Force and Torque on Current Loops

Hey, guys. So this week we're gonna talk about magnetic forces and magnetic torques. When you have current loops, let's check it out. All right. So remember that if you have a current carrying wire, in other words, A Why that has current running through it on. Do we have that wire in a magnetic field? It will feel a magnetic force. So it current in an existing in the presence of an existing field field, a force. Okay. And that force was given by Bill. Sign of theta. Okay, this is old news now. Realize that wires don't have to be straight, Obviously. In fact, in circuits they turn around all the time, But you can actually get a wire to also look like a loop. So imagine a long wire that you just bent so that it looks like this. So you maybe you have some current going in this way. Which means currents going that way, this way, all the way around and current leaves this way. And who knows? Maybe this is connected to a battery somewhere, right? So that it produces a current anyway. So you have a wire there with the currents and in some cases. When you have this arrangement, you're going to get a torque. Not always, but in some cases, first things first. The net force is always going to be zero. So you should remember that whatever forces exist in this wire will cancel. Um, well, cancel each other out, and then we're gonna get a magnetic torque. But first, let me show you how these forces air going to work so that the torque can make sense. So this is a wire. So we're gonna use the right hand rule to look at directions off forces. This is the magnitude of the force right here. But we're gonna look at the direction of the forces. So I'm gonna follow this path here, and I have my right hand and I want to go, right, Okay. And the area that we're actually interested is sort of like the square area. So let's start over here. Let's start over here. Let me delete that green right there. So we're gonna start with the first arrow and I'm going up right? And the magnetic field at this point is also up. So the only way I could do that is if I do this. And then I put this up right or this. Right? And this is bad news, If you remember, right hand rule, You're supposed to be like this. This is ideal. Maximum force. You can do this, right? That's cool, too. That's a little bit less force, but this gives you a force of zero, and that's okay. That's actually what's going on here in this wire segment over here. Let's call this current in part one. All the current will be the same, but this is just part of one I'm gonna say, Actually, it's just called left. So I'm gonna say that the force on the left segment of this wire will be zero, because the current is parallel with the magnetic field magnetic field, the blue lines here. Okay. And that's cool. But now we're gonna go this way, which means my thumb is gonna go to the right, and if I want my beef fields to go up, Look what happens. My hand. Please do this and follow me. My hand is I'm looking at my hands on my palm is going this way, which means it's coming out of the page. So the force on this wire will be out of the page and out of the page is represented by little dots. Okay, so the force on the top segment of the wire will be out of the page when you go down, what you're gonna happen? What you're gonna have is something similar on the right side and you had on the left. So I'm going down. But then my, But then my beef lines are supposed to go up. And the only way to do that is to do this, make a 1 80 degree there. And we know that there is no force in that situation either. So the force on the right side of the segments of this wire will be zero. And then on the bottom here, I'm going to the left, and I want my Byfield to go up, which means my hand is away from me and into the page, into the page, into the page. Looks like this. Okay, so the force on the bottom is into the page, So look what happens. Look at this rectangle or the square. Whatever. This top part is being pushed out while the bottom part is being pushed in. So what you then do is imagine that you can fix this around a axis of rotation, right? You can fix the square in an axis of rotation. The top part of the square is being pushed out, which means out of the page means towards me when the bottom is getting pushed into the page, which means away from me. So the top comes in and the bottom comes out. So this thing does this right, which means it's going to rotate. That's why we get torque. Remember, Torque is a force away from a axes of rotation in such a way that you have a rotation eso That's what this thing is gonna do. This thing is going to start spinning and you have a torque. So that's how that works. And if you want to do some calculation, you would arrive at the equation, which I'm just going to give it to you in the equation. Um, looks different from what I'm gonna write, but I write in this way so that it's easy to remember which is N. B. A. I sign of data N b. A basketball. I sign of data. Somehow I have to remember that there is an I sign of data there. Don't forget that part, but torque equals n ba and is the number of loops. I will talk about that in a second. Just right. That there be is the strength of the magnetic field. That's a factor A is. The area is the area that is made by the wire arrangements. Right. So this is the area, Um, I is the current and sign of theta theta is the angle between the A and the B. Okay, so data the angles data is between the normal off the area and be we'll talk about the normal of the area and the second number of loops. What's the deal with this number of loops? Well, you can