 A proton and an electron have the same kinetic energy. If the mass of the proton is 1800 times the mass of the electron, find the ratio of their De Broglie wavelengths. Where do we begin? Well, since we're dealing with De Broglie wavelengths, we could probably start by De Broglie's wavelength equation. We've seen already in previous videos that De Broglie wavelength, that is the wavelength associated with any object having some momentum P can be written as lambda equals h the Planck's constant divided by the momentum of the object. And this is a pretty cool equation because it's saying that if you throw a ball which has some momentum, it will have some wavelength, a ball, a moving cricket ball has a wavelength. It behaves like a wave. And that's the whole idea behind the wave particle duality that all things can behave like both waves and particles. And where does this equation come from? Well, we've derived this equation in a previous video and you can feel free to go back and check that out. But I would recommend remembering this equation as a fundamental equation, mainly because the derivation requires us to know about Einstein's relationships of energy, mass, and momentum, which is also beyond the scope of our syllabus. So it's one of those few equations in physics that you can treat it as a fundamental equation and I've just recommend remembering it. So this could be a starting point for whenever we're dealing with De Broglie wavelengths. All right, so now we are, what are we given? We are asked to find the ratio of the De Broglie wavelengths. So we have to find the ratio of wavelength of an electron to the wavelength of a proton. And what are we given? We are given that they have the same kinetic energy and we are given the ratio of their mass. We have given that the mass of the proton is 1800 times the mass of the electron. So what I'm thinking is we need to convert this equation in terms of kinetic energy and mass. Once we do that, then we can compare them, we can divide them and we can solve it. So I want to get rid of momentum and bring mass and kinetic energy. How do I do that? We know what momentum is. It is h divided by momentum is m times v. So we brought mass into the picture, so that's good. But we don't want velocity. We're not given any information about velocity. We are given kinetic energy. Can I somehow get rid of velocity and bring kinetic energy into the picture? I think we can. And I want you to try to do this yourself first before we do it together. All right, let's see. We know that kinetic energy of any object is half mv squared. And so from this, if I want to isolate v, I can just rearrange and I get v square as 2k divided by m. That is our v square. But then I can take square root on both sides and this is v. So what I'll do is I'll just substitute that for v here. And so v becomes square root of 2k over m. And if I simplify this, what do I get? I get m divided by root m is just root m. So I get root of 2k into root of m, which is root of 2km or root of 2mk. And there we have it. We now have the equation for wavelength in terms of the mass and in terms of the kinetic energy. And now we can build two equations and we can divide and we can see what we get. So again, get ready to pause and try to do it yourself before we continue. All right, let's do it. So let's divide wavelength of proton into wavelength of electron. That will equal wavelength of proton would be, I'll use the same equation. It's going to be h divided by square root of 2 times mass of the proton times the kinetic energy of the proton divided by for electron, I would get the same thing. It'll be 2 times mass of the electron times the kinetic energy of the electron. So h cancels, the two cancels. We know we're given that they have the same kinetic energy, so that means the kinetic energy also cancels, yay. And so what we end up with now is this goes on top. So we'll end up with both are under square root, so I'll write a common square root over here. We'll end up with mass of the electron divided by mass of the proton. Divide by mass of the proton. And that happens to be, let's see. We know mass of the proton is 1800 times the mass of the electron. So this would be, I just keep mass of the electron as it is, mass of the proton, I can write that as 1800 times mass of the electron. This cancels, and I get this answer to be one over square root of 1800. And I can use my calculator. I'll just take the square root of 1800, so 1800 square root, that gives me about 42.4 something. So I'll just write this as one by 42.4. So this will be one divided by 42.4. And so now I know the ratio. I can say the wavelength of electron has to be 42 times four times the wavelength of proton. And there we go. So we see that in this case, the electron ends up having a higher wavelength compared to the proton. And let's see if this makes sense. If you look at the basic equation, because they have the same kinetic energy, whichever has smaller mass ends up having more wavelength. We know electrons have tiny mass, so we would expect their wavelengths to be higher. And so this kind of makes sense. And so it's like a sense check to make sure that I haven't made any mistakes over here in, you know, while solving it. Let's try another one. A proton and alpha particle are accelerated through the same potential difference. Find the ratio of their de Broglie wavelength. Again, we can start with the basic equation. Lambda is equal to H divided by P with the momentum. And this time we are given a proton and an alpha particle. So we have to compare their wavelengths. But