Force and Torque on Current Loops - Online Tutor, Practice Problems & Exam Prep

Magnetic forces act on current-carrying wires in a magnetic field, producing torque when arranged in loops. The net force is zero, but torque arises from opposing forces on different segments of the wire. The torque can be calculated using the equation τnet = NBAIsinθ, where N is the number of loops, B is the magnetic field strength, A is the area, I is the current, and θ is the angle between the area and the magnetic field. Understanding these concepts is crucial for applications in electromagnetism and engineering.

Force and Torque on Current Loops

Hey guys, so in this video we're going to talk about magnetic forces and magnetic torques when you have current loops. Let's check it out. Alright, So remember that if you have a current-carrying wire, in other words, a wire that has current running through it, and we have that wire in a magnetic field, it will feel a magnetic force. So a current existing in the presence of an existing field feels a force. Okay? And that force was given by F=BILθ. Okay? This is old news.

Now realize...
imagine a long wire that you just bent so that it looks like this. So you maybe have some current going this way, which means current's 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 a current 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 0, So you should remember that whatever forces exist in this wire will cancel, will cancel each other out. And then we're gonna get a magnetic torque.

But first, let me show you how these forces are 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 of 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 in 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 can 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 the right-hand rule, you're supposed to be like this. This is the 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 0. And that's okay. That's actually what's going on here. In these wire segments over here, let's call this, currents in part 1. All the currents will be the force on the left segment of this wire will be 0, because the current is parallel with the magnetic field. Magnetic fields are 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 B fields to go up, look what happens. My hand is, I'm looking at my hand so my palm's 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 have, what you're gonna have is something similar on the right side than you had on the left. So I'm going down, but then my B lines are supposed to go up and the only way to do that is to make a 180-degree turn there and we know that there's no force in that situation either. So the force on the right side of the segments of this wire will be 0. And then on the bottom here, I'm going to the left and I want my B field 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 this 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 an axis of rotation, right? You can fix the square on an axis of rotation. The top part of the square is being pushed out, which means out of the page means towards me, while 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 an axis of rotation in such a way that you have a rotation. So that's what this thing is gonna do. This thing is gonna start spinning and you have a torque. So that's how that works.

And if you were to do some calculation you would arrive at the equation, which I'm just going to give it to you. And the equation looks different from what I'm gonna write, but I write it this way so that it's easier to remember, τ=NBAIθ. NBA Basketball, I sine of theta. Somehow you have to remember that there's an I sine of theta there. Don't forget that part. The torque equals NBA. N is the number of loops. And we'll talk about that in a second. Just write that there. B is the strength of the magnetic field. That's a vector. A is the area, is the area that is made by the wire arrangements. Right? So this is the area. I is the current and sine of theta. Theta is the angle between the a and the b. θ, the angle theta is between the normal of the area and B.</p>>

The 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 can get a wire and make 2 loops on top of each other. It would look exactly like this because they're sort of like the first floor and then a second floor of loops. The only difference is that here N=1 and here N=2. A lot of the times you first learn this equation without the N and then they say hey you can have multiple loops and then they add the N. I'm just putting the N already in there. And that way you get the NBA, you don't have to go back to this equation. Cool? So that's that. By the way, most of the time this N is going to be 1 because you're gonna have a single loop. In fact, if they don't specify that you have multiple loops, N will be 1. So N=1 unless otherwise stated.

The normal of A. Remember whenever you have an area, 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 an area, right? 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 do, but something like that. Okay? So that's the area, that's the area. Always 90 degrees to the surface.

