 All right, well, we're going to kick this off. So I have a lot of things I'd like to get through today. And some of you have commented to me that magnetism hasn't been so bad so far. Well, brace yourselves, because B.O. Savard is about, it's the wall, usually, for this course, all right? All right, so Jean-Baptiste B.O., Felix Savard, they came up by doing lots and lots of experiments and using mathematics to describe the outcome of the experiment. They figured out what the relationship is between current flowing through a wire and the direction of that current flow and the magnetic field that the current appears to set up. So we have a couple of really key things that we have discovered so far about magnetism. And let me just jot them down here. Chemistry has real staying power on this board. All right, so one, magnets are affected by, well, let me write this by magnetic fields, B vector, just like charges are affected by E vectors, OK? So by electric fields. And the key difference here is that there is no evidence that there are so-called magnetic monopoles anywhere in nature. There are no isolated north and south poles. They always come as a pair, north and south together. And the reason for this has to do with the third thing I'll write down. But the second thing is that electric charge is affected by magnetic fields. And specifically, maybe electric charge, that is current. Current is affected by magnetic field, OK? So if you have a current traveling at a right angle to a magnetic field, then it will be bent at right angles to both of those things. So that's the observation number two. This is f equals qv cross B, f equals il cross B. Finally, we have the thing we're looking at right now, which is that electric current creates magnetic field. And this is what the law of Theo and Savard describes. So I'm going to kill the lights here because I think the bulb is going in this projector. So imagine that we have some conductor, this orange thing. It's carrying a current, and the direction of the current is denoted by a little i. I'll use a little i and capital I interchangeably. I know that when we have i-hats and j-hats and k-hats, that little i can be confusing. We only have so many letters in the alphabet that we can play with for this, OK? But I might use a capital I for current. So watch out for that. You have a current flowing through the wire. It's coming in from the bottom and going out the top. And if we were to pick a little section of this squiggly wire and think about the magnetic field that's generated by it, what Theo and Savard discovered by doing experiments is that if you're observing the magnetic field from that little piece of current at some distance away, r, OK, and you're just looking at the little magnetic field d-d vector created by that little piece of current, OK, i-d-l, then the magnetic field points at right angles to both the current direction and the direction of the r vector, OK? So r vector points from the source of the magnetic field to where you're observing it. Just like r vector for electric charges points from the source of the electric field to where you're observing it. So the same rules of defining r vector apply just like they did before, OK? So every little piece of current generates a little piece of magnetic field. And just like with electric fields, Coulomb's Law Problems, you have to sum the pieces to get the total magnetic field. And so you wind up summing over some path. So you do some integral over some path, like stitching all the little pieces of the wire together along this squiggly path. You sum up all the little magnetic fields and you can get the total magnetic field at that point due to all the little elements of current, all the little pieces of current in the wire. So it's a calculus problem. You break the moving charge into little pieces of moving charge, and you analyze each one. And in general, this gets very complicated. Just like Coulomb's Law gets very complicated very fast. It's much more convenient in the modern era to, for instance, use a computer to do this kind of integration here. Not analytically, that is not exactly by using an exact formula like the ones you look up in a book, but numerically by letting the computer actually break the wire into pieces and physically summing each contribution of the magnetic field db vector physically in the computer. And by doing this, you can build up, for instance, very accurate maps of magnetic field predicted for different devices. And I'll show you some of these today. Now, what B.O.S.Sivar also discovered is that there's a new constant of proportionality between the vector stuff and the final magnetic field that you get out. And they wrote it this way, or at least it's written in modern courses on this stuff in this way. It's written as a number mu naught over 4 pi. And mu naught, I'll have a slide here in a second, it's not so bad to have to remember. It's fantastic. 4 pi times 10 to the 7th, Teslas over amps meters squared, let me turn on the lights here, all right? So it's 4 pi times 10 to the 7th, Tesla, that's units of magnetic field, because that's what we want out of this thing at the end is magnetic field. So we want Tesla out of this. But we've got amps, I, and we've got, let's see, yeah, meters squared in, yes, none of this thing. OK, so we need to, actually, hang on a second, it should be Tesla meters squared over amps. There we go, Tesla meters squared over amps. So this is another number that you can commit to memory or write on the formula sheet. It has a name, let me just skip to the next slide here, it has a name. Epsilon naught, which we've seen already in electricity, so that's related to k, 1 over 4 pi epsilon naught is equal to k, that constant we encountered at the beginning of the course. Epsilon naught is actually considered to be the fundamental number. It's the number that physicists talk about when they talk about electricity. And it has a name. It's called the permittivity of free space. I think I've mentioned this before, but I'll formally mention it now. It's given its own name. It's so important in electricity that we named it. And from a capacitor, if you remember the field inside of a capacitor, an ideal parallel plate capacitor, it was proportional to epsilon naught, right? It was the area of the plates divided by the separation of the plates times epsilon naught. And so epsilon naught is meant to describe somehow if you have nothing in between the plates. It sort of describes the freedom with which electric fields have to propagate through empty space. And if you add any material into that empty space, if you put a dielectric inside the capacitor, you will weaken the electric field. So you'll change the permittivity of free space to something else by multiplying it by a number kappa that's 1 or greater. So this number physically represents what is known as the dielectric properties of empty space. And similarly, mu naught represents, in a magnetic sense, the dielectric properties of empty space. So mu naught is the equivalent of epsilon naught but for magnetic fields. And it tells us how permeable free space is to magnetic field propagation in the same way that epsilon naught tells us how much empty space permits electric fields to propagate through it. Now this might seem kind of random. What the heck is with these numbers? I mean, they're just numbers that some French guys figured out in a lab, so be it. But it turns out that they actually have a deep connection to a much more important number. And I won't talk about it today, but in one of the lectures that you're going to get in the next couple of weeks, there is going to be a lecture that talks about the nature of light. What is light exactly before we get into optics? And there isn't going to be any reading for that lecture. It's going to be just kind of a nice, fun lecture to break up the course a little bit. And it will unite absolutely everything that we'll have talked about up to that point in the course before we get into optics. And optics is just the transportation of light. That's all. One thing I want to say now that there are more of you here, so I'm going to be in Israel for 10 days starting on Monday. I'm giving two seminars while I'm there and attending three workshops. So it's on a vacation, I wish. That means that you have an exam next Thursday while I'm out of town. So rather than making, OK, so first things first, rather than making my teaching assistant teach my lectures, which I don't want to have him do, I'm going to record lectures in advance, and then you'll just have them to watch over the 10 days