weak, the electrons going around the wire are going to be stronger, so they just go through it, but they get slowed down a little bit. If he made the battery really strong, then the battery would win the battle, and the current would go around that way, so we wouldn't see anything on the ammeter. Basically, what he did was, he put a variable battery that he could slowly adjust. He turned it from zero, which was showing maximum current, and then he would increase the voltage. So, the electrons are going to go opposite to the electrons coming around. He would slowly increase that until they balanced out. Once the voltage increased and I got zero current, then at that point, we know the two have to be equal. So, he called up the stopping voltage, which is the amount of voltage required to stop the electrons from flowing. Then he just modified an old formula. Again, we use voltage equals energy over Q. So, energy equals QV. We rearrange that and say, "Well, the Ek max, the energy of those electrons, the maximum kinetic energy, is equal to the charge of an electron times whatever that stopping voltage is." This formula says determine Ek max, so it says electrons from a big zinc surface, if they give us an electric field and the distance. Domain, we have to do with this remember voltage equals electric field. Its electric field equals voltage over distance, or voltage will be the collection field times distance. So we just have to do that in our formula. Sorry, K max would be charge of electron times electric field, which is 16 n times our distance of 0.3 meters. And that gives us the total energy that those electrons have. So what we're going to do is we're going to put all of these formulas together now into sort of a new formula that we use quite often. So we basically, in this situation, we're really dealing with conservation of energy. We have energy coming in in terms of the light, and then we have energy going out in terms of the electrons. So if we think of, you kind of redraw this here, so we have some sort of light coming into a metal surface. So that energy will be HF equals HF. And we have the electron getting shot off, so that energy then would be one-half MV squared. But remember, there's also that work function so much energy is going to be used to get that electron off the surface. So we can use any variety of these formulas. We can have HF for light coming in, we'd have our work function of the metal plus re K Max. And sometimes we'll have to even go further. Our work function remember is HF oh, and re K Max could be 1/2 and V squared. So we can do any of these, or you could even use 3K Max if it was a stopping both it's question. You could use Q times V stop as well. So either of those formulas would work. So all you have to do is plug in whatever situation they give you. So for this next example, it says threshold frequency and we have EMR coming in. It's asking for our speed of electrons going out. So we need to go HF coming in, HF-O for the work function, and then one-half MV squared for the electrons going out. So we just need to plug all those in. So 6.634 H, our frequency in this case is to give us wavelengths. We're actually going to use HC over lambda instead, thank you. So we still have HC over lambda. We have 6.63 times 10 to the 8th and we divide that by 1.25 times 10 to the -7. And then on the other side, we have H again, so 6.63 times 10 to the -34 times our threshold frequency, which we give us is 1.14 times 10 to the -15. And then we'd have one-half times mass of electron times V squared. And then we just solve it for velocity. So I won't bother doing this calculation, you can see you just plug in all the numbers, solve it for V, make sure you square root it, and that's it. So quite simple when you're dealing with these formulas. Sometimes you might just say we have 6 electron volts of energy coming in, it requires 2 electron volts of work to get it off the metal, and they might ask what's the K max? So we just go 6 minus 2 is 4. So that's as simple as it gets. Or it might become too complicated, we have to plug in all the numbers and solve it for velocity or whatever, like we just did in this example. One last thing to look at when we're dealing with the photoelectric effect is quite often, just like any other thing, graphing will sometimes help us understand what's going on. So if we graph for electric. Effect, and we think about it. It's not going to have any energy. The electrons are gonna do nothing till we hit a threshold frequency. So if we draw a graph, we just get a nice straight line. And what happens is we graph energy versus frequency. So we know that threshold frequency has to be the x-intercept. Anything beyond that frequency, our electrons would leave, and anything less than that is just going to be used up by the work function. So we get a couple of key points. The first one is our x-intercept is the threshold frequency, and the y-intercept ends up being our work function. So those two values are pretty important. And then because we're dealing with energy equals hf as a whole, if we do rise over run, so the energy over frequency, that'll end up giving us Planck's constant. And it doesn't have to be that specific formula, but it works for any frequency. So you can see that rise over run, which is slope, which, in this case, is energy over frequency, ends up being Planck's constant. So the slope of the line will always be Planck's constant. And because it's constant, it will always be the same. So what that means is if you graph different metals, like in this example here, it shows copper, selenium, and calcium, and it's showing three different lines. What that means is each one has a different threshold frequency. So we have different values here. That means each one's gonna have a different y-intercept as well, a different work function. But what you do know is that they all have the exact same slope. They're all parallel lines. So that means Planck's constant has to be the same for each, which it is. So you're always going to get parallel lines. So you can see in the formula, we have hf equals EK max plus work. We rearranged that for energy through y equals hf minus W. And you can see that we basically get a slope of the equation of a straight line. So our H is our slope. Our X is our threshold frequency. Our y-intercept is our energy, which in this case is our work function. And that's it. So to answer this question, now you can see it's asking which line is which. So if you look at the three numbers, selenium has the largest work function. So that means it's going to be the highest threshold frequency. So that means the green one has to be selenium. And you can see the next highest is copper. So that means that's the red one. And our last one, the least work function, would have the least amount of threshold frequency. So that would be our calcium. And that's all for this topic.