Magnetism Lecture Notes

>> So, magnetism is something that you're all familiar with. You've played with magnets in your life, right? You're a little kid and you grab the magnet, you go down to the sandbox, and you start digging around in the sandbox. And you surprise yourself by finding a whole number of sharp objects in the sandbox. Everybody notice that? It's like nails and staples and stuff coming out of the sandbox. It's kind of scary. So, don't ever do that. Okay. What does a magnet look like? A bar magnet looks like the following. Bar magnet, let's see if we can spell correctly. It looks like this. It's a bar. And as Professor Dollakash [phonetic] already told us, a magnet has a north pole and a south pole. Okay. You're all familiar with this. So, let's say we take that bar magnet and we cut it in half. Okay. So, I'm going to cut the thing in half, right there. And if I do that, that south pole was down there, the north pole is up there. Is that it? Is that all that happens? The answer is no. You in fact develop two new magnets. Both of those have a north pole and a south pole. And now if you do it again, and I cut this little piece and that little piece, I cut them in half again, each one has a north pole and a south pole. Okay. North and south, north and south, and so on. Okay, and you can keep doing that all the way down until you get to essentially nothing. And at no point do you get just a north pole or just a south pole. And this is the statement: there are no magnetic monopoles. Okay. You can't have just a north pole. You can't have just a south pole. Anytime you have a north pole, you automatically have a south pole. Okay. North and south always come together. And this is very different than charge. When we talked about charge, you can have a positive charge sitting all by itself, a proton. You can have an electron sitting all by itself. But you can't have a north pole just sitting all by itself or a south pole sitting all by itself. Now, there is a little bit of a caveat which Professor Dollakash [phonetic] really alluded to. There are theories that magnetic monopoles should exist. That there should be a north pole all by itself. And there should be a south pole all by itself. But currently, there has been no experimental proof that those exist. Okay. So, this is the current understanding of the universe. But ten years from now, fifty years from now, this may no longer be true. Remember, physics, like any science is an evolving field, okay. We are continually evolving our models of the universe. And this is the current understanding. But ten years from now, fifty years from now, maybe in your lifetime, maybe it will say ah, there are magnetic monopoles. We found them definitively, and here they are. Okay? All right. What are these magnets do? Well, I have a magnet here. Let's put another magnet right next to it. So, if I put another magnet, right there, something's going to happen. What is going to happen? You've probably done this experiment. If I take two magnets and I try to push the north poles together and I try to push the south poles together, they push apart. There is a repulsive force, F, which is pushing them apart. Okay. But if I flip one of them over, now they are attracted together. Okay. So, in the first case they pushed apart, in this case they attract. They're going to pull together. And so, this is a statement that we talked about yesterday with magnetism: likes repel, opposites attract. Which is very consistent with what we've already been talking about with electrostatics. Positives repel each other. Those would be likes. Negatives repel each other. Okay? But opposites attract. If I had a positive and a negative, they would attract. If I had a negative and a positive, they would attract. So, likes repel and opposites attract. Okay. Let's see what the B field looks like for a bar magnet. Okay. So, here's our bar magnet. And before we do that, let's redraw the dipole. We remember what the dipole field looks like. Looks like this. Okay, and you can draw as many as you like. And now if I draw the bar magnet, it looks extremely similar. Comes out of the north goes into the south. Okay, it's a dipole field, just like the electric dipole. This is a magnetic dipole. So, one question that you may have is the following. If north and south always come together, then I must always have a dipole field for magnetism. And that is true. There is no monopole field. There is no just north charge. So, lines of B, in fact, are continuous. And this is an important statement. They always connect back on themselves. Whereas lines of E do not. They start on positive charge, they end on negative charge. And so now, here's the question. What is the direction of the magnetic field inside this thing? Inside the bar magnet? And let's go back to the dipole for a second. The field in between these, E, is going down from the positive to the negative. But over here, the fields of B in fact are going up through the center of the magnet. And that continuous line is the B field. The B field is a continuous loop. When it comes down here, it doesn't just stop at the south pole. It continues back up to where it started and goes back around again. Okay. So, B fields are a little bit different than E fields because B fields are continuous. Okay. Let's talk about the magnetic force. What we learned yesterday is if a charge is moving, and it's in a B field, it feels a force. What force does it feel? It feels this: qvBsin(theta). What is this stuff? This is obviously the force that it feels. q is the charge. v is the speed as measured by you in the laboratory rest frame. What speed v do you measure? B is the magnetic field. And theta is the angle between the velocity, v, and the B field, B. The magnetic field B. So, force we know is a vector. This is the magnitude. How do we determine the direction? Well, let's write it down again. qvBsin(theta) and then we put an n hat on there, which is a direction. That makes it a vector. And this direction, we have to use something called the right-hand rule to determine the direction. Okay. q is the charge. So, it can be positive or negative. You always use the right-hand rule to determine the direction and then if it's a negative, you flip that direction. v is a speed. B is a magnetic field. Theta is the angle between those two. So, we need to say a little bit about this, the right-hand rule. Okay. And this is slightly tricky because of this learning glass approach. So, I'll talk you through it. And hopefully, we can make some sense of it. So, let's write this down again. The magnetic force we said was qvBsin(theta) n hat. And now let's determine this n hat. What direction is it? Well, turns out that direction is given by your thumb if you take the velocity, v, and you put your fingers in the direction of v, and you take your fingers bent into the direction of B. Okay. F, v, B. All right. Fingers. Let's clarify this one a little bit. We're going to say, fingers straight for that first one and then fingers bent for B. All right. So, let's see if we can figure that out. I'm going to draw your hand. Okay. There's your hand. Okay. And we'll put a put a thumbnail on it so you know which way your hand is pointing. Nice pink shade. There we go. Okay, this is your right hand. The force is the direction of your thumb. The fingers straight are the direction of the velocity, v. The fingers bent are the direction of the B field lines. Okay? And so, here's how you do it. Okay. Let's say that we have the following system. We've got an XYZ coordinate system x y z. And we're going to have a particle that is moving along to the right, velocity v. It is in a magnetic field B pointing up. And now we need to figure out the direction of the force on this particle. Okay. And so we can use the right-hand rule to do that. So, now I'm going to demonstrate this to you. But because we're flipping the image, I got to use my left hand. And so don't look at me through the glass. Look at the monitor over there and see if you can follow along. Okay. So, hold up your right hand. Okay. Everybody got the right hand up? The direction of v is to the right. So, put your fingers straight in the direction of v. There we go. I'm going to curl my fingers into the direction of B. B is going up. So, curl your fingers until they're going up. Your thumb is now the direction of the force. And if you look at the monitor, it should look like that force is coming towards you. Okay. Which in our coordinate system would be coming towards you out in the x direction. v cross B gets me a force that's coming out. And hopefully everybody is familiar with this vector sign. A circle with a dot is coming out of the screen towards you. Okay. So, if you're looking at this on home on your computer, it's coming out of the screen towards you. And this is going into the screen. Okay. And this is like an arrow, right? If you see a dot with a circle, that's the arrowhead coming right at you. If you see a circle with an X, that's the tails of the feather going away from you. Okay. So, think of an arrow. Is it coming at me to do I see the tip? Or is it going away from me? Do I see the feathers? Let's take v, we said it was going up, and B, was going to the right. So, v cross B, should get me something that is going, I think maybe right into the screen. Let's see. v going up. B to the right, that's this way, which is away from you guys, which is into the screen. I think I agree with you. Good. Let's try it for a different orientation. Let's say we do B going down and v going to the right. All right. v fingers straight. B curl it. I still get something that's going into the screen. Good. Let's try one more. B is going to the right. v is going down. Okay. Now how do I do that? Well, I got to put my fingers down, okay, for v. And then I curl them into the direction of B. So, v cross B gets me something coming towards you guys, which is out of the screen. Good. So, let's say we do the following. Let's say we have this. Let's draw on XYZ coordinate system x y z. And now let's do this. We're going to take a positive q, and we're going to send it flying in the y direction. And now it's also in a B field that is also pointing in the y direction. What is the force? What is the force on our charge? Okay. We