 Let's explore how to calculate the path taken by a charge particle in a magnetic field. And then we'll be able to answer why these beautiful auroras are only seen by people at the poles. That's right. At times, these people can see spectacular light shows in the sky, but only happens at the poles. We'll be able to answer that question later. All right. So let's start with a case where you have magnetic field, which is into the board. Place an uniform into the board and let's consider a charge particle. Let's say a proton that's moving perpendicular to the field. That's what we'll start. So let me just write that down. Let's consider a case where particles are moving. Let me just let me just draw it over here. So let's say we have a proton that is moving perpendicular to the field. So the field is inwards. Let's consider the proton going upwards. The case we're dealing with is velocity of the charge particle is perpendicular to the magnetic field. I want to now know in this case, what's going to happen to the proton? How will it move? Will it keep moving straight or will it turn? What's going to happen? What is the path that it'll be taking? How do we calculate that? Well, to calculate the path of any object, we'll have to figure out the force acting on it. And we've seen before that the force acting on any charge particle in a magnetic field is given by the Lorentz equation. The Lorentz force is f equals q, the charge of the particle, times v cross b. And if you're not familiar with this, we've talked about this in our previous videos on Lorentz force. Feel free to go back and check that out. So let's figure out the direction of the force acting on this using the cross product. And then from there, let's see what will be the path taken. So why don't you pause the video and see if you can figure out the direction of the force? Remember, to calculate cross product, you use your right hand rule. So why don't you try using your right hand rule and see the direction of the force? All right, if you're given a shot, let's see. I'm going to bring in my right hand. I orient my right hand to make sure that my four fingers, the four fingers are in the direction of the velocity because velocity is my first vector. And then I curve them in the direction of the magnetic field. Now the magnetic field is inwards. So I've curved my four fingers also inwards, see? And so the thumb now represents the direction of the force. So my force acting on the charged particle will be this way. This is the direction of the magnetic force. So what's going to happen to this particle? Well, because it's pulled inwards, the particle will sort of turn. It's now new velocity after some time is going to be this way. It's important to notice that the speed will be unchanged. Why? Because the force is perpendicular. Remember, vectors, vectors don't have a component in the perpendicular direction. So this force will be unable to change the speed. So the speed remains the same. Okay, what's going to happen now? What direction will the force act now? Will the direction of the force be the same? Well, let's see. Again, if we do a V cross B, let me uncurl the fingers. Okay. Now I'll have to align my finger again in the direction of the velocity. And now again, I will curve them this way. And notice now my thumb is pointing at an angle. So the force now is going to be pointing this way. Interesting. So again, it's going to turn a little bit inwards, and this will continue. And as a result, you can kind of see that the force is always going to be pointing at a some common point, which is going to be the center. And as a result, you can kind of predict now the charge particle is going to go in a circular path. So let me draw that circle for you. So charge particle is going to go like this in a circular path. This Lorentz force is going to act as the centripetal force of the charge particle. And so what we now find is when the velocity of the charge particle is perpendicular to the magnetic field, we find that particle travels in a circle, goes in a circle. And it's going to go this way in this example. So that's how you figure out the direction. Let me give you one. Let me give you an example. What if we have magnetic field to the right like this? This is our magnetic field. And let's say we have a proton that is moving downwards. Again, notice, in the same case, we have velocity perpendicular to the magnetic field. But now I want you to use your right hand rule. Think about which direction the force is going to be. And think about how that circular path is going to look like. This is going to be in three dimensions. So you'll have to visualize a little bit. So why don't you pause the video and visualize what will be the part taken by the particle? All right, hopefully you have tried. Let me bring in my beautiful right hand. And I'm going to orient my right hand this way. Again, notice, my forefingers must be pointing in the direction of the velocity. And I'm pointing in such a way that I'm going to curve my forefingers in the direction of the magnetic field. My magnetic field is like this. So now I'm going to curve like this. Can you see that? Here we go. Curve this way. And as a result, notice the thumb will point out of the screen. And so from what we learned over here, this means this particle is also going to go in a circle. But the center of that circle is going to be somewhere out of the screen. So can you