And the last morning sequence, we talked extensively about the force that a magnetic field applies to a moving charge, and we talked a little bit about how magnetic fields are also made by moving charge. We had a little bit of current in a wire that's a moving charge, we had a field going around the wire, if we had current in the loop, you made a field inside we talked about natural magnets. They're actually little atomic currents flying going around the atoms, but now what we're going to do in this warning sequences talk about making a magnetic field in more detail with math. Okay, so you do that with a law called Biot savart, and they were both French, so the T is silent if I had to define the Biots of our law, I would say that it is the B field due to a current. Also, I'm going to just say current all the time and that's just going to mean a wire carrying a current, okay, so the wire is implied really, we're just speaking generally about really any current, but I'll draw them like their wires. So let's imagine then we have some wire here, doesn't have to be straight right now, we're doing just the general biots of our law for any current like that. There's a wire, it's got a current I flowing into it, and you want to know what's the magnetic field somewhere like here at point P what magnetic field do these moving charges create? You're not asking much that's all you want to know, so what is BP and you think we have some simple formula, like columns are, but we don't because there are no point currents, right? And we're doing electric fields point charges can add up and give you this nice simple expression, we went to continuous charges we had to do an integral. We have a differential, well with the magnetic field we have to go straight to the differential because there's no point currents, any current you have has to be some distributed current has to be flowing through a wire. You can have one place where there's a current because where did the charge come from and where did it go? So here we have to go straight to a differential definition so let's do that, So Biot savart would say then that you have to think about a little piece of the wire. Get to divide it up into little imaginary differential elements, so remember these things, differential elements are just imaginary little chunks, and when we're thinking about these little things then you can treat it like a point current. So one might be sitting here and we give them a little vector differential ds that just means the path, that's really the length in that case, along the direction of the wire ds. And we want to know what differential magnetic field dB is created by this little differential element Ds and then we add them up for the whole wire, so the answer is Db. At point P is u note over four pie it's a new constant will define it in a minute, times the current I it's a big I, time's Ds crossed with our hat. My gosh over R squared that looks horrible, it's got everything that you don't like it, It's got across product inside of an integral with a vector differential and then just a unit vector. Just to make it extra horrible okay, but let's go through and figure out what they are, one at a time. So Ds we already talked about, we would call that the differential element of the wire, and its direction is of course along the wire that's usually how you set up a differential element R let's look at our R Is the distance from the Ds. To the point P and you'll see it written this way and books a lot with just an R but if you really want to understand the whole setup, it's actually better to make it a vector and say it's the magnitude of the vector. It's the more formal way to say that it's the distance that way we can define the vector here okay, if we do it that way I can show you our R is just the vector from Ds two P. R is just the separation so it's the vector DS to P and now this makes our hat easier to identify and to draw, r hat would then just be the unit vector along that direction. So if I had to draw our r hat, assuming unit was only that far there's our r hat right? There are hat unit vector along r and let's see four is the fourth integer and pie now what pi is and u note, that's the one that is left you not is the permeability of free space. It's a lot like epsilon note for the electric field, the permeability of free space, how much electric field do you get for some charge? Permeability of free space helps you figure out how much magnetic field you get for some current, It's equal to four pi times 10 to the 7th, It must be tesla meters per amp, that's the constant. You want to use four pi times 10 to the seventh, tesla meters per amp, so now we're going to do is keep going and think about how to do this integral.