can do the right-hand rule again. So, you put your finger straight in the direction of v. And then you have to curl them into the direction of B. But B is in the same direction as v. So, I can't curl my fingers that way because they won't be pointing towards B. And I can't do them this way, because it won't be pointing towards B. And so, none of those directions make sense. And if none of the directions make sense. What does the force have to be? What is it 90% of the time when I asked you to estimate? Zero. The force has to be zero. Right? If the physics says you can't determine the direction, then no direction is preferred. It has to be zero. Okay, but how do we see that mathematically? Well, let's go back to our definition. F equals qvB at a direction which we determined from the right hand rule. Okay, we've got a q. We've got a v. We have a B, but then we have the sine of the angle. And remember the angle here is between v and B. What's the angle between v and B? What do you think the angle is between v and B? Zero, there we go. What's the sine of 0? The sine of 0 is 0. There is no force. Okay? If v is in the direction of B, there's zero force on the particle. All right. So, v is to the right still, but we're going to let our B be up everywhere. And now we do the right-hand rule for qv cross B. So, v hold -- everybody hold your right hand up. v finger straight curl into the direction of B. That's telling me that my thumb is the direction of the force, which is coming towards you guys. So, there is a force in the -- sorry, the dot right -- coming towards you. Which would be in our 3D picture in the negative -- or I'm sorry, in the positive x direction. Okay. So, that force will be coming towards you. What is its strength? Well, it's qvBsin(theta) n hat. We just said that it's going to be in the x direction. We know that the x direction means i hat. Okay. i j k, x y z. So, n hat becomes i hat. What is the angle? What is the angle between v and B? It's 90 degrees. Right? They're at a right angle. v is in the y-axis, B was in the z-axis. And so, you get 90 degrees. And sine of 90 degrees is of course one. So, what's the force? It is qvB i hat. It's coming out of the page, and it is a strength of qvB. So, if you are in some other arbitrary direction, then you have to include that theta. So, if this is x y z and let's say that our particle is going to the right, but now our B field is up here at some angle theta. Now, what is the direction of the force? Well again, we use a right hand. So, we do a qv cross B and that tells me that it's coming out towards you, right? Just like we had here. It's still coming out towards you and the strength is going to be with the sine theta. Okay. Sine theta is of course smaller than one so, it's going to be a little less than this force. So, if they're at a right angle, you get the maximum amount of force. If they're parallel, you get zero. If they're somewhere in between, you get sine of theta. B, we said is this: qvBsin(theta). So, B we can solve. It's going to be force over qvsin(theta). And so, the units here are what? Well, we know force is Newton. We know that charge is Coulombs. We know that v is meters per second. Sine theta is unitless. That's just a number. And this is Newton second per coulomb. And a newton per -- a Newton second per coulomb sorry, meter in the bottom is called a Tesla. Okay. And it's named after Nikolai Tesla. You might think of something else when you hear that word Tesla. What do you think of when you hear that word Tesla? You're thinking of the car, right? The electric car company Tesla. Okay. It's all related to magnetism. Why is an electric car related to magnetism? Why didn't they call it the Coulomb? Maybe that would be a better name. Okay. The batteries produce the electricity but then they use the electricity to in fact drive the wheels. And you can't just take that electricity and do that directly without using magnetic fields. So, there is an electric motor in there that has magnets in it. And when you use electricity, you can in fact drive those magnets, drive the wheels. And so, that's why it's called a Tesla, which I think is a great name. So, let's say that we're going to think about an E field first. Okay. Let's say I have the following picture. Here's my 3D coordinate system x y z and let's draw our E field everywhere to the right. Okay. And let's say that we have a charge, which is initially moving in the x direction. Okay. Here comes my charge. It's a positive charge and it's going to move in the x direction. What is the electric field going to do to this charge? Well, we know that if it is a positive charge, plus q, it's going to feel a force due to that electric field. And so eventually, it's going to start moving in the positive y direction. Okay. It's going to bend around and eventually come in the positive y direction. Okay, it will eventually be parallel to E. But let's do the same thing now and think about the motion of q in a B field. So, let's draw the same thing. Here's our coordinate system x y z. And now let's have this q initially moving in the x-axis direction. Okay? Well, if my B field is now pointing everywhere to the right, then there has to be a force on this thing. Right? And the force on it, we determined from our right-hand rule. So, we've got a v coming out in the x direction. We've got a B going back to the right. So, I got to figure out how to rotate my hand such that I can do that properly. And if I do that just right, you guys can try it. My arthritis is kicking in. That's how you know physics majors, right? Because they walk around the hall and they go like this all time. What you should get is the following. v cross B is going to get me a force that's going up. And so, it's going to start bending up. And now when it's going up, it's going to feel a force to the right. And then when it's going back, it's going to feel the force down and eventually we'll come back around to where it started. Okay? And so, you're going to get a v cross B force that makes it move in a circle and that circle is always perpendicular to the B field. Okay? The force is perpendicular to the velocity and the B field and therefore this circle is perpendicular to the B field. In other words, the circle lies in the xz plane. Okay. That's what we're trying to draw here. We're trying to draw a circle in the xz plane. Okay. So, let's say we have a region of space here. And in this region of space, we have a B field that is everywhere going into the screen. Okay. And B field is everywhere pointing into the screen. And now you want to put a charge in this thing and send it flying through. Okay? So, I'm going to take a q and let's see. Which way do I want to send it? I want to start right here. We'll make it a positive q and let's send it this way. All right. This thing is going to feel a force and it certainly has to feel a force either up or down. Okay. And remember, B is going into the screen. So, which way is that force going to be? Is it up or is it down? Look at the computer monitor over there, pick up your right hand, put your finger straight in the direction of v. Curl them into the screen for the direction of B. Which way is the force? It's got to be up or down. What do you guys think? Anybody get an answer on that one? Let's try it. v is going to the right. B is into the screen. So, I'm going to curl my fingers into the screen, right? That's to you what looks like into the screen. Okay. And so, I'm going to end up with a force that is up and that means that this thing is going to move in a circle. Okay. That particle that came flying in is in fact in a move around in a circle. Okay. Is that right? Did everybody see that okay? v is to the right. B is into the screen. Thumb is the direction of force. So, it's towards the center of a circle at this point. Later on, v is going up. B is into the screen. And so, I again get a force towards the center of the circle. And everywhere around, it's always a force towards the center of the circle. Okay, but we know what that force is. It's qvBsin(theta). Let's just worry about the magnitude of it. We already figured out the direction. q v B. What's the angle between those two? Well, v was to the right, B was into the screen and so, we get 90 degrees. But 90 degrees is just 1. So, what's the force? qvB. All right. But this thing is moving in a circle now of radius r. And what we know from last semester is if things are moving in a circle, we can consider the forces adding up to something very specific. Namely, m v squared over r, right? This is uniform circular motion. The centripetal force is mv squared over r. But the only force is this magnetic force. Okay? And so that becomes the left side of this equation. So, we get qvB equals mv squared over r. And now we can quickly solve this thing for the radius. Right? I multiply by r, I'm going to divide by qvB. One of the v's is going to cancel out and I get mv over qB. Okay? What's the radius of that circle? It's that. It's the momentum of the particle divided by the charge times the magnetic field. So, when you are doing a high energy physics experiment and you create all these charged particles that are moving very quickly, how can you determine something about them? Well, you put them in a big magnetic field and you watch them spin around in a circle. And when you do that, you can determine this quantity right here. All you have to do is determine the radius of the circle. And so, when you see those pictures from CERN and places like that, where they have particles coming in and then doing this sort of stuff, and this sort of stuff, and all these circles and beautiful pictures like that, what those are are charged particles moving through magnetic fields. And from those pictures, all they have to do is measure the radius. And if they know the magnetic field that was in there, they can figure out something about how fast it's moving. What's the mass? What's the charge on it? Positives are going to go this way. Negatives are going to go the other way. Okay? Ones that are very lightly charged -- have a small q have a bigger radius. If they have a bigger mass, they have a bigger radius. Okay? And so, you can determine a lot just from taking these sorts of pictures. And it's really