Hi guys, happy Monday. How's everybody feeling today? >>Good. >>Got a case of the Mondays. [Laughter] Office Space, anyone? Office Space. It's a great movie you gotta see Office Space. Um alright welcome back. Hopefully that weekend wasn't, uh, too short. Seemed a little short to me. Um, but we had a good time, we went down to Comic-Con and walked around a bit. Did anybody go down to Comic-Con? Yeah, it was kinda- We didn't go into the convention center, we just walked around near the convention center. Lots of interesting characters down there, yeah fun place. My boy is like three and a half so he was walking around in an outfit because that's what you do, right? When you're three and a half or 35 or however old you want to be, but he had a Superman hat on. He had a Batman shirt on and he had a cape that said "Shooter Man" so he was looking good and confusing everybody. All right we, uh, have actually quite a bit to cover today and we in fact might not finish up this chapter today. We might work on it a little bit more tomorrow. So the first thing that you might notice is that the homework that was due tonight is no longer due tonight. It is due tomorrow night. Okay I see a sigh of relief over there. Several people just looked up to the heavens so that's good. Um, so let's get going on this. We'll talk for about 45 minutes, we'll take a break, and then we'll talk for another, uh, three or four hours. Okay so just get comfortable and let's have a discussion. Uh, let's talk a little bit more about this idea of flux, okay. Magnetic flux through a surface. And let's ask, let's ask this question: Let's say I have lines of B that are pointing to the right and now I have a surface like that. Okay so it's a plane parallel surface that is perpendicular to those lines of B. And we want to ask what is the flux through that surface? And remember, flux is lines of B through a surface. All right, well here comes one line. Goes through the surface. Here comes another line. Goes through the surface. Here comes another line and it goes through the surface. And so we would say what's the flux? It's three. Okay we know that it's B times the area A but whatever units you end up with, we can just say that is three. Three lines of B are going through that area. But now let's do the following: let's say we're going to take our lines of B still pointing to the right. But we are going to tip the area sideways, okay, so this is supposed to represent a vertical plane but now it is a horizontal plane. And it's still area A. What is the flux in this case? What do you guys think sitting here in my studio? What's the flux through this surface? >>Zero. >>Zero. Right? The top line skims right on past. The middle line skims right on past. The bottom line skims right on past. None of them actually go through this infinitely thin surface and so the flux is zero. Okay, so there is some relationship between B and the area which is important. Let's look at the intermediate case. So let's say my lines of B are pointing to the right. Now I'm going to draw this area but it's tilted. Okay and this is the surface normal to the area, n-hat. And let's say that this thing is tilted down at an angle, theta, relative to the horizontal. It's still area A, but now what is the flux? It's not the full three. It's not zero. It's something in between. Okay and so flux in this case. Remember we write flux with a capital Phi. Is the following. It's B times the area times the cosine of that angle theta. When theta is zero, you're back to this case. And therefore the flux, three, is just B times A times the cosine of zero degrees. Which is just B times A. In this case, it would be B times A times the cosine of what? Well the surface normal would be pointing up. So the angle between B and the surface normal is 90 degrees. And we know that 90 degrees when you take the cosine of it, you get zero. So for some angle in between, like we have here, it's just B times A times cosine of that angle. Right? Cosine of theta. All right, so let's try this for an example. And let's try the following situation. So let's say we draw an xyz coordinate system. Right handed xyz. And now let's have our B field pointing up at an angle of 35 degrees. And this is in the yz-plane. Okay so it's in the plane of the glass here. And now let's draw two different areas. Let's draw this one. And we will call this one area A xz. Because it's in the xz-plane. And then we'll draw this one and that's area A xy because it's in the xy-plane. And we want to find the flux through the xz-plane. And we want to find the flux through that xy area. Okay. How do we do this? Well we go back to our definition, right? Flux is going to be B times A times the cosine of the angle between the two. All right, so for Phi xz. What do we have? Well we've got B, we have some area A, and we're going to say that both these areas are equal. So A xz is equal to A xy and we're just going to call that capital A. Now we have the angle between this area and this B field. And now we have to be a little bit careful, right? Because the area has a surface normal to it and that's the relevant angle here. So the surface normal to A xz is in fact directly to the right. Let me make a little space here. It is directly to the right. That is the normal to that surface so if that's pointing in the y-axis and B is pointing up in this direction, what is the angle between those two? Well the other angle is 35 so this one has to be 90 minus 35. 90 minus 35 is 55. So this is 55 degrees so that's the relevant number that you want to put right there. B times A times the cosine of 55 degrees. What about, uh, the other one Phi xy? Well, A xy is pointing straight up, right? That's the normal to it. So that is along the z and we already know that the angle between z and B is 35 degrees. So that one becomes B times A times the cosine of 35 degrees. Okay so this is how you attack flux problems when you're dealing with B fields in areas that are not necessarily at right angles to each other or parallel to each other. It's somewhere in between the two.