 The man standing next to Einstein is Robert Millikan. He's pretty famous for his discovery of the charge of the electron, but he also has a very nice story in photoelectric effect. Turns out when he looked at the Einstein's photoelectric equation, he found something so weird in it that he was convinced it had to be wrong. He was so convinced that he dedicated the next 10 years of life coming up with experiments to prove that this equation had to be wrong. And so in this video, let's explore what is so weird in this equation that convinced Robert Millikan that it had to be wrong. And we'll also see eventually what ended up happening. Okay, so to begin with, this equation doesn't seem very weird to me. In fact, it makes a lot of sense now. When an electron absorbs a photon, it uses a part of its energy to escape from the metal, the work function, and the rest of the energy comes out as its kinetic energies. Makes a lot of sense. So what was so weird about it? To see what's so weird, let's simplify a little bit and try to find the connection between frequency of the light and the stopping potential. We'll simplify, it'll make sense. So if we simplify, how do we calculate the energy of the photon in terms of frequency? Well, it becomes h times f, where f is the frequency of the incident light. And that equals work function. How do we simplify work function? Well, work function is the minimum energy needed. So I could write that as h times the minimum frequency needed for photoelectric effect. Plus, how, what can we write kinetic energy as? We can write that in terms of stopping voltage. We've seen before in our previous videos that experimentally, kinetic, maximum kinetic energy with electrons come out is basically the stopping voltage in electron volt. So we can write this to be e times v stop. And if you're not familiar about how, you know, why this is equal to this, then it'll be a great idea to go back and watch our videos on this. We'll discuss this in great detail. But basically if electrons are coming out with more kinetic energy, it will take more voltage to stop them. So they have a very direct correlation. All right. Again, do you see anything weird in this equation? I don't. But let's isolate stopping voltage and try to rearrange this equation. So to isolate stopping voltage, what I'll do is divide the whole equation by e. So I'll divide by e. And now let's write what v s equals. V s equals, let's see, v cancels out. We get equals h f divided by e. I'm just rearranging this. h f divided by e minus, minus h f naught divided by e. Does this equation seem weird? Well, let's see. In this entire equation, stopping voltage and the frequency of the light are the only variables, right? This is a Planck's constant, which is a constant. Electric charge is a constant. Charge on the electron is a constant. Threshold frequency is also a constant for a given material. So for a given material, we only have two variables. And since there is a linear relationship between them, both have the power one, that means if I were to draw a graph of, say, stopping voltage versus frequency, I will get a straight line. Now again, that shouldn't be too weird because as frequency increases, stopping potential will increase. That makes sense, right? If you increase the frequency, the energy of the photon increases and therefore the electrons will come up with more energy and therefore the stopping voltage required is more. So this makes sense. But let's concentrate on the slope of that straight line. That's where all the weird stuff lies. So to concentrate on the slope, what we'll do is let's write this as a standard equation for a straight line in the form of y equals mx plus c. So over here, if the stopping voltage is plotted on the y-axis, this will become y and then the frequency will be plotted on the x-axis. So this will become x. And whatever comes along with x is the slope and so h divided by e is going to be our slope minus this whole thing becomes a constant. For a given material, this number stays the same. And now, look at the slope. The slope happens to be h divided by e, which is a universal constant. This means, according to Einstein's equation, if you plot a graph, if you conduct photoelectric effect and plot a graph of stopping voltage versus frequency for any material in this universe, Einstein's equation says the slope of that graph has to be the same. And Millikan is saying, why would that be true? Why should that be true? And that's what he finds so weird. In fact, let us draw this graph. It'll make more sense. So let's take a couple of minutes to draw this graph. So on the y-axis, we are plotting the stopping voltage, and on the x-axis, we are plotting the frequency of the light. So here's the frequency of the light. Okay, let's try to plot this graph. So one of the best ways to plot is plot one point, especially the straight line is you put f equal to zero and see what happens, put vs equal to zero and see what happens, and then plot it. So if I put f equal to zero, this whole thing becomes zero and I get vs equal to minus hf naught by e. So that means when f is equal to zero, vs equals somewhere over here, this will be minus hf naught by e. And now let's put vs equal to zero and see what happens. When I put vs equal to zero, you can see these two will be equal to each other. That means f will become equal to f naught. So that means when vs equal to zero, f will equal f naught. I don't know where that f naught is. Maybe somewhere over here. And so I know now the graph is gonna be a straight line like this. So I can draw that straight line. So my graph is gonna be a straight line that looks like this. Let me draw a little thinner line. All right, there we go. And so what is this graph saying? The graph is saying that as you increase the frequency of the light, the stopping voltage increases, which makes sense. If you decrease the frequency, the stopping voltage decreases. And in fact, if you go below the stopping voltage, of course the graph is now saying that the, sorry, below the threshold frequency, the graph is saying that the stopping voltage will become negative, but it can't, right? Below the threshold frequency, this equation doesn't work. You get stopping voltage to be zero. So of course, the way to read this graph is you'll get no photoelectric effect till here. And then you will get photoelectric effect, stopping voltage. So this is like, you can imagine this to be hypothetical. But the focus over here is on the slope of this graph. The slope of this graph is a universal constant, h over e, which means if I were to plot this graph for some other material, which has, say, a higher threshold frequency, a different threshold frequency, see somewhere over here, then for that material, the graph would have the same slope. And if I were to plot it for some other material, which has, let's say, a little lower threshold frequency, again, the graph should have the same slope. And this is what Millicon thought, how, why should this be the case? He thought that different materials should have different slopes. Why should they have the same slope? And therefore he decided to actually experimentally, you know, actually conduct experiments on various photoelectric materials that he would get his hands on. He devised techniques to make them, make the surfaces as clean as possible to get rid of all the impurities. And after 10 long years of research, you know what he found? He found that indeed, all the materials that he tested, they got the same slope. So what ended up happening is he wanted to disprove Einstein, but he ended up experimentally proving that the slope was same. And as a result, he actually experimentally proved that Einstein's equation was right. He was disappointed, of course, but now beyond a doubt, he had proved Einstein was right. And as a result, his theory got strengthened and Einstein won a Nobel Prize, actually, for the discovery, you know, for his contribution to photoelectric effect. And this had another significance. You see, the way Max Planck came up with the value of his constant, the Planck's constant, was he looked at certain experimental data. He came up with a mathematical expression to fit that data. And that expression, which is called Planck's law, had this constant in it. And he adjusted the value of this constant to actually fit that experimental data. That's how he came up with this value. But now we could conduct a completely different experiment and calculate the value of H experimentally. You can calculate the slope here experimentally. And then we know the value of E, you can calculate the value of H. And people did that. And when they did, they found that the value experimentally conducted over here, calculated over here, was in agreement with what Max Planck had originally given. And as a result, even his theory got supported and he too won the Nobel Prize. And of course, Robert Millikan also won the Nobel Prize for his contributions, for this experimentally proving the photoelectric effect. All in all, it's a great story for everyone. But turns out that Millikan was still not convinced. Even after experimentally proving it, he still remained a skeptic. Just goes to show how revolutionary and how difficult it was to adopt this idea of quantum nature of light back then.