not that complicated, right? We just did it with the force law -- magnetic force law and the centripetal force law. Let's talk a little bit about wires carrying current. So, let's say I have a wire here and it's going to carry some current. Okay, let's say that current I is to the right. And let's take this wire and let's put it in a magnetic field. And let's see if there is a force on it. All right. How do we do this? Well, let's say this is our wire and let's put a magnetic field everywhere pointing up. Is there a force on that wire? Well, let's go back to the magnetic force for a second. Magnetic force is the following: F equals qvBsin(theta) and then we have to worry about the direction from the right-hand rule. So, we certainly have a B. We probably have some angle theta. What about this right here? q and v? Do I have a charge moving? Well, we have a current, right? Current is charged moving. We know what current is. It's delta q in some amount of time, delta t. So, if I think about a wire carrying current, I can consider that is consisting of charges moving at speed v. Charge q moves at v through the wire. That is current. Okay. So, F we could really say is the following. Let's just call this thing delta qvBsin(theta) n hat. The delta q just means some amount of charge. Okay. We're going to look at some chunk of charge. But what if I divide by a delta t? I can't just divide by a delta t arbitrarily without changing the value. But if I divide by a delta t, I can multiply by a delta t and then I'll get the same thing. Okay? So, I haven't changed the relationship. It's the exact same relationship. But, now we can write delta q over delta t as I. What about v delta t? What is v delta t? Well, if the charge is moving along at v and it does that for some amount of time, delta t, then that corresponds to some length of wire. The length of the wire is L. And that is just v delta T, right? Speed times time gets me distance. So, this v delta t just becomes L. It's some length of this wire. We have an I, we have an L, and then we have of magnetic field B, and then we have sine theta n hat. So, a wire carrying current, if it is sitting in a B field, it will feel a force. It will feel a force given by that ILBsin(theta). And then the n hat again is determined by the right-hand rule. So, let's say we do this and let's see if we can figure out the direction. And let's simplify our wire a little bit. Let's say our wire is this. Okay? We don't have to draw it all 3D. It's just a single line. Now let's again say that the B field is everywhere pointing up. Okay? What is the force? Well, we know the magnitude. It's ILBsin(theta). The direction is going to be the right-hand rule. Theta now is the angle between I and B. Okay. I becomes like our qv. B is still the magnetic field, of course. So, in this problem, what is the angle between those two? Well, I is to the right, B is up. So, the angle between them is 90 degrees, which is just one. But now we got to figure out the direction. Remember, the direction is always going to be perpendicular to I and perpendicular to B. So, you only have two choices here. It's either out of the screen, or it's into the screen. And the way we do it is that we again use the right-hand rule, but the fingers go in the direction of I. So, hold up your right hand, put your fingers in the direction of I. B we said is everywhere going up. So, I cross B gets me something that is coming out of the page. Right? My thumb is coming out of the screen towards you. And so, there is a force here which looks like that. I cross B is coming out of the screen. So, if you want to write this direction, you could say out of the screen or out of your computer monitor. Okay. If you increase the length of the wire, you increase the total force. If you increase the current, you increase the total force. If you increase B, you increase the total force. So, let's ask you this question. Let's say I have current, I, going to the left. And now I have B going up? What direction is the force on that wire? Into the screen or out of the screen? Okay. Into the screen. Let's see that. Put my fingers in the direction of I. Rotate my fingers bent into the direction of B. My thumb is pointing into the screen -- into your computer monitor. And so, the force is in fact into the screen. All right. What about this? Let's say I draw current going up. B field going up. What's the direction of the force on that? Or what is the force on that? Remember, I fingers straight in the direction -- your fingers straight in the direction of I. Now you curl your fingers into the direction of B. But why don't I can't curl them in any direction. Right? That doesn't work. And that doesn't work because bees in the same direction. So, the force is zero. And that's because theta is of course zero. You get sine of 0 which is 0. How about this? Let's say I have I going down, but B is going up. What's the force there? Well, I put my fingers in the direction of I and now I curl it in the direction of B but I got to come back all the way around. Should I go that way? Oh, Should I go this way? I don't know. So again, it's zero. And that's because this is 180 degrees. Sine of 180 degrees is equal to 0. Okay? So, just like we had with qv, anytime the current is parallel or antiparallel to the magnetic field, there's no force. All right. But this looks like a section of wire. This looks like a section of wire. These two are sections of wire. Can we just put them all together into one loop? Okay. So, we're going to do a square circuit and it's in the B field. And so, let's draw that picture again. And we'll do the following: current I going around like this. Okay. Everywhere going around. Here's our square wire carrying current I. And now let's put it in a B field. And we're going to say that B is up everywhere. Okay. We have some B field that's pointing up everywhere. And now let's see if we can calculate the forces on each side here. So, we know that the force on the top side is either going to be out of the screen or into the screen. Right? It has to be perpendicular to I. It has to be perpendicular to B. So, on this top side, what should I draw for the force? Should I put a dot in there or should I put an X? Well, let's try it. I is the direction of your straight -- finger straight. You curl it into the direction of B. Okay? And if you do I cross B, you should get something coming towards you. Right? That's what you see. I cross B, it's coming back towards you coming out of the screen. Good. What about this set? The force on this side has to be just the opposite of what we just did. Right? We got a dot. This one has to be an X. Let's just confirm. I is going to the right. B is going up. My thumb is going into the screen. Okay? And so, you get a force that is right there with an X into the screen. What about the force on either side? The force on this side is of course zero, because they're parallel. The force on this side is also zero because they're anti parallel. And so, that entire square looks like this. It's got a force up here that's going in. It's got a force down on that side that is going out. Sorry, out of the screen -- into the screen. And so, the net force on this whole loop is equal to zero. Okay. The whole loop doesn't move up or down or left or right. But there is a twist. There is a torque because one side is trying to come out of the screen, the other side is trying to go into the screen. And so, this entire thing is going to rotate in this B field. Let's try it with some numbers here and see if we can calculate what's happening. This is a square of side L and let's say it's 32 centimeters, which is 0.32 meters. Let's say that the current running around in this thing is 12 amps. And let's say that the B field that it's sitting in is a quarter Tesla. Okay? All right. Let's do each side individually and calculate the force on it. So, the top side, we said there is a force coming out of the screen. And that force is going to be qvBsin(theta). But we know that qv turns into IL. We give you I. We give you L. We give you B. What is the angle between those two? Well, I was to the right. B was up. Those are 90 degrees to each other. Sine of 90 is just 1. And so, we get ILB and we know what those numbers are. I is 12 amps. That's SI units. L is 0.32 meters. And B, we said was 0.25 Tesla. Okay, and let's run those numbers and see what we get and why don't you just punch it into your calculator and tell me what you get. I'll approximate it here. We got a quarter of 12. A quarter of 12 is 3. 3 times 0.32 is got to be really close to 1. Maybe it is more like 0.96. Is that what you get if you plug it into your calculator? Okay, 0.96. And it's a force. Right? So, these SI units, amps times meters times Tesla gives you Newtons. That's the force on the top wire. The sides we said were zero. So, we don't have to worry about those. The one on the bottom is into the screen. And guess what? That's going to have the exact same strength as the one on the top. Because it's a square wire, it's the same length, we have the same angle theta, which is 90 degrees. And so, we also get 0.96 Newtons. Force on the top is exactly the same as the force on the bottom. Means it doesn't move anywhere. But it does rotate. It does twist. What is that twisting? Torque. Right? When we talk about twist, we talk about torque. What is that torque? I know. You guys thought you could forget about torque, right? That was -- that was so last semester. Do we have to really remember everything from last semester? Yeah, you kind of do. Torque is written with a tau. Okay? And torque, tau, is what? Well, its force times lever arm, right? Torque is equal to force times lever arm. Okay. This whole thing wants to rotate, right? The whole square wants to rotate towards you on the top. Away from you on the bottom. What's a good axis of rotation to pick? Well, how about halfway through? Seems like this thing might want to rotate right about its center. So, let's make this our axis of rotation. And if I make that the axis of rotation, then the lever arm here is just L over 2. And the lever arm on the other side is L over 2. All right. We want the net torque. So, there is a force on the top side of IL times B. There is a lever arm on that top side