 All right, so when you're done with the quiz, I'll do a little blathering here for a second and then when you're done with the quiz, you can grab a styrofoam cup and go get some coffee if you need it. Okay, so just to review from last time, we were looking at the effect that an external magnetic field has on a current carrying wire. So you've got electric charge, it's being moved in the wire under the influence of an electric potential difference, a battery or something like that. There is a magnetic field to which this wire is exposed and let's for now just assume it's a nice constant uniform magnetic field, all the B vectors point in the same direction. So remembering our symbol for magnetic field is this B vector. Magnetic field and then B vector is our old friend, the electric field. It's possible that in a region of space, maybe like in this conductor, you've got both a magnetic field and an electric field and they are acting on charges. So our electric field is accelerating our charges in the wire and our magnetic field is causing them as long as the magnetic field is at some angle to the current direction, the drift velocity of the electrons, it will cause them to bend their trajectories and because they're trapped on this physical conductor and they can't leave it, the whole conductor moves with them. So there's this immense force that's actually put on the conductor itself that's induced merely by the motion of charge in the presence of an external magnetic field. And I showed this last time, let me kill the lights here. All right, so just again to demonstrate this nice little picture here, we have a big battery, it's going to be kind of loud, okay? We have a switch, a wire draped through a really big magnet. We actually have a magnet like this in the back, I'll bring it up for a later demonstration, it's really strong and you really hurt a student with it once. All right, so you hook up the battery, now you have a circuit, you throw the switch and you get a current through the wire and it jumps and if you flip the switch the other way and reverse the direction of the current, see it's jumping up there, but now you make the current go the other way and the force is down and it pulls the whole assembly down. And the point I was making at the end of last class is if you're in a situation where there are strong magnetic fields and also significant electric currents, you see you can get motion, you can get force that causes acceleration just due to moving charge particles going through a magnetic field region. And so I mentioned at the end of last class that there can be potentially dangerous situations that even medical professionals might be in where there's a lot of current in a strong magnetic field and that's around an MRI machine. And I'll show a video a little bit later on after we talk about solenoidal magnetic fields, which we're going to sort of lead into today. But if you have a big magnetic field and there's a whole bunch of cables plugged into the device nearby that are carrying lots of current for the device like an MRI machine is very current hungry. You can get a potentially dangerous situation if those cables are not bolted down, strapped down in some way. If their currents ever go perpendicular to any leaking magnetic fields from the MRI machine, it can rip them physically out of their locations. They can flap in the magnetic field as they move and they change their orientation with respect to the magnetic field and then the wire will continue to kind of bend in different directions. This can cause physical harm to somebody and this could rip the cable out and then you could have an exposed high voltage electric cable that kind of somebody touches, they could die. So, yeah, Rachel. All right, fun fact. I volunteered in the radiology unit once and I could never go back with the patients to the room of the machines because I hadn't gone to training because you have to be trained to be in the room with them and like to be brought to a jewelry because it's like we're earring out or something and it's like close to the map. I'll give you guys a problem on that eventually. What's the force on an earring if you were to accidentally walk into an MRI room? It's not good. You can rip the earring out of the ear. But you have to have serious training at least to be around them. Yep, yep, yep. I haven't had the training, so. Yeah, no, and then the reason, I mean, you probably would have been okay if you'd gone in with no jewelry or anything, but they want to make sure to cover their legal butts first that they've made. We told you, no jewelry near the MRI. If your ears are gone, it's your own fault, right? So the staff has to go to training to, again, pick the ability for you there or the hospital. But that's got some great videos. You can find these on YouTube. Just go search for like MRI mishap or something like that. People do this intentionally. They'll have like an old MRI machine around and then just to demonstrate how potentially dangerous they can be if you don't respect protocol. They'll do things like, they'll take an oxygen gas bottle, which is usually made of metal, and you'll find those in hospitals all over the place. If you were to accidentally have one of those near an MRI machine and you flip the machine on, that bottle can actually be lifted physically off the ground. It will try to get inside the magnetic field and it can punch through anything that's between it and the place it wants to be. So it's a very dangerous device. One is not at least versed in walking around it. Even something as simple as a metal ruler, a piece of jewelry, a ring, anything like that, if it can carry a current, as you'll see, that can be a bad thing. Metal that can carry current around magnetic fields is bad, whether it's carrying a current already or not. So as we continue to explore magnetism, you'll see why even just having an inert loop of metal in your ear, even though you don't have a battery hooked up to it, there's no current in it, you can actually get a current by just moving through a magnetic field. So it's a fast, this is, electromagnetism is really fascinating, and now that I think we have everybody here, let's go ahead and do the quiz. There's only a little bit. All right, well the penalty of getting coffee is I'm going to start talking, so, don't catch up, don't worry. Yeah, no, it's not a perfect system, but hey, they walked out, so. All right, so, I want to build on this equation. I want to go one step further and I want to introduce another concept that will be helpful going forward in the discussion of electric currents, magnetism, and so forth. So let's start from the force equation on this current carrying wire. So the force is given by the magnitude of the current, I, times a vector, L, which is units of meters, it's a length vector, and it represents the length of the wire that's exposed to the magnetic field. And its direction, it's where it points is the direction that current points. So, for instance, if we were to look at this loop of current that I've drawn here, the bottom section points to the right. So here, down in this section, that's where L vector for the bottom points to the right. For this section over here, L vector for the right-hand side points up. And then on the top, it points to the left. And on the left, it points down. So L vector is just following the current direction in each section of this square loop that I've drawn here. So, well, square or rectangular. So we'll call this a square loop just to keep it simple, square loop. So each side of this has a length L. So I didn't do a very good job of making this look square, but each side of this has a total length L. And it's immersed in a magnetic field, B, which is uniform in strength. And it points always in the same direction. And that's to the right. So B vector in this picture are these sort of fainter lines with arrows that I've drawn here pointing