get a wire, you can get a wire and make a little loop something like that, right? Or you could get a wire and make two loops on top of each other. It would look exactly like this because they're sort of like a first floor and then the second floor of loops. The only difference is that here and the equals one and then here is n equals two Ah, lot of the Times. You first learned the situation without the end, and then they say, Hey, you can have both loops and then they at the end. I'm just putting the end already in there. Um, and that way you get the N b A. You don't have to go back to this equation. Cool. So that's that, by the way, most of the time, the end is going to be one because you're gonna have a single loop. In fact, they don't. They don't specify that. You have multiple groups. Um, and it will be one so one, unless otherwise stated. Um, the normal of a Remember, whenever you have an area whenever you have an area. So imagine this. This isn't like a surface or a plane or a area. Right? Um, the normal to the area is normal means perpendicular, which means 90 degrees is 90 degrees away from the surface. So if this is your area, the normal of a is this right? So I'm gonna try this in a bunch of different angles. If you get something like a textbook and you have something come out of it, imagine if I poke a hole through this which I'm not gonna dio but something like that. Okay, so that's the area. That's the area, always 90 degrees to the surface. Hopefully, that helps the normal of a and the direction of be We're gonna talk about this a little bit more. But this is the big equation that really matters here. Lastly, there's this property this quantity called magnetic moment. I'm not gonna get into it, But I just want you to know the magnetic moments given by the letter mu not to be confused by the constant mu Not right given by the letter mu, which is the same letter for coefficient of friction. But it's totally unrelated. Um, is the equation for this guy is n a I Okay, so this is n B a. I and this is ni are nay or something like that. So mu is n a I. And if you look at this equation, this is an end. This is an A and this is an I. So, technically, you can rewrite torque as mu because it's gonna take over for these three guys. Be sign of data. That's another equation for torque. You don't really have to remember that equation. Because if you know these two, you should be able to manipulate and come up with that third one. Okay, so that's the magic moment. Magic moments. The magnetic moment. Rather, Um anyway, this is simpler than it looks. It looks Harry, but it isn't. Let's do an example here. So it says here a loop which looks like this with magnetic moments. Oh, that's that guy Mu equals 0.5 carries a current of of 0.1 So I equals It is placed in the presence of a magnetic field of strength. 005 0. or just 05 And then which points in the plane of the loop. We want to know what is the magnitude of the torque? OK, on day, this is the hard part. Here is this which points in the plane of the loop. So it didn't tell you exactly direction of loop. But let's just draw a loop. By the way, the loop doesn't have to be spectacular. It could actually have looked something like this, and it works just the same. Um, it's saying here that the magnetic field points in the plane of the loop. The plane of the loop is this right? Is this so it means that the magnetic field looks something like this or something like this or something like this? We don't know. It doesn't say so. We're just gonna pick one. I'm actually drug. Just draw them up so that it looks more similar to what we already have here. Um, in the plane of the loop. So back to my little surface here, the plane of the loop is this right? If it's horizontal, it's the floor. Or if it's like this vertical, it's the wall. So it means that the magnetic field is pointing in this direction here. Okay, in this direction. And remember, a is perpendicular to that. So what's the angle between the plane off the plane of the of the loop and the normal of the loop? It's always 90 degrees. Okay, so let's write that down. The angle between the plane and it's normal is always 90 degrees, because it's if you have the plane down here. This is the direction of the plane. And this is the direction of the normal cake. So this is my baby my A is out of the plane. So my a is either into the page or out of the page. We don't know. So I'm just gonna say that it's out of the page, right? What matters is that the angle between B and the normal of a is 90. Okay, so the angle I'm gonna use in the equation is going to be the sign of 90. Which, by the way, is one hope that makes sense. I spent a lot of time in the angle there because that's the hard part. Everything else is easy. How do we find torque? So you might be tempted into writing the big equation MBA. I sign of data and I have data. By the way, this whole thing is just one, so let's just get rid of it. I have it. Didn't say anything about loops. It just said a loop A loop. So And is just one because didn't say otherwise be I have it. A I don't have it. I have it. We're stuck. We don't have a we can go find a or because we have mu. We could just plug it into this equation instead. So this was a little tricky, but just super basic physics problem solving. Good hustle. Here. Um, you be sign of data. Just one mu is 10.5 b is 0.5 Um, this is 0.0 25. The unit for torque. You may remember from a long time ago torque is F R. That's like the general equation for torque. So the units are Newton. Times are is a distance. Newton Times meter. Cool. That's it for this one. Let's get going.

