### Video Transcript

The diagram shows a direct current motor. The coil of the motor is simultaneously shown at four different angles to the magnetic field of the motor. At which position is the torque on the coil of the motor at its maximum? At which position is the torque on the coil of the motor at its minimum?

Let’s start with the first part of this question. Now this part of the question asks us to work out when the torque on the coil of the motor is at its maximum, given four positions to choose from. To figure this out, we’ll start by looking at which direction force acts on the coil. And to do this, we can use a right hand rule. The right hand rule states that if you point the fingers of your right hand in the direction of current in a wire and then you bend your fingers so that they point in the direction of magnetic field across the wire and then you outstretch your thumb, then your thumb will be pointing in the direction of force that acts on the wire.

To apply this to the motor shown in our diagram, we can first see that the magnetic field is from left to right, from north to south. Working out the direction of current in the coil is slightly harder because we’re shown the coil at four different positions. However, we do know that the positive terminal of the motor is on the left-hand side and the negative terminal is on the right-hand side. To work out which part of the coil makes contact with which terminal, we can look at how the orientation of the commutator changes in each position and which of the brushes it is in contact with.

Starting with position one where the coil is parallel to the magnetic field, if we label the two halves of the commutator one and two, we see that the first half of the commutator is connected to the positive terminal and the second half is connected to the negative terminal. So we can draw the route that current takes around the coil, which is through this left-hand half of the commutator and into the coil. And then on the left-hand side of the coil, the direction of current is into the diagram, then across the coil at the back, then traveling out of the diagram on the right-hand side of the coil, finally out of the coil through the right-hand half of the commutator and to the negative terminal.

Next, let’s look at position two. Now position two is nearly identical to position one, except it is rotated clockwise by approximately 45 degrees, meaning the commutator has also rotated. However, the first half of the commutator is still connected to the positive terminal and the same for the second half and the negative terminal. So the route that current takes around the coil is very similar to how it was in position one: into the diagram on the left-hand side of the coil and out of the diagram on the right.

Next, we’ll look at position three. Position three is rotated another 45 degrees clockwise, meaning this time the coil is perpendicular to the magnetic field. Here we can see there is a gap on the commutator on both sides, which means that neither of the halves one or two are connected to the terminals, meaning there will be no current in the coil in position three.

Finally, let’s take a look at position four. Again, we see that the commutator is rotated a further 45 degrees clockwise. Meaning the first half of the commutator is now connected to the negative terminal and the second half is now connected to the positive terminal. However, the second half of the commutator is now on the left. So the direction of current on the left-hand side of the coil is still into the diagram and on the right-hand side of the coil is out of the diagram.

So we’ve established that for any of the positions, the direction of current on the left-hand side of the coil is into the diagram and on the right-hand side of the coil is out of the diagram. And we can use this to work out the direction of force on each side of the coil. Let’s use the right hand rule, looking at the side of the coil on the right-hand side of the diagram.

Here, we know that the magnetic field is pointing from left to right and the direction of the current is out of the diagram. And this is actually very similar to the example we had for the right hand rule, except the current and the magnetic field have both rotated by 90 degrees. If we rotate the hand to match this, we will actually see that the thumb is still pointing upwards. So the direction of force is also upwards. And we can draw this on our diagram. This also means that because the current is in the opposite direction on the left-hand side of the coil, the force must also be in the opposite direction, which is downwards. So we’ve worked out that on the right-hand side of the coil, the direction of force is upwards. And on the left-hand side of the coil, the direction of force is downwards.

Next, we need to work out the magnitude of the force, which we can work out using the following equation, which tells us that the magnitude of the force is equal to the strength of the magnetic field multiplied by the current in that side of the coil multiplied by the length of that side of the coil. Now, in our motor, we will assume that the magnetic field strength is constant at any point. Also, the side length of the coil is fixed, so this is also constant. Finally, the magnitude of the current in the motor does not change either, with the exception of position three where the coil is not connected to the terminals. So for the other positions, we can say that the current in the coil is also constant.

So what we’ve just found out is that the force acting on either side of the coil is constant in any of the positions where the commutator makes contact with the brushes. So this is positions one, two, and four. So far, we’ve worked out two things. We’ve worked out that the force is acting in the vertical direction on the left- and right-hand sides of the coil, so down on the left and up on the right. We’ve also found out that the magnitude of the force does not change in each position with the exception of position three. And we can use this to work out which position has the maximum torque on the coil of the motor.

The formula for calculating torque states that torque about the axis of the coil is equal to the force acting on the coil multiplied by the perpendicular distance of the force to the axis. If the force is acting vertically on each side of the coil, then the torque on the coil would be biggest when this horizontal distance is biggest, which is achieved when the coil is horizontal. And this happens in position one. So the position at which the torque on the coil of the motor is at its maximum is position one.

Now, let’s take a look at the second part of this question.

At which position is the torque on the coil of the motor at its minimum?

And we’ve actually already done all of the hard work for this part of the question when we did the first part of the question. So we’re trying to find out which of the four positions produces the minimum torque on the coil of the motor. And we’ve already worked out in position three there is no current in the coil of the motor. Therefore, in position three, there is no force acting on the coil. So there is no torque acting on the coil. Every other position, we can clearly see that there is some force acting on the left- and right-hand sides of the coil which generates a torque. So position three must have the minimum torque.

The position at which the torque on the coil of the motor is at its minimum is position three.