Lesson Video: The Torque on a Current-Carrying Rectangular Loop of Wire in a Magnetic Field Physics

Video Transcript

In this video, we’re talking about the torque on a current-carrying rectangular loop of wire in a magnetic field. We’re going to learn why a torque acts on such a current-carrying wire, how to calculate its magnitude, and also how to determine something called the magnetic dipole moment for such a current-carrying loop. We can get started on this topic by thinking at first of just a rectangularly shaped wire. So here’s one side, here’s a second side, a third side, and then a fourth side. And then we’ll say that this dotted line right here represents an axis that goes right through the center of the rectangle.

Now what if, perpendicular to this axis, we set up a uniform magnetic field? We can call that field strength 𝐵. Even with our rectangular wire in this uniform magnetic field, nothing would happen unless we put current in the wire. If we do this, things would start to change because now we have electrical charge in motion in a magnetic field. Those charges when they’re moving in certain directions relative to the field will experience a force. And since the charges that make up this electrical current are in the wire, the wire itself will then experience a force.

A bit earlier, we identified the four sides of our rectangular wire. We said that here was side one, here was side two, and then three and then four. As electric charge moves along through these four different straight segments of wire, only two of these sides will experience a force, side one and side three. That comes down to the direction of the moving charge relative to the external magnetic field. On sides one and three, there will be a net magnetic force that acts along that whole segment of wire. And those two forces, the one on side one and the one on side three, act in opposite directions.

Considering that magnetic force as it acts on side one, let’s say that it points in this direction, upward. That means that the corresponding force on side three will point the opposite way, downward. Given these two forces acting on opposite sides of our rectangular wire, we can see what’s going to happen. These forces combine to create a torque on the wire, which tends to make it rotate around this axis we drew through its center. Now, the symbol to represent this torque is the Greek letter 𝜏. And what we’d like to do is write a mathematical relationship for this torque. It turns out that in a case like the one we’re looking at here, this torque depends on several factors.

First, it depends on the strength of the magnetic field that our current-carrying wire is in. The stronger that field is, the more torque the wire experiences. It also depends on the magnitude of the current in the wire. The more current, the more moving charge, and therefore the greater the force sides one and three of our wire will experience and so the greater the torque. Something else the torque depends on is the cross sectional area, we’ve called it 𝐴, of our loop. And along with this, we’ll want to consider the possibility that a rectangular coil of wire has more than one loop.

The way we’ve drawn it right here, it has only one. But in general, there could be any number of turns in this coil. There could be some integer value we just call 𝑁. So the torque on this rectangular current-carrying wire depends on 𝐵, the magnetic field, the current 𝐼, the cross sectional area 𝐴, and the number of turns in our coil. And for all four of these factors, the bigger they get, the greater the torque. That tells us that all four will be in the numerator of our equation for torque.

Our equation now is almost complete, but there’s one other factor we’ll want to add here. Remember, we noted earlier that this coil in this field would experience a torque, and that torque would tend to rotate the wire about this axis. When that happens, it will make the overall orientation of our current-carrying loop change. And this change then leads to a change in the torque it experiences. We can get a better sense for what’s going on here by looking at our setup from a different perspective. If we watch the edge of our rectangular coil, as the coil turns, we would see it start out horizontally in the magnetic field, but then turn like this, then like this, and so forth as we continue to rotate clockwise as we’re looking at the coil.

Well, the angle between our coil and the external field that it’s in affects the torque that the coil experiences. The way we measure that angle is by considering the plane in which the coil lies. We represent that by our cross sectional area 𝐴, and we picture a vector that’s perpendicular to that plane. To be clear about what that looks like, if our coil looked like this and on relative to the magnetic field, then that vector perpendicular to the cross sectional area of the rectangular loop would look like this pink one. Once we have this vector, which is normal to the area of our loop or perpendicular to it, then we measure the angle in between this vector and the direction of the external magnetic field. If we call that angle 𝜃, then we can enter, over on our equation for torque, the last factor we need to complete the equation.

When 𝜃 is the angle between the external magnetic field that our coil is in and the vector that points perpendicular to the area of our coil, then the overall torque that our rectangular coil experiences is 𝐵 times 𝐼 times 𝐴 times 𝑁 times the sin of 𝜃. Let’s consider now how this factor, sin 𝜃, affects the torque on our coil. Say that our coil of wire was oriented like this with respect to the magnetic field. In this case, the vector perpendicular to the area of the coil would point in this direction. And we can see that this is perpendicular to the external magnetic field. In this case then, 𝜃 would be 90 degrees, and we know that the sin of 90 degrees is one. It’s the maximum value the sine function attains.