that I'm gone. So there will be three lectures, one on light and two finishing up magnetism that will be assigned. What Govinda is going to do is he's going to use the class time to show you how to set up and solve problems. And then you can ask him questions about the homework. So you can treat them like office hours. This is something called a flipped classroom, where the burden of learning is firmly placed on the student. You have to go and watch the lecture videos ahead of time. And then you come in ready to ask questions about the material, because Govinda's going to start solving problems based on the material in the lecture videos. So you'll be utterly lost if you don't do things in advance before coming into the class. So now, regarding the exam next week, it's only on two homeworks. It's a short exam, because we haven't had that much time since the second exam. So the third exam is going to cover RC circuits and everything up to the B.O. Savart law. So the B.O. Savart law will not be on this exam. All the consequences of the B.O. Savart law, you haven't had any homework on it. So homework seven and eight will be the only homeworks that are covered on this exam. So it's not a whole lot of stuff to have to worry about. But anyway, you should come and ask me questions this week. If you'd like to meet with me on Friday, for instance, to talk about the exam or preparations for the exam, that's fine. But I'll be flying to Israel on Sunday. So I won't be here at all next week. Will we still get the extra credit problems to work on? Yes. Yeah, that's right. So this will be extra credit, just like always. The other thing that's going to happen is that the homework assignment that I assign this Thursday won't be due until either the, well, probably I will have to do two Thursdays later. So not next Thursday, because there's an exam, the Thursday after. But the Thursday after that is Thanksgiving. So I might just push the deadline for the next homework all the way to the Tuesday before Thanksgiving and just have one big homework assignment that's due over like two and a half weeks. All right, so. So you mean the one that's due this Thursday? The one that's due this Thursday is due this Thursday. But the one assigned on Thursday may not be due until the Tuesday before Thanksgiving. I need to kind of look and see what material is going to get squeezed in there, but I think the way it's going to work is that we'll have this lecture. We have an in-class exercise on Thursday where we're going to use all the stuff from the B.O. Savart to actually predict what the strength of terrestrial magnetism is. So we're going to predict what the strength of one of these little refrigerator magnets is, just using a very simple model of nature and the B.O. Savart law. So that'll take up a good part of Thursday. And then after that, we're going to talk about the fourth thing, the fourth feature of magnetism, which I haven't gotten into yet, which is known as induction. And I'll show you a little bit, I think I'll have time today to show you a little demonstration of induction. But there'll be more demonstrations in the lecture videos. Yeah. So is chapter 29 covered or no? I can't remember if it's on. I forget what chapter we're in right now, honestly. But it's seven and eight, whatever homework seven and eight cover. That's what will be on the example, period. That's it. Okay. So you're not responsible for anything you haven't been tested on in a homework yet. Let me put it that way. All right. Okay. So going back to B.O. Savart, let's do an exercise with B.O. Savart. All right. And the exercise that I'm interested in doing is the magnetic field from a long straight wire. Christian? Yes. Yeah. We're going to get our homework back. Yeah, yeah, yeah. So Govind is still working on the homework from last week. I asked him to grade one of two problems. And so he's kind of flipping between those, getting that one. Yeah. Okay. Okay. Let's imagine we have a long straight current carrying wire. All right. So I'm going to just draw a little section of it. But I want you to imagine that it continues on to infinity in both directions. So let me just say that that goes up to plus infinity. And that goes up to minus infinity. I'm really under-representing this wire. Okay. So really long straight wire. So we don't have to worry about what's going on at the end. They're very far away from us. And we're up here at this point, P. And we're going to try to calculate the magnetic field at that point. So far, this is looking a whole lot. Like Coulomb's law, trying to figure out the electric field from a long line of charge with some charge density lambda. Well, there's no lambda anymore for this. There's just current, I. Constant current. So in place of linear charge density for this problem, we'll have I to deal with, okay? In place of DQ, okay? In place of DQ, we have to deal with DL vector. The little piece of the wire where the direction of DL vector points in the direction of the current. All right, so let's go ahead and pick a little piece of the wire. I'll shade it in here, okay? So this right here is our little so-called current element. Current element, I DL vector. I haven't done anything with coordinate systems yet. I'm just labeling pieces using VOS of R. Well, that little piece of current, that little current element, generates a little magnetic field up here. Okay? And we have one more vector to write. So actually, let's write the vector here. So as I said, we have to say where the current is going. So let's say current is going to the right. And actually, in anticipation of using Cartesian coordinates, let me make a quick little notation change here. And let me switch this to big I. So we have a current big I going to the right because I'm gonna need I hats and J hats and K hats in the second. Yeah? I just see how you got the I DL from that equation. How did you know what I DL means or stands for? It's the current. So I, that's just whatever the current is. But how that piece is I, N, D, and L. Well, it's this piece right here, usually I and DL or DS, or however the book wants to note it, I think the book uses DS vector. They're written together. So it's current times distance. And that's known as a current element. So I'm just labeling that we should. Current distance and length. I don't know the DL is a derivative. DL is a piece of length. So it's not a derivative, it's a differential. It's a tiny infinitesimal piece of length and it has units of meters. Okay. And I all together, that's just what a current element is, is an I and L? Yeah. Exactly. Yeah. So I mean in a principle we've been kind of looking at quote unquote current elements when we were looking at the force on a current in a wire I L cross B that I L appeared in there already. Now we're just looking at I in a little piece of the length DL. That's all. So we've kind of migrated to looking at an infinitesimal little slice of the current in the wire. Okay. And then finally, just like with Coulomb's law problems, the R vector, where does that go from and to? Well, it goes from the thing creating the magnetic field to the place where we're measuring the magnetic field. So it goes from the current element to the point P. So that convention remains exactly the same as before. Okay. So we're missing a coordinate system here. We need to start writing things down in terms of stuff we can actually maybe integrate, for instance. Okay. So we need some coordinates. And I'm going to choose to put my observation point right on the Y axis. And I'm going to choose my wire to lie along the X axis. So positive I hat is to the right. Positive J hat is up. And again, using this little right hand rule, X, Y, Z, Z points out of the board. So let me just go ahead and draw. There's the Z, the Z axis points out of the board like that at a 90 degree angle to both X and Y. So classic Cartesian coordinate system. All right angles. Okay. Well, we can already basically use the right hand rule for a cross product to anticipate where this magnetic field db vector is going to have to point from this current element. So for instance, we can take our index finger, we can point it in the direction of dl vector, which points in the direction of the current. Okay. I can then point my index finger in the direction of our hat, which is the direction that our vector points in. So I'm actually already doing that. And then my thumb indicates the direction of the magnetic field. The little piece of magnetic field db vector will point in this picture. So without doing any calculations, I can already create a situation that I