 So now we have enough information to look at the motion of particles in magnetic fields. And we're going to start by looking at our equation for the magnetic force, which we can express either as QVB sine theta or QV cross B if we're using the vector cross product notation. But regardless of which equation we use here, what we see is that the direction is always perpendicular to the velocity and the magnetic field. So let's look at an example of what's going to happen then. I'm going to have a uniform magnetic field pointed into the screen. That's where I've got all my little X's showing the back sides of the arrows. I'm going to take a positive charge particle and shoot it over towards the left for its velocity. Well, if you do your right hand rule on this, what you see is that the force is actually downward for this case. And that means that as the particle is moving, it's not going to go straight left, but it's actually going to be deflected downwards just a little bit. Well, now at this point, because it's got a velocity moving down and to the left, that also changes my force to be down and to the right. So as it's deflected, it's going to continue to move sort of along an arc. I can keep using my right hand rule at different points around the circle, but what becomes really obvious is very quickly is that I'm going to end up with circular motion as I go around here, where the force is always pointing in towards the center. Now I did this right hand rule using a positive charge. If I had a negative charge like an electron, it would still have circular motion, but it would circulate in the opposite direction. So to summarize, if I start out with a velocity which is perpendicular to the magnetic field, I have a particle which follows a circular path. And electrons and protons are both going to follow circular paths, positive and negative charges, but they're going to circle in opposite directions. And you'll be able to use your right hand rule for any particular case to figure out which way it should go, clockwise or counterclockwise. If I start off with a velocity which is partially along the field and partially perpendicular, I end up getting a helical path. In other words, it's going to spiral along the magnetic fields and around the magnetic fields at the same time. So it looks something like this. Now coming back to our magnetic force equation, I've got a perpendicular force. Well, from what we've studied in Physics 1, when I have a perpendicular force, that means I've got a centripetal force. It's causing circular motion. Our equation for centripetal force from last semester was mv squared over r. And our equation for the magnetic force from this semester is qvb sin theta. Now remember what we talked about, though. I could take that v sin theta and represent that as the perpendicular part of the velocity. In the same way for my centripetal force, I want to make sure that I'm talking about the perpendicular component of the velocity there as well as I'm going around the circle. So since both of these are forces, if I have a situation where the only force causing that circular motion is the magnetic field, then I can equate those two forces. And if I do just a little bit of algebra to rearrange these terms, I could solve it for the radius of that circle. And what I see is the radius of that circle depends on the mass and charge, but also on the velocity and the magnetic field. We'll use this equation in several problems. So that gives us an overview of the motion of free particles in uniform magnetic fields.