to the right. And so we can start thinking about what the force is going to look like on each piece of this square loop. And then the total force is just the sum of the forces. So the forces are going to be, they're going to add just like vectors because they are vectors. So the total force is just the sum of the individual forces on each of the four sides of the loop. So I equals side of loop drawn over here. So let's start looking at the forces qualitatively. I'm not going to calculate anything quite yet, but we can look qualitatively at the forces. Let's start with the right-hand side of the loop. So to figure out the direction that the force due to the magnetic field is pointing, what we can do is we can apply one of the two versions of the right-hand rule that I've been talking about. We can take the fingers on our right hand, flatten out our hand, and make sure that that points in the direction that current is flowing because that's the direction L vector points. And now what we're going to do is we're going to reorient our hand until my palm basically faces the direction that the magnetic field is pointing. That is, that's the direction I would have to curl my fingers to point and be. And wherever my thumb is pointing, if I do thumbs up at this point, that's the direction the force would then point on this current. And since current is defined as the direction that positive charge flows, we already are dealing with positive charges here. So we don't have to worry about a weird sign change like you do with Qv cross B, where it could be positive or negative. I is always defined in the direction that positive charge is moving. So that sign is already taken care of for us. So L, I'm going to have to tip my palm to the right to point in the direction of B. And then my thumb indicates the direction of the force on this wire. So the force over here, and I'll represent that, is these little arrows with X's in them. So the tail feathers of arrows fired into the board. Those are the directions that F right points. So the force on the right hand side of the loop points into the board. Let's look at the top. So the top of the loop, you now have a current going to the left. So I take my fingers, and I point in the direction of the current. And now I'm going to orient my palm in the direction of the magnetic field. Well, the good news is I don't really have to sweat this one too much, because what's true about the magnetic field and L vector in that particular case? They're parallel. And so the cross product is anyway 0, right? So up here, you have no force in this particular case. Because L vector, which is the direction that current points, is in this case, it's anti-parallel. So it points in the opposite direction of B. But nonetheless, it means that L and B lie on the same line. And there is no cross product. The cross product of two parallel or anti-parallel lines is always 0. So the top is easy. And then let's just do the bottom while we're at it. So the bottom L vector points to the right. The B field points to the right. So the force is 0. So there is no force down here, either. So let's then focus on the left. Yeah. It's hard to clarify, is it the cross product of those things is 0? Yeah. Is there no force? No force. Any time if L is ever parallel or anti-parallel to B, the force is 0. Because the force is the result of taking the cross product. If the cross product is 0, the force must be 0. So parallel and anti-parallel, do you mean the lines are anti-parallel or anti-parallel? The vectors are parallel or anti-parallel to B. Yes. Well, these two are to each other. These are the only two vectors involved in the cross product. So if L is parallel or anti-parallel to B, then it's 0. So let me just make this a bit more concrete by reminding you that the magnitude of the force is I, L, B sine of the angle between them. And if theta equals 0 or pi, so pi is 180 degrees, the sine of 0 and the sine of pi are 0. So that's the other way you can remember this, is if you remember, OK, the magnitude of the force is I, L, B sine theta. Then if theta is 0 or pi, force magnitude is 0. There's no force. So that's another way you can do this. All right, so for the left, so rather than playing this game where I have to kind of get my fingers to the point, what I'm going to do is I'm going to use the other right-hand rule, which is this one. You make sort of coordinate axes out of your index, middle, and finger, and thumb. This is x. This is y. This is z. That's the way I remembered, at least. So you're going to take your finger and you're going to point your index finger in the direction that L vector points. You want to point your middle finger in the direction that B vector points, and your thumb will indicate the direction of the resulting force. So it's not very comfortable, but it's not too bad. So here I'd have to do this. So my index finger points in the direction of current, which is down. My middle finger points to the right, which is the direction of the magnetic field. My thumb indicates the direction of the force, which is out of the board over here. So draw these as circles with dots in them, arrows flying at your face. So this is F vector left. So to do the total force, we just need to add F vector left plus F vector right. So let's just look at the magnitudes of those for a second. We know that their directions are opposite. We know that F left points out of the board and F right points into the board. So they're opposing each other. What are their magnitudes? What are the magnitudes of F left and F right? The same. So these vectors oppose each other, and there's no net linear force. This is net linear force on this loop of current. So that's not so bad. But you've got the forces applied at different locations on the loop. So if I take an object, and I apply, so I don't have a good example here, but I can do with this, if I have an object like this chalk, which is very rolly, and I apply opposing forces to the ends of the chalk, what happens to it? It tilts, yeah? So I can apply equal but opposite forces. I can let gravity do the force on the left, and I can push up on the right. It tilts. It rotates. This is causing rotational motion to put opposing equal magnitude forces on the ends of the chalk. Similarly, on this loop of current, then you could make one of these by taking copper wire and bending it very sharply, and then hooking it together with a little battery at one end. And then you could actually make a little loop like this. Lizzie, did you have a question for her? I saw this, and I wasn't sure if this was like a wait stop. So yeah, Catherine. And then Laura, do you have a question after Catherine? So are those anti-parallel then? The forces? Which directions of what? Oh, I see. Do the wires, we haven't gotten there yet. Oh, we haven't gotten there yet. Yeah, yeah. That's the next step. So right now we're still investigating what does any magnetic field do to a wire. The next thing we're going to do is we're going to find out what exactly causes magnetic fields in the first place, and then we'll get into what happens when you have two wires that are next to each other. So you guys are a little ahead in the reading. That's why I've slowed down a little bit. I'm going to lighten the reading load for next week. So I'm going to skip Ampere's Law. So we'll catch up to where I want to be. But I'm trying to go carefully through this so you can see the steps. So the price is I have to slow down a little bit. So let's just focus on the forces. So we've got one force going in over here, pushing on that side of the loop. We've got one force coming out here, pulling on the left side of the loop. The loop is going to begin to rotate. So we have a torque. We have a force that's displaced from the center of rotation, and this causes a rotation. So just like you can put your force on the end of the door, and the hinges are fixed, so you cause a torque that opens