Magnetic force on a charge

We know a little bit about magnets now. Let's see if we can study it further and learn a little bit about magnetic field and actually the effects that they have on moving charges. And that's actually really how we define magnetic field. So first of all, with any field it's good to have a way to visualize it. With the electrostatic fields we drew field lines. So let's try to do the same thing with magnetic fields. Let's say this is my bar magnet. This is the north pole and this is the south pole. Now the convention, when we're drawing magnetic field lines, is to always start at the north pole and go towards the south pole. And you can almost view it as the path that a magnetic north monopole would take. So if it starts here-- if a magnetic north monopole, even though as far as we know they don't exist in nature, although they theoretically could, but let's just say for the sake of argument that we do have a magnetic north monopole. If it started out here, it would want to run away from this north pole and would try to get to the south pole. So it would do something, its path would look something like this. If it started here, maybe its path would look something like this. Or if it started here, maybe its path would look something like this. I think you get the point. Another way to visualize it is instead of thinking about a magnetic north monopole and the path it would take, you could think of, well, what if I had a little compass here? Let me draw it in a different color. Let's say I put the compass here. That's not where I want to do it. Let's say I do it here. The compass pointer will actually be tangent to the field line. So the pointer could look something like this at this point. It would look something like this. And this would be the north pole of the pointer and this would be the south pole of the pointer. Or you could-- that's how north and south were defined. People had compasses, they said, oh, this is the north seeking pole, and it points in that direction. But it's actually seeking the south pole of the larger magnet. And that's where we got into that big confusing discussion of that the magnetic geographic north pole that we're used to is actually the south pole of the magnet that we call Earth. And you could view the last video on Introduction to Magnetism to get confused about that. But I think you see what I'm saying. North always seeks south the same way that positive seeks negative, and vice versa. And north runs away from north. And really the main conceptual difference-- although they are kind of very different properties-- although we will see later they actually end up being the same thing, that we have something called an electromagnetic force, once we start learning about Maxwell's equations and relativity and all that. But we don't have to worry about that right now. But in classical electricity and magnetism, they're kind of a different force. And the main difference-- although you know, these field lines, you can kind of view them as being similar-- is that magnetic forces always come in dipoles, soon. while you could have electrostatic forces that are monopoles. You could have just a positive or a negative charge. So that's fine, you say, Sal, that's nice. You drew these field lines. And you've probably seen it before if you've ever dropped metal filings on top of a magnet. They kind of arrange themselves along these field lines. But you might say, well, that's kind of useful. But how do we determine the magnitude of a magnetic field at any point? And this is where it gets interesting. The magnitude of a magnetic field is really determined, or it's really defined, in terms of the effect that it has on a moving charge. So this is interesting. I've kind of been telling you that we have this different force called magnetism that is different than the electrostatic force. But we're defining magnetism in terms of the effect that it has on a moving charge. And that's a bit of a clue. And we'll learn later, or hopefully you'll learn later as you advance in physics, that magnetic force or a magnetic field is nothing but an electrostatic field moving at a very high speed. At a relativistic speed. Or you could almost view it as they are the same thing, just from different frames of reference. I don't want to confuse you right now. But anyway, back to what I'll call the basic physics. So if I had to find a magnetic field as B-- so B is a vector and it's a magnetic field-- we know that the force on a moving charge could be an electron, a proton, or some other type of moving charged particle. And actually, this is the basis of how they-- you know, when you have supercolliders-- how they get the particles to go in circles, and how they studied them by based on how they get deflected by the magnetic field. But anyway, the force on a charge is equal to the magnitude of the charge-- of course, this could be positive or negative-- times, and this is where it gets interesting, the velocity of the charge cross the magnetic field. So you take the velocity of the charge, you could either multiply it by the scalar first, or you could take the cross product then multiply it by the scalar. Doesn't matter because it's just a number, this isn't a vector. But you essentially take the cross product of the velocity and the magnetic field, multiply that times the charge, and then you get the force vector on that particle. Now there's something that should immediately-- if you hopefully got a little bit of intuition about what the cross product was-- there's something interesting going on here. The cross product cares about the vectors that are perpendicular to each other. So for example, if the velocity is exactly perpendicular to the magnetic field, then we'll actually get a number. If they're parallel, then the magnetic field has no impact on the charge. That's one interesting thing. And then the other interesting thing is when you take the cross product of two vectors, the result is perpendicular to both of these vectors. So that's interesting. A magnetic field, in order to have an effect on a charge, has to be perpendicular to its you velocity. And then the force on it is going to be perpendicular to both the velocity of the charge and the magnetic field. I know I'm confusing you at this point, so let's play around with it and do some problems. But before that, let's figure out what the units of the magnetic field are. So we know that the cross product is the same thing as-- so let's say, what's the magnitude of the force? The magnitude of the force is equal to? Well, the magnitude of the charge-- this is just a scalar quantity, so it's still just the charge-- times the magnitude of the velocity times the magnitude of the field times the sine of the angle between them. This is the definition of a cross product and then we could put-- if we wanted the actual force vector, we can just multiply this times the vector we get using the right-hand rule. We'll do that in a second. Anyway we're just focused on units. Sine of theta has no units so we can ignore it for this discussion. We're just trying to figure out the units of the magnetic field. So force is newtons-- so we could say newtons equals-- charge is coulombs, velocity is meters per second, and then this is times the-- I don't know what we'll call this-- the B units. We'll call it unit sub B. So let's see. If we divide both sides by coulombs and meters per second, we get newtons per coulomb. And then if we divide by meters per second, that's the same thing as multiplying by seconds per meter. Equals the magnetic field units. So the magnetic field in SI terms, is defined as newton seconds per coulomb meter. And that might seem a little disjointed, and they've come up with a brilliant name. And it's named after a deserving fellow, and that's Nikolai Tesla. And so the one newton second per coulomb meter is equal to one tesla. And I'm actually running out of time in this video, because I want to do a whole problem here. But I just want you to sit and think about it for a second. Even though in life we're used to dealing with magnets as we have these magnets-- and they're fundamentally maybe different than what at least we imagine electricity to be-- but the magnitude or actually the units of magnetism is actually defined in terms of the effect that it would have on a moving charge. And that's why the unit-- one tesla, or a tesla-- is defined as a newton second per coulomb. So the electrostatic charge per coulomb meter. Well, I'll leave you now in this video. Maybe you can sit and ponder that. But it'll make a little bit more sense when we do some actual problems with some actual numbers in the next video. See