can use to cross check my answer later. I expect that little piece of magnetic field that when I write it down, I expect it to point out in the positive z direction, the k hat direction. So let's just mark that for later. It appears that db vector will be written as something like a magnitude times k hat. We don't know what db is yet. We've gotta write that in terms of i and l and r and hopefully some Cartesian coordinates in there someplace, okay? We'll try to rewrite this in terms of our coordinate system. But we can anticipate that that's what we expect to see. Something that points in the k hat direction, something that points along the z axis. And that makes sense because the Beaus of Art law says that the magnetic field element from that current element had better point perpendicular to both dL and r hat. And since dL is in the plane of the blackboard and r is in the plane of the blackboard, the only place where you can be at right angles to both of those is either out of the plane or into the plane. So perpendicular to the plane like this, okay? So that's just, let's file that away for a second and let's do some math. Just like Coulomb's law, we now seek to represent r, r hat, r vector, dL in terms of coordinates. So let's do i dL first, okay? i dL vector. Well, a couple of things we notice. First of all, this is supposed to point in the direction that the current is pointing. And what direction is the current pointing by construction in my coordinate system? S, yeah. So it's pointing in the positive i hat direction. In fact, any current element I look at in here, the current element vector will always point along i hat. So I can write this one, for instance, as i dL i hat. So I've got the direction I'm taking care of in terms of my coordinate system, that's done. dL, all right, well what is this? This is a little piece of the length of this wire which goes out to positive infinity that way and negative infinity that way. It's just a little piece of the x-axis. So dL can be identified merely as dx. And if I were to look at another piece of this, it would be a dx over here or maybe a dx over here or over here. But they're all little dx's. They're all pieces of the x-axis for this conductor. So that's about as far as I can go. I mean, I would assume in a problem like this that you'd be given the current. Okay, so you'd be given a number for i. Now dx is something we can integrate. You can imagine that our integral, when we sum up all the little current elements to get the magnetic field, we're gonna be summing over dx's. And dx's are gonna go from negative infinity to positive infinity. So we already have our limits of integration figured out too. Let's move on to r. Okay, so r vector, well, r vector is the vector that we have to take. We're gonna break it into Cartesian components, x's and y's and z's as needed, okay? It's the path that we wanna take to go from the current element to p, all right, along our axes. So to go from our current element at some distance x out from the origin of the coordinate system back to the origin and then from the origin up to p, okay, which we'll write as being some distance d above the line. So we're some distance d above the line in this. It's actually not too bad. We have to go a distance x in the negative i hat direction and then we have to go a distance d in the positive j hat direction. And that's our r vector in terms of our coordinate systems. Okay, x, well, x is the thing we're gonna integrate over later. So that's good. We've got x in here. d is just a constant. We're just at a fixed height above the wire. In fact, no matter which current element you pick anywhere on this wire, you'll always be this height d above the line. Okay, that shortest distance from the wire to the point p is always d. So d never changes for any of the current elements. x changes, but d doesn't. And we're gonna integrate over x, so that's good news. And then, great, so magnitude of r is just going to be, well, r squared is gonna be x squared plus d squared. Okay, that's easy enough to do from Pythagorean theorem where you can square, you can take the dot product of r vector with itself to get r squared, whatever you like. But Pythagorean theorem works just fine for this. It's a right triangle. So you wanna get the length of the hypotenuse, you take x squared plus d squared, that's the hypotenuse length squared. Done. I guess rewriting terms are not using the formalization of the same thing. I'm just getting things into coordinates. Yeah, using my coordinate system, which is usually the longest step. Getting things into coordinates. Outline, and then we'll use the formalization. And then we'll get this into an integral, and then we'll integrate it. All right, so finally, we have r hat. Well, that's just r vector divided by its length. And we already have r vector, and we already have the length. So we just have to write an ugly-looking ratio here. So this is going to be negative x i hat plus d j hat, all over the, I'll write it this way, x squared plus d squared to the one-half power, or the square root of x squared plus d squared. That's what r is, it's the square root of this equation. Okay, I'm gonna write that as just to the one-half, because in a moment it will be convenient to combine exponents with other parts of the equation. Okay, well, let's see. We've got ideal vector all sorted out, r hat all sorted out, r squared all sorted out, and now we can plug this into the BIOS of our law. And I'm going to assume that you're, at this point, if you weren't at the beginning of the course, you certainly are now, that you're capable of popping up in a book and figuring out what an integral's supposed to be, or flipping to the formula sheet on the exam and reading off the result of an integral, or asking Wolfram-Alfra, how do I integrate this, and then getting the answer from it, okay? We're gonna kind of skip steps for the integrals, and then a miracle's going to occur, and we're just gonna move on with our lives, okay? So, we need the little db vector due to this little current element I've written here. Well, the easy part of the constants out in front, you're not over four pi, they never change. Now, let's deal with the numerator up here. So we've got ideal vector. Well, that's i dx i hat cross product with r hat. Well, r hat's a little more complicated, so. Let's see here. So this is going to be, let's put the numerator of r hat up here. It's this thing, negative x i hat plus d j hat, okay? And then down in the denominator, we've got r squared in the denominator of r hat, which is x squared plus e squared for the one half. So I can already simplify this by multiplying these two denominators together, okay? So let me rewrite this, mu naught over four pi. In fact, let me just group mu naught and i together up here. Mu naught over, mu naught i over four pi, put the dx here on its own, and then we have i hat cross negative x i hat plus d j hat all over x squared plus d squared for the three halves. And in fact, the integrals that we have to do here are eerily similar to the integrals that one has to do for the electric field due to a line of static charge, or any line of charge, okay? So if you dig way back in your notes, you'll see that we had to integrate something just like this. All right, so let me go ahead and now let's see. I think I've got all the pieces here, yes. So it says r squared over here, but did r get squared over here? It did, so r squared is x squared plus d squared, and that's right here, x squared plus d squared, and then I multiplied that by the denominator of r hat. I thought you found r and then put a square. No, well, no, no, I just left that as r squared. Okay. Now, we get a, oh, yeah, I'm sorry, go ahead. So is the reason that you group, like I had cross, like all that, so would i dx be a constant? Would we probably be given dx, or no? dx we're gonna integrate. dx is gonna go away once we do the integral. We're gonna sum up over all the little dx's and then it's gone. Okay. So that's just integration. But what made you group that way? Because the next thing I want to do is evaluate those cross products in the numerator. So I wanted them off on the run. So the i hat is necessary, not really the i dx part, or that's the part you wanna keep separate from that? Well, dx for now is just a number that sits out in front of all of this until we integrate it. And i is just a number, mu naught's just a number, four, and pi is numbers. So I've just grouped all the numbers together and I couldn't disentangle that x and that b from up here without making this thing look really weird. So