 When electric charges move in the presence of a magnetic field, something incredible happens. Something that defies your natural expectations of what a field of force ought to do on an object it can interact with. Let's explore the behavior of electric charges in the presence of external magnetic fields. This study will build us a foundation for a fuller understanding of electric and magnetic phenomena. What we're up with today is the force on a whole bunch of charges. So awesome. You take one electron, you fire it into a magnetic field, it bends. That's super cool. But what happens if you take a whole bunch of electrons and fire them into a magnetic field? Let's say through a conductor. So the question we want to kind of answer here is if I were to take a material, expose it to an electric potential difference and drive a current through it, which is just charge moving through a volume, and then run that current through a magnetic field, what would I expect a current to do? More importantly, because the current is trapped on the conductor, it can't escape the conductor, what will the conductor do in response? And I'll show you a video of what this looks like. But we can sort of figure it out. So let's begin with the microscopic picture. So we have a conductor. Let's think of it as a cylinder of material, conductor. And it's got charge moving in it. I'm going to draw them as positive just to stick with our convention that current is the direction that positive charge moves, although the electrons will be moving in the other direction from where I draw these. These positive charges, if I have managed to put some kind of delta v, some kind of change in potential between the two ends of the conductor, like looking at it through a battery, for instance, I would expect then that the charge would drift in response to this electric field set up by the electric potential difference. Well, the conductor, all right, fine, let's think about the conductor. It's got a length L. It's got a length L. It's got an area. Let's just call that A. Nothing remarkable here. I'm just writing letters down. And our charge, it's some charge q. And it has a velocity on average that is the drift velocity, e drift. Let's imagine now that I expose this bunch of charge to an external magnetic field. So I immerse the conductor in an external magnetic field. And what I'm going to do is I'm going to, just because I have pluses already written up here, I don't want to put crosses in here. So what I'm going to do is I'm going to say that the magnetic field points out of the board. So it's arrows coming out at you. So this is B. And it's everywhere. I'm just going to draw a few magnetic field lines in a few places. But it's everywhere. It's uniformly envelopes this entire material. So every charge in here, everywhere it goes, there's a magnetic field that points out of the board. OK. Well, let's just analyze. Oh, yeah. I have a picture. We talked about positrons, but if you drop plus lines from now on, do we have to assume? Electrons are moving to the left. Right. We don't need to worry about, is that a positron, or is it a photon, or are you sharing that with us? We're going to conduct it again. So I should have said earlier that there's not a whole lot of antimatter in the world around us. If there were, we'd all be dead. Actually, there was a whole lot present just after the Big Bang. And most of it went away, and we don't know why. We shouldn't actually even be here having this conversation today. Because equal amounts of matter and antimatter were forged in the Big Bang. And yet, the universe appears to be like 99.99999999999999999% matter-matter. We don't know why. That's a big mystery that we're trying to figure out. So we're talking about conductors that made of matter. They made of protons and neutrons and electrons. The electrons are what do the moving, but we talk about the direction of positive current is being the direction positive charge is moving. So if the electrons are going to the left, the positive charge is going to the right. That's all. OK. That's it. So, yeah, so don't worry too much about antimatter. There's not a whole lot of it around. We can make it, but not in large quantities. Anyone read the book or see the movie Angels and Demons? Yeah? So, OK, about no one. OK, one person. All right, so the hallmark of the book Angels and Demons is an antimatter weapon, right? And the terrible thing is that in the first 10 pages of that book, the main character gets a demonstration of the annihilation of electrons and positrons in the antimatter storage containment device. And he says there's a big flash of light. Well, that flash of light is a flash of gamma rays. They should have all died. Right there at the beginning of the book, everybody should have been dead. So it should have been a short book, but unfortunately it wasn't. It was not a great book. But the movie was punchy. So go see the movie. You have nothing else to do with your life. Go see the movie Angels and Demons. It's a fun ride. The very beginning of the movie is the Atlas experiment where I work. Although we are never allowed to sit that close to the Atlas experiment. Again, we'd all be dead because there's so much radiation coming off that thing when the beams are in operation. OK, more mundane. Conductor, positive charge, magnetic field. So if we're just talking about the magnetic force on a single charge. OK, so here's a single charge q. We would just say no problem. That's just qv cross b. And this is v drift. OK, and then the v field comes out of the board. So right hand rule. See if we can do this one. So we've got velocity, magnetic field, forces down. So I would expect the force on a single particle, any one of these, due to the magnetic field to point down. Again, velocity curl toward the magnetic field, thumb indicates the direction of the force. This gets a little tricky on exams. OK, so just practice. I don't want you to throw a shoulder out doing this. This is the most physical activity you'll get in E and M the whole semester. So enjoy it. So we could just do that. But we want to know what's the consequence of the aggregate of all of this charge. And what happens to the material in which this charge is trapped? So to do this, we need a few more pieces of our microscopic picture. For instance, current density. Current density is the number density of charge carriers n times the charge of each carrier, q, times the velocity of drift, so v drift. Current density is just current per unit area. And I'm going to, for now, put a little unit vector over here. Remember, unit vectors, you can write it. No penalty. They have a length of 1. I'm just going to write little v hat drift there to indicate that this j points in the direction of v drift. Catherine? Yeah? I thought that you said that the velocity was the velocity before it was affected by the magnetic field. So shouldn't that just be regular velocity, not v drift? Because v drift is not affected by the magnetic field. No, no, v drift is when it's affected by the electric