visualize what that circle is going to look like? Let me get rid of this hand. So this particle is going to go somewhat like this. Let me draw. It's going to go this way. It's going to come out of the screen. Let's go like this. And again, it's going to go into the screen. It's going to go in a circle like this. It doesn't look like a circle, but that's what it is. So here's will be the center of that circle. So again, circular path particle will go in a circular path. So this is how you figure out the path taken. But now let's consider a more general case. Can let me draw over here. What if I have a magnetic field like this? And let's consider that same proton, charge particle, proton, whatever that is. But it's not perpendicular to the magnetic field. Let's say it's moving at an angle. Interesting. What's going to happen now? How do we figure this out? Well, the way I like to think about it is I already know that if the particle was going perpendicular, I know it's going to go in a circular path. So what I want to do is divide this velocity into two components. One that is perpendicular to the field and one that is parallel to the field. And look at what will be the path for each of the component. So we already know for the one that is perpendicular is going to be a circular path. Let's now figure out what happens to the path when the charge particle is moving parallel to the magnetic field. And again, we're going to do the same thing. We're going to first figure out what the direction of the force is. And then from that, we're going to figure out the path taken. And again, why don't you pause the video, use the Lorentz force, and think about this. What would be the force acting on a particle if it was going parallel to the magnetic field? Pause and try. All right, let's see. Because we're doing a cross product, remember that cross product becomes zero when vectors are parallel or anti-parallel to each other. So because this is going parallel to the magnetic field, it experiences no force. So this component will just go in the straight line with the same speed, unaffected. This component will try to make the particle go in a circle. So what's going to be the net effect? Well, one is trying to make it go in a circle. Another one is trying to make it go forward like this in the opposite direction. And so together you're going to have a circle that's going forward. Let me try and draw that. So together we're going to have a circle. Let's use blue, okay. A circle that is going forward. Something like this. In other words, this is going to be a helical motion. So in general, what we find, in general, what we find is particles tend to go in a helical path. And look at that helical path. The axis of the helical path is the magnetic field. Can you see that? And the circle, the plane of that circle is going to be perpendicular to the magnetic field just like what we found over here. All right, let's do one last example. What if we have magnetic field this way? And imagine there's now an electron. An electron is going in some random direction. Let's say an electron is going this way. Can you pause and figure out what would be the path taken by this electron? All right, let's see. We're going to do the same technique. I'm going to decompose this velocity into two parts. One perpendicular to the magnetic field, one parallel to the magnetic field. I know this parallel component tries to keep this electron going in the same direction. Let's see what the perpendicular component tries to do. Well, velocity is upwards. Magnetic field is towards the right. And so if you use your right hand rule, I hope you can do that yourself now. I don't have the picture of that available. But if you use your right hand rule, the force, the thumb would point inwards. But remember, this is an electron. And so if you have an electron, there'll be a negative charge over here. And so you have to flip the force. That means the force will act out of the screen. And so the center of that circle is also going to be out of the screen. And as a result, the way I visualize this, the circle is going to be like this. The electron will come out of the screen, go into the screen, come out of the screen. But at the same time, it's going to go forward. And so the net is going to be, the net motion will be a helical path that will look like this. This is not a great helix, but you get the point. This is how it's gonna go. All right, now let's see why we only get auroras at the poles. It's got something to do with the magnetic field of the earth. So it turns out that sometimes our sun, everybody's sun, it launches charged particles. And these charged particles, once they enter into the magnetic field, they get trapped. Because once they enter the magnetic field, they start following the field in a helical path like this. So they will start following the field like this in a helical path. And eventually they will follow all the way towards the poles and notice that's where they enter into the atmosphere. The same thing could happen on the other end as well. So if the charged particle gets trapped over here, it'll follow the field and enter into the atmosphere only at the poles because that's where the magnetic field starts and ends. And when these charged particles enter the atmosphere, they interact with the atmosphere causing these beautiful auroras. And that's why we only see them at the poles. Beautiful, isn't it?