Question Video: Torque Acting on a Current-Carrying Rectangular Loop Physics

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° 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 0° to 90°?

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Video Transcript

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.

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