 If a charged particle, like say a proton, is moving with some speed in a magnetic field, we've seen that it will experience a force given by this expression which we call the Lorentz force. And the direction of this force is in the direction of v cross b and we get that using our right hand thumb rule, where we cross our hands from v to b and the thumb represents the direction of the force. And as a result, the force on this would be in this direction. And we saw that this force acts like a centripetal force, making that charged particle go in a circular path like this. And we've talked a lot about this in great detail in previous videos, so if you need a clarity, feel free to go back and check that out. But what we're going to do in this video is figure out the properties of this motion. For example, we want to figure out what will be the expression for the radius of this circle. So what would be the radius of that circle and how much time would it take to complete that circle? And trust me, this has an application, it's a pretty cool application that we'll talk about in a few minutes. So let's start with the radius. How do I figure this out? Well, whenever I want to calculate or whenever I want to connect force and any properties of motion, like the radius or the time or whatever that is, I go back to mechanics, Newton's second law. Newton's second law says force F equals mass times acceleration. We know the force. We can calculate the magnitude of the force from here. Mass we can take it as m. What about the acceleration? Well, this is going in a circular motion. Ooh, circular motion, uniform circular motion, centripetal acceleration. I think I've said too much. So can you pause the video? I'm sure you'll pump right now. Can you pause the video and see if you can put all of that together and figure out what the radius is going to be? Remember, the formula is not important. What's important is how you get that. That's the fun part of it. So pause the video and see if you can figure this out. All right, let's make some space. Let's start with the force. What's the magnitude of that force? The magnitude of this is going to be Q times what's the magnitude of V cross B. A cross product is VB sine theta. So V times B times sine of the angle theta. And what is theta here? Theta is the angle between these two vectors, V and B. And B is into the board. V is this way. So that's 90 degrees and sine 90 is 1. So this number is going to be 1. That equals mass times acceleration. Mass is m. What is acceleration? This is the centripetal acceleration. Acceleration is towards the center. And therefore, centripetal acceleration, I hope you remember the formula, that's V squared over R. And that's where the R comes. V squared over R. And now we can just do the algebra. So 1V cancels out. And if I just rearrange, I'll get this to be mV divided by QB. This is divided by Q times B. And there we go. We have the expression for our radius. And let's quickly see if this makes sense to us. So it's saying that if the mass increases, it's saying that the radius is going to increase. Does that make sense? Well, yeah. If the proton of this particle had more mass, then it would have more inertia. And so we tend to have a more straight line velocity increasing the radius. That makes sense. It says if you have more charge or more magnetic field, then the radius is going to be smaller. Why would that be? Well, that's because if you have more charge or you have more magnetic field, your force is going to be stronger. If you have more force, it tends to curve more strongly, giving you a smaller radius. That makes sense. What about the velocity? This is the tricky one. Velocity has effects in both directions, in both cases. With a high velocity, you tend to have more force, but you also tend to, because it's also in your centripetal acceleration formula, you also tend to have a larger radius if you think about this. Now, because there is a V squared over here, and you can literacy derivation, because you have a V squared over here, what we're seeing is with more velocity, we are seeing that the radius tends to become larger. You can also think of this as momentum. With more momentum, we tend to get a larger radius, because it becomes harder to curve it. All right, before we go forward, a quick question for you. Along with the proton, imagine we also were to throw a deuteron. A deuteron is basically a proton and a neutron stuck together. We are to throw a deuteron inside the magnetic field with exactly the same velocity. Same magnetic field, same velocity. My question is, which one would have a bigger radius? The proton, or the deuteron, or both would have the same radius. Can you pause the video and think about this? All right, well, they have the same charge, because neutron has neutral, so there's still only same charge, one proton's charge. Magnetic field is same, velocity is same, which means the only difference between them is their mass. The deuteron, because it has an extra neutron over here, will have twice or more mass. Think of it that way, more mass. Which means that will have more radius. Ooh, you know what that means? That means imagine we had thrown both a proton and the deuteron together into this magnetic field. You know what would happen? Because the deuteron has more mass, it'll end up having a bigger, let me try and draw this properly, it'll end up having a bigger radius. And that is really interesting. I'll tell you why. Because protons will have smaller radius and deuterons will have a bigger radius, this means we can now separate these two particles. I mean, think about what an amazing application we can have. This means as long as we have charged particles with us, and if they have different masses, we can just throw them in the magnetic field and we can separate them out. We can have some kind of a collection chamber somewhere over here, somewhere over here. And the lower masses will get collected over here. And once we have higher masses, we'll end up getting collected over here. We can separate particles out, wow, just by using magnetic fields. And you might say, okay, but where would we actually use that? Let me give you an actual application of this. In the production of atomic bomb, that's right, nuclear bomb, we required a particular kind of uranium isotope. It turns out that when you actually mine uranium, the uranium ore contains lots and lots of different kinds of isotopes. And so to separate them, one of the methods you can use is this. You can heat up that uranium, you can shoot ions, uranium ions in a magnetic field, and the lighter ones will get separated in one bucket, if you're using a bucket. The heavier ones will get separated in another bucket. And that's how you can separate the isotopes. And then you can use the ones that are required to build a bomb. And in fact, that's exactly one of the methods which was used in the production of the first atomic bomb by the US. This method where we are able to now separate masses out and sort of create a spectrum. A spectrum means an arrangement, an ordered arrangement according to the masses. This method is called mass spectroscopy. Pretty cool name, but it's all it's saying is that you can separate out particles with different masses. Okay, one last question. Imagine this time I throw two protons, but one with twice the speed as the other. Then this one would have twice the radius because radius is proportional to velocity. Everything else remains the same for them. Which means in this case, let me get rid of this. The second one would have twice, twice the radius. Okay, I can't see that, but it's gonna come over here. The question I have for you is, which one would take more time to complete the turn, okay, the time period? If you're allowing them to go in circles, which one would take more time? Again, can you pause the video and think about this? All right, now my common sense is telling me that since the second particle is taking twice the radius, it's gonna travel twice the distance. Because circumference is 2 pi r, twice the radius is twice the distance. Which means that's gonna take longer, right? But wait, this is not only going twice the distance, it's also going twice the speed, twice the distance, twice the speed compared to this. What's gonna happen to the time period? Well, let's go back to our speed formula. Speed equals distance over time, so time equals distance over speed. And so if distance is doubled and the speed is also doubled, then that means the time taken remains the same. Of course, twice the distance and twice the speed, same time period. Ooh, what an amazing, another amazing thing that we're seeing is that the time period taken by both these charged particles remain exactly the same. What if this was three times the speed? Now it would be a three time the big circle, but three times the speed, again the time period would remain the same. Which means, which means the time taken, the time period is independent of the speed. Ooh, it does not depend upon the speed. It also has an application in a device called Cyclotron, which we'll not talk about right now. But that's an important result. And let's see if we can now derive the expression for time period just by using this. Again, I encourage you to pause for one last time and see if you can quickly derive the expression for time period yourself. All right, so the time period, let's call that as T, is gonna be the distance that the particle travels. The distance is gonna be two pi R. So that's going to be two pi times R and we know what R is, that's MV by QB. So let me quickly substitute MV by QB divided by speed, which is V. And the V cancels, and sure enough, time period is independent of the velocity. So we also got the expression for time period. And again, you don't have to remember these expressions. You can just derive them. So what we see is important is that the time period is independent of the velocity or the speed.