the door, this magnetic field is causing two torques, one that tries to rotate the right end down into the board, and one that tries to rotate the left side of the board up and out. So what I want to do now is I want to look at the sum of those torques and get a sense of what the total torque is on this loop. Once we do that, we can do all kinds of things like figure out how fast this thing's going to spin in a circle, and so forth. And this comes in handy. Let me preview why this is useful. If you can get a rigid mechanical loop exposed to a magnetic field, and I hinted at this in the video lecture that I gave you guys on just magnetic phenomena, if you can expose a loop like this to an external magnetic field, it will begin to rotate. And if you time your device just right as the loop rotates so that it's now perpendicular to the magnetic field but spinning, and you flip the sign of the magnetic field, it will then continue to rotate around. And then you flip the magnetic field again, and now you have a device that's spinning, and if you were to hook into that rigid mechanical loop, something like a drive shaft, you could move a car. You could run an electric generator. And in fact, this is the exact principle with a few extra pieces we'll get to in a lecture or two. This is the exact principle behind electric motors. So you use a current and a magnetic field. You can make the magnetic field from permanent magnets, big, strong, permanent magnets, and all you have to do is get something to change sign whenever the loop rotates so that there's no more force due to the magnetic field. In the case of an electric motor, you change the direction of the current. If you flip the direction of the current, now the force points in the other direction and it keeps spinning the loop. And if you time this just right, you can make a motor. And so mechanically, electric motors are set up to make that timing work perfectly. They do this with various tricks, but that's how you get a motor, basically. So basically the significance of not having a force is that it doesn't translate, it just rotates? Exactly. So this loop isn't like doing this. It's just spinning in place. Now, of course, there's no perfect loop. There's no perfectly uniform magnetic field. So engineers have to engineer those motors with some mechanical assembly that keeps the loop from tilting, because that can cause the whole thing to just rip itself apart. So there's, of course, good engineering that goes into this to take account of the fact that there is no such thing as a perfect system. If you design systems and assume that they're perfect, you will fail. And the early, like, airline industry illustrates this perfectly. The early commercial airline industry in Britain was building airplanes with materials that when received from factories were assumed to be perfect. And so when they manufactured their planes, the assumption was that they're perfect and that there's no tolerance for failure. But these planes started flying, and then after a while, they started ripping themselves apart in the sky and falling out of the sky, killing people on board. So there were at least two major airline accidents of planes tearing themselves apart in mid-flight before they stopped and tried to figure out exactly what was going on. And the flaw, essentially, was their reasoning. They reasoned that when the material shows up from the factory, it's flawless. And so they can assemble anything out of it and it will be flawless. But no material is flawless. There are always micro fractures, tiny cracks, imperfections, whatever. And so it's those imperfections that cause tears to begin. And what you have to do is engineer your aircraft so that if a tear begins, the material can absorb the spiderweb and crack and stop it before the fuselage tears itself apart. So modern aircraft are designed with failure in mind. They have multiple redundant fail-safe systems in the fuselage construction, and they use multiple kinds of materials so that if a crack develops, it stops. And so you have this kind of philosophy about designing now, which is much more, it's accepting of faults and tolerant to faults. And interestingly, you can use electric currents and magnetic fields to scan the surface of a plane and detect microcracks. And this is in fact how airline safety professionals, when they're assessing the fuselage of a plane, they don't take a magnifying glass and look for little cracks. Your eyes are not that good. But you can take a little coil of wire with a current running through it and you can run it over the surface of the plane and you can measure the current coming out of the loop. And from that, you can figure out whether or not there's a crack in the plane, even one you can't see with your visible eyes. You can take the plane out of production and you can fix the crack. So fault is very important. Whenever we do physics problems, we're setting up perfect situations, but as you go into the world, the world is imperfect and one has to be tolerant of that. So engineers have to mechanically fix this loop so that it doesn't wobble like this because if it does this, the engine will tear itself apart. My electric lawn mower failed because the rotor tilted out of alignment and one day I was just mowing the lawn and I went, and that was it. That was the last sound it ever made because at that point, the wire, the loop of wire inside of it came in contact with the magnets and ripped the whole thing apart. How would you know? That's a, oh, you should take courses in the engineering school. They spend immense amounts of time on how do you build in tolerances, bridge failing, fuselage failing, car failing. How do you assess the faults that are possible in materials? There's a whole branch of material science focuses on this whole thing, development and failure of materials. So that's a great question and it's a zero time for it in an introductory physics course, but the engineering school focuses a lot on things like this. So it goes into quality assurance and so forth. So, yeah, Jasmine. Okay, so kind of back to what you were saying about the tour. Yeah. So you said something about it rotating one way, but if you flip something, it still rotates the same way. Right, so let's say we have this loop and we turn on the magnetic field or we start a current running through the loop. So now there's a force and the loop starts to spin like this. Now, when it gets up here, it doesn't stop going. It keeps going, right? So all you have to do now is when it gets up to the apex of the rotation and it starts heading into the second half of the rotation, you flip the sign of the current and the force will flip direction and it'll just keep spinning like this faster and faster and faster. So as you cycle the current in different directions in the loop, you can make it spin faster or slower. It's all in the timing of the flip. I'll show a motor demo later because we're not quite yet ready to talk about motors. Flip the current? Yeah. Yeah, if you want it, there are two things you can do. You could flip the direction of the magnetic field, but it's really tough to do that if you're gonna use big permanent magnets in your motor. It's cheaper to just have a circuit so that as the rotor comes around, it makes a different contact with the battery and changes the direction of current. So the battery might be hooked in in this mechanical assembly one way and then as it spins half a cycle, the connections to the battery reverse. And so plus becomes minus and minus becomes plus and the current flows in the other direction. And this is mechanically how they design motors. I'll show you guys some videos of this, but we're not quite there yet. I just wanted to preview that, okay? Let's just do the blase mundane thing of calculated torque. I know I really sold that, right? On a Thursday morning. God, it's hot in here. Why don't you people complain about it being