 Before we start applying the concepts that we learned about magnetic forces to start building something cool, we need to first get some practice done. So here's our first practice question. A proton and a helium nucleus are shot into a magnetic field at right angles to the field with the same speed. Find the ratio of the radii of their part taken. So we have protons and helium which are thrown into a magnetic field perpendicular. As you can see the magnetic field is coming out of the screen. And they're thrown with the same speed. And we have to find the ratio of the radii of their part taken. So apparently they go in circles. They take circular parts. So this is the part of a circle and we'll see why that is. We'll recall that. And so we basically need to find out what the ratio of those radii are. So let's begin. Before we begin, the first question I have is why does it go in circles? Let's try and remind ourselves why does that happen? To find that out we need to think about how to calculate part taken by any object. How do you do that? Well for that you need to first investigate the forces acting on it. See if there are no forces acting on a particle then it would just continue to go in a straight line. But if there are forces acting on it then we can use Newton's laws to figure out how does the force affect the path. So what's the force acting on these particles? Well there's only one force because it's entering a magnetic field. That is the magnetic force. This means that the magnetic force, the Lorentz force, is given by the expression f equals q times v cross b. This is the force that's acting on them. Let's look at the direction very quickly and I want you to pause and see if you can quickly figure that out. We've done this before. The direction of the magnetic force will be in the direction of v cross b. So if you take the proton, the velocity is to the right, the magnetic field is coming out of the screen. And so if you cross from the velocity vector to the magnetic field vector v cross b, the thumb points downwards with your right hand. And so the magnetic force will be acting downwards. So over here the magnetic force acts downwards and that makes the proton turn downwards. But the beauty about this magnetic force is that it will always be perpendicular to the velocity vector. Right? As a result of that, over here when the velocity is this way, the magnetic force will be perpendicular to that. Over here the magnetic force will be perpendicular to the path. So it will always be perpendicular to the path, the velocity vector. And as a result you can see it will always point towards the center. And so this magnetic force acts as a centripetal force. And so immediately we know that's why it goes in a circular path. And of course if you need more clarity on this, we've talked about this in great detail in previous videos. So feel free to check those videos out. Okay, now that we understand why it goes in a circular path, the next step is how do you calculate the ratio of the radius? For that I need an equation for the radius. Now this is also something that you've derived in previous videos. But if you're someone like me, you won't remember those results. And I actually recommend you not to remember the results. Physics is not about remembering equations. It's about taking some fundamental equations and deriving everything else. That's where the fun lies. So let's see if we can re-derive the expression for the radius very quickly. How do I do that? Well, some of the fundamental equations I know is, one is I know the expression for the magnetic force. But I also know the expression for centripetal force from mechanics. From mechanics I know that centripetal force has to equal mv squared over r. And since I know that this centripetal force is provided by the magnetic force, I can now figure out what the magnitude of that is because the direction is already taken care of. And I can plug that in over here and then I can solve for the radius. And that's how I get the radius. And then I can take the ratio. So it'll be a great idea if you're excited as I am. Great idea to pause and see if you can derive it yourself first. Okay, let's do it. So let's first take the magnitude of the magnetic force. So that's going to be magnitude of this. That's qvb sin theta. Where theta is the angle between the velocity vector and the magnetic field vector. And that angle is 90 degrees. It's given to us. You can see it from the diagram as well. Velocity is to the right. Magnetic field is coming out. So it's 90 degrees. Sin 90 is 1. So this thing is just 1. And so now I can plug this in over here and what I end up with is qvb sin this is 1. That is our magnetic force. That's our centripetal force. That equals mv square over r. And therefore if I simplify, I get r to be mv divided by qb. And there you go. That's the expression for the radius. And look, I don't need to remember this equation. I can always derive it. That's what I love about this. Okay. Now all I have to do is figure out what the radius for the helium for the proton is and divided by the radius of the helium. So let's do that. So radius of the proton divided by the radius for the helium. That will be this for the proton divided by this for the helium. So this is going to be mass of the proton times the speed of the proton divided by the charge of the proton times the