 So we know quantitatively where electric fields come from and what they do. And we've talked about where magnetic fields come from quantitatively. Now we need to talk about what they do quantitatively. And first let's summarize what we've learned already. So we've learned about two kinds of things, electric fields and magnetic fields. And we know that electric fields are made by charges. And Coulomb's law lets us be precise about that. So if we have a point charge Q and we want to know the electric field a distance r away from that point charge, then it points away from Q if Q is positive. And the strength of the electric field is given by KQ on r squared. And for a negative charge where the sign on that Q is opposite, you just get the opposite vector, same size by pointing the opposite direction. So that's what electric fields are made by. What do they actually do? Well, electric fields affect charges. So again, if I have a charge and this time it's in electric field made by something else and put a bar under that symbol to denote that it's got direction as well, then there is a force on that charge. And the force is just proportional to the electric field and the charge. So Coulomb's law talks about the force between two charges. And so it just kind of skips electric field between here. So we can sort of put Coulomb's law in between the two. And if you have a charge Q1 and a charge Q2, a distance r apart, then the force between them is just given by this. And you see that's exactly the same as you get if you talk about the electric field, maybe a charge Q1 and the effect of that electric field on a charge Q2. So if you take this result and throw it into there, then you get exactly Coulomb's law. So Coulomb's law is not an extra law. It's just the effect of the electric field of one charge on a second charge. And so what makes magnetic fields? Well, the answer is moving charges. So we saw that if we have an infinitely long wire carrying a current, then the magnetic field, a distance r away from that would have a strength given by... But the direction's more complicated. You have to use the right hand rule. So you put your right hand thumb in the direction of the current and then your fingers point in the direction that the magnetic field's going to go. So in this case, it's going to go into the page, which we usually denote by a circle with a cross in it, which looks like the back of an arrow. The arrow is going into the page, so we see the fletching at the back of the arrow. And a common example of something that is definitely not an infinitely long straight wire that we've looked at is the solenoid. With a magnetic field due to each piece of current in each piece of wire here, so we've got all these little bits of wire that we have to take into account separately because they're all pointing in different directions. But if we add up the magnetic field due to all those pieces of wire, then we end up with a surprisingly constant magnetic field through the middle of this solenoid. And the field is otherwise very similar to a bar magnet. Now a bar magnet's magnetic field is also made by moving charges, though in this case it's the microscopic quantum mechanical spin of the individual particles inside it. And if we were to isolate just a single moving charge, and it was moving at a velocity v, and we wanted to know the magnetic field at a position that was a distance r away, then again we'd have to do a right hand rule. And this moving charge here looks exactly like a current. And so we use essentially the same right hand rule. For a positive charge we put our right thumb along the velocity and then our fingers curl down and so we can see that the magnetic field in that case would be going again into the page. Now calculating the strength of that magnetic field or the strength of magnetic fields around magnets or the strength of magnetic fields in solenoids can be done, of course. It does require a little bit more complicated calculation, so we're not going to worry about that now. As long as we can figure out what direction we expect the magnetic field to be going, we're pretty happy. And we know that we have a nice long wire, we can figure out how strong the magnetic field is away from that. But we have not yet addressed what magnetic fields do. So we saw that we made electric fields from charges and we made magnetic fields from moving charges. We discovered that electric fields apply forces to charges, and so it might be a little bit unsurprising to discover that what magnetic fields do is they apply forces to moving charges. So if I have a charge in a magnetic field, then if that charge isn't moving, it doesn't actually see any force due to that magnetic field. And so I have to have a moving charge, and if I have a moving charge in a magnetic field, then it does see a force. So you get the direction from another right-hand rule. You take your right hand, you point your fingers in the direction of the velocity that the charge is moving, you swing your fingers round to the magnetic field, and then your thumb is the direction of the force. So V, swinging around to B, thumb is F. So we look at the velocity, I put the fingers of my right hand in the direction of the velocity, I swing them round the magnetic field, and my thumb is pointing straight up at my face, and so the force is coming up at us. And we describe that by an arrow with a tip, so the force is coming out of the screen there, straight up at us. And the strength of that force is given by the charge times the velocity times the magnetic field, and there's a sign of the angle between the velocity and the magnetic field. So when they're at right angles, it's just charge times velocity times the magnetic field. And the current is just a whole bunch of charges moving together, and so if we know what happens to one charge, we can figure out what happens to a current. So if we have a current traveling along in a magnetic field, then we should be able to figure out what the force is on that. So is there moving charges in the current? Absolutely. That's exactly what a current is. So if this distance is a length L, so we've got a length L of wire, say, then we know that the current is telling us how much charge is going along that length of wire every second. So if we want to know the charge times the velocity, well, velocity is just length divided by time, and so that's just going to be the charge times that length of wire divided by the time it takes for charges to travel along that wire. And if we rearrange and note that the charge per unit time is just exactly what we meant by the current, then instead of having a charge times the velocity, we'll end up with the current times the length. And if you look at that, that has exactly the same unit. So we expect the strength of the force to be, and indeed it is. And if we want to get the direction, we can get it from the same right-hand rule we had before. Before, remember we have our right-hand fingers in terms of the velocity, and we swing it round to B. So now we have them in the direction of the current, which is the same as the velocity of the charges and swing it round to B, and so the current, we swing them round I through to B, and I have to have my right-hand thumb pointing down into the screen. And so the direction of the force is down into the screen from our right-hand rule. So if the current is perpendicular to the magnetic field, if they're at right angles, then the force is just the current times the length of the wire times the magnetic field.