# Lesson Explainer: Direct Current Motors Physics

In this explainer, we will learn how to describe the use of a commutator in producing uniform circular motion from the output of a direct current source.

A direct current, or dc, motor is a device that transfers electrical energy to kinetic energy. It does this using the principle of electromagnetic induction. We are going to look at the design of a dc motor and see how it works.

Let’s begin by seeing how a dc motor is built. The basic design is shown in the diagram below.

The device is called a dc motor because it runs off a direct current source. A direct current source, such as a battery, is a current source that always sends current in the same direction.

The dc current source is connected by wires to two brushes. The wires and brushes are illustrated in blue in the diagram. These brushes are curved to help maintain electrical contact to the commutator that sits between the two brushes.

The commutator is shown in orange in the diagram. Commutators generally look like a circle or solid ring that has been split into two halves. They are made of metal, and so they conduct electricity. However, the gap between the two halves means that they are electrically separated from each other—that is, charges cannot flow directly from one half of the commutator to the other.

Each half of the commutator is connected to one end of a loop of wire. This loop of wire, shown in pink in the diagram, is called the coil. It is sometimes also referred to as the armature. The loop of wire has been drawn such that it is oriented in a horizontal plane. However, it is capable of rotating, along with the commutator, about an axis through its center. This axis is illustrated in the diagram as a dashed gray line.

Around the armature, there is a permanent magnet. This is shown in gray in the diagram. This magnet is often referred to as the stator. The name is chosen to emphasize the fact that this part of the motor remains stationary, in contrast to the rotating coil.

The commutator and brushes are shown in a close-up view in the diagram below. Two different commutator designs are illustrated: the commutator may be formed of either two D-shaped halves, as in the left-hand diagram, or two halves of a split ring, as in the right-hand diagram. These diagrams are shown from “behind” the commutator, as compared to the previous diagram. It is important to notice that each end of the conducting loop of wire is connected to one half of the commutator. As the commutator and wire loop rotate, the ends of the wire remain fixed to the commutator halves.

We initially drew the diagram of the dc motor with all the different parts of the motor highlighted using different colors. However, now that we have identified the various components, it is perhaps more helpful to draw it as follows.

In this second version of the diagram, we have used the color gray for all parts of the motor that remain stationary, and we have used the color orange for all parts of the motor that can rotate.

Let’s consider the path that the current follows. This is illustrated in the diagram below where the coil is horizontally oriented.

We can recall that a conventional current goes from positive to negative. This means that we have a current coming from the positive terminal.

The gap between the two halves of the commutator blocks the current from being routed directly to the negative terminal. However, since each end of the coil is connected to one half of the commutator, the current instead goes through the coil. The current follows the loop made by the coil until it reaches the other half of the commutator.

This second half of the commutator is in contact with the brush connected to the negative terminal. This gives the current a route to follow to reach the negative terminal, hence completing the circuit.

Now, let’s think what this current actually does to cause this device to act as a motor.

The basic principle underlying the operation of the dc motor is that an electric charge moving in a magnetic field experiences a force.

In this particular case, we are considering the flow of charges in a wire, in other words, an electric current. We have a wire of a certain length carrying a current in the presence of a magnetic field. Since the wire contains moving charges, we know that there will be a force acting on it.

### Equation: The Force on a Current-Carrying Wire in a Magnetic Field

Consider a wire of length carrying a current of magnitude in the presence of a magnetic field .

If the direction of the wire is perpendicular to the direction of the magnetic field, then the magnitude of the force on the wire is given by

The direction of the force is perpendicular to both the current in the wire and the magnetic field and may be found using the left-hand rule.

The force on the wire acts perpendicularly to the direction of the current in the wire and the direction of the magnetic field. So, let’s look at the directions of the current and the magnetic field.

The direction of the magnetic field is indicated in the diagram above. We know that the magnetic field between two poles of a magnet goes from the north pole to the south pole; in our case, that is from left to right on the screen.

The direction of the current in both parts of the coil that are perpendicular to the magnetic field is also indicated. We can recall that only the current that is perpendicular to the field will result in a force. On the left-hand side of the coil, that current is directed into the screen. On the right-hand side, the current is directed out of the screen toward us.

Let’s focus on the left-hand side of the coil. Here, the current is directed into the screen. The magnetic field is directed from left to right. We know that the force must be perpendicular to both of these quantities, but that still leaves two options: up or down.