what I'm gonna do is just try to isolate these until I can get the cross product down. And then it will actually greatly simplify in about two seconds. So let's look at just, let me just look at, so look at the cross product by itself. I'm gonna just isolate that little piece up here. So we have i hat cross negative x i hat plus, I'm just distributing the cross product, i hat cross d j hat. So all I did was I took i hat cross that term plus i hat cross that term. There's a gift. i hat cross i hat is zero. Awesome, don't even have to worry about that stupid term. Okay, well we have this one left now to deal with. So that thing's zero, so we have zero plus whatever this thing is. So I can pull the d out in front, that's just a number that multiplies either the i hat or the j hat. Again, cross product is just like a multiplication sign and the same algebra like rules that apply to algebra distribution and so forth association. That's all that holds here. This is d is just a number. It's not a vector, it has no direction. So you can just pull out in front and just leave behind this cross product. And what's i hat cross j hat? k hat d k hat. Okay, well let me go ahead and rewrite this thing now. db vector using what I've just learned. So I've got mu not i over four pi. I got dx and now I have just the distance d k hat up in the numerator. x squared plus d squared for the three halves. And that's about as far as I can take this thing without integrating it. So let's take a look at this. This has got the key thing that we kind of anticipated at the beginning. So right now we've got mu not positive number, i positive number, four pi positive number, dx we're gonna integrate that away in a second. We got x squared and d squared. Even if those are negative, they become positive by squaring. So this whole thing in the denominator is positive number and d by construction is a positive number. So we have a positive number times k hat. Number times k hat, we've done a good job. We can use the right hand rule to anticipate what the answer should look like and we can check to make sure the answer looks like that when we get through all the math. That's the nice thing about the right hand rule. It gives you a chance to think about things slowly, write a few notes down and then proceed blindly into a blazing calculation of mathematical stupendous tricks or whatever this is and then hopefully check your answer at the end, which is more valuable than anything else. Okay, awesome. So now we have to do an integral. So let's do an integral. Is it okay if I clear this out? Yeah, not a whole lot going on here anymore. Okay, so now to get the total magnetic field due to every current element in that liar, I have to do some integral over the little db vectors. So that's gonna be an integral over mu not i over four pi dx d over x squared plus d squared to the three halves. Ah, okay, well, I only need to integrate over things that have a dependence on x. And so I can take mu not put it out in front. Can take i put it out in front, no dependence on x, four pi out in front. That d, that d can go out in front because it doesn't depend on x. Now this nasty thing in the denominator depends on x. So I gotta leave it in the integral. And k hat doesn't depend on x, so I can move it out. So this winds up being an integral of mu not i over four pi d k hat integral of dx all over x squared plus d squared to the three halves. And remember I have my limits of integration from before. I'm going from x equals negative infinity to x equals positive infinity, infinitely long wire. Now that's an integral you can look up in a book, ask school for a mouthful for the answer, whatever you feel like you need to do, okay? So I'll just write down the answer. It turns out that the answer to this is just two over d squared. That's it. Not a bad integral to have to write down the answer to at the end. And so we can just plug that in for that integral and we're done. We've got the magnetic field around a wire at a distance d from the wire is just mu not i all over. We've got a two in the numerator and a four in the denominator. So we're left with a two in the denominator, pi and d. And I feel like that pi should be gone. Let me check. Nope, that's correct. That is it, ladies and gentlemen. That is the, oh, that's not quite it. Okay, that is the direction and magnitude of the magnetic field around a long straight wire, which essentially is what Biot and Savart used to understand the law they wrote down later in the beginning. Let's look at the elements of this. If I double my distance away from the wire, if I go from d to 2d, I cut the strength of the magnetic field in half. So it has a linear dependence on the distance you are from the wire. So it goes just one over the distance from the wire. Similarly, if you remember for the electric field from a line of charge, that also fell off linearly with distance. So if we were to draw what these magnetic fields and electric fields look like from a current. So let's say I have a current coming out of the board. Okay, so it's coming at you out of the board. Let's just think about the electric field lines from the charges that make up this current. They are going to radiate outward from the wire. So the electric field lines, okay, so there are the E's. And then through the number of technology, the magnetic field lines on there at the hand, they go in circles around the wire. We saw this from an experiment. Let's go back here, here. So that's what the magnetic field lines look like. And let's, too far, there we go. So again, let's kinda look back at this wire picture for a second here. If I'm up here at this distance d, viewing the magnetic field, and I use the right hand rule to figure out what direction the magnetic field points. Remember, you do current, that index finger, middle finger points along our vector, thumb indicates the direction of the magnetic field. If I go out here and I observe the magnetic field from, again, a distance d, but now along the positive z-axis, okay? So now what I have to do to get out to that point is I have to point my finger along i. I have to point my middle finger out toward that point p, and my thumb indicates the direction of the magnetic field. So up here, the magnetic field points out. Over here, it points down. If you go below the wire, it points into the board. If you go behind the wire, it points up. It's circulating. And so the magnetic field around a current carrying wire looks like this. This is a rich set of phenomena. Every time you elect chord on the floor, that thing in principle is emitting electric field lines radially outward. So charges could be accelerated by it along the radii. And it's got a circulating magnetic field around it, okay? Such that if a charge was up here and started moving toward the wire, its velocity would be at right angles to the magnetic field and it would start to bend in the magnetic field. All right, so apart from the various reasons you'd want to shield a lamp cord from being touched. You don't want to electrocute yourself. Shielding, like putting plastic around this, is also useful for basically dimming the electric field lines that come from a wire and the magnetic field lines that come from a wire. So that there aren't stray electric and magnetic fields all over the room, yeah? Can you do the right hand rule for the two points inside? The two points? Yeah. I missed my cue. Okay, so what I want to do is let me actually, or it doesn't matter where I do this as long as I get my directions right. So if I put my hand up at this point P and I point my index finger in the direction of I, which is along the positive X axis. How does that always? You point your index finger in the direction? Well, you always point your index finger in the direction of the first vector and then you point your middle finger in the direction of the second vector, so A cross B equals C, always. Yeah, so A, B, C, or in this case, D, L, R hat, B. Okay, so up here, that's where the magnetic field points. Out here, you have to rotate your hand and this is really painful, I promise you. You still have your index finger pointing in the positive X direction, but now to get from the current element out to this point, you have to follow a line indicated by my middle finger and so your thumb points down and that's about as far as I can, as long as I can hold that pose, so. Your middle finger points out, kind of toward my face, like this. It's really hard for me to do this. So it's very painful. But I thought your middle finger pointed in the direction of the R vector. Let me do this. That's