Everything else being equal then, our coil will experience a maximum torque due to the magnetic field it’s in when it’s oriented to that field this way. But then, what about this? What if our coil has rotated to this position? At this point, the vector perpendicular to the area of the coil would point this way. And we can see this points in the same direction as the external field. These two vectors are parallel, and so 𝜃 is zero degrees. And then, the sin of zero degrees is zero. So when our coil is oriented this way, it experiences no torque. So if our coil is perpendicular to the field like this, it experiences zero torque. And if it’s parallel to the magnetic field like this, 𝜃 is 90 degrees and the coil experiences a maximum torque.

Now, let’s focus for a moment on this case where our coil experiences that torque maximum. What we can do is write a specific version of our torque equation for this maximum value, we’ll call it 𝜏 sub 𝑚. And the only difference between this equation and our original general equation for torque is that now we’re assuming that 𝜃 is 90 degrees. So the sin of 𝜃 is one. If we consider this maximum torque our coil can experience and the strength of the field that causes that torque, we can identify what’s called the magnetic dipole moment of our current-carrying loop. This term, magnetic dipole moment, refers to the tendency of an object to interact with an external magnetic field.

So given our external field, we’ve called that capital 𝐵, the more torque our coil experiences, the more it interacts with that field we can say. And the magnetic dipole moment measures the extent of that interaction. If we represent magnetic dipole moment symbolically using 𝑚 sub 𝑑, mathematically, it’s equal to this ratio, the maximum torque that our current-carrying coil can experience divided by the strength of the field that the coil is in. This shows more clearly what we mean by saying the magnetic dipole moment measures the response of an object to the magnetic field it’s in. Given a magnetic field strength 𝐵, the more torque our object experiences, the greater its magnetic dipole moment.

Just as a side note, because 𝜏 sub 𝑚 down here is equal to 𝐵 times 𝐼 times 𝐴 times 𝑁, that means that specifically for a current-carrying rectangular loop of wire, we can also write the magnetic dipole moment as 𝐼 times 𝐴 times 𝑁. This shows us that if we were to keep everything the same in our scenario, but increase the current 𝐼, then we would also raise the magnetic dipole moment of our coil. Or, likewise, if we kept everything the same but increased the cross sectional area of our coil or increased the number of turns in it, those are also ways that we could increase the coil’s magnetic dipole moment. Knowing all this about the torque on a current-carrying rectangular loop of wire in a magnetic field, let’s get some practice now with these ideas.

The diagram shows a rectangular loop of current-carrying wire between the poles of a magnet. The longer sides of the loop are initially parallel to the magnetic field, and the shorter sides of the loop are initially perpendicular to the magnetic field. The loop then rotates through 90 degrees so that all its sides are perpendicular to the magnetic field. Which of the lines on the graph correctly represents the change in the torque acting on the loop as the angle its longest sides make with the magnetic field direction varies from zero degrees to 90 degrees?

Okay, in our scenario, we have a rectangular current-carrying loop of wire that’s shown in our diagram here. And our loop, we can see, is positioned between the poles of a permanent magnet, which means it’s exposed to a uniform magnetic field. Because a rectangular loop carries current and is in a magnetic field, it experiences a torque about its axis of rotation. This torque leads to the rotation of the coil that we see here in a clockwise direction. We’re told that, initially, our coil is oriented like this, where the longer sides — that is, the side of the front and the back we could say of the coil — are parallel to the magnetic field and the shorter sides are perpendicular to it.

But then, under the influence of torque, our coil rotates 90 degrees so that in this position, all four of the sides of the coil are perpendicular to the magnetic field. And we know that that field points from the north pole of our magnet to the south pole, so left to right as we’ve drawn it here. The other part of our diagram is this graph right here. Looking at it carefully, we see that on the vertical axis, the torque acting on a rectangular coil is plotted against the angle of orientation of that coil to the magnetic field. That angle varies from zero degrees, which is the position of our coil relative to the field when it starts out in this horizontal plane, all the way up to 90 degrees, which is the coil’s position relative to the field once it’s rotated.

On this graph, we see these lines of different colors. There’s a red curve, a yellow curve, a blue one, and a green one. What we want to do here is identify which of these four curves, which color, correctly represents the change in the torque that acts on a rectangular current-carrying loop as this angle that its longest sides make with the magnetic field direction varies from zero to 90 degrees. In other words, which of the four lines on our graph correctly shows the torque acting on our coil as it changes from this position here to this position? Notice that that change in position is defined on our graph by a change in this angle called 𝜃. 𝜃 goes from zero up to 90 degrees.