magnetic field in which the proton was thrown divided by mass of the helium nucleus into the velocity of the helium nucleus divided by the charge of the helium nucleus into the magnetic field in which the helium nucleus was thrown. Okay. Now a few things are common. They're coming in the same speed. They're thrown with the same speed. So the speed cancels out. They're throwing the same magnetic field. So these magnetic fields are the same. That cancels out. I know their masses are different. I know all their charges are different. That's the final piece. So if I simplify, I get mp by qp into qhe, sorry, qhe divided by mhe. Now let's look at their masses and their charges. So I know that the helium nucleus is made of what? It's made of two protons and two neutrons. And protons and neutrons pretty much have the same mass. Unless you're doing nuclear physics and you're talking about their precise processes, we don't have to worry about the difference. Pretty much have the same mass. That means helium has four times the mass of the proton because it has four particles. And so I can write that over here. This is four times the mass of the proton. So if I cancel this, I get four. What about the charge? Well, helium, neutrons don't have charge. Helium has two protons. Therefore it has twice the charge of a single proton. And so this should be twice as much as this. So I can cancel this and I get two. And we're done. If I cancel this, I get two shear SN. So one over two is the answer. So the radius of the proton divided by the radius of the helium nucleus happens to be one over two. And you can indeed see helium ends up taking a larger circle compared to the proton. Let's solve one more example. And let me take this radius equation. It might be handy again. All right, this time we are given there are two protons that are short into the magnetic field at right angles with their velocities in the ratio one is to two. Now this time both are protons, but their velocities are different. Find the ratio of the time periods of their circular paths. So what's going on over here? Well, now we already know that these particles are going to go in circular paths and we can now assume that the magnetic field is large enough that they can actually traverse these circles. Then what we need to find out now is the expression for the time it takes for them to complete one full circle and find the ratio of that. And again, this is an expression that we have derived before, but I don't remember the result. I don't recommend you to remember those results as well. Let's start thinking from first principles again. Okay, if I want to figure out how long it takes for this to complete a full circle, how do I do that? Well, because we are dealing with uniform speeds and uniform motion, I go back to my speed equals distance over time, and therefore to calculate the time, I need to figure out what this distance is. Speed is known to me. How do I calculate the distance? Hey, this distance is just the circumference of this circle, and I know the circumference is 2 pi r and the radius is given to me. Okay, cool. So again, great idea to see if you can put it together and figure out what the time period is and then find the ratio. All right, let's do this. So I know that speed equals distance over time. So time, let me use red for that, time or the time period over here, that's going to be speed equals distance over time, time equals distance by speed. That's going to be distance divided by speed. So what's distance going to be? Distance is 2 pi r and r is mv by qv. Divided by what's the speed? Well, speed is just v. So what I get is that the v cancels out when I simplify, and immediately I have my time period equation. And again, this is what I love. See, I don't need to remember any equations. Just imagine how horrible it is to try and remember all these equations. No, you don't need them. So that's the expression for time period, and now I can compare them. What I find is interesting is that there is no speed in the time period, which means if I compare them, they have the same mass, they have the same charge because they are both protons, and they are thrown in the same magnetic field. Their speeds doesn't matter, which means the time it will take for this is exactly the same as the time it will take for this. And so immediately the answer is 1. The ratio is just 1. So t1 to t2, or t1 divided by t2, whatever that is. It's just 1 is to 1. But I'm a little curious as to why is this happening? Let's see if we can intuitively make sense of it. Why? This is thrown at twice the speed, right? So why is it taking the same time as this one to complete a circle? Well, because if it's thrown at twice the speed, if you look at the radius equation, you can see the speed doubles, and everything else remains the same, the radius doubles. So that means this thing is going to go in twice the big, this thing will go in a circle which is twice as big as the other one. Okay? So now think about it. You're traveling twice as much distance, but which twice as much speed, so the time it will take for this person to complete will be the same as this person to complete, and therefore the time period is the same. So it's a wonderful consequence that we see is that in the magnetic fields when charges are thrown, the time it takes for them to complete a circle is independent of the speed.