good. Okay, there we go. All right, so let me construct the same here, like this. All right, so we're gonna do like, this is where having one of these all pre-built would be really handy, so. All right, so there's my little coordinate system, okay? So D L vector point this in the direction of R hat, that indicates the direction of the magnetic field. So if I apply that here, okay, I just have to adjust that back so that it points along R and then the magnetic field is at right angles to both of them, okay? So I can translate that up here. That's what that vector would look like. And if I go out here, I have to rotate this so the green still indicates the direction of the current. The R vector points out this way, okay? And magnetic field now points down, okay? And then if I rotate that around one more time, I keep going and this is a circulating magnetic field line, okay? There's actually an easy way to remember. There's another right-hand rule you can bust out. There's always another right-hand rule. You can remember which way the magnetic field lines are circulating around a segment of a wire. So even if you have a really complicated wire, like the one I showed you here, if you just isolate a tiny little piece of it and you wanna know where its little magnetic field lines are circulating before they're summed up with all the others, you just take your thumb, you point it in the direction of the current and the way your fingers bend, they tell you the direction of the magnetic field. So thumb out current and look at that. They curl in the direction of P. If the current pointed in, the magnetic field lines would circulate in the opposite direction. So this is another way you can remember. Current, sense of rotation of the magnetic field, okay? Questions on the BIOS of our law. I'm gonna use it for one more thing today. So I wanna make sure people are semi-hoover comfortable with this. I know it's warm and sleepy in here. Cross-products are hard. See, you should never complain about the climate in this room, cause you'll never get what you want, right? It's like a deal with the devil. It always seems like a good idea at the beginning and then you're like, oh, unforeseen consequences, right? So asking for facilities to alter the temperature in a room like this is really bad. If anyone needs some candy to get through the class, come on down and help yourself, okay? So this is not a process. It's new. No, you can ignore absolutely everything I'm saying today and pretend you never know if I'm gonna do that, okay? It's just, there's no time to assign homework on this before the exam, so. Nonetheless, it's important material, okay? You look skeptical, can't you? What's that? The final will cover this material. If the final is cumulative, so, yeah, yeah. Actually, so, it's a game which turns you on what BIOS of art is for. Absolutely, BIOS of art tells you what does the magnetic field look like that comes from moving current? So the thing we're probing right now is that based on experiments and then the work done by the BIOS of art, moving charge creates magnetic field. Moving charge appears to be one of the primary sources of magnetic fields in nature. So by, you know, URSTED, Hans Christian URSTED accidentally discovered this while prepping a lecture demonstration for his students and BIOS of art in that same year did lots of experiments and figured out what the rule is, what that equation is, that describes it. So anytime you have a current, any moving charge at all, it will create magnetic field and that is how you figure out what that magnetic field looks like, okay? So moving charge makes magnetic field. Magnetic fields bend moving charges. So there's an elegant symmetry to this which is no accident as you will see in a few lectures. There is, none of this is an accident. It may seem weird and strange, but there's a very good reason why magnetic fields influence the motion of charged particles and the motion of charged particles create magnetic fields, okay? We'll have one more piece that we have to set up. But I'll get to that in the video lecture, okay? I'll preview it a little bit today. All right, we're gonna use this one more time. So in case you're feeling sad, like you're not gonna get to see BIOS of art used one more time, you're gonna get to see it used one more time. All right, so commit this ultimate memory because it's going away. Going once, going twice. Now, a line of charge is all well and good. A line of current is all well and good. But we've looked at in the previous section of this class, circuits. Circulating currents, okay? Moving in resistors, being pushed out by batteries. And we can imagine a very simple circuit. The simplest circuit of all, which is just a circular loop. And in fact, on the homework, you have problems involving circular loops. So we've already kind of looked at what happens if you have a loop of current like a square loop or a circular loop, so some current I, and you immerse it in an external magnetic field. And we looked at that in class and we saw that a torque gets created and it causes the loop to rotate, okay? And you can describe the response of the loop using something called its magnetic dipole moment. And the magnetic dipole moment turns out to be a very fundamental concept in nature introduced for convenience sake, but it turns out it's applicable to even things like a singular electron. A singular electron contains its own internal magnetic dipole moment as if it's a little bar magnet. We can think about now, if you have a loop of current, what the magnetic field looks like that it's generating. I'll show you a picture in a minute, well, not a minute, maybe a few minutes, of what this looks like in its totality, but we can, for instance, ask, what does the magnetic field look like at dead center in the loop? The reason I picked dead center is it's an awfully easy place to do the Theosovar law. If I go off that point, let's say, if I were to go up along this axis that cuts through the center of the loop, it gets more complicated anyway. I'll show you a picture of what these magnetic fields look like, but what's really kind of cool about this is we'll use Theosovar, we'll get the magnetic field in the center, and then we'll use that for something else. I'll show you a picture of what the magnetic field looks like from a loop of current, and I want you to tell me what it reminds you of. So keep that in mind. I'm gonna ask you to tell me in a bit what does it remind you of to see the magnetic field from a loop of current? But let's right now focus on the magnetic field dead center in this loop. Okay, well, let me redraw this so that it's a bit more conductor-like. So, I'll give this thing some thickness. So this is like a hoop of conductor with a current moving through it, and let's say the current is going like this. It's circulating this way. So the current up here points that way. So it's going in a circle like that, and we're dead center in this loop. So it almost looks like a little wedding ring, like a little male wedding band. So it's exactly like a little platinum thing on our finger. So you're at the dead center in this, and there's a current circulating in it, and we wanna figure out what the magnetic field looks like at the center of the hoop using Theosovar. So we already know, we gotta write this silly thing down, U knot over the four pi, and then we're gonna have I, the dL vector, we gotta figure out what that is, our half, we gotta figure out what that is, and our squared, we gotta figure out what that is. Okay. So, let's start by drawing a current element in this circle. Okay, so I'm just gonna pick one up here. All right, so here's my little I, I, dL vector, and dL vector points tangent to the circle at that little segment, all right? So dL vector is pointing in the direction of the current, and the current is a circulating thing. So if I redraw this from overhead, all right? So here is, here's the point P. Some coordinate axes on this thing. So we're looking at this from overhead now, okay? I've picked a little current element up here, and since the current is circulating like this and I counterclockwise sense, up here, this little dL vector points tangent to the circle at that segment of the circle, okay? So it's, it's like the derivative, it's just a little tangent there. Any other element I pick, it points tangent to that element. So if I pick an element down here, it points that way, over here it points up until we get back to where we are and it points out again, okay? So again, this looks weirdly familiar. This looks just like a circular ring of charge