field that I said. So I'm making a current with an electric field, and now I'm affecting that current with an magnetic field. So it's sort of two pieces. So actually, there's a sum of two forces on these charges. There's a force, q, e, due to the electric field in the conductor that's causing them to drift in the first place. And there's a force due to the magnetic field. And so the net force actually has to sum up those two things. OK? That's a good question, though. But there is a net force. And as with any other force, you do the vector sum of the forces if you were asked to find the total force due to e and b on these particles. Yeah, but there's an e. We're just not focusing on it right now. It's made the current. It's done its job, and now we're focusing on what happens in response to b for these charges. OK, so j can be related to a macroscopic thing like electric current, which we can measure with instrumentation, the area of the conductor, which we can measure with calipers, and something like that, and then the direction the charge is drifting. So this is a penalty-free addition of a unit vector on to the end, just to remind us that there's direction in j. OK? All right. Well, that's nice. What we would like to do is we would like to kind of relate the stuff that's going on in this equation to the current and other features of the conductor that's been immersed in the magnetic field. So the thing that we need, the thing that we're missing here, if we want to know the total force on the charges in this conductor, we obviously need the total charge q. And v drift, which is the same for all the charges, cross b, which is the same for all the charges. But we need big q. We've got little q. That's the charge that's actually doing the moving and the material. But we need big q. Well, that's actually not so hard to get. n is the number of charges per unit volume. So how might I get the total number of charges from little n? I want the total charge, or total number of charges here. So how would I go from number of charges per unit volume to just number of charges total? I don't know if this is right. But since we're slimming a lot of little charges, would you take it into account? Don't say the c-word. Don't say the c-word calculus? Yeah. You could do that. But actually, we could go much simpler here. OK. Can we use the A3 and then? Oh, would you like to take this and rearrange it? Yes. Let's save that for the next step afterwards. I'm asking for something much simpler. I've got number of charges per unit volume. And I just want the total number of charges in the volume. So I would do what? Well, out of all the charges, yes. But I don't know how many there are. I just know the number of per unit volume. So what's that? You want charges? I want the total number of charges present in the conductor. So what do I do with this number? Multiply by volume. Yeah, exactly. So I'm just going to take n and multiply by, I'm going to write my volume as a V with little hats on top. So not to be confused with voltage. What's the volume of a cylinder? Area times length. Exactly. So this is n, a, l. And now I'm like a hair's breadth away from the total charge. Because I know the charge carried by each of the little charges. And to get the total charge, all I have to take is q times n times the volume. And that's big q. So let me write that out. Big q is little q times the number density of charges per unit volume times the volume total. And this is just q, n, a, l. OK. Now that the total force, f, is q V drift cross B. And I can write this as q, n, a V drift cross B. And now we're going to use j. Or more to the point, we're going to use i over a. So if I go back to this equation, I can write that i is equal to, well, i in the direction of V drift is equal to n, a, q. So let's just rearranging the current density. Current density is i over a. I added that little unit vector into reminders that has direction in the direction of the drift velocity. That's equal to n, q, V drift. So if I want to solve for i, I just have to move area to the other side of this equation. So I wind up with n, a, q, V drift. Well, this is nice. I've got n here. I've got a here. I've got q here. Oh, and look, I've got V drift here. So I can actually substitute current into the force equation. So I'm going to do that. FB is equal to L, which is the only thing I haven't used so far, times i, V hat drift cross B vector. So now it's conventional at this point to do the following. The book does this. It's convention to define a vector, L vector, the length of vector, which is just L times the unit vector that points in the direction of the drift velocity. Totally legit to do that. You can always define a vector if you like. So the way the convention usually works is we define L vector. And so we have FB equals i L vector cross B. So if I know the current passing through a conductor, and I know its length, and the direction in which, say, charges drifting, and I know the magnetic field, I can not only calculate the force on each of those individual little positive charges. I can calculate the force on the entire conductor. And to figure out the force, it's easy. You just take your fingers and you point them in the direction of the drift velocity. You curl in the direction of B. And that's the direction that the force on the conductor points. And I mean the conductor. I mean the whole damn thing. Because, OK, so that's just an illustration of what I just said. So basically, if you know current is flowing this way, you know the direction of L vector. It's just the direction current flows, so V drift. So check this out. This is the so-called jumping wire. So you have a 12-volt battery hooked up with a switch. It's open right now. You have the long wire that goes between the poles of a very strong magnet, a north and a south pole. So that wire, that conductor, is exposed to a strong magnetic field. You plug in the battery and you throw the switch. Boom. Currents in magnetic fields don't play nice together. If you're designing something like a power distribution system involving high current and you've got magnetic fields present, here come the lights, you need to be very much aware of IO cross B. Because if you're not, and you can kill yourselves or other people. So for instance, I think I have a video of this. I'll show you guys later when I talk a little bit about superconductivity. But if you have loose cables next to the MRI magnet in the hospital where you're working, and you switch on the MRI magnet, and those cables are carrying significant current to drive the magnet, they can not only flail, like they look like they're in a breeze, I'll show you a video of this, they just flail, like they look like they're in a wind, like a flag in a wind. But if they rip free, carrying that current for that moment, they can become deadly projectiles, and you don't want your patients exposed to that. So always have your techs check your MRI cabling to make sure it's bolted down. IO cross B, it's not just an equation, it's a way of life.