 Let's look at what a magnetic force does to a single moving charge. So there's our charge. It's moving with some velocity, which means it's going to experience a force if there's a magnetic field. Let's say there is a magnetic field, let's have it pointing up out of the screen. So what direction is the force going to be? All right, so we need the direction of the velocity, so right hand pointing direction the fingers, and then I need to have my fingers swing round to the B, so that means I have to swing up, and that means that my force has to be in this direction. Note that there's no way you can have the force going in the direction of the velocity or even backwards. It always has to be at right angles. Now what kind of motion do you get if your force is never in the direction of velocity? It means it never does any work. It means it never changes the kinetic energy of that particle. So that particle is never going to change speed. It's going to keep on going at speed of the magnitude of V the whole time. But what's going to happen because there's force? What's going to happen is it's going to change direction. And then as soon as it changes direction a little bit the force is going to move and so it changes direction. So after a little amount of time you'll find that this velocity is changed and then you'll find that the force changes and so on and so on. And so what's going to happen is that we're going to have this charge move in a circle. And we can figure out how big a circle because we know that the centripetal acceleration has to be provided by the magnetic force. And so the magnetic force is going to be equal to the mass of that particle times its velocity squared divided by the radius of that circle. And we also know that that force is being generated by the magnetic field and so we know how big that is. So we know that the force is also equal to the charge times the velocity. And because everything's at right angles it's the simple version so it's just the magnetic field. And so therefore these two things must be the same and so we can equate them. And then solve for R. So the stronger the magnetic field the tighter the circle you can swing it round and the faster it's going the bigger the circle be. So that all makes sense. May come as a surprise but actually this phenomenon is something that you owe your very life to and indeed probably all life on earth. Because coming from the sun is an enormous stream of charged particles high energy protons and electrons and ions. And if they were to hit the earth then they would do enormous damage to anything that was living. So we owe our lives to the fact that the earth has a very strong magnetic field. If a charged particle comes flying from the sun as though it was going to hit the earth it would hit these magnetic fields and then they would start spiraling around the magnetic field lines just going in circles and they in fact form quite complex behavior and they end up getting trapped in belts around the earth. And here's a zoom in there's sort of loosely two belts as an outer belt and an inner belt and they each are complicated and changing as the solar winds change and as various events happen to our magnetosphere but this is a big area of highly charged particles. If those particles hit us we'd be sad. In fact if they hit satellites the satellites tend to cook too. So you have to be very careful if you're going to have satellites flying through these regions. And these regions of charged particles trapped by the Earth's magnetic field are known as the Van Allen belts. The sun is so hot and energetic because of the fusion reactions going on inside it. When we try and build fusion reactors we also need to find a way to control incredibly hot plasmas of high-energy charged particles. And all modern designs work very much like the Van Allen belts in that they generate very strong magnetic fields in order to try and trap the charged particles going in circles around the field lines. And you can also use a no magnetic field to figure out what charge a particle has because the different charged particles will move with different radii and this is used extensively in a device called a cloud chamber. So in a cloud chamber you have a super saturated gas so it's full of a super saturated gas and if you give it a little jiggle then it tends to form a little cloud. And so what you do to learn about particles is you pass them through this cloud chamber and wherever they go they tend to form a little line of clouds. And you can see here we have a particle that's formed a line of clouds that's moving in a curved trajectory. And the reason it's moving in a curved trajectory is that it's in a magnetic field. And from the direction of the magnetic field and the direction that the particle is moving you can tell what charge it has. And indeed this is a famous cloud chamber picture because this is the picture in which they saw something that looked just like an electron except it was curving the wrong direction so it had the wrong sign of charge. And so this trajectory here is the trajectory of the positron. This is the first discovery of the positron which is just like an electron except it has a positive rather than a negative charge.