cold? I hate you all now. I don't know. See, I told you, you can't win with facilities. It's either too hot or too cold. There is no Goldilocks zone for this building or this room. It's like a cursed room. Happy Halloween. All right, torque. I'm gonna stop trying to crack terrible jokes. Yeah, that one worked. So we have two torques caused by two forces in this problem. So we need to figure out what each of the torques is. Let's focus on, we'll call one, here, let me re-label this. So we'll call this the right-hand torque and we'll call this, that is not how you spell that, the left-hand torque, okay? So let me rotate the picture for you so that we can see the forces and we can picture the torques. So to rotate the picture, all I'm gonna do is I'm gonna draw the loop. So this is now the side. So what I've done is I've taken this loop and I've just tipped it like this so that we're looking at the end here, all right? Where there's no force. And so we have the current on this bottom side is now facing us and it's moving to the right and it's coming out of the left side. The current is coming toward us. So I'll put a little circle of a dot there and as the loop bends over here, the current goes into the board. So we have I out of board over here and I into board over on the right. So and then in this section, it is going to the right, all right? So here it goes into the board, here it comes out of the board and this is the bottom of the loop. The magnetic field still points this way, okay? But now we've revealed the z direction. So this, if I label that up here, the B field is pointing along the x direction. Here the current was going in the positive y direction. Here the current was going in the negative y direction and the force here is going into the negative z direction. It's going into the board. So it's going down, all right? The positive z direction is indicated here with this dot coming out of the board and the force here is coming out of the board in the positive z direction. So negative z direction there, positive z direction there. So out of the board is positive z direction. Now I've flipped my coordinate system around. I've got x going that way. I've got z going that way. And I have positive y coming out of the board. So again, I've just tilted the picture so that we can see the forces now. All right, so the forces here, I'll draw this over here, i into board. So the force here points down. This is f right. And the force here points up. This is f left. So again, I've just tilted the picture. So now we can see the force arrows. The current, the L vectors, are coming out of the board here and into the board here. So let's think about what happens. We have equal forces on either end of this loop. And so we get this rotation. And the center of the rotation, if this is a nice uniformly distributed loop of wire board. So the distance from where the force is applied to the center of rotation is half the length of either side of this loop. So the whole side is of length L. That's half L and half L. So let me just redraw this here. Here's the center of rotation. This is a length of 1 half L. That's a length of 1 half L. We have a force down here. We have a force up here. Now for torques, let's do the torque on the right-hand side first. This is equal to r cross f. So we need to know the radius vector that points from the center of rotation to where the force is applied. And we need to take the cross product of that with the force that's applied. Bethany, question? It looks from the picture like it would be rotating the opposite way. Rotating the opposite way. Well, what I did was I tilted this way. So the bottom is now facing us. So that force points down that way and that force points up. So this loop is rotating out of the board like this. And in this picture, it's rotating up. So I did that consistently. There's always a danger that I'll flip a sign here. So be vigilant. But I think that's OK. I think that those are both consistent pictures. Because all I've done is if I could take this board and tilt it like this so we could reveal the z direction, that's basically what I did is I tipped the picture so that the bottom is facing us this way. And that's what you see here. It's important to play around with this on your own. I'll give you torque problems. You'll have lots of fun with this, don't worry. All right, so we need to know the vector for the right-hand side that points from the center of rotation out to here. So that's this vector. So this is our right. Points from the center of rotation out to where the force is applied. The force is applied on the end of this arm here of the loop. And we have the force already. So let's write down some vectors. So keeping in mind that, again, to the right is still the x direction, just like it was there. Our right is a magnitude times a direction. Well, the magnitude of that r is just the length of the conductor between the center of rotation and the end. That's a half l. We need a direction. And for that, we just need a unit vector that points in the direction that r vector points. That's in the positive x direction, so that's going to be i hat. That's it. We're done with r. Now we need f. f, right. Well, we know that that is going to be i l cross b. So we need some vectors here. We need to write in terms of our coordinate system these vectors. So let's see. So for this, what I'm going to do is, what we know when l is already, it's the total length of the wire that's exposed to the magnetic field. So this is going to be i l. And the direction in which l points is the direction of positive current flow, positive charge flow. That's to the right. That's the positive x direction. So that's just i hat. And then we have to cross that with the b vector. Oh, hang on a second. Hang on. Hang on. Nearly went off the rails on that one. The force is acting on this length here, which we can't see anymore because it's hidden behind the bottom of the loop that's now facing us. So this is the direction of the current flow that's experiencing the force. And it points in the positive y direction. Positive y direction. Positive y direction. That's where current is flowing. So we have to consider the length that's exposed to the magnetic field that's getting the force put on it. So that's l. That doesn't change. But now we're in the positive y direction, which is j hat. And then finally, we have to get a vector in for magnetic field. Well, the magnitude of that vector is just b. And it points in the positive x direction. That's where I drew b originally, positive x. So this is i hat. OK. So let's do some cross product here. Let's get that cross product sorted out, and then we'll do the torque cross product. So let me pull the constants out of here. We've got l. We've got b. They don't participate in the cross product. They're just numbers. It's the length of the wire, whatever it is, a millimeter, or 2 millimeters, or 10 centimeters. And then the magnetic field string, b, whatever it is, a tesla, a milli-tesla, we don't care. It's just a number. So we pull it out in front, i lb. And then we just have this cross product, j cross i. OK, so let's dig back a lecture. i cross j is k hat. So i hat cross j hat is k hat. So what is j hat cross i hat going to be? Negative k hat. Great, thank you. So we have i lb. So negative i lb k hat. So that is the exact expression for the force acting on the right-hand side of the loop. And that's the force that enters the torque equation here. OK. And as usual, I'm running out of board. So what I'm going to do is I'm going to just use this space here and finish off the cap. So continue from left board. OK, so now we're going to do with r, which we have a vector for here, cross f, which we have a vector for. So the torque on the right is equal to 1 half l i hat cross negative i lb k hat. OK, so let's get all the constants out in front. The negative i lb, the 1 half l. Let's just get that out in front