that we dice up into little pieces of the circle and we wanna figure out not what the electric field is now, but the magnetic field due to the totality of all the pieces of the circle. All right, so I've got coordinate axes. I got an x-axis, I got a y-axis. The z-axis points out of the board, all right? Again, x, y, z, okay? I'm missing something, what am I missing? Yeah? Well, I have a question. So why is it that it points tangent? Is it similar to the radius how it has the x and y components? Well, let me put it this way. Imagine you have a circular race track and you're, you have a car that's driving in a circle around the track, much like the charge is going in a circle around the hoop, okay? And you wanna describe the velocity, instantaneously the velocity of the car at any time. What direction, so if you have a hoop, all right, we have a track, the track has a radius, r, and you need to write down the velocity vector at an instant in time for the car going around the hoop. Where does it point relative to the radius? Yeah, it's 90 degrees to the radius, so it's tangent to the circle always, and that holds no matter where you are on the race track. So it's the same thing here. If you imagine the electrons that are moving or the ions that are moving in the other direction, they're moving like a car around a race track. Instantaneously, at any moment, the current points tangent to the circle. Yeah, well, remember that, well, that length vector, it points in the direction current points in that segment. So that, the vector part of DL vector is defined entirely by the direction I points. Okay, I would have thought that the current would still be in a certain direction. Is that what it's meant to be? Exactly, yeah, and if you think, what we're kinda doing here is we're kinda telling the current, you know, freeze for a moment, and then we're looking at what direction it was moving in, just as it froze, and we're kinda writing that static picture down using these characters. Okay, in reality, this thing is continuing to circulate, right? We could continue to follow that little piece of the current around, but we kinda freeze frame it in a photo for a moment, and then see where it was going at that instant in time. And then we analyze the problem by then adding up all the pieces, so. It's equivalent to sort of following the current all the way around, but it's easier to do it this way than to get the board to rotate, so. Okay, so what am I missing from the picture? Very crucial part of the setup. Do you have to do your point or your I has j of x? Yeah, okay, but specifically, what major piece is missing? R vector. R vector, yeah, I'm missing R vector. So where's R vector gonna point? What's that, 2p from? From IDL. From IDL, from the current element. Yep, exactly, so I need to put that in here, okay? So humor me for a second, and let me denote this as big R vector, all right? So big R vector is our radius vector, okay? All right. We're pretty much ready, oh, sorry, actually, yes. Sorry, where did the 2p come from? Where did the, where did the what now? Towards p. Oh, towards p, yes, yeah, thank you. Yeah, yeah, okay, yeah, no, that's good. I hadn't thought about that. Yes, that could sound wrong. Yeah, points towards p. Okay, so let's start writing down stuff in terms of coordinates. I'll keep using this space over here, okay? So IDL vector, well, the length of DL is a little chunk of the circle, okay? So it's a little arc length of the circle. So this is, again, this looks scarily like a circular ring of charge. So we could write this as IDS vector, so just a little arc length. And if you remember, S is equal to R theta, okay? So I'm gonna do something now that may look a little frightening, but I promise you it's not as bad as it seems. I want you to see this at least once. I could write this as now DS vector equals R. Well, R is fixed, no matter where I am in this hoop, in this ring, I'm at the same physical radius from the center. But my angle changes, right? So if I pick a segment of the hoop over here, I've gone from maybe 45 degrees to 135 degrees. So theta's changing as I go through the circle. So I already know that I can write this in terms of D theta. But the thing that might freak you out for just a moment is I'm gonna write a little vector hat over that because I feel like it today. And I'm gonna introduce to you something which is formally circular coordinates. And you'll see why, because it makes solving problems like this like ridiculously easy. There's a reason why mathematicians use this, okay? For a coordinate system, you only ever need three perpendicular directions, all right, sort of very Cartesian-like coordinate system. You just have to pick three independent axes so that none of the axes overlap with each other. And this is what's known as a basis, and you can choose any basis you want in math as long as you follow the rules of picking them to be so-called orthogonal, that is 90, 90, 90, okay? All 90 degree angles. This direction theta vector, or more to the point, theta hat, okay? The direction of theta. It just points in the counter-clockwise direction. So if I have a circle like this, okay, and I draw a radius out like that, this is just another circle, let me just draw this for a second, okay? So I can describe any point, any point on the circle by its distance from the center and, of course, its angle theta. So what the heck is the theta hat direction then? Well, theta hat at all times points 90 degrees to the r vector and is positive in the counter-clockwise direction, all right? So theta hat is perpendicular in this plane, perpendicular to r vector and positive, because you have to define the positive direction when moving counter-clockwise. This is a standard math trick to define circular coordinates. So that's what this is called, circular coordinates. The reason I go through the effort of writing this down is because we can use this to get through this problem like super quick, okay? We need one more thing, and that is with Cartesian coordinates, i hats and j hats and k hats, we had these cross-product rules, right? So in Cartesian, we had i hat cross j hat equals k hat, do the conveyor belt thing and get the other positive cross-products, swap any two on the left and you get a minus sign on the right, okay? What's the equivalent in circular coordinates? Well, let's think about it a second. We can actually use the right hand rule, okay? So if I point my finger in the direction of r and I point my middle finger in the direction of theta hat at that moment, which again is 90 degrees to r and positive in the counter-clockwise direction, my thumb indicates the direction of the third spatial coordinate axis, which we'll just call z, okay? So z, why not just keep that? z points out of the board, just like it did before with x and y, okay? Circular coordinates can be related to Cartesian coordinates very easily, although they're a lot easier for this problem. And the cool part of this is remember, once I have defined my i, j, k, I know that i hat cross j hat is k hat. So in circular coordinates, I know that r hat cross theta hat is k hat, the vector that points out at the board. Do the conveyor belt. K hat cross r hat equals theta hat. Theta hat cross k hat equals r hat. Swap any two of these, put a minus sign on the right hand side. You've got all six rules for your cross products like that, okay? You only need to be able to write down one of them. You can figure out all the other five. That's why I like this way of doing it. So let's keep these in mind as we now go back to this picture. So I have defined ds vector is rd theta hat, and I can rewrite this as r d theta theta hat. That is, it's a little differential piece of angle, a little wedge of pizza pie with angle d theta pointing in the theta hat direction. So positive this way. Great. So in circular coordinates, we've done our job. You were saying r or theta hat, but you wrote vector. What do you mean? When you wrote rd theta vector. Yes, and so I wrote that as a magnitude and a direction. So I just took d theta and I decompose it into d theta and theta hat. You were just saying theta hat instead of theta vector up there. Oh, there? Oh, that should be d theta vector. Yeah, and I wrote it with a little vector thing over it, not the little hat symbol. If I said hat, I meant vector, but I wrote it correctly. Okay, well let's get r vector. So r vector is it has a direction that points not in the positive r hat. Remember, okay, so this is positive radii directions. These are negative radii directions back toward the center. So I have to take a