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Anderson Video - Magnetic Torque

Professor Anderson
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We're gonna do a square circuit and it's in a B field. Okay, and so let's draw that picture again. 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 gonna 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 -- fingers 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 0. 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 0. Okay? And, 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 1/4 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 qVB sine theta. Okay? 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 1/4 of 12. A quarter of 12 is 3. 3 times 0.32, it's got to be really close to one. 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 0, 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 a force on the bottom. Means it doesn't move anywhere but it does rotate. It does twist. Okay. What is that twisting? Torque. Right? When we talk about twists, we talk about torque, What is that torque? I know. You guys thought you could forget about torque, right? That was so last semester. Do we have to really remember everything from last semester? Yeah, you kinda do. Torque is written with a tau. Okay? And, torque tau is what? Well, it's 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 bottommwire. 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 1/2 an L and another 1/2 an L. And so, they combine to give me another one L. And this is IL squared times B. Aha! 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 0 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 1/4. 12 times 1/4 is 3. And then, 0.32 is really close to 0.33 which is 1/3 squared. And if I do 3 times 1/3 squared, I get 3 over 9 which is pretty close to 1/3, right? So, I'm gonna say this thing has got to be very close to 0.33. Anybody get a real answer for that one? >> (student speaking) 0.372 Okay. 0.3 -- what? >> (student speaking) 072 >>072. Okay. And that's the torque. And, what are the units of torque? Remember, torque is force time to lever arms. So 0.3, force is newtons, lever arm is meters. And so, the units of torque are newton meters.