that is L over 2. That's the torque due to the top wire. But I also have to add the torque due to the bottom wire. And the torque due to the bottom wire is force ILB also times the same lever arm because it's trying to rotate it the same way. This force is pushing the top towards you. This one is away from you. And so, they're both trying to rotate the object in the same direction. And so now look what happens. We have I times L times B. I have a half an L and another half an L. And so, they combined to give me another one L. And this is IL squared times B. Ah ha! But L squared, that's just the area of this loop. So, this whole thing becomes I times A times B where this thing is the area of the loop. Okay? And this is torque on a loop if it is at a right angle to the magnetic field. And in general, we can write the following: torque is IAB times the sine of phi where phi is the angle between B and the planes normal. Okay, what do we mean by that? Let's say I draw a square loop. Okay. The normal would be that. So, if the B field is to the right, then phi in this case would equal 90 degrees. But if I draw it again with the B field going up, then the normal to the plane and B are in the exact same direction. In that case, phi equals zero degrees. Okay. So, this phi is between the normal to the loop and the B field. And we said that torque was equal to I times A times B in our example. And we gave you those numbers. We said that the current was 12 amps. We said that the area was -- let's see. It's a square and it's 0.32 on a side. So, I have to square that. And then we're multiplying by B which we said was 0.25 Tesla. So, somebody punch that into your calculator and tell me what you get. We can approximate it here because 12 times a quarter -- 12 times a quarter is 3. And then 0.32 is really close to 0.33 which is 1/3 squared. And if I do 3 times a 1/3 squared, I get 3 over 9, which is pretty close to 1/3, right? So, I'm going to say this thing has got to be very close to 0.33. Anybody get a real answer for that one? Okay. 0.3 what? >> 072. >> 072. Okay. And that's the torque. And what are the units of torque? Remember, torque is forced time to lever arm. So, 0.3 force is Newtons, lever arm is meters. And so, the units of torque are Newton-meters. Let's take a look at this problem. So, this one is a 2028 on your homework. And it says a vertical straight wire carrying an upward 22 amp current exerts an attractive force per unit length of 8.8 times 10 to the minus 4 newtons per meter on a second parallel wire 5 centimeters away. What is the magnitude of the current in the second wire? All right. To do this, we need to know just a little bit more. And what we need to know is something that was introduced yesterday, but let's say it again. Currents themselves produce B fields. Okay. B fields affect currents but currents also produce B fields. So, if I think about a wire and it's carrying current I, there is a B field around that wire. And the B field around the wire looks like this. The dashed line indicates going behind the wire. The solid indicates coming out in front of it. And we can figure out the direction of B based on the right-hand rule. Okay so, if you take your thumb and put it in the direction of I, then your fingers wrapped around in the direction of B. Okay? So, put your look at the computer monitor, put your thumb in the direction of I, wrap your fingers around, that should be the direction of B. Okay? Now, if that thing is making a circle, then the B field has to have the same strength at any radius from the line. And we know what that strength is. We talked about it yesterday. The strength of B is this: mu naught I over 2 pi r. The direction is determined from the right-hand rule but it's a slightly different right-hand rule. It's a slightly modified right hand rule, which is the following. Put your thumb in the direction of the current and your fingers curl around in the direction of B. Okay. So, my thumb is going up in the direction of the current. My fingers are wrapping around in the direction of B. Okay? And we call this the right-hand rule for a reason. It works with your right hand, it doesn't work with your left hand. Okay. If you do your left hand, you will get the exact opposite of what you intended. So, you have to use your right hand to do it. I'm, of course using my left hand in reality, but then when we flip it on the monitor, it looks like my right hand. Okay. And so, I'm finding this challenging over here to do this. All right. What is this mu naught? Mu naught is the permeability of free space. Just like with the electric field, we have the permittivity of free space. Now we have mu naught which is the permeability and it's 4 pi times 10 to the minus 7. And the funky units on it are Tesla meter per amp. All right. So, that's the strength of the B field at any radius -- at any distance from a current carrying wire. All right. So, maybe that'll help us because it's telling us we have a long current carrying wire and then there is some force per unit length on the second parallel wire. All right. So, let's hang on to some of this. We're going to say B equals mu naught I over 2 pi r. And now let's add a second wire and see if we can make some sense of it. I've got a B here that is going around, as we indicated in that picture. Now we're going to take a second wire and run current through it. And now the question is, are those wires attracted together or are they pushed apart? And let's think about this B field for a second at the point of this wire. Okay? This B field is not just at that radius. It's at any radius you want to pick. And so, at this wire, this B field is in fact doing what? It is going into the page going into the page. Okay? If it's going into the page, then there must be a force on this current carrying wire. And the force has to be ILBsin(theta) and I determine that from the right hand rule. So, I hold up my right hand and I put my finger straight in the direction of I and then the B field is going into the screen. Which way? Is it towards the other wire? Or is it away? It's towards. There is a force pulling it towards the other wire. Okay? And that's exactly what they said in the problem, right? They said an upward current exerts an attractive force on the second wire. Okay. And now we need to figure out what the current is in the second wire. And what they tell us in the problem is the following. They give us the force per unit length. Force per unit length is -- we can use my numbers 8.8 times 10 to the minus 4 newtons per meter. And they also tell us that the distance between the wires, which is our r, is five centimeters. So, 5 centimeters equals 0.05 meters. And we need to figure out what this current is in this wire. Okay? What's the magnitude of the current and the second wire? So, to be clear, we better call these two different things. Let's call this one I1 and this one I2. And that means this down here is I1. And this B field here is B1 What is the B field due to the first wire at this location of the second wire? Okay. So, let's see if we can put this stuff together. What we know is force is ILBsin(theta). That's the magnitude of the force. All right. Let's divide by L. F over L equals IBsin(theta). And now we can start being specific about what currents and what magnetic fields we're talking about. If we're talking about the force on this second wire, then we're worrying about the current I2 in that wire. But it feels a force due to the B field from the other one. So, that should be B1. And we know that sine of theta -- let's see. Theta is going to be 90 degrees, right? I is up. B is into the screen and so theta is going to be 90 degrees. And so, now we can plug in for B1. The one was mu naught I1 divided by 2 pi r. We have sine of 90 degrees, but that's just 1. And so, we get mu naught over 2 pi times I1 I2 all over 2 pi r -- I already wrote the 2 pi just over r. Okay. And now we can solve this equation for I2 Okay? And that's what we're looking for. So, let's move it over here and we'll solve it for I2 All right. So, we had F over L equals mu naught over 2 pi. I1 I2 all over r. And we want to solve this for I2 So, let's see. I got to multiply by 2 pi r. I have to divide by mu naught and I have to divide by I1 And that should be my I2. And so, now we can calculate I2. I2 is F over L, which we're given. r we're given. I1 we're given. F over L was this: 8.8 times 10 to the minus 4. We have a 2 pi. We have r which is 0.05 and then we have mu naught, which we know is 4 pi times 10 to the minus 7 in SI units. And then I1 for our problem is 22 amps. All right. So, somebody punch that into your calculator and tell me what you get. And let's see if we can approximate it. It's kind of fun to approximate these things. So, what do we got? We got the pi's. Those are going to drop out. That's nice. We got a 2 that drops out with that if I give that a 2 right there. And so, we're going to get 8.8 -- 0.05 is 1/20. Then we have a 10 to the minus 4. And we're going to divide the whole thing by 2 times 22, which is 44. And then we're going to end up with another 10 to the 7 up on top. So, let's see. What do we get? 8.8 over 20. That is really close to 1/2. If I divide a half by 44, what do we get? That's 1/88, which is pretty close to 0.01. A little bit more. How about 0.01 -- let's take a guess -- 0.012? And then we've got a 10 to the minus 4 and a 10 to the 7 which we get a 10 to the 3. So, I'm going to say that I2 is approximately one, two, three -- 12 amps. That's my guess. Anybody get a real number? Nobody got an answer yet? >> I got 10. >> 10? Okay. That's pretty close to 12. Anybody else get 10? 10? Okay. All right. So, our guess was a little off. But that's fine. Let's try 10. 10 units are amps. Should we click submit? Let's click it. Correct! All right. I got one point on my homework. Excellent. Okay. The part B of this is what is the direction of the current in the second wire? If you remember our picture, we had current in the first wire and the current in the second wire was up and it was an attractive force. So, we're going to say upward for the second part -- part B. Submit. Also correct. Okay. Good problem. All right. Why don't we wrap it for today? And I will see you guys tomorrow. Online. Cheers.