In our problem statement, we’re told that 𝜃 represents the angle between the longest sides of our coil, those would be the sides that initially are parallel with the magnetic field and then end up perpendicular to it, and the magnetic field direction itself, which we’ve seen goes from left to right between the north and south pole of our magnet. So the angle between this line here, which is one of the longest sides of our coil, is zero degrees. That corresponds to this point on our graph. And then, after our coil has turned 90 degrees, the angle between the longest side of the coil, which is now here, and our external magnetic field is, we can see, 90 degrees. And that represents this data point here on our curve.

To answer our question then, of which of these four lines correctly represents the torque experienced by our loop as it goes through this rotation, we’ll need to know how the torque on our rectangular current-carrying wire varies with the angle 𝜃. To figure that out, we can recall a general mathematical equation for the torque on just such a current-carrying rectangular loop of wire in a magnetic field. That torque 𝜏 is equal to the strength of the magnetic field the coil is in times the magnitude of the current in it multiplied by its cross sectional area 𝐴. On our diagram, that area would be this area we’re showing here times the number of turns in our rectangular coil all multiplied by the sin of an angle we’ll call 𝜙.

Now it’s important to note that 𝜙, this angle here, is not equal to the angle 𝜃 identified on our graph. The angles are different. But nonetheless, now that we have this equation written out, we see how the torque on a rectangular current-carrying wire in a magnetic field varies with the angular orientation of the coil. And that’s really what we need to know to answer our question. So even though the torque 𝜏 depends on all these various factors, really we’re interested in the fact that it is directly proportional to the sin of the angle we’ve called 𝜙. And now just what is this angle?

Going back over to our diagram, if we draw a vector that’s perpendicular to the cross sectional area of our loop, then 𝜙 is the angle between this vector, that we’ve drawn in blue, and the magnetic field lines. And in this case, it’s worth noticing that that angle is 90 degrees. So this is very important. It tells us that when the angle 𝜃 is zero degrees, the angle 𝜙 is equal to 90 degrees. So it’s true that 𝜃 and 𝜙 are not the same, but this is how they correspond when 𝜃 is zero. And if we then consider the orientation of our coil after it’s gone through its 90-degree rotation, in this case, a vector drawn perpendicular to the cross sectional area of our coil would look like this.

Once again, the angle 𝜙 is the angle between this blue vector and the direction of the magnetic field. But now we can see these two vectors are parallel. In other words, the angle between them is zero degrees. And note that this corresponds to the point where 𝜃, the angle between the longest side of our coil and the magnetic field, is 90 degrees. So it’s a bit confusing, but when 𝜃 is zero degrees, 𝜙 is 90 degrees. And then when 𝜃 is 90 degrees, 𝜙 is zero degrees. We do all this because once we know what 𝜙 is, we can take the sine of that angle. And then we’ll know how the torque on our coil will vary over the course of this rotation, specifically, a rotation from 𝜙 equals 90 degrees over to 𝜙 equals zero degrees.

In order to see what happens to the sin of 𝜙, when we change 𝜙 this way, let’s recall the shape of the sine function. If we plot the sin of an angle 𝜙 as 𝜙 varies from zero up to 360 degrees, we see that the graph reaches its maximum value at an angle of 90 degrees. And then when the angle is zero, back at the origin, the sine of that angle is zero. So as we transition from 𝜙 equals 90 to 𝜙 equals zero degrees, we’re covering this portion of our graph. And this tells us what kind of curve to look for among our four candidates. It should be a line that starts out at its maximum value when 𝜙 is equal to 90 degrees and then tails off to zero when 𝜙 goes to zero degrees.

Looking at our four curves then, we see that just one of them starts out at the maximum value it attains over this range of angles and then, over the angular change from 𝜙 equals 90 to 𝜙 equals zero degrees, tails off to zero. That’s the red curve on our graph. This is the only line that consistently decreases whereas the other three options at some point increase in value. And so this is the answer we’ll give to our question. It’s the red curve that correctly represents the change in torque acting on a loop as it rotates.

Let’s take a moment now to summarize what we’ve learned about the torque on a current-carrying rectangular loop of wire in a magnetic field. In this lesson, we saw that if we have a current-carrying rectangular loop in a uniform magnetic field, then that loop experiences an overall torque equal to the magnetic field strength times the current magnitude in the coil times the cross sectional area of the coil multiplied by the number of turns in the coil loop. And all of this is multiplied by the sin of an angle we’ve called 𝜃, where 𝜃 is the measure of an angle between a vector perpendicular to the cross sectional area of the coil and the magnetic field lines.

Along with this, we learned the term “magnetic dipole moment”, which indicates how strongly a coil such as this interacts with an external magnetic field. The magnetic dipole moment 𝑚 sub 𝑑 is given by the maximum torque experienced by the coil divided by the strength of the field it’s in. This is a summary of the torque on a current-carrying rectangular loop of wire in a magnetic field.