Hey, guys. So let's check out this example. So here I have a wire arranged in a loop on That's this blue line here. This is the loop current loop on. Do we want to know what must be the currents running through that loop? Given all this information. So I wanna look for the currents. Cool. So it says here that it is arranged as a rectangular, um, to arrange the rectangular 4 m wide and 2 m deep. I forgot to say that this is a loop, but these two pieces of information 4 m and 2 m will allow you to find the area so right away. I know that the area is eight square meters. It is placed in the plane. Shown were a constant five t magnetic field exists. B equals five. It says the wire loop is parallel to the plane. Parallel means side by side. So imagine the X axis right here. And I'm gonna do this just to sort of try to support that image. Imagine the X axis your parallel. So you're sitting on the plane. If it's a horizontal plane, which it is here, it looks like it is, um then you're essentially on the floor. So the cable is on the floor. Imagine that This is the floor of the X Y plane, and this is sort of the height here, So it's parallel to the plane and the magnetic field is directed 30 degrees above the plane. So if this is your plane, then the magnetic field are lines that go like this. So instead of it going like this, they're going like that. Okay, so it's a little weird to draw this here, but basically you're gonna have lines, magnetic field lines that look like this and they make an angle of 30 degrees. This is like the positive X axis. It says you have to make an angle of 30 degrees above the plane. Okay, above the plane theme. The loop experiences a network of 10 and we want to know what is the current. So how do we do this? There's two equations for torque. You can use n ba I sign of data, or you can use theme magnetic moments. Be a sign of theta. Okay, um, here we don't have the magnetic moment, we can calculate it, but there's no point in doing that it's easier to just plug everything here. I'm looking for. I I equals torque. Divided by N b. A sign of data. I have torque, which is 10 and it is the number of loops. It didn't specify how many loops we have, so we're gonna assume that is one. That's what we're supposed to do. B is five. Area is eight, and as usual, the tricky part is finding the angle. So which angle should we use? Well, remember, the definition is that it's the angle between A and B. Now, whatever we talk about, um, the direction oven area, it's always the normal of that area, meaning if you have an area, it's the direction that is perpendicular to that area, the direction that makes a new angle of 90 degrees with its Okay, So it's this Okay, this area. So if this is the area here in the area enclosed by the wires that is sitting on an X on the on the floor, the perpendicular to that will be going up like this. Okay, going up like this. In other words, the A vector points in this direction. Here, this is the normal of the surface. Okay, normal. The surface Diego I want is between the normal of the surface and be and this is be right here. Okay, so they sort of cross over here. So what I want is I want this angle here. This definition, By the way, you might have noticed that I almost always give you the wrong angle, because I want you to be, uh I want you to be very careful. I want to be, um, paranoid about this stuff. But just know that it's not the case that it's all that you're always going to get the wrong angle, right? You could very well just get looking at the right angle. Or maybe some sort of like, you know, backwards. Tricky question that you actually get the right angle. And maybe you changed it. So the angle here is 16. Because that's the one that I want. Onda. We're gonna put it here 60. So if you do all of this and I have it here, the answer will be 0.29 or just 0.30. Our 0.3 amps, co. Uh, that's it for this one. Do we want to know the direction of the current? Um, we're not. We're not told the direction of the torque we can find out. Never mind. So that's it for this one. Let's keep going.