&gt;&gt; So, magnetism is something
that you&#39;re all familiar with. You&#39;ve played with magnets
in your life, right? You&#39;re a little kid and
you grab the magnet, you go down to the sandbox,
and you start digging around in the sandbox. And you surprise yourself
by finding a whole number of sharp objects in the sandbox. Everybody notice that? It&#39;s like nails and staples and
stuff coming out of the sandbox. It&#39;s kind of scary. So, don&#39;t ever do that. Okay. What does a
magnet look like? A bar magnet looks
like the following. Bar magnet, let&#39;s see if
we can spell correctly. It looks like this. It&#39;s a bar. And as Professor Dollakash
[phonetic] already told us, a magnet has a north
pole and a south pole. Okay. You&#39;re all
familiar with this. So, let&#39;s say we take that bar
magnet and we cut it in half. Okay. So, I&#39;m going to cut the
thing in half, right there. And if I do that, that
south pole was down there, the north pole is up there. Is that it? Is that all that happens? The answer is no. You in fact develop
two new magnets. Both of those have a north
pole and a south pole. And now if you do it again,
and I cut this little piece and that little piece, I
cut them in half again, each one has a north
pole and a south pole. Okay. North and south,
north and south, and so on. Okay, and you can keep
doing that all the way down until you get to
essentially nothing. And at no point do you
get just a north pole or just a south pole. And this is the statement:
there are no magnetic monopoles. Okay. You can&#39;t have
just a north pole. You can&#39;t have just
a south pole. Anytime you have a north pole, you automatically
have a south pole. Okay. North and south
always come together. And this is very
different than charge. When we talked about charge, you can have a positive charge
sitting all by itself, a proton. You can have an electron
sitting all by itself. But you can&#39;t have a north
pole just sitting all by itself or a south pole sitting
all by itself. Now, there is a little
bit of a caveat which Professor Dollakash
[phonetic] really alluded to. There are theories that
magnetic monopoles should exist. That there should be a
north pole all by itself. And there should be a
south pole all by itself. But currently, there has
been no experimental proof that those exist. Okay. So, this is the current
understanding of the universe. But ten years from now,
fifty years from now, this may no longer be true. Remember, physics, like any
science is an evolving field, okay. We are continually evolving
our models of the universe. And this is the current
understanding. But ten years from now,
fifty years from now, maybe in your lifetime,
maybe it will say ah, there are magnetic monopoles. We found them definitively,
and here they are. Okay? All right. What are these magnets do? Well, I have a magnet here. Let&#39;s put another
magnet right next to it. So, if I put another
magnet, right there, something&#39;s going to happen. What is going to happen? You&#39;ve probably done
this experiment. If I take two magnets and I try
to push the north poles together and I try to push the south
poles together, they push apart. There is a repulsive force, F,
which is pushing them apart. Okay. But if I flip one of them over,
now they are attracted together. Okay. So, in the first
case they pushed apart, in this case they attract. They&#39;re going to pull together. And so, this is a statement
that we talked about yesterday with magnetism: likes
repel, opposites attract. Which is very consistent with
what we&#39;ve already been talking about with electrostatics. Positives repel each other. Those would be likes. Negatives repel each other. Okay? But opposites attract. If I had a positive and a
negative, they would attract. If I had a negative and a
positive, they would attract. So, likes repel and
opposites attract. Okay. Let&#39;s see what the B field
looks like for a bar magnet. Okay. So, here&#39;s our bar magnet. And before we do that,
let&#39;s redraw the dipole. We remember what the
dipole field looks like. Looks like this. Okay, and you can draw
as many as you like. And now if I draw
the bar magnet, it looks extremely similar. Comes out of the north
goes into the south. Okay, it&#39;s a dipole field,
just like the electric dipole. This is a magnetic dipole. So, one question that you
may have is the following. If north and south
always come together, then I must always have a
dipole field for magnetism. And that is true. There is no monopole field. There is no just north charge. So, lines of B, in
fact, are continuous. And this is an important
statement. They always connect
back on themselves. Whereas lines of E do not. They start on positive charge,
they end on negative charge. And so now, here&#39;s the question. What is the direction of the magnetic field
inside this thing? Inside the bar magnet? And let&#39;s go back to
the dipole for a second. The field in between these, E, is going down from the
positive to the negative. But over here, the fields
of B in fact are going up through the center
of the magnet. And that continuous
line is the B field. The B field is a
continuous loop. When it comes down here, it doesn&#39;t just stop
at the south pole. It continues back up
to where it started and goes back around again. Okay. So, B fields are a little
bit different than E fields because B fields are continuous. Okay. Let&#39;s talk about
the magnetic force. What we learned yesterday
is if a charge is moving, and it&#39;s in a B field,
it feels a force. What force does it feel? It feels this: qvBsin(theta). What is this stuff? This is obviously the
force that it feels. q is the charge. v is the speed as
measured by you in the laboratory rest frame. What speed v do you measure? B is the magnetic field. And theta is the angle
between the velocity, v, and the B field, B. The
magnetic field B. So, force we know is a vector. This is the magnitude. How do we determine
the direction? Well, let&#39;s write it down again. qvBsin(theta) and
then we put an n hat on there, which is a direction. That makes it a vector. And this direction, we have to use something called
the right-hand rule to determine the direction. Okay. q is the charge. So, it can be positive
or negative. You always use the right-hand
rule to determine the direction and then if it&#39;s a negative,
you flip that direction. v is a speed. B is a magnetic field. Theta is the angle
between those two. So, we need to say a little bit
about this, the right-hand rule. Okay. And this is
slightly tricky because of this learning
glass approach. So, I&#39;ll talk you through it. And hopefully, we can
make some sense of it. So, let&#39;s write this down again. The magnetic force we said
was qvBsin(theta) n hat. And now let&#39;s determine
this n hat. What direction is it? Well, turns out that direction
is given by your thumb if you take the velocity,
v, and you put your fingers in the direction of v, and
you take your fingers bent into the direction of B. Okay. F, v, B. All right. Fingers. Let&#39;s clarify
this one a little bit. We&#39;re going to say, fingers
straight for that first one and then fingers bent
for B. All right. So, let&#39;s see if we
can figure that out. I&#39;m going to draw your hand. Okay. There&#39;s your hand. Okay. And we&#39;ll put a put a
thumbnail on it so you know which way your hand is pointing. Nice pink shade. There we go. Okay, this is your right hand. The force is the
direction of your thumb. The fingers straight are the
direction of the velocity, v. The fingers bent are the
direction of the B field lines. Okay? And so, here&#39;s
how you do it. Okay. Let&#39;s say that we
have the following system. We&#39;ve got an XYZ coordinate
system x y z. And we&#39;re going to have a particle that is
moving along to the right, velocity v. It is in a
magnetic field B pointing up. And now we need to
figure out the direction of the force on this particle. Okay. And so we can use the
right-hand rule to do that. So, now I&#39;m going to
demonstrate this to you. But because we&#39;re
flipping the image, I got to use my left hand. And so don&#39;t look at
me through the glass. Look at the monitor over there
and see if you can follow along. Okay. So, hold up
your right hand. Okay. Everybody got
the right hand up? The direction of
v is to the right. So, put your fingers straight in
the direction of v. There we go. I&#39;m going to curl my fingers into the direction
of B. B is going up. So, curl your fingers
until they&#39;re going up. Your thumb is now the
direction of the force. And if you look at the
monitor, it should look like that force is
coming towards you. Okay. Which in our coordinate
system would be coming towards you out in the x direction. v cross B gets me a
force that&#39;s coming out. And hopefully everybody is
familiar with this vector sign. A circle with a dot is coming
out of the screen towards you. Okay. So, if you&#39;re looking at
this on home on your computer, it&#39;s coming out of the
screen towards you. And this is going
into the screen. Okay. And this is
like an arrow, right? If you see a dot with a circle, that&#39;s the arrowhead
coming right at you. If you see a circle with
an X, that&#39;s the tails of the feather going
away from you. Okay. So, think of an arrow. Is it coming at me
to do I see the tip? Or is it going away from me? Do I see the feathers? Let&#39;s take v, we said
it was going up, and B, was going to the right. So, v cross B, should get
me something that is going, I think maybe right
into the screen. Let&#39;s see. v going up. B to the right, that&#39;s this way,
which is away from you guys, which is into the screen. I think I agree with you. Good. Let&#39;s try it for
a different orientation. Let&#39;s say we do B going down
and v going to the right. All right. v fingers straight. B curl it. I still get something that&#39;s
going into the screen. Good. Let&#39;s try one more. B is going to the right. v is going down. Okay. Now how do I do that? Well, I got to put my
fingers down, okay, for v. And then I curl them
into the direction of B. So, v cross B gets me something
coming towards you guys, which is out of the screen. Good. So, let&#39;s say
we do the following. Let&#39;s say we have this. Let&#39;s draw on XYZ
coordinate system x y z. And now let&#39;s do this. We&#39;re going to take a
positive q, and we&#39;re going to send it flying
in the y direction. And now it&#39;s also in a B field that is also pointing
in the y direction. What is the force? What is the force on our charge? Okay. We can do the
right-hand rule again. So, you put your finger
straight in the direction of v. And then you have to curl
them into the direction of B. But B is in the same
