Physics · 4 Electricity and magnetism

4.5.5 The d.c. motor

Before you start

A current-carrying wire across a magnetic field feels a force, and reversing either the current or the field reverses that force (4.5.4).
Fleming's left-hand rule for the relative directions of force, field and current (4.5.4).
Between two facing poles, the magnetic field lines run from the N pole to the S pole (4.1).
The moment of a force is the force multiplied by the perpendicular distance from the pivot to its line of action (1.5.2).

Syllabus objectives covered (2)

Know that a current-carrying coil in a magnetic field may experience a turning effect and that the turning effect is increased by increasing: (a) the number of turns on the coil (b) the current (c) the strength of the magnetic field
Describe the operation of an electric motor, including the action of a split-ring commutator and brushes

Part 1 of 5

Why a coil in a magnetic field turns

Start from what you already know. A wire carrying a current across a magnetic field feels a force. Fleming's left-hand rule gives the direction: First finger along the Field (N to S), seCond finger along the Current (+ to −), and the thuMb gives the Motion, which is the direction of the force.
Two facts from 4.5.4 carry this whole lesson. Reverse the current and the force reverses. Reverse the field and the force reverses.
Now bend the wire into a rectangular loop, called a coil, and put it between the poles of a magnet. An axle runs through the middle of the coil so that it can rotate.
Follow the current round the loop. It flows along one long side, across the short end, and back along the other long side. So the current in the two long sides flows in opposite directions.
apparatus diagram — not drawn yet
A simple d.c. motor drawn in perspective: a rectangular coil lying horizontally between an N pole on the left and an S pole on the right, straight field lines from N to S; an axle along the middle of the coil; the two ends of the coil joined to the two halves of a split-ring commutator on the axle; two fixed brushes pressing on the commutator, with leads to a d.c. supply; arrows show the current going one way along one long side and the opposite way along the other, with a force arrow pointing up on one long side and down on the other.
Opposite currents in the two long sides, the same field, so opposite forces: one side goes up, the other goes down.
Both long sides are in the same field. Opposite currents in the same field give opposite forces, so one side is pushed up and the other is pushed down.
These two forces are equal in size and act on opposite sides of the axle. They do not move the coil sideways. Together they turn it about the axle. That is the turning effect.
The two short ends of the coil add nothing to the turning effect. Any force on them acts along the line of the axle, and a force along the axle cannot turn the coil about it.
The direction of rotation follows from the two facts above. Reverse the current, or reverse the field, and both forces reverse, so the coil turns the other way. Reverse both at once and the two changes cancel, so it turns the original way.

Part 2 of 5

Why the objective says "may": the coil's angle matters

The objective says the coil may experience a turning effect. The word "may" is there because the size of the turning effect depends on the angle of the coil in the field.
Use moments from 1.5.2. The pivot is the axle. The moment of each force depends on how far its line of action is from the axle, measured at $90 °$ to the force.
$moment = force \times perpendicular distance from the axle$
The forces on the long sides are always at $90 °$ to the field. In the usual picture the field is horizontal, so the forces are always vertical, whatever angle the coil is at.
When the plane of the coil is parallel to the field (the coil is horizontal in the usual picture), the long sides are as far out to the sides of the axle as they can be. The perpendicular distance is largest, so the turning effect is largest.
forces diagram — not drawn yet
Two end-on views along the axle, side by side, with a horizontal field from an N pole on the left to an S pole on the right. Each view shows the coil as a straight line through a dot marking the axle, with a small circle at each end for a long side: a cross in one (current into the page) and a dot in the other (current out of the page). Left view: coil horizontal, a downward force arrow on the left side and an upward force arrow on the right side, with the perpendicular distance from the axle to each arrow marked. Right view: coil vertical, a vertical force arrow on each side lying along the line through the axle.
Horizontal: the forces act as far from the axle as possible, so the turning effect is largest. Vertical: the forces act through the axle, so the turning effect is zero.
As the coil turns towards vertical, the forces stay vertical but their lines of action move in towards the axle. The perpendicular distance gets smaller, so the turning effect gets smaller.
When the plane of the coil is at right angles to the field (the coil is vertical), each force acts straight through the axle. The perpendicular distance is zero, so the turning effect is zero.
Here is the problem that the next part of the lesson solves. Suppose the coil overshoots vertical. The side that was on the right is now on the left of the axle, but it still carries current the same way, so it is still pushed the same way. On this side of the axle, that push turns the coil backwards.
So a plain coil does not keep spinning. It swings to and fro about the vertical position and comes to rest there. To get continuous rotation, something must reverse the current in the coil every half turn.