now. So we have negative 1 half l squared b. And then we have the cross product, i hat cross k hat. What happened to that i? That is an outstanding question, doesn't it? Current. Missing the current. OK. Yeah, negative 1 half i, the current, l squared b. OK. And then we have unit vector in the x direction cross unit vector in the k direction. So again, if we dig back to last time, we had i hat cross j hat equals k hat. k hat cross i hat equals j hat. And j hat cross k hat equals i hat. Flip any of those two and you put a minus sign in front of the right hand side of the equation. So we have i cross k. Here we have k cross i is j hat. So this is negative j hat. Excellent. Negative j hat. So we're done with the torque on the right. The torque on the right is negative 1 half i l squared b negative j hat. And let's combine those minus signs so that they cancel out and we just wind up with 1 half l squared b j hat. Half the problem is now done. I have a question. Do we have an i going and an i hat going? Yeah. This is where notation gets annoying. Would everyone like me to switch to capitalized for the currents? Yeah. I'm getting confused because I see talk in hats. That's fine. So big i, big i, big i. Anyway, I wanted to do this and then forgot to do it. So it was just a change. A big i. Didn't we use a negative sign now? What shouldn't it have been? We lost an i. So we had a negative j hat and a negative 1 half here. So on the torque on the right, so you have negative i l b and then you have negative, but then negative. Oh, I pulled it. Oh, sorry. It got crammed in here. But there's a minus sign hiding right there. Yeah. So what about the crammed in there? Oh, and then yes. Thank you. And this is why it was a good idea for me to switch to capitalized because I keep dropping the little i's thinking that they are unit vectors. So big i, big i. I think that's all of them. Big i, big i. There we go. So how's the problem done? Hooray. We need the other torque. The torque on the left side. All right. So I'm going to slide things down a little bit here. All right. Come back to those in a second. So the torque on the left is r vector left cross f vector left. OK. Well, r vector to the left will point out to where the force is applied. So it goes from the center of rotation out to where the force is applied. And the force is applied right on the end of the loop here. So this now points in the negative x direction. So our vector for r left is negative 1 half l i hat. The force on the left is, again, i, l vector left cross b vector. Now, l vector left is this. This is the side that actually gets the force that causes the torque. It has a length of l. And it points in the negative y direction. So negative j hat. It points down. So this is just equal to i negative l j hat cross b still points in the same direction, b positive i hat. So we can group these terms together and we get negative i l b j hat cross i hat. OK. j hat cross i hat is negative k hat. So we wind up with just i l b k hat. So how did I screw that up, if at all? I did not. These torques, yeah, this is fine. So the forces should point in the opposite direction. This one points in the negative k hat direction. This one points in the positive k hat direction. We figure that out anyway from the right hand rule. The equation reflects that fine. So we're good. So now we just need the torque. And that is going to be the cross product of this vector. Negative 1 half l i hat cross i l b k hat. So I can again pull all the constants on the front. Negative 1 half i l squared b. And then I just have i hat cross k hat. Which is negative j hat. So I just wind up with 1 half i l squared b j hat. And oh look, torque on the right plus torque on the left. They add up. You get a rotation. There's a net torque. That's good because we kind of assume that that was going to happen and the math bears that out. We get a net torque. The total torque is the sum of the two. And it is just 1 half plus 1 half. Those equations are the same otherwise. So we just get i l squared b k hat. J. Excellent. You're learning to question the teacher. My mission here is complete. I can quit and retire now. Yes. I doubt it's quick. But go ahead. Do you see left on the right board? Yeah. Bottom section there? Yeah. I understand where the second term came from because it's literally right above. Where did the left turn come from? Good question. So that is going to be our left vector. And our left vector is a vector that points from the center of rotation out to where the force is applied. So it's arrowhead points to the left. This is the x-axis. Positive x is that way. So this is a negative i hat direction. And it has a magnitude of 1 half the length of the loop. So that's where the negative 1 half l i hat comes from. And so that's that negative 1 half l i hat. So you solved that equation for f left first and then plugged it back in to get our left? No. I plugged in. No, no, no, no. I wrote down our left. Then I figured out what f left was. And I used that in the cross product to get torque. And then that's what I got. I want to unpack this for a moment because in order to introduce the key concept here that I'm using this calculation to motivate, I need to kind of step back for a moment and put a cross product back into this equation. And I know that's going to seem a little odd, but let me go ahead and do it and you'll see where I'm going with this. One thing that I want to point out, what is l squared equal to for a square loop? You have a square and you calculate l squared. What's that? The area. The area. Yeah, it's the area of the loop. So let me just make a quick substitution here and write this as i times the area. This is the area times b j hat. And then let me go one step further and let me unpack j hat into a cross product. So what cross what gives me j? See if we can do this backward. K cross i. Yep. So this is i a b k hat cross i hat. Let me unpack this one step further, okay? What I'm going to do is I'm going to put b back here in front of i hat. Okay, and I'm running out of board space again. So who gets sacrificed? You. What should I get rid of? In the box. Okay, does anyone know what needs this? Great. So we'll just reclaim this for space. Excellent. Okay, so we have torque total equals i a b and then I have k hat cross i hat. So I'm going to put the b back with the i hat. No reason I can't do this. Totally legit. What is b i hat equal to? What was it originally? The b vector. Yep. So let me go ahead and put that in. Okay, we're nearly there. The new concept is nearly upon us. What I will now do is I will define this thing i a current times area gets a new name, mu. And it stands for magnetic dipole moment. And it is analogous just like an electric dipole in an electric field will execute rotation and it has a moment that moment is equal to its length times the magnitude of the charge on either end. A current loop behaves just like an electric dipole but in a magnetic field. Remember I mentioned that the simplest fields in nature we've ever seen are dipole fields. So it's very convenient to simply define the equivalent for the magnetic field in force that we had for the dipole and the electric field in force. There's a dipole moment for electric charge. There's a magnetic dipole moment for magnetic rotations in a magnetic field due to occurrence. Okay, and what is it? It looks very similar in construction to the electric dipole moment. Electric dipole moment was q times d, charge times the length of the separation. This is i moving charge current coulombs per second times the area of the loop. So it is very similar in its construction and this is no accident. As you'll see in a bit, I'll demonstrate it in a moment. There is no accident that there's a weird symmetry here between current loops in magnetic fields and two charges bound together in an electric field. There's a reason for this and it's very convenient to define this quantity