direction as v. So, I can&#39;t curl my fingers that way because they won&#39;t be
pointing towards B. And I can&#39;t do them this way, because it won&#39;t be
pointing towards B. And so, none of those directions
make sense. And if none of the
directions make sense. What does the force have to be? What is it 90% of the time
when I asked you to estimate? Zero. The force has to be zero. Right? If the physics says you
can&#39;t determine the direction, then no direction is preferred. It has to be zero. Okay, but how do we see
that mathematically? Well, let&#39;s go back
to our definition. F equals qvB at a
direction which we determined from the right hand rule. Okay, we&#39;ve got a q. We&#39;ve
got a v. We have a B, but then we have the
sine of the angle. And remember the angle
here is between v and B. What&#39;s the
angle between v and B? What do you think the
angle is between v and B? Zero, there we go. What&#39;s the sine of 0? The sine of 0 is 0. There is no force. Okay? If v is in
the direction of B, there&#39;s zero force
on the particle. All right. So, v is to the right
still, but we&#39;re going to let our B be up everywhere. And now we do the right-hand
rule for qv cross B. So, v hold -- everybody
hold your right hand up. v finger straight curl
into the direction of B. That&#39;s telling me that
my thumb is the direction of the force, which is
coming towards you guys. So, there is a force in the -- sorry, the dot right
-- coming towards you. Which would be in our 3D
picture in the negative -- or I&#39;m sorry, in the
positive x direction. Okay. So, that force will
be coming towards you. What is its strength? Well, it&#39;s qvBsin(theta) n hat. We just said that it&#39;s going
to be in the x direction. We know that the x
direction means i hat. Okay. i j k, x y z. So,
n hat becomes i hat. What is the angle? What is the angle
between v and B? It&#39;s 90 degrees. Right? They&#39;re at a right angle. v is in the y-axis,
B was in the z-axis. And so, you get 90 degrees. And sine of 90 degrees
is of course one. So, what&#39;s the force? It is qvB i hat. It&#39;s coming out of the page,
and it is a strength of qvB. So, if you are in some
other arbitrary direction, then you have to
include that theta. So, if this is x y z and let&#39;s
say that our particle is going to the right, but
now our B field is up here at some angle theta. Now, what is the
direction of the force? Well again, we use a right hand. So, we do a qv cross B and
that tells me that it&#39;s coming out towards you, right? Just like we had here. It&#39;s still coming out towards
you and the strength is going to be with the sine theta. Okay. Sine theta is of
course smaller than one so, it&#39;s going to be a little
less than this force. So, if they&#39;re at a right angle, you get the maximum
amount of force. If they&#39;re parallel,
you get zero. If they&#39;re somewhere in
between, you get sine of theta. B, we said is this:
qvBsin(theta). So, B we can solve. It&#39;s going to be force
over qvsin(theta). And so, the units here are what? Well, we know force is Newton. We know that charge is Coulombs. We know that v is
meters per second. Sine theta is unitless. That&#39;s just a number. And this is Newton
second per coulomb. And a newton per -- a Newton
second per coulomb sorry, meter in the bottom
is called a Tesla. Okay. And it&#39;s named
after Nikolai Tesla. You might think of
something else when you hear that word Tesla. What do you think of when
you hear that word Tesla? You&#39;re thinking of
the car, right? The electric car company Tesla. Okay. It&#39;s all related
to magnetism. Why is an electric car
related to magnetism? Why didn&#39;t they call
it the Coulomb? Maybe that would
be a better name. Okay. The batteries
produce the electricity but then they use
the electricity to in fact drive the wheels. And you can&#39;t just take that
electricity and do that directly without using magnetic fields. So, there is an electric motor
in there that has magnets in it. And when you use electricity, you can in fact drive those
magnets, drive the wheels. And so, that&#39;s why
it&#39;s called a Tesla, which I think is a great name. So, let&#39;s say that we&#39;re going
to think about an E field first. Okay. Let&#39;s say I have
the following picture. Here&#39;s my 3D coordinate
system x y z and let&#39;s draw our E field
everywhere to the right. Okay. And let&#39;s say
that we have a charge, which is initially
moving in the x direction. Okay. Here comes my charge. It&#39;s a positive charge
and it&#39;s going to move in the x direction. What is the electric field
going to do to this charge? Well, we know that if it is
a positive charge, plus q, it&#39;s going to feel a force
due to that electric field. And so eventually, it&#39;s
going to start moving in the positive y direction. Okay. It&#39;s going to bend
around and eventually come in the positive y direction. Okay, it will eventually
be parallel to E. But let&#39;s do the same
thing now and think about the motion
of q in a B field. So, let&#39;s draw the same thing. Here&#39;s our coordinate
system x y z. And now let&#39;s have
this q initially moving in the x-axis direction. Okay? Well, if my B field
is now pointing everywhere to the right, then there has
to be a force on this thing. Right? And the force
on it, we determined from our right-hand rule. So, we&#39;ve got a v coming
out in the x direction. We&#39;ve got a B going
back to the right. So, I got to figure out
how to rotate my hand such that I can do
that properly. And if I do that just
right, you guys can try it. My arthritis is kicking in. That&#39;s how you know
physics majors, right? Because they walk
around the hall and they go like this all time. What you should get
is the following. v cross B is going to get
me a force that&#39;s going up. And so, it&#39;s going
to start bending up. And now when it&#39;s going up, it&#39;s going to feel a
force to the right. And then when it&#39;s going back,
it&#39;s going to feel the force down and eventually we&#39;ll come
back around to where it started. Okay? And so, you&#39;re going
to get a v cross B force that makes it move in a circle and that circle is always
perpendicular to the B field. Okay? The force is perpendicular
to the velocity and the B field and therefore this circle is
perpendicular to the B field. In other words, the circle
lies in the xz plane. Okay. That&#39;s what we&#39;re
trying to draw here. We&#39;re trying to draw a
circle in the xz plane. Okay. So, let&#39;s say we have
a region of space here. And in this region of
space, we have a B field that is everywhere
going into the screen. Okay. And B field is everywhere
pointing into the screen. And now you want to put
a charge in this thing and send it flying through. Okay? So, I&#39;m going to
take a q and let&#39;s see. Which way do I want to send it? I want to start right here. We&#39;ll make it a positive q
and let&#39;s send it this way. All right. This thing is going to feel
a force and it certainly has to feel a force either
up or down. Okay. And remember, B is
going into the screen. So, which way is that
force going to be? Is it up or is it down? Look at the computer
monitor over there, pick up your right hand,
put your finger straight in the direction of v.
Curl them into the screen for the direction of B.
Which way is the force? It&#39;s got to be up or down. What do you guys think? Anybody get an answer
on that one? Let&#39;s try it. v is going to the right. B is into the screen. So, I&#39;m going to curl my
fingers into the screen, right? That&#39;s to you what looks
like into the screen. Okay. And so, I&#39;m going to
end up with a force that is up and that means that this thing
is going to move in a circle. Okay. That particle that
came flying in is in fact in a move around in a circle. Okay. Is that right? Did everybody see that okay? v is to the right. B is into the screen. Thumb is the direction of force. So, it&#39;s towards the center
of a circle at this point. Later on, v is going up. B is into the screen. And so, I again get a
force towards the center of the circle. And everywhere around, it&#39;s always a force towards
the center of the circle. Okay, but we know
what that force is. It&#39;s qvBsin(theta). Let&#39;s just worry about
the magnitude of it. We already figured
out the direction. q v B. What&#39;s the angle
between those two? Well, v was to the right,
B was into the screen and so, we get 90 degrees. But 90 degrees is just 1. So, what&#39;s the force? qvB. All right. But this thing is moving in
a circle now of radius r. And what we know from last
semester is if things are moving in a circle, we can
consider the forces adding up to something very specific. Namely, m v squared
over r, right? This is uniform circular motion. The centripetal force
is mv squared over r. But the only force is
this magnetic force. Okay? And so that becomes the
left side of this equation. So, we get qvB equals
mv squared over r. And now we can quickly solve
this thing for the radius. Right? I multiply by r,
I&#39;m going to divide by qvB. One of the v&#39;s is going to
cancel out and I get mv over qB. Okay? What&#39;s the
radius of that circle? It&#39;s that. It&#39;s the momentum of
the particle divided by the charge times
the magnetic field. So, when you are doing a high
energy physics experiment and you create all
these charged particles that are moving very quickly, how can you determine
something about them? Well, you put them in
a big magnetic field and you watch them spin
around in a circle. And when you do that, you
can determine this quantity right here. All you have to do is determine
the radius of the circle. And so, when you see those
pictures from CERN and places like that, where they
have particles coming in and then doing this sort of
stuff, and this sort of stuff, and all these circles and
beautiful pictures like that, what those are are
charged particles moving through magnetic fields. And from those pictures,
all they have to do is measure the radius. And if they know the magnetic
field that was in there, they can figure out something
about how fast it&#39;s moving. What&#39;s the mass? What&#39;s the charge on it? Positives are going
to go this way. Negatives are going
to go the other way. Okay? Ones that are
very lightly charged -- have a small q have
a bigger radius. If they have a bigger mass,
they have a bigger radius. Okay? And so, you can
determine a lot just from taking these
sorts of pictures. And it&#39;s really not
that complicated, right? We just did it with
the force law -- magnetic force law and
the centripetal force law. Let&#39;s talk a little bit
about wires carrying current. So, let&#39;s say I have a
wire here and it&#39;s going to carry some current. Okay, let&#39;s say that
current I is to the right. And let&#39;s take this wire and let&#39;s put it in