To work out which of these directions the force points in, we can use Fleming’s left-hand rule.

### Rule: Fleming’s Left-Hand Rule

Fleming’s left-hand rule allows us to find the direction of the force on a current-carrying wire in a magnetic field, provided that the field and current directions are perpendicular.

The rule is shown visually in the diagram above. It works as follows:

• With our left hand, we point the first, or index, finger along the direction of the magnetic field.
• We then point our second finger, at to the first, along the direction of the current.
• Our thumb, at to both fingers, then points along the direction of the force on the wire.

Let’s apply this left-hand rule to the coil of wire in our motor.

We will begin by considering the left-hand side of our coil of wire. Here, the current is directed away from us. The magnetic field is directed to the right. Applying the left-hand rule, we point our first finger along the field direction and our second finger along the current direction. This is shown in the diagram below.

We find that our thumb ends up pointing downward. This tells us that the force acting on the left-hand side of the coil is directed downward.

We can apply the same process to the right-hand side of the coil. In this case, the field direction is still to the right, but the current is now directed toward us. It may easily be verified using the left-hand rule (and it is a useful exercise to try this) that the force on this right-hand side of the coil acts vertically upward.

So, the forces acting on these two sides of the coil are as shown in the diagram below. In this diagram, we have shown a top-down view on the left in which the current direction is indicated. On the right, we have shown a side view in which the forces are indicated. On this side view diagram, we have also indicated the current direction using the symbols (into the screen) and (out of the screen).

It is worth reiterating at this point that the other two sides of the coil do not experience a force. The reason for this is that the current in these sides runs either parallel or antiparallel to the direction of the magnetic field.

Let’s have a look at an example to help familiarize ourselves with dc motors and get some practice at using the left-hand rule.

### Example 1: Finding the Direction of Current in the Coil of a dc Motor

The diagram shows a direct current motor. The pink arrows shown represent forces that act on the coil. Which of the terminals or is the positive terminal of the motor?

### Answer

The question is asking us to find which of the two terminals labeled and is the positive one. In order to do this, we need to identify the direction of current, since we know that conventional current is directed from positive to negative.

We are given the direction of the force on the left- and right-hand sides of the coil. We also know that the magnetic field is directed from the north pole to the south pole of the permanent magnet; this is from left to right.

Now, we may appeal to our left-hand rule. We will consider the left-hand side of the coil. In this case, we know the direction of the magnetic field (along which we point our first or index finger) is to the right and the direction of the force (along which we point our thumb) is upward.

We find that our second finger, which gives the current direction, points toward us. This means that the current in the left-hand side of the wire is directed toward us, out of the screen.

Since the direction of current flow cannot just change partway around a circuit, we can see that the current must then follow this path in the motor:

Then, since current is directed from positive to negative, we see that the answer to the question is that the positive terminal is .

Near the beginning of this explainer, we said that the coil was able to rotate (along with the commutator). We have now seen that the combination of the current in the coil and the magnetic field from the stator (the magnets around the coil) results in forces acting on two sides of the coil.

It turns out that it is this force that causes the rotation of the coil. More precisely, it is the torque resulting from this force that causes the rotation.

The torque resulting from a force is defined as the product of the magnitude of that force and the perpendicular distance of the line of action of the force from the axis of rotation. In other words, whenever we have a force that is acting on an object some perpendicular distance away from the axis about which that object can rotate, there will be a torque.

In the diagram below, we can see the axis about which the coil is able to rotate, that is, the axis of rotation. We can also see that the two forces are not acting along that axis but rather some distance away from it.

The distance of the forces from the axis has been highlighted in the diagram using two black dashed arrows. Since the forces are not along the axis, they will indeed result in a torque on the coil.

In this case, the left-hand force is acting downward while the right-hand force is acting upward. So, as we might expect, the torque causes the coil (along with the commutator) to rotate in the direction shown in the diagram, that is, counterclockwise from the direction we are looking at it.

All of our analysis so far has been when the coil is in a horizontal plane. However, we have just shown that the forces acting on the coil at this point produce a torque that causes it to rotate. This means that we also need to consider what is happening when the coil has rotated to other angles.

Let’s look at the case where the coil has rotated by some amount less than relative to the initial horizontal position we considered. This is shown in the diagram below.

We can notice from the diagram that the commutator has rotated around along with the coil, but each of the two halves of the commutator is still in electrical contact with the same brush. For clarity, we have labeled the commutator halves 1 and 2. Then, we can say that, at this point, commutator half 1 is still in contact with the positive terminal and commutator half 2 is still in contact with the negative terminal.