journey to go from this element to the point P using my circular coordinate system. The good news is this is why circular coordinates are so awesome. I'm only moving along one coordinate axis in circular coordinates. I'm moving along the radius. And I'm doing it however in the negative direction. So I'm just gonna move a distance r in the negative r hat direction. Done. This is why circular coordinates are awesome. Okay, so now I can start doing things like, all the magnitude of r is just big r. And our hat, okay, the unit vector that points back toward the center here is just going to be, oh, you know what, hang on a second, let me, yes, that's fine. So this is where I'm being naughty, okay. I'm trying to figure out what the components of this r vector are. So I should have written this as big r vector. I'm trying to leave this as my circular coordinate axis. So little r is circular coordinate axis of this thing here, okay. So the magnitude of r vector is just r. That's not rocket science. And then we just need r hat. Well, r hat is just going to be r negative r hat all over r, which is just negative r hat. Okay, I have all my pieces. I've got everything in circular coordinates. IDL is r d theta theta hat. r vector is just the magnitude times negative r hat. Magnitude of r is just r. And then the hat direction of the big r is just negative r hat, the negative circular coordinate radial direction. I'm ready for a cross product. So I'm going to do a cross product. And I'll keep those rules uncovered. So dv vector at the center of the circle is write the constants down, u naught over four pi i d theta theta hat cross, let's see, our big r hat is negative little r hat. And this whole thing is over the magnitude squared, which is just r squared. All right, well, mu naught and four pi and i, they don't change in an integral. And in fact, we know we're going to be integrating over theta because I got d theta in here. So let me pull them out in front. Mu naught i all over four pi. And I'll put the d theta here. Actually, I can pull the one over r squared out too. So I got d theta over r squared. And now I'm just left with this cross product. Theta hat cross negative r hat. Let me pull the minus sign out of this whole thing. Mu naught i over four pi d theta over r squared times negative theta hat cross r hat. Consult r tables. r hat cross theta hat is k hat. If I swap these two and I compute theta hat cross r hat, I get negative k hat. So this is negative k hat. I have two minus signs. Negative k hat is just positive k hat. So I just have this. Okay, so that is db vector. That's it. Okay, so why do I screw up? I must have screwed something up. That went too easily. Anyone have a problem? So the last step is to sum up the pieces. So we have to do an integral. All right, so let me slide this over now. So to get b vector, we have to integrate the little pieces, which is in this case the integral of mu naught i over four pi d theta over r squared k hat. And we're integrating over theta, which goes from zero to two pi radians. Easy to use integral in the world. This is why circular coordinates are awesome for circular problems. Because usually at the end of a circular problem, you just wind up integrating over something like theta and there's no dependence on theta anywhere in the problem. So you have like the integral of dx, which is just x, and then you just evaluate it at the limits. Okay, so this winds up just giving us mu naught i over four pi r squared k hat two pi. The integral from zero to two pi of d theta is just two pi. Pies cancel, you're left with the two down here. And so at the center of the circle, just at the center of the circle, two r squared. In the, keep doing that in the k hat direction. Okay, so the magnetic field right at the center of this circle points up the positive z direction right out of the center of the circle. Yeah, thank you. A b center, sorry, I couldn't touch your circles. You didn't spot my mistake. What did I leave out? IDL, I knew something felt wrong. Units were off on this. R d theta, theta hat. But when I put it up here, I left the R out. So there should have been an R in the numerator from R d theta, theta hat. You knew something felt wrong about this. So there should be an R here, R here, R here, R here, R here. And then finally, that R in the numerator cancels with one in the denominator. That's the correct formula. That feels much better. If you teach a course enough, you have a nagging sense, you've done something stupid. So there's my stupid thing for the day. Sorry, I just forgot that R, which cancels with one of the R's down here. I just carried it through here just to keep this simple R, R squared, R, R squared. And then finally I canceled it in the last step. Yeah, yes. So do we have to go through this whole process every time we're asked this question to get to the B center? Well, you know, for instance, you could be asked set up the integral but not actually integrate it. So I'd expect you to be able to write down the coordinate system, write down the pieces of the BIOS of art law in terms of the coordinate system, do some cross products, get an integral, and don't do anything with it. Or I just give you the result and say, okay, now that's the result of doing an integral. What's the final answer you get? Yeah. For that, maybe they're like. Come up with it? Yeah. I took math classes and they showed me how to do this. I mean, honestly, yeah. It's, the way you can make up a coordinate system is your rule has to be, is that your coordinate axes, whatever they are, they just have to be at right angles to each other. And so if you pick the, I mean, unhide this here, if you pick the radius to be one of your coordinate axes, you know, the other axis can just point out of the circle. So you've got two of them already that are defined at right angles. You've got radius, you've got z direction. You just need a third one that's at a right angle to both of them. And so you can define this thing called theta hat that always points at a 90 degree angle to the radius no matter where you are in the circle. So basically you're just respecting the fact that a circle is circular. And anywhere you are on it, you just reorient your coordinate axes so that it always points out along the radius to the left of the radius for the theta hat direction. And then up out of the circle for the z direction. It was advantageous because otherwise, I would, if I had used like dx and dy, I would have had to rewritten ds in terms of dx and dy. I would have had to have written r hat in terms of x and y and i hat and j hat. And I would have had a two dimensional integral to do at the end. So I would have an integral over dx and an integral over dy that I'd have to do. And that's calculus of multiple variables, which I'm trying to avoid in this course. So it seems weird, but it lets us do a one dimensional integral at the end because the only coordinate direction that's changing is theta hat. Because we're fixing our radius in the circle to be constant. A little bit, I didn't formally introduce circular coordinates largely because we didn't need them. And because when we were doing the electric field due to a circle of charge, we didn't have this funny cross product. We didn't need a third direction. When you do magnetic fields, you always need all three dimensions because if two of your vectors lie in one plane, a third vector has to lie out of the plane. We're stuck doing three dimensional physics, no matter what. So I protected you guys from it for electric fields, but I can't do that any longer with magnetic fields. It's unavoidable. And so you could always say foo on me for doing that earlier in the course, but I tried. I tried to ease the flow at the beginning of the course. The way this course usually works is you feel demoralized at the beginning because you're like, what the hell is going on? I don't understand electric fields. And then you get the circuits, and you're like, well, this ain't so bad. This is algebra. I feel like a mathematical God or goddess right now. Then you get the magnetic fields and you're like, holy, there are three dimensions and there's like cross products and oh my God, I could have done that to you earlier, but I feel like you need that confidence boost in the second part of the course before we get into this, okay? It's a teaching style. My secrets are revealed to you now, okay? All right, so let me show you some pictures. Pictures are worth a thousand words. So I have to kill the lights here again because I gotta talk to facilities or classroom folks about this projector. All right, so