a magnetic field. And let&#39;s see if there
is a force on it. All right. How do we do this? Well, let&#39;s say this is our wire and let&#39;s put a magnetic
field everywhere pointing up. Is there a force on that wire? Well, let&#39;s go back to the
magnetic force for a second. Magnetic force is the following:
F equals qvBsin(theta) and then we have to
worry about the direction from the right-hand rule. So, we certainly have a B. We
probably have some angle theta. What about this right here? q and v? Do I have
a charge moving? Well, we have a current, right? Current is charged moving. We know what current is. It&#39;s delta q in some
amount of time, delta t. So, if I think about a wire
carrying current, I can consider that is consisting
of charges moving at speed v. Charge q moves
at v through the wire. That is current. Okay. So, F we could
really say is the following. Let&#39;s just call this thing
delta qvBsin(theta) n hat. The delta q just means
some amount of charge. Okay. We&#39;re going to look
at some chunk of charge. But what if I divide
by a delta t? I can&#39;t just divide by
a delta t arbitrarily without changing the value. But if I divide by a delta t,
I can multiply by a delta t and then I&#39;ll get
the same thing. Okay? So, I haven&#39;t
changed the relationship. It&#39;s the exact same
relationship. But, now we can write
delta q over delta t as I. What about v delta t? What is v delta t? Well, if the charge is
moving along at v and it does that for some amount of time,
delta t, then that corresponds to some length of wire. The length of the wire is L. And
that is just v delta T, right? Speed times time
gets me distance. So, this v delta t just
becomes L. It&#39;s some length of this wire. We have an I, we have
an L, and then we have of magnetic field B, and then
we have sine theta n hat. So, a wire carrying
current, if it is sitting in a B field, it
will feel a force. It will feel a force given
by that ILBsin(theta). And then the n hat
again is determined by the right-hand rule. So, let&#39;s say we do
this and let&#39;s see if we can figure
out the direction. And let&#39;s simplify
our wire a little bit. Let&#39;s say our wire is this. Okay? We don&#39;t have
to draw it all 3D. It&#39;s just a single line. Now let&#39;s again say that the B
field is everywhere pointing up. Okay? What is the force? Well, we know the magnitude. It&#39;s ILBsin(theta). The direction is going to
be the right-hand rule. Theta now is the
angle between I and B. Okay. I becomes like our qv. B is still the magnetic
field, of course. So, in this problem, what is
the angle between those two? Well, I is to the
right, B is up. So, the angle between them is
90 degrees, which is just one. But now we got to
figure out the direction. Remember, the direction is
always going to be perpendicular to I and perpendicular to B. So,
you only have two choices here. It&#39;s either out of the screen,
or it&#39;s into the screen. And the way we do it is that we
again use the right-hand rule, but the fingers go in
the direction of I. So, hold up your right
hand, put your fingers in the direction of I. B we said is everywhere
going up. So, I cross B gets me something
that is coming out of the page. Right? My thumb is coming out
of the screen towards you. And so, there is a force
here which looks like that. I cross B is coming
out of the screen. So, if you want to write
this direction, you could say out of the screen or out
of your computer monitor. Okay. If you increase
the length of the wire, you increase the total force. If you increase the current,
you increase the total force. If you increase B, you
increase the total force. So, let&#39;s ask you this question. Let&#39;s say I have current,
I, going to the left. And now I have B going up? What direction is the
force on that wire? Into the screen or
out of the screen? Okay. Into the screen. Let&#39;s see that. Put my fingers in
the direction of I. Rotate my fingers bent
into the direction of B. My thumb is
pointing into the screen -- into your computer monitor. And so, the force is in
fact into the screen. All right. What about this? Let&#39;s say I draw
current going up. B field going up. What&#39;s the direction
of the force on that? Or what is the force on that? Remember, I fingers
straight in the direction -- your fingers straight
in the direction of I. Now you curl your fingers
into the direction of B. But why don&#39;t I can&#39;t curl
them in any direction. Right? That doesn&#39;t work. And that doesn&#39;t work because
bees in the same direction. So, the force is zero. And that&#39;s because
theta is of course zero. You get sine of 0 which is 0. How about this? Let&#39;s say I have I going
down, but B is going up. What&#39;s the force there? Well, I put my fingers in the
direction of I and now I curl it in the direction of B but I got
to come back all the way around. Should I go that way? Oh, Should I go this way? I don&#39;t know. So again, it&#39;s zero. And that&#39;s because
this is 180 degrees. Sine of 180 degrees
is equal to 0. Okay? So, just like
we had with qv, anytime the current is
parallel or antiparallel to the magnetic field,
there&#39;s no force. All right. But this looks like
a section of wire. This looks like a
section of wire. These two are sections of wire. Can we just put them all
together into one loop? Okay. So, we&#39;re going
to do a square circuit and it&#39;s in the B field. And so, let&#39;s draw
that picture again. And we&#39;ll do the following: current I going around
like this. Okay. Everywhere going around. Here&#39;s our square wire
carrying current I. And now let&#39;s put
it in a B field. And we&#39;re going to say
that B is up everywhere. Okay. We have some B field
that&#39;s pointing up everywhere. And now let&#39;s see if we
can calculate the forces on each side here. So, we know that the force on
the top side is either going to be out of the screen
or into the screen. Right? It has to be
perpendicular to I. It has to be perpendicular
to B. So, on this top side, what should I draw
for the force? Should I put a dot in
there or should I put an X? Well, let&#39;s try it. I is the direction of your
straight -- finger straight. You curl it into the
direction of B. Okay? And if you do I cross B, you should get something
coming towards you. Right? That&#39;s what you see. I cross B, it&#39;s coming
back towards you coming out of the screen. Good. What about this set? The force on this side has to be just the opposite
of what we just did. Right? We got a dot. This one has to be an
X. Let&#39;s just confirm. I is going to the right. B is going up. My thumb is going
into the screen. Okay? And so, you get a
force that is right there with an X into the screen. What about the force
on either side? The force on this side
is of course zero, because they&#39;re parallel. The force on this
side is also zero because they&#39;re anti parallel. And so, that entire
square looks like this. It&#39;s got a force up
here that&#39;s going in. It&#39;s got a force down on
that side that is going out. Sorry, out of the screen
-- into the screen. And so, the net force on this
whole loop is equal to zero. Okay. The whole loop
doesn&#39;t move up or down or left or right. But there is a twist. There is a torque because
one side is trying to come out of the screen, the
other side is trying to go into the screen. And so, this entire thing is
going to rotate in this B field. Let&#39;s try it with some
numbers here and see if we can calculate
what&#39;s happening. This is a square of side L and let&#39;s say it&#39;s
32 centimeters, which is 0.32 meters. Let&#39;s say that the
current running around in this thing is 12 amps. And let&#39;s say that the B
field that it&#39;s sitting in is a quarter Tesla. Okay? All right. Let&#39;s do each side individually
and calculate the force on it. So, the top side, we said
there is a force coming out of the screen. And that force is going
to be qvBsin(theta). But we know that
qv turns into IL. We give you I. We give you L. We give
you B. What is the angle between those two? Well, I was to the right. B was up. Those are 90
degrees to each other. Sine of 90 is just 1. And so, we get ILB and we
know what those numbers are. I is 12 amps. That&#39;s SI units. L is 0.32 meters. And B, we said was 0.25 Tesla. Okay, and let&#39;s run those
numbers and see what we get and why don&#39;t you just punch
it into your calculator and tell me what you get. I&#39;ll approximate it here. We got a quarter of 12. A quarter of 12 is 3. 3 times 0.32 is got to
be really close to 1. Maybe it is more like 0.96. Is that what you get if you
plug it into your calculator? Okay, 0.96. And it&#39;s a force. Right? So, these SI units, amps times meters times
Tesla gives you Newtons. That&#39;s the force
on the top wire. The sides we said were zero. So, we don&#39;t have to
worry about those. The one on the bottom
is into the screen. And guess what? That&#39;s going to have
the exact same strength as the one on the top. Because it&#39;s a square
wire, it&#39;s the same length, we have the same angle
theta, which is 90 degrees. And so, we also get
0.96 Newtons. Force on the top is exactly the
same as the force on the bottom. Means it doesn&#39;t move anywhere. But it does rotate. It does twist. What is that twisting? Torque. Right? When we talk about twist,
we talk about torque. What is that torque? I know. You guys
thought you could forget about torque, right? That was -- that was
so last semester. Do we have to really remember
everything from last semester? Yeah, you kind of do. Torque is written with a tau. Okay? And torque, tau, is what? Well, its force times
lever arm, right? Torque is equal to
force times lever arm. Okay. This whole thing
wants to rotate, right? The whole square wants to
rotate towards you on the top. Away from you on the bottom. What&#39;s a good axis
of rotation to pick? Well, how about halfway through? Seems like this thing might want to rotate right about
its center. So, let&#39;s make this
our axis of rotation. And if I make that
the axis of rotation, then the lever arm
here is just L over 2. And the lever arm on the
other side is L over 2. All right. We want the net torque. So, there is a force on
the top side of IL times B. There is a lever arm on that
top side that is L over 2. That&#39;s the torque
due to the top wire. But I also have to add the
torque due to the bottom wire. And the torque due to the bottom
wire is force ILB also times the same lever arm because
it&#39;s trying to rotate it the same way. This force is pushing
the top towards you. This one is away from you. And so, they&#39;re both
trying to rotate the object in the same direction. And so now look what happens. We have I times L times