we are given that they're accelerated through the same potential difference. And I'm wondering how do I bring potential difference into the picture here? I mean, what's potential difference got to do with momentum? Like, what is this? Well, let's think about this. They're accelerated through the same potential difference, meaning there is an electric field probably and they're accelerated. So let's try and draw, let's try and make sense of it. Let's assume that there are two plates over here. This is positive plate. This is negative plate. We are given that the potential difference between the two plates is V. So probably there's an electric field like this over here. You know, drawing like this will actually help us make sense of what's going on. And now we are given that both of these particles, let's consider any of these particles. Let's say there's a particle, maybe there's a proton. We are given that this is accelerated. And that kind of makes sense. If I take a proton and leave it in the electric field, it will experience a force and it will start accelerating. It will become faster and faster and faster and faster and faster and faster. And by the time it comes over here, it would have some kinetic energy. And I know that if I know the kinetic energy, I can substitute like what it did in the previous problem. There is a connection between momentum and kinetic energy. Root of two mk, I can do that. So that means I need to find out, I need to figure out how to calculate kinetic energy from potential difference. If I can do that, I'm done. Is that making sense? So how do I do this? How do I figure out what the kinetic energy is going to be? Well, again, why don't you think about this? This is going back to, you know, basic electrostatics. So maybe try to give it a shot. Okay. So I think of this as a ball falling down in gravitational field. As its potential energy reduces, its kinetic energy increases. So this kinetic energy that it gained must be equal to the potential energy lost. And so what is the potential energy lost? Well, potential energy, there is potential difference. Potential difference literally tells you how much is the potential energy lost per coulomb. So if this was one coulomb, the potential energy lost is V. If this was two coulombs, the potential energy lost would be two V. Make sense? So this is a number, this tells you how much is the potential energy lost for one coulomb. So if this was Q coulombs, the potential energy lost would be Q times V. So the potential energy lost in this case must be Q times V and that must be the kinetic energy gained. And so now I can put this in this equation just like before. And do I know the charge of the proton and alpha particle? Yes. I don't need to know their charge. I need to know their ratio. And I think we do. And we are given that they are excited to the same potential difference. I think we have everything we need. I think it will be a great time again to pause one last time and see if you can substitute everything and try. Okay, let's do this. Again, I don't remember any equation besides this. So let me do it, let me redo everything. So momentum is mass times velocity. And then I know that kinetic energy is half MV square. Just redoing everything we did before very quickly this time. So V would be two K divided by M. So square root of that. So when I substitute over here, I get M divided by square root of two MK, two M divided by K. So that ends up giving me root of two MK. Just like what we got last time, very similar. And now for kinetic energy, I will substitute Q times V. So this time I'll get H divided by square root of two M Q V. And now I can do the same thing as before. I can divide the two wavelengths. I can say wavelength of the proton divided by, let's say wavelength of the alpha particle should equal H divided by square root of two times mass of the proton, charge of the proton, and the potential difference through which the proton was accelerated divided by H divided by square root of two times mass of the alpha particle, charge of the alpha particle, and the potential of the alpha particle, potential through which the alpha particle was accelerated. Cancel stuff, H cancels, two cancels. The potential difference through which they are excited was the same, we can cancel that, let's make some space. And so this means we will get, let's make some more space, okay. So this means we'll get this divided by lambda of alpha equals, this comes on the top, whole thing under root, mass of alpha particle, charge of the alpha particle, divided by mass of the proton, times charge of the proton, okay. Now, let's say if you know the ratio, I know that alpha particle is basically the helium nucleus, right? So it has two protons and two neutrons, a total four particles, and this is one particle, and mass of proton and mass of neutron are almost same. So that means the alpha particle should be four times as heavy as the proton. And so this should be, so let me just write that. This should be mass of the alpha particle should be four times the mass of the proton. What about the charge of the alpha particle? I know it has two protons, so you should have twice the charge of the proton. So charge of the alpha particle will be two times the charge of the proton. And there we go, we can cancel stuff now and we get root eight. And we can calculate what that is, but I'm just gonna leave it as root eight. So that means we get the wavelength of the proton to be root eight, root eight times the wavelength of the alpha particle. And there we go, that's our final answer.