I already went through that. All right, so we went through this. So the, well, okay. So this is just the wire again. I already went through the right hand rule. Okay, so something I wanna talk about now. What are the consequences of the BO Savart law? Let's go back to the wire again. All right, so the magnetic field from a wire. If, let's focus on wire A up here. So I have a long wire labeled A. It's carrying a current that way, so I'll point my thumb in it. And if I curl my fingers around my thumb, that tells me the direction that magnetic fields are circulating around this wire due to the moving charge. So the magnetic field lines are circulating out of the, this is sort of a funny angle, around at right angles to the wire. What are those magnetic field lines doing? There's another wire over here, and it too is carrying a current, I, B, let's say. So wire B points in that direction, and suddenly it encounters a magnetic field due to the first wire. What happens when you have a current traveling through an external magnetic field? What does the current do? I failed as a teacher. I failed. That's it, they won't pay me this year. External magnetic field, B, caused by something else, we don't care what. I, traveling through the magnetic field along some line L. And hint. What's that? Yeah, there's a force due to that external magnetic field on the other current. So if the source of the magnetic field is another wire, which we know from B.O. Savart is absolutely possible, in fact, very likely, this wire will feel a force due to the magnetic field from wire A penetrating the current on wire B. All right, so where does that magnetic field point? Well, it's circulating around wire A. So over here, it points down, and that's what's indicated here. B vector magnetic field due to wire A points down. And so if we think about the forces involved, we have a current, a magnetic field, I L cross B, and then the resulting, well, I'm trying to do this way, the resulting force points toward the other wire. This wire exerts a force due to its magnetic field on this wire that pulls them together. And if you look at the problem the other way around, well, what about B? Doesn't B cause a force on A? I mean, B carries a current. It's got a magnetic field that circulates around it. That magnetic field penetrates the conductor A, which is carrying a current, I A. It too feels a force, and that force points from A to B. So they pull each other together. When you have currents pointing in the same direction by two neighboring wires, they exert mutual forces on each other that cause the wires to move together. If the currents point in opposite directions, they repel. So this is a bit like, you know, to a north and a south pole next to each other, a north and a north pole next to each other. Has anyone ever been under high voltage power lines before? Anyone ever gone running or hiking in the wilderness around the city, just walked, like maybe by the Katy trail, or on the Katy trail, I think there's some high voltage lines that go over that. Has anyone ever done that before? Morgan? Yeah? Or does that just depend in the air? Okay. Lynn? Yeah? What have you noticed? What have your senses picked up when you're around high voltage power wires? Is something really obvious? Loud crackling. Yeah, loud crackling sound. Actually, it's a buzzing sound. It sounds some, well, okay. So that's what a high voltage, a high tension wire looks like. All of these conductors are stuffed together in a single bundle, and they're all carrying current in the same direction. And actually, they carry alternating current. So we're not gonna cover this in the course, but alternating current is sort of just what it sounds like. For one moment in time, the current points one way. And then another moment in time, the current goes to zero, and then it points the other way. And then it goes back to zero and points the first way. This was, this way, you get a very good transmission of electricity at low loss over long distances, which is important in an electric power transmission system like in the U.S. or any country, okay? So alternating current is just the current goes one way in, let's say, one sixtieth of a second, and it goes the other way, the next sixtieth of the second, and then back and forth and back and forth and back and forth, and you can still do work with that current. And after all, that's what we want. Alternating current is what powers the lights in here. Alternating current is what comes out of this wall socket here, and you can analyze it. It's a whole chapter in the book on it, we're not doing it, okay? But it is consequences. And that means that for one sixtieth of a second, all the currents in these wires point one way. So what do the wires do when they're at max current going one way? What do they do? They attract. Then the current goes to zero. What happens to the wires? They pull apart. Now the current flips the other way. Again, all going in the same direction. What do the wires do? Attract. Then they relax, then they attract, they relax, and they attract. And they do that about 120 times a second, okay? So if I could do that 120 times a second, this is what you'd hear. It's kind of paralyzed, making like a buzzing noise. Aw, I crashed the internet. All right, so the power lines make a little buzzing sound. Dzzzz, like that. That buzzing sound is from the mechanical force of the wires actually colliding with one another in the bundle. Okay, so this is why it's important to, when you design systems like this, or if you're working with systems like this, you need to be aware that any high current carrying wire with other wires near it will cause a deflection of the neighboring wires toward the high current carrying one because of its strong magnetic field. Let me bring these slides back up. So there's one more thing I want to show you. And this is something I hinted at before, but now you can really see it. And that is an MRI machine. And what happens if there are strong magnetic fields and high strength current carrying wires? So I'll show you this image first. Give it a second. There we go. So I'll go into this thing next lecture. This thing is called a solenoid. And it is nothing more than a bunch of loops of wire placed right next to one another. I'll show you one in class next time. They all carry current I. And together they make a big magnetic field in the center of the loops. It's very uniform. In fact, it's almost perfectly uniform if you design your solenoid correctly. Got two minutes left here, folks. And this is great for human beings. Once we learned that there's something called nuclear magnetic resonance, that is that you can expose nuclei to alternating magnetic fields and get them to sympathetically flip back and forth in the magnetic field. And by doing that, looking at the frequency, you can figure out, oh, that's calcium. That's fluorine. That's carbon. So you can actually identify atoms in a substance, in a material. You can then put a human being inside of a very long, straight, powerful magnetic field, get all of the little electronic and nuclear, most importantly, the nuclear spins in their body to all point in the same direction because they're all little dipole bar magnets. And then you can blast the body with radio frequency waves, which causes the bar magnets to flip in the magnetic field. And the frequency of the flip can be used to image the body. So that's what a computer aided imaging of the magnetic field of one of these MRIs look like. Note the scale. The red is one Tesla. These are really big magnetic fields. They go up to two Tesla. In contrast, the magnetic fields in my experiment, the Large Hadron Collider, which we used to steer protons, they go up to like six, seven, eight Tesla. They're really powerful magnets. When they explode, it's catastrophic. And that happened in 2008. So here's why you should secure your high current carrying cables going into your MRI magnet because there's leaking fields outside of this really powerful solenoid, this really powerful big loop of current. So there's the MRI magnet. And this is a trailer. This is one of those portable MRIs. And someone's done a crap job of securing the high current cables that feed into the wires that create the magnetic field in that device. And so it looks like, let me go back and do that again. It looks like they're flapping in a breeze. But they're actually whipping back and forth. I mean, those are really thick cables, whipping back and forth almost effortlessly in the stray magnetic fields of the solenoid. OK, we'll take this up next time. Thank you.