B. I have a half an L and another half an L.
And so, they combined to give me another one L. And this is IL squared
times B. Ah ha! But L squared, that&#39;s just
the area of this loop. So, this whole thing
becomes I times A times B where this thing is
the area of the loop. Okay? And this is torque on a
loop if it is at a right angle to the magnetic field. And in general, we can
write the following: torque is IAB times the sine
of phi where phi is the angle between B and the planes normal. Okay, what do we mean by that? Let&#39;s say I draw a square loop. Okay. The normal would be that. So, if the B field
is to the right, then phi in this case
would equal 90 degrees. But if I draw it again
with the B field going up, then the normal to
the plane and B are in the exact same direction. In that case, phi
equals zero degrees. Okay. So, this phi
is between the normal to the loop and the B field. And we said that
torque was equal to I times A times
B in our example. And we gave you those numbers. We said that the
current was 12 amps. We said that the area
was -- let&#39;s see. It&#39;s a square and
it&#39;s 0.32 on a side. So, I have to square that. And then we&#39;re multiplying by
B which we said was 0.25 Tesla. So, somebody punch that
into your calculator and tell me what you get. We can approximate it here
because 12 times a quarter -- 12 times a quarter is 3. And then 0.32 is really close
to 0.33 which is 1/3 squared. And if I do 3 times a 1/3
squared, I get 3 over 9, which is pretty close
to 1/3, right? So, I&#39;m going to say
this thing has got to be very close to 0.33. Anybody get a real
answer for that one? Okay. 0.3 what? &gt;&gt; 072. &gt;&gt; 072. Okay. And that&#39;s the torque. And what are the
units of torque? Remember, torque is
forced time to lever arm. So, 0.3 force is Newtons,
lever arm is meters. And so, the units of
torque are Newton-meters. Let&#39;s take a look
at this problem. So, this one is a
2028 on your homework. And it says a vertical straight
wire carrying an upward 22 amp current exerts an attractive
force per unit length of 8.8 times 10 to the
minus 4 newtons per meter on a second parallel
wire 5 centimeters away. What is the magnitude of the
current in the second wire? All right. To do this, we need to know
just a little bit more. And what we need to
know is something that was introduced yesterday,
but let&#39;s say it again. Currents themselves
produce B fields. Okay. B fields affect currents but currents also
produce B fields. So, if I think about a wire
and it&#39;s carrying current I, there is a B field
around that wire. And the B field around
the wire looks like this. The dashed line indicates
going behind the wire. The solid indicates
coming out in front of it. And we can figure out
the direction of B based on the right-hand rule. Okay so, if you take your thumb
and put it in the direction of I, then your fingers wrapped around in the direction
of B. Okay? So, put your look at the
computer monitor, put your thumb in the direction of I,
wrap your fingers around, that should be the
direction of B. Okay? Now, if that thing is making
a circle, then the B field has to have the same strength
at any radius from the line. And we know what
that strength is. We talked about it yesterday. The strength of B is this: mu naught I over 2 pi r.
The direction is determined from the right-hand rule but it&#39;s a slightly
different right-hand rule. It&#39;s a slightly modified
right hand rule, which is the following. Put your thumb in the
direction of the current and your fingers curl around
in the direction of B. Okay. So, my thumb is going up in
the direction of the current. My fingers are wrapping around
in the direction of B. Okay? And we call this the
right-hand rule for a reason. It works with your right hand, it doesn&#39;t work with
your left hand. Okay. If you do your left hand, you will get the exact
opposite of what you intended. So, you have to use your
right hand to do it. I&#39;m, of course using my
left hand in reality, but then when we flip
it on the monitor, it looks like my right hand. Okay. And so, I&#39;m
finding this challenging over here to do this. All right. What is this mu naught? Mu naught is the
permeability of free space. Just like with the
electric field, we have the permittivity
of free space. Now we have mu naught
which is the permeability and it&#39;s 4 pi times
10 to the minus 7. And the funky units on it
are Tesla meter per amp. All right. So, that&#39;s the strength of
the B field at any radius -- at any distance from a
current carrying wire. All right. So, maybe that&#39;ll help us because it&#39;s telling us we have
a long current carrying wire and then there is some
force per unit length on the second parallel wire. All right. So, let&#39;s hang on
to some of this. We&#39;re going to say B equals
mu naught I over 2 pi r. And now let&#39;s add a
second wire and see if we can make some sense of it. I&#39;ve got a B here
that is going around, as we indicated in that picture. Now we&#39;re going to take a second
wire and run current through it. And now the question is, are
those wires attracted together or are they pushed apart? And let&#39;s think about
this B field for a second at the point of this wire. Okay? This B field is
not just at that radius. It&#39;s at any radius
you want to pick. And so, at this wire, this B
field is in fact doing what? It is going into the
page going into the page. Okay? If it&#39;s going into the
page, then there must be a force on this current carrying wire. And the force has to be
ILBsin(theta) and I determine that from the right hand rule. So, I hold up my right hand
and I put my finger straight in the direction of I and
then the B field is going into the screen. Which way? Is it towards the other wire? Or is it away? It&#39;s towards. There is a force pulling
it towards the other wire. Okay? And that&#39;s exactly what
they said in the problem, right? They said an upward current
exerts an attractive force on the second wire. Okay. And now we need to figure out what the current
is in the second wire. And what they tell us in the
problem is the following. They give us the
force per unit length. Force per unit length is -- we
can use my numbers 8.8 times 10 to the minus 4 newtons
per meter. And they also tell us that the
distance between the wires, which is our r, is
five centimeters. So, 5 centimeters
equals 0.05 meters. And we need to figure out what
this current is in this wire. Okay? What&#39;s the magnitude of
the current and the second wire? So, to be clear, we better call
these two different things. Let&#39;s call this one
I1 and this one I2. And that means this
down here is I1. And this B field here is
B1 What is the B field due to the first wire at this
location of the second wire? Okay. So, let&#39;s see if we
can put this stuff together. What we know is force
is ILBsin(theta). That&#39;s the magnitude
of the force. All right. Let&#39;s divide by L. F over
L equals IBsin(theta). And now we can start being
specific about what currents and what magnetic fields
we&#39;re talking about. If we&#39;re talking about the
force on this second wire, then we&#39;re worrying about
the current I2 in that wire. But it feels a force due to
the B field from the other one. So, that should be B1. And we know that sine
of theta -- let&#39;s see. Theta is going to be
90 degrees, right? I is up. B is into the screen and so theta is going
to be 90 degrees. And so, now we can
plug in for B1. The one was mu naught I1
divided by 2 pi r. We have sine of 90 degrees, but
that&#39;s just 1. And so, we get mu naught
over 2 pi times I1 I2 all over 2 pi r -- I already
wrote the 2 pi just over r. Okay. And now we can solve
this equation for I2 Okay? And that&#39;s what we&#39;re
looking for. So, let&#39;s move it over here and
we&#39;ll solve it for I2 All right. So, we had F over L
equals mu naught over 2 pi. I1 I2 all over r. And we want to
solve this for I2 So, let&#39;s see. I got to multiply by 2 pi r.
I have to divide by mu naught and I have to divide by I1 And that should be my I2. And so, now we can calculate I2. I2 is F over L, which
we&#39;re given. r we&#39;re given. I1 we&#39;re given. F over L was this: 8.8
times 10 to the minus 4. We have a 2 pi. We have r which is 0.05
and then we have mu naught, which we know is 4 pi times
10 to the minus 7 in SI units. And then I1 for our
problem is 22 amps. All right. So, somebody punch that
into your calculator and tell me what you get. And let&#39;s see if we
can approximate it. It&#39;s kind of fun to
approximate these things. So, what do we got? We got the pi&#39;s. Those are going to drop out. That&#39;s nice. We got a 2 that drops out with that if I give that
a 2 right there. And so, we&#39;re going to
get 8.8 -- 0.05 is 1/20. Then we have a 10
to the minus 4. And we&#39;re going to
divide the whole thing by 2 times 22, which is 44. And then we&#39;re going to end up with another 10
to the 7 up on top. So, let&#39;s see. What do we get? 8.8 over 20. That is really close to 1/2. If I divide a half by
44, what do we get? That&#39;s 1/88, which is
pretty close to 0.01. A little bit more. How about 0.01 -- let&#39;s
take a guess -- 0.012? And then we&#39;ve got a 10 to
the minus 4 and a 10 to the 7 which we get a 10 to the 3. So, I&#39;m going to say that
I2 is approximately one, two, three -- 12 amps. That&#39;s my guess. Anybody get a real number? Nobody got an answer yet? &gt;&gt; I got 10. &gt;&gt; 10? Okay. That&#39;s pretty close to 12. Anybody else get 10? 10? Okay. All right. So, our guess was a little off. But that&#39;s fine. Let&#39;s try 10. 10 units are amps. Should we click submit? Let&#39;s click it. Correct! All right. I got one point on my homework. Excellent. Okay. The part B of this
is what is the direction of the current in
the second wire? If you remember our picture, we
had current in the first wire and the current in
the second wire was up and it was an attractive force. So, we&#39;re going to say upward for the second part
-- part B. Submit. Also correct. Okay. Good problem. All right. Why don&#39;t we wrap it for today? And I will see you
guys tomorrow. Online. Cheers.

Transcript for:Magnetism Lecture Notes

Transcript for:
Magnetism Lecture Notes