This means that electrical charge still moves around the circuit in the same way as it did before when the coil was horizontal. The current has the same direction in the left- and right-hand sides of the coil as it did previously.

Since the current directions are still the same and the direction of the magnetic field has not changed either, this means that the forces acting on each side of the coil are still in the same directions as they were before. That is, the force on the left-hand side acts downward, while the force on the right-hand side acts upward.

As before, these forces do not act on the line through the center of rotation of the coil. This means that they act to produce a torque. However, we can see from the diagram above that the perpendicular distance of these forces from the rotation axis is smaller than when the coil was horizontal. Because these forces are acting closer to the axis of rotation than they previously were, the magnitude of the torque produced by them has decreased.

As the coil rotates further from the horizontal position and approaches the vertical position, the magnitude of the torque on that coil gets smaller and smaller as the distance of the forces from the axis of rotation decreases.

Let’s now consider what happens at this vertical position of the coil, which is illustrated in the diagram below.

We can see from the diagram that any forces acting on the sides of the coil in this position would be acting along the axis of rotation. Therefore, no torque would be produced by these forces. In other words, when the coil is oriented vertically like this, there is no net torque acting on it. The only thing that keeps the coil turning at this point is that it has some rotational inertia; since the coil was already moving counterclockwise, it will continue to do so unless there is some resistance.

There is also something else that it is important to notice in this diagram: the position of the commutator. Up until this point, the half of the commutator labeled 1 has always been in electrical contact with the brush connected to the positive terminal. Similarly, the commutator half 2 has always been in contact with the brush connected to the negative terminal. This vertical position of the coil represents a switchover point. As the coil rotates beyond this point, commutator half 1 will be in contact with the negative terminal, while commutator half 2 will be in contact with the positive terminal.

Let’s consider what happens to the current in the coil after rotating past the vertical. The charges now flow from the positive terminal to commutator half 2. They flow through the coil until they reach commutator half 1. Then, they flow through the right-hand brush to the negative terminal. This is shown in the right half of the diagram below.

The left half of the diagram shows the coil before it has rotated past the vertical. For clarity, we have labeled the sides of the coil 1 and 2 in accordance with the commutator half that each is connected to.

We can see that as the coil passes through the vertical orientation the direction of the current in the coil itself changes. Before passing through the vertical, the current in side 1 was directed away from us (into the screen) while the current in side 2 was directed toward us (out of the screen). Now, after passing through the vertical, the current in side 1 is directed toward us while the current in side 2 is directed away from us.

However, the current direction in the circuitry outside of the coil is unchanged. The current is still directed from the positive terminal to the left-hand brush and from the right-hand brush to the negative terminal. It is the addition of the commutator that causes the change of direction of the current in the coil.

We have seen what happens to the current in the coil as it rotates past the vertical. Now, let’s also consider the forces acting on each side of the coil. These forces are shown in the diagram below.

We have again shown the coil in two positions: before and after the coil rotates past the vertical orientation. In addition to indicating the direction of the current in the sides of the coil in each case, we have labeled the forces acting on each side of the coil. The directions of these forces may be verified by applying the left-hand rule.

Before passing past the vertical position (left-hand diagram), the force on side 1 was directed downward and the force on side 2 was directed upward. Previously, we described this as the force on the left-hand side of the coil being directed downward and the force on the right-hand side being directed upward.

Looking at the right-hand diagram, we see that after the coil rotates past the vertical, the force on the left-hand side of the coil is still directed downward and the force on the right-hand side is still upward. However, side 1 is now the right-hand side and side 2 is now the left-hand side. Because the direction of the current through the coil has changed, the direction of the forces on each side of the coil has also changed.

Let’s look at an example.

### Example 2: Identifying the Positions of Maximum and Minimum Torque in a dc Motor

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.

1. At which position is the torque on the coil of the motor at its maximum?
2. At which position is the torque on the coil of the motor at its minimum?

### Answer

Part 1

The diagram presents us with four different angles of the coil in a dc motor. In position I, the coil is horizontal. In positions II and IV, the coil is at a angle to this horizontal. In position III, the coil is vertical.

We can recall that there is a force on the two sides of the coil that are perpendicular to the direction of the magnetic field. Those are the sides that are directed toward or away from us (left and right sides when the coil is horizontally oriented).