as mu and here's why. If I want to know the total torque on a loop, I just have to know mu vector cross b vector and mu vector is just ia in this case k hat. How do you figure out the direction of mu in an arbitrary problem involving a current loop? Well, it's actually not as hard as it seems. Where does k hat point in this picture? Out of the board or into the board? Out of the board. k hat points in the positive z direction, positive z comes out this way. So if I wanted to just draw mu, that would be mu and its magnitude would have been the current times the area of the loop. That's it. You don't have to do this nonsense garbage with writing r cross f and then I'm going to sum the torus and then... No, it's easier than that. You need to know the area of the loop. You need to know the current in the loop and you need to know one direction. The direction of a vector perpendicular to the area, which is what k is, right? The area is in the plane of the board, mu vector points out of the board and the way you figure it out, take your fingers, curl them in the direction that current is flowing. Current is flowing counterclockwise in this loop. Your thumb indicates the direction of mu vector. That's it. Whole lot easier. Whole lot easier to figure these things out. With this trick, you can figure out, for instance, how a microscopic current loop that might be present, for instance, in the electrochemical processes in a cell would respond to the cell being exposed to a magnetic field. With this information, you might be able to figure out how currents in the brain, current loops in the brain, would be affected by magnetic fields. Does anybody remember that video I showed on the first day of the guy from the BBC? I'll show it again in a bit. He was talking. It was Humpty Dumpty sat on a wall, Humpty Dumpty had a great fall. It wasn't stuttering because he stutters normally. He was stuttering because they had a giant magnet next to the speech center of his brain. All they did was they screwed with the currents in that section. He couldn't talk anymore. It's very important to have these concepts down because you can do all kinds of research with them, mischievous or otherwise. Actually, that research is very useful because what that research revealed, which I think have been known in other ways, but you can test it directly in the lab, was that singing and speaking are different things. I have a colleague who suffers from a very strong stutter when he talks. He's brilliant. He's a fantastic physicist. When he gives talks, though, you have to be patient with him and just let him talk because he'll catch on words and not be able to keep going. But he sings in a choir in Europe. I asked him once, when you sing, do you stutter? He said, no. It's well-known that extreme stutterers can sing just fine. The demonstration of the video I showed you on the first day, which I will show again at some point, was the guy could sing Humpty Dumpty, Humpty Dumpty had a great fall, and he could get through it even though they were knocking out his speech center with the magnet. This kind of tool is useful for revealing the behavior of the brain in the way that basically its functions are distributed. So neat little things you can do with magnets and currents and so forth. So to solve force problems like this, though, you just need to know what the magnetic dipole moment is. Any time you have a loop, and that loop has an area, and there's a current in that loop, all you do is take your fingers, curl them in the direction the current is flowing. Your thumb indicates the direction of the magnetic dipole moment. So you know it. You can just write it down. In this case, it's k hat, because it points perpendicular to this area, and you just put ia k hat. If I increase the current by adding more loops, let's say I add a second loop in here carrying the same current with basically the same area, I've doubled the current. I have two i instead of i, but the same a. So I can double the magnetic moment by doubling the current. I can have the magnetic moment by halving the current. I can control the behavior of a loop of wire by altering the current flow, and that is how you control an electromagnetic motor. And again, I'll keep illustrating that. We have other principles in magnetism and electricity we need to illuminate, but this is a really fundamental principle. And in fact, what we know now is that even subatomic particles like electrons behave like little current loops and they have an irreducible little magnetic moment inside of them, which is related to something called spin. And so the actual origin of magnetic fields in a terrestrial magnet like those craft magnets I showed you were like in this compass needle. The spins of the electrons are all pointing, let's say 1% or 10% of them are all pointing in one direction. And those little current loops are all oriented. And you'll see why that matters in a moment because I'm going to demonstrate the phenomenon. But this makes magnets. So each electron is a little magnet and it's a little dipole magnet. It has an north end and a south end. And that's really useful because we can then, for instance, protons have this too, neutrons have this as well. So if you can get those magnetic moments to flip in response to an external magnetic field, you can do things like image the chemicals in the body in different slices of the human body. That's what a magnetic resonance imaging scan is. You take the spins of the atoms and you flip them in resonance with an external magnetic field or an electric field. And then by doing that, you can figure out what chemicals are present in different parts of the body and build up these gorgeous 3D images without ever cutting a single human body part. Which is really important, right? I mean, when you think about the way we used to learn about human beings, you wait until they die and you cut them open. But you can't learn about a human being who's breathing when they're dead and you're cutting them open. You can learn a lot of things, but there are some things like the functioning of the human body. You can't learn that unless you can cut open a human non-invasively. And that's the technology that we as a species have been developing now for decades. Non-invasive human imaging. And MRI and spin and magnetic fields are essential to that. So let me demonstrate the next important principle here. Any questions on this before I move on? This will all tighten together at some point. So we did the loop. On this slide, I'll put this all up on the webpage today. So here's the example of how you calculate the direction of the magnetic moment. That little n vector is the vector that's so-called normal to the area. Normal means at a 90-degree angle. So the only way, if you've got your area in a plane like this, the only way to get a normal vector is to point perpendicular to the plane. And to figure out the direction that n points, as a point out or as a point in, you just curl your fingers in the direction of the current flow. That's it. So another right-hand rule you'll have to absorb. And then this is just pointing out what I said here. You can align loops of current along or against a magnetic field, just like you can align dipoles, electric dipoles with or against the field. And this is the aligned, is the low-energy configuration and anti-aligned against the field is the high-energy configuration, just like electric dipoles. So let me demonstrate something here. Now, I have to be very careful because as it turns out, these stands are magnetized, which I didn't appreciate the first time I did this. So I have here... Let me do this. What's up here? Terrifying. I'm going to just take this camera. The camera is primarily made of plastic, non-conductive, non-magnetizable material. So I have here a compass. And we'll try moving it here. So if I let that settle for a second, you'll see that it points in the direction that north