The diagram represents a rectangular loop of wire at three different rotational positions in a uniform magnetic field. The wire loop carries a constant current supplied from an external circuit that is not shown in the diagram. Which color arrows correctly represent the variation in the magnetic force on the loop as it rotates?
In our diagram, we see this rectangular loop of wire at one, two, three different orientations between the poles of a magnet. We’re told that this wire carries current. And since it’s in a uniform magnetic field, the wire therefore experiences a magnetic force. We want to identify which set of arrows, the blue arrows or the black arrows, correctly represent the variation in magnetic force on the loop.
As we compare the blue arrows to the black arrows, there are two differences between them that will turn out to be important. First of all, notice that the black arrows act at the center of rotation of our loop of wire. If these represent magnetic force, then that force would tend to translate our wire rather than rotate it about some axis. On the other hand, the blue arrows act in different directions on different parts of the wire so that they would tend to make it rotate. That’s one difference.
The second difference, and it’s a little bit hard to see at first, is that for the black arrows, we have arrows for each of the three orientations of our loop. But notice that for the blue arrows, those only exist for two of the three orientations. When the loop of wire is in its vertical position here, there are no blue arrows, potentially representing magnetic force acting on the wire.
To see which of these two sets of colored arrows is correct, let’s begin by looking more closely at the loop of wire in its horizontal orientation. Clearing some space on screen, something interesting about our wire is that while we do know there’s a constant current in it, we don’t know the direction of that current. It could be traveling this way, what we’ll call clockwise, or in what we’ll call the counterclockwise direction. Either way, because our coil of wire is in a uniform magnetic field, the wire will tend to experience a magnetic force.
The direction of this force is determined by what’s called a right-hand rule. Say that we have a wire carrying a current 𝐼 in this direction. And imagine further that the wire is in a uniform magnetic field pointing in this direction. If we take the fingers of our right hand and point them in the direction of the current 𝐼 and then curl those fingers so that they point in the direction of the magnetic field 𝐵, then our right thumb will point in the direction of the magnetic force on the current-carrying wire. On this wire then, that force will be pointed out of the screen towards us.
Going back then to our coil of wire, we can see that the direction of the magnetic force on each of the four sides of the wire — we can call these side one, side two, side three, and side four — will depend on the direction of current in that side length. We see that the current in side lengths two and four is either parallel or antiparallel to the external magnetic field. Whenever this is the case, no magnetic force results. However, in the side lengths one and three, the current, in whichever direction it points, will be perpendicular to that magnetic field. Therefore, these sides will experience a force.
We’ve said that we don’t know the direction of current in this loop, and that’s true. But in whichever of the two possible directions the current does actually point, that will result in current pointing in opposite directions, we can say, in side lengths one and three. What we mean by that is say the current travels in what we’ve called a clockwise direction, indicated by our pink arrows. In that case, in side length one, the current points like this, while in side length three, it points like this, due to the right-hand rule, which tells us the resulting direction of magnetic force.
The force on side length one — which, if current did point in this direction, would point downward toward the bottom of the screen — is opposite the direction of the force on side length three. If current in that side length pointed like this, then the resulting magnetic force would point upward. These particular force directions assume that the current really does point clockwise in this loop. We might’ve gotten that wrong though. The current might point counterclockwise. But even if it does, the resulting force vectors on sides one and three will again point in opposite directions from one another. They won’t point in the same direction.
That’s because, like before, current in these two side lengths will travel in what we could call opposite directions. Our takeaway, then, is that regardless of current direction, there will be a net torque acting on our loop in this orientation. That’s one piece of evidence in favor of the blue arrows indicating the magnetic force variation rather than the black ones.
Let’s think now about what we call the second important difference between these two sets of arrows. Earlier, we noticed that while there are black arrows for all three of our loop of wires orientations, there are blue arrows only for what we could call the first and second orientations of the loop but not the third. We found so far that there will be a nonzero torque acting on a rectangular loop of wire. The magnitude of that torque is given by this equation. It’s equal to the product of the magnetic field magnitude, the current magnitude in the loop of wire, the cross-sectional area of the wire, and the sine of this angle we’ve called 𝜙.
If we go to our loop of wire and we draw a vector that’s perpendicular to the wire’s surface, then the angle between that vector and the external magnetic field — where here that angle is 90 degrees — is 𝜙. When our loop is in its horizontal orientation then, 𝜙 is 90 degrees, so the sin of 𝜙 equals one. But then, let’s think about our loop after it’s gone through a rotation, so now it’s vertically oriented. A vector perpendicular to its plane now points to the right in the same direction as the external magnetic field. In this case then, the angle between these two vectors is zero degrees, and the sin of zero degrees is zero. Therefore, when our loop of wire is perpendicular to the magnetic field, we expect the magnitude of torque on that wire to be zero. That would explain why there are no blue arrows for this vertical orientation of the wire.
So in both points of difference between the blue and black sets of arrows, it’s the blue arrows that appear to correctly represent the variation in magnetic force on the loop.
Let’s look now at the second part of this question.
Which color arrows correctly represent the variation in the magnetic dipole moment of the loop as it rotates?
Given a loop of wire carrying a current 𝐼 and having cross-sectional area 𝐴, the magnetic dipole moment 𝜇 of that loop equals 𝐼 times 𝐴. Here, though, rather than knowing the magnitude of 𝜇, we really want to know its direction. That direction is indicated by yet another right-hand rule. If we curl the fingers of our right hand in the same direction as current in a loop of wire, then our right thumb points in the direction of the magnetic dipole moment.
Notice that this direction doesn’t change depending on which part of our current-carrying wire we’re considering. It’s an overall direction that takes into account the entire loop. We expect, therefore, that the set of arrows which correctly represent the variation in 𝜇 will only show one arrow for a given orientation of our loop of wire. That’s what we see with the black arrows. There’s one arrow for each wire orientation. Therefore, it’s these arrows, rather than the blue ones, that represent the variation in magnetic dipole moment.
By the way, based on this result or the result for part one of this question, we now know the direction of current in our loop of wire. In its horizontal orientation, for example, current will point like this. This would lead to the blue arrows representing magnetic force and the black arrows representing magnetic dipole moment.