The torque on the coil depends on the size of the force itself but also on the distance of the line of action of this force from the rotation axis.

The size of the force is calculated using , where is the magnetic field strength, is the current, and is the length of the wire. Since none of these quantities are changing as the coil rotates, the magnitude of the force will not change. Therefore, any changes in the torque will be a result of the changing distance of the line of action of the force from the rotation axis of the coil.

When the coil is oriented horizontally, this distance is maximized. Therefore, the torque on the coil is greatest when the coil is oriented horizontally, as in position I.

Part 2

The torque will be a minimum for the minimum distance between the line of action of the force and the rotation axis of the coil.

This will occur when the coil is vertical. In this case, the distance from the axis to either of the two sides of the coil that are perpendicular to the magnetic field direction is zero. So, when the coil is vertically oriented, the torque is not only minimized, but is actually zero.

Therefore, our answer is that the torque on the coil is minimized when the coil is oriented vertically, as in position III.

Every time the coil rotates through the vertical orientation, the direction of the current in the coil changes. This means that the direction of the forces on sides A and B will also change each time.

The result of this is that the force on the side of the coil to the left of the vertical (whether that side is 1 or 2) will always be directed downward, while the force on the right-hand side of the coil will always be directed upward.

This means that the torque from these forces will always act to turn the coil in the same direction. The coil will therefore continue to rotate in this same direction.

Let’s look at one more example.

### Example 3: Identifying the Direction of Rotation of a Coil in a dc Motor

Which of the diagrams showing a direct current motor correctly represents the direction of the rotation of the motor? The coil of the motor is simultaneously shown at four different angles to the magnetic field of the motor.

### Answer

This question is asking us which of the two diagrams shows the correct direction of rotation of the motor. In order to answer this, let’s remind ourselves what causes this rotation to occur.

We can recall that the rotation is caused by a torque on the coil and that this torque is the result of a force induced on the current-carrying wires of this coil.

Let’s consider a simplified diagram showing just one coil angle. We have drawn this below. We can recall that because of the way that the commutator links the brushes to the coil, if we know which direction the torque causes the coil to rotate at any one angle, then we know that the effect of this torque will remain the same for all coil angles. In other words, the coil will continue to rotate in the same direction.

We can recall that the direction of the force can be found from the current direction and the direction of the magnetic field using the left-hand rule.

We have indicated the direction of the magnetic field on the diagram. This direction is toward the right since the magnetic field between two poles of a magnet goes from the north pole to the south pole.

Since a conventional current is directed from positive to negative, we know that the current in the coil will be directed as shown in the diagram.

Let’s consider the left-hand side of the coil. We see that the current is directed away from us, while we know that the magnetic field is directed toward the right.

Using the left-hand rule, we point our first finger along the field direction (to the right) and our second finger along the current direction (away from us). This is illustrated in the diagram below.

As shown in the diagram, this leads our thumb to point downward. Hence, the force on the left-hand side of the coil acts downward.

If we apply the same left-hand rule to the right-hand side of the coil, we find that the force on this side acts upward, since, in this case, the current is directed toward us.

The forces are illustrated in the diagram below.

Since the forces are acting to push the left-hand side of the coil downward and the right-hand side upward, we see that they are going to give us a torque that causes the coil to rotate counterclockwise.

Comparing against the diagrams given to us in the question, we see that the correct direction of rotation, counterclockwise, is shown in diagram B.

We have now seen all of the fundamentals underlying the operation of a dc motor. There is just one part left: how does this rotating coil actually function as a motor?

The answer is that a bar is placed along the rotation axis of the coil. As the coil turns, this bar also turns along with it.

This rotating bar is then capable of causing a gear or other mechanical object to rotate, and this rotating object can do mechanical work. In this way, a dc motor uses electrical energy from the circuit in order to generate mechanical work.

Let’s finish up by summarizing what we have learned.

### Key Points

• A direct current, or dc, motor uses electrical energy in order to generate mechanical work.
• The motor consists of a coil of wire attached to a commutator that is free to rotate in a magnetic field produced by a permanent magnet (the “stator”).
• The two sides of the coil perpendicular to the magnetic field (the “left” and “right” sides when the coil is horizontal) experience a force that is given by , where is the magnetic field strength, is the current in the wire, and is the length of the wire (in this case, the length of the side of the coil).
• The direction of the force on each of these two sides of the coil may be found using the left-hand rule.
• These forces cause a torque, leading the coil to rotate.

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