is supposed to be. So let me line up N on the compass roughly if I can. So now we know which way is east and south and west and so forth. So there we go. I'm going to get the reflection off this. I'm going to kill the lights. So brace yourselves. A little easier to see now. So I have this assembly here now with a coil of wire. That's the orange stuff. So it's just wire that's been looped many times around a piece of plastic. And it's coated with a thin plastic coating so that the wire doesn't actually come in electrical contact with itself. So those loops are actually electrically isolated from one another by just a thin little clear plastic coating. This is a very common way of making coils of wire like this. If you didn't do this, you short the loop. So the current would not flow through each cycle of the loop. It would just run the shortest path to the other end and come back to the power source. So I have a switch here. And I have power coming in from this low voltage power supply over here on the right. So it's plugged in and at this point I can switch this on. You should see a green light come on. Now I'm only going to get to do this experiment a couple of times and switch very carefully. So we have here again the compass still pointing north loop of wire over here and I'm going to throw the switch. Watch the compass. All I'm doing is running an electric current through the loop of wire. There are no magnets anywhere else near this. There's no magnets in my hand. Okay. All I have to do is close the switch below and I can change the direction of north locally. This in and of itself was an astounding discovery. For almost 2,000 years human beings have been playing around with magnets not at all understanding where magnetic fields came from. They seem to be something apart as chemists and physicists and mathematicians started studying the electric phenomenon. The magnetic phenomenon seemed eerily similar. In fact there were scientists that suspected a connection between the electric phenomenon and the magnetic phenomenon but no one could quite prove it until one day quite fortuitously while doing a lecture demonstration for his students in a physics class Hans Christian Ursted closed a switch on a battery and noticed that a very thin metal compass needle landed nearby. And he did it again and the compass needle deflected again and it was a eureka moment for him because he realized it's not static electric charge. I could put for instance that device I showed you at the beginning of class that can hold charge. For instance I could rub PVC pipe and hold it next to this very still and this compass won't respond but if you move charge it creates a magnetic field that affects permanent magnets. In this case it made the compass needle deflect. Terrifying, let me get rid of that. So let me reinforce this a bit. So Hans Christian Ursted, here he is Danish physicist and chemist. There we go. And as I said he observed in 1820 that electric current was a compass needle to deflect. That was his basic simple observation of the natural world and it caused quite a stir when he published his results. I mean anybody could reproduce it if they had an electrochemical cell what we call a bowery now and a compass needle some wire and a compass needle. Now the reason I only get to do this a few times is this is copper wire this is copper wire there's not a whole lot of resistance in this circuit so there's a big current of order and amp or two that's flowing through this copper wire straight back into the power supply I'm effectively shorting the power supply through the circuit. So eventually if I leave this on long enough it's actually warm to the touch now there's a safety mechanism that when it gets too hot it pops a little switch and the power supply doesn't work anymore. If I do that four or five times in a row quickly I can short the power supply and it will save itself by breaking the circuit. When it cools down again I can pop the switch back in so I didn't actually get it to flip this time but I did this while I was practicing the demonstration. It's actually quite difficult to get this to work you need a lot of current and so, you know, Ersted needed really strong chemical cells and he wasn't the one that developed them other people developed them and he was using them in experiments. So as I said he actually first observed this amazing phenomenon while doing a lecture demonstration of electricity in front of his students in his class so this marks the discovery of what we now call electromagnetism which is a relationship between electricity the electric force and magnetism the magnetic force that's intimate and related through the motion of charge. So what we learn at this point is magnets like these permanent magnets will respond to other permanent magnets like the earth so the north will line up with the south magnetic pole of the earth which is the geographic north pole. We also know that electric charge we can take a magnet we can put it in the path of electrons and they will deflect if their velocity vector is at any angle to the magnetic field from the permanent magnet. Magnets affect electric charge but we also see a beautiful symmetry here electric charge in motion affects magnets they generate magnetic fields and this was a stunning revelation at the time and we have benefited immensely from this very basic observation about the natural world so let's watch this this is awesome stuff this is a magnetic fluid by the way has anyone ever seen this before? let me go back here so magnetic fluids are just like solid permanent magnets except liquid form and they're neat because if you spritz them onto a permanent solid magnet you can see the magnetic field lines see the dipole field lines shooting out of the top coming out of the sides you can suck the fluid back into the plunger it was a famous car commercial a couple of years ago when I first started teaching the course it was a car commercial about the fact that this car company uses a magnetic fluid as part of their brake system to create a more responsive brake system so that fluid is actually used by car manufacturers to improve the quality of safety systems on a vehicle like brakes this is awesome stuff I can sit here and stare at this for a moment he's going to do another demonstration in a moment and it's similar to what I just did I used to have to only show the video because I couldn't get enough current to do this but this year I got it to work but the standard traditional way of showing magnetic fields is with a compass I've got a wire big ass wire I've got it surrounded by compasses compass points towards a magnetic field if you were to bring a strong magnet near it they would point towards it and an interesting phenomena takes place you get lined up when we put electricity so they're all pointing north and then they all start to point around the wire it's going around that wire and if you wrap the fingers of your hand they go in the direction of the magnetic field you curl your hand around the wire another right hand rule like this curl the fingers around for your hand around it the thumb indicates the direction of current it's got this picture upside down okay so we'll come back to that in a minute but let's just take a look at a photograph of that experiment not using compass needles which are big but using little iron filings and each piece of the iron filing each iron filing is a little north pole and a little south pole a little bar magnet and if you take a wire and run a couple of amps through it in this case they cut a hole in a piece of paper so non-conductive paper they ran a conductive wire perpendicular through the paper okay so out of the plane of the paper they ran an electric current through the wire and the iron filings on the paper which were originally just shaken randomly and uniformly on the paper they line up and isn't it gorgeous right you can see the current is going up the