Torque on a Loop at an Angle

Hey guys. So let's check out this example. Here, I have a wire arranged in a loop, and that's the blue line here. This is the current loop. And we want to know what must be the current running through that loop given all this information. So I want to look for the currents. Cool. It says here that it is arranged as a rectangular, 2 meters arranged as a rectangular, 4 meters wide and 2 meters deep. I forgot to say that this is a loop, but these two pieces of information, 4 meters and 2 meters, will allow you to find the area. So right away, I know that the area is 8 square meters. It is placed in the plane shown where a constant 5 T magnetic field exists, B equals 5. It says the wire loop is parallel to the plane. Parallel means side by side. So, imagine an x-axis right here, and I'm going to do this just to sort of support that image. Imagine the x-axis you're parallel, so you're sitting on the plane. If it's a horizontal plane, which it is here, then you're essentially on the floor. So, the cable is on the floor. Imagine that this is the floor, the xy 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 lines go like this. So instead of them going like this, they're going like that. Okay? So it's a little weird to draw this here. But basically, you're going to 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 here they make an angle of 30 degrees above the plane. Okay? Above the plane. The loop experiences a net torque of 10, and we want to know what is the current. So how do we do this? There are 2 equations for torque. You can use nBAIsin⁡θ or you can use the magnetic moment Bsin⁡θ. 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=tnBAsin⁡θ. I have torque which is 10. N is the number of loops. It didn't specify how many loops we have, so we're going to assume that it's 1. That's what we're supposed to do. B is 5. Area is 8. 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, whenever we talk about the direction of an area, it's always the normal of that area. Meaning if you have an area, its direction is perpendicular to that area. The direction area if this is the area here and the area, enclosed by the wires, that is sitting on the floor, the perpendicular to that will be going up like this. Okay? Going up like this. In other words, the vector A points in this direction here. This is the normal of the surface. Okay? The normal of the surface. The angle I want is between the normal of the surface and B. And this is B 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 very careful. I want you to be paranoid about this stuff. But just know that it's not the case that you're always going to get the wrong angle. Right? You could very well just get lucky at the right angle. Or maybe some sort of you know, backwards tricky question that, you actually get the right angle, and maybe you change it. So, the angle here is 60 because that's the one that I want. And we're going to put it here 60. So, if you do all this, and I have it here, the answer will be 0.29 or just 0.30 or 0.3 amps. Cool? That's it for this one. Do we want to know the direction of the current? We're not told the direction of the torque. We can't find out. Never mind. So let's hit for this one. Let's keep going.

Here’s what students ask on this topic:

What is the formula for calculating torque on a current loop in a magnetic field?

The formula for calculating the torque (τ) on a current loop in a magnetic field is given by:

$\tau =NBAI\sin \left(\theta \right)$

where:

• $N$ is the number of loops
• $B$ is the magnetic field strength
• $A$ is the area of the loop
• $I$ is the current
• $\theta$ is the angle between the normal to the area and the magnetic field

How does the right-hand rule apply to current loops in a magnetic field?

The right-hand rule helps determine the direction of the magnetic force on a current-carrying wire. For a current loop:

1. Point your thumb in the direction of the current.
2. Extend your fingers in the direction of the magnetic field (B).
3. Your palm will face the direction of the force (F) on the wire segment.

For a rectangular loop, forces on opposite sides will be equal and opposite, leading to a net torque that causes rotation. The top and bottom segments experience forces in opposite directions, creating a rotational effect around the loop's axis.

What is the significance of the angle θ in the torque equation for current loops?

The angle $\theta$ in the torque equation $\tau =NBAI\sin \left(\theta \right)$ represents the angle between the normal to the area of the loop and the magnetic field. This angle is crucial because:

• When $\theta$ is 0° or 180°, $\sin \left(\theta \right)$ is 0, resulting in zero torque.
• When $\theta$ is 90°, $\sin \left(\theta \right)$ is 1, resulting in maximum torque.

Thus, the angle determines the effectiveness of the magnetic field in producing torque on the loop.

What is the magnetic moment of a current loop and how is it related to torque?

The magnetic moment (μ) of a current loop is a vector quantity that represents the loop's tendency to align with a magnetic field. It is given by:

$\mu =NAI$

where:

• $N$ is the number of loops
• $A$ is the area of the loop
• $I$ is the current

The torque (τ) on the loop can also be expressed in terms of the magnetic moment:

$\tau =\mu B\sin \left(\theta \right)$

This shows that the torque is directly proportional to the magnetic moment and the magnetic field strength.

How do multiple loops affect the torque on a current-carrying wire in a magnetic field?

Multiple loops in a current-carrying wire amplify the torque experienced in a magnetic field. The torque equation $\tau =NBAI\sin \left(\theta \right)$ includes the number of loops (N) as a factor. Each loop contributes equally to the total torque, so:

• If $N$ doubles, the torque also doubles.
• More loops mean a stronger magnetic moment, enhancing the loop's interaction with the magnetic field.

This principle is used in devices like electric motors and generators to increase efficiency and output.