wire and the magnetic field is circulating around that wire and it was this observation that two physicists used now write down a law that we understand deeply predicts the magnetic fields caused by electric currents Catherine did you have a question? Yeah I was wondering why they wouldn't all line up like on top of each other because wouldn't they want to get as close as possible to the kind of thing? Absolutely and the paper has friction so their rotating is pretty easy but sliding that whole little bit of so they're just like well they're just shaking on the paper and then you kind of just kind of tap the paper to get them to settle uniformly and then you turn on the current and wait and the little magnets will all line up and they'll line up along magnetic field lines so what you're seeing here is the little bar magnets are each of them is sort of like a little you're kind of looking at calculus here but done physically each little bar magnet is like a piece of a much larger magnet that's lining up along the magnetic field lines which are big swooping circles that go around the wire so the magnetic fields appear to be in the plane of the paper and they are circulating perpendicular to the current flow again you can't get away from the cross product with magnetic fields the current goes this way the magnetic field goes at an angle to that that's perpendicular we'll see what Bo and Savart did in a moment with experiments using compass needles to figure out the form of the rule ok but yeah so if you waited long enough and there was a low enough friction that would allow motion in the plane as well yeah eventually those little ferromagnets the little iron filings would clump closer and closer to the wire that would be dangerous because you'd start making electrical connections between the wire and iron which is conductive well eventually you could cause shorts or sparks or things like that again that's like 2 or 3 amps of current going through that that's a lot of current you don't want to short that or touch that with your finger because we know what just a few milliamps will do to a human being so these are they actually I thought it was kind of interesting as I was doing a little bit of historical homework for this slide that there's a photograph of Jean-Baptiste Biot in 1851 or so so maybe about a decade before he died photography was only commercially available as of about 1839 or so it had been invented in the decades prior to that but in terms of commercial photography that wasn't really something you could get you couldn't get a commercial photograph until almost 1840 so you know within a decade after commercial photography first appeared people were photographing other people and there's a picture of a very old Jean-Baptiste Biot and this is probably closer to when he was doing his work in the what 18 well 1820 actually the year that published his results was the year that Savard and Biot did their research with compass needles and currents to figure out what the rule was for having a current in some length of wire from resulting magnetic field from that they figured out the mathematical relationship for that so they worked together in about 1820 and came up with this thing we now call the Biot-Savart Law so let me show this to you and I'll spend the last five minutes talking a little bit about this it looks nasty dust off your calculus because here we go what I'll do in the next lecture is to figure out the magnetic field from a straight wire and if there's time I'll also do a circle and then we're going to do an exercise next week that synthesizes everything that we've been learning about magnetism so far and those things are magnets cause other magnets to repel or attract we know that from just playing with magnets we know from experimenting I showed you the cathode ray tube we know that electric charge can be deflected by magnetic fields so magnetic fields affect moving charge moving charge generates magnetic fields that's what the Biot-Savart Law encapsulates we observe the phenomenon in nature we describe it mathematically and it requires calculus so just like Coulomb's law this is the Coulomb's law of magnetism this law tells you that thanks to moving charge I, Coulomb's per second has a magnetic field B resulting from that and in order to figure out the magnitude in the direction of the field given the current and the length of the conductor there's a relationship so the magnetic field so let's take a look at this bendy wire I have over here let's focus on just a little piece of the wire which I'll call DL vector so DL vector is just a little piece of the length of the wire and the direction that that little vector points is the direction that current I is flowing so this is current I comes in the wire, snakes around goes out that end over there I'm only showing you a little piece of it right now and if we were over at some point P which I've indicated with the black dot we would experience a little bit of magnetic field due to that little bit of conductor now of course the total magnetic field some of the effects of the current here and here, here, here, here all the way up there but if we can calculate using coordinate systems and vectors one piece of the wire and then we can integrate over the whole wire using some trick we can get the total magnetic field we can sum all the little DBs that result from these little IDLs these little current times these little length vectors in the conductor now some experiment so like the demonstration I did here it's easier to see in the demonstration in the video or the little paper with the wire going through it and you can see the circles of magnetic field around the wire we know from that that the direction that the current flows is perpendicular to the direction that the magnetic field points so if the current points this way up here the magnetic field might point that way over here it might point down it's always perpendicular to the direction of the current flow and that's what gives you a circulating magnetic field around the wire so that's an empirical observation that's when you do the experiment that's what nature does and so mathematics you use to describe that phenomenon and try to figure out the exact precise relationship so beyond Sivar worked really hard to help us needles to do this they mapped out magnetic fields from wires and they came up with the rule that the magnetic field direction is at right angles to the current flow which we already knew but the trick is if you draw a vector from the little piece of wire out to where you're observing the field and you use a right hand rule DL points that way R points that way the magnetic field at point P according to the direction your thumb is pointing so again, D L R D B that's what they figured out and the only way you can describe that is with a cross product you have to know the D L you have to write down the R and you have to then from that compute the direction of B and the magnitude of B now the magnitude of B comes from multiplying I D L and then the unit vector R hat and dividing that whole thing by the distance squared much like with Coulomb's law the magnetic field strength seems to decrease as a function of distance and the rule that they worked out is just like Coulomb's law it's an inverse square law which is already in and of itself fascinating that that happens to be the same and then it turns out you have to multiply the number you get from this by a constant mu knot over 4 pi and mu knot is well I'll define it next time but it's actually really easy to remember 4 pi times 10 to the 7 that's it 4 pi times 10 to the 7 it has units but we'll talk about those next time okay so I'll be assigning the next assignment and it'll be focused on you know magnetic fields and forces on wires and things like that and then I'll probably put a little bit of a subart law problem in there too to get you started on that so thanks everybody