Part 3 of 5

Three ways to make the turning effect bigger

The objective lists three changes. Each one works the same way: it makes the force on each long side bigger, so the moment about the axle is bigger.
(a) More turns on the coil. Each turn is another loop of wire carrying the current, and each loop feels its own pair of opposite forces. With the current kept the same, the forces from all the turns add together.
(b) A larger current. The force on a wire in a magnetic field is larger when the current in it is larger. For the same coil, a supply with a larger p.d. gives a larger current.
(c) A stronger magnetic field. The force on each wire is larger in a stronger field, so use a stronger magnet. Say stronger, not bigger: a bigger magnet is not necessarily a stronger one.
In a working motor, a larger turning effect makes the coil spin faster, or lets it turn a heavier load.
Keep size and direction apart. The three changes above change how big the turning effect is. Reversing the current or the field changes its direction. Swapping the supply connections does not make the turning effect bigger. It only makes the coil turn the other way.

Part 4 of 5

The split-ring commutator and the brushes

The plain coil stopped because, after half a turn, each long side was on the other side of the axle and still being pushed the same way. The fix is to reverse the current in the coil every half turn.
Here is why that works. If the current reverses just as the coil passes vertical, the force on each side reverses too. The side that has just arrived on the left is now pushed the same way that the left-hand side was pushed before, so the coil keeps turning in the same direction.
The part that does this is the split-ring commutator. It is a metal ring cut into two halves, with an insulating gap between them. Each half is joined to one end of the coil. The ring is fixed to the axle, so it rotates with the coil.
apparatus diagram — not drawn yet
Close-up end view of a split-ring commutator in two positions, half a turn apart: a ring split into two halves labelled X and Y with an insulating gap between them, each half joined to one end of the coil; two fixed brushes press on the ring from opposite sides, one wired to the + terminal and one to the − terminal. In the first position X touches the + brush; in the second position X touches the − brush.
The halves swap brushes every half turn, so the current in the coil reverses while the supply stays the same.
The brushes are two fixed contacts that press against the commutator from opposite sides. They are often blocks of carbon. They are connected to the d.c. supply.
The brushes are needed because the coil spins. Wires fixed to the coil would twist round the axle. The brushes make a sliding contact instead, so the current can reach the coil while it rotates.
Every half turn, the two halves of the ring swap brushes. The end of the coil that was connected to the + terminal is now connected to the − terminal, so the current in the coil reverses.
The trap here: the supply does not change. It is d.c., and its current always flows the same way. Only the current in the coil reverses, and the commutator is what reverses it.

Part 5 of 5

One full turn of the motor, step by step

Start with the coil horizontal. The current in the two long sides flows in opposite directions, the forces on them are in opposite directions, and the turning effect is at its largest.
The coil turns. As it approaches vertical, the turning effect gets smaller, but it is still in the same direction, so the coil keeps turning that way.
At vertical, the brushes are at the gaps in the split ring, so for an instant no current flows in the coil. The turning effect at this angle is zero anyway.
The coil does not stop at vertical, because it is already moving. Its momentum carries it past the vertical position.
Just past vertical, each half of the ring touches the other brush. The current in the coil reverses, the force on each long side reverses, and the turning effect is in the original direction again.
The second half turn repeats the first. So the current in the coil reverses twice in every revolution, and the coil keeps rotating in one direction.
$reversals per second = 2 \times revolutions per second$
When you describe the motor, give the chain in this order: current in the coil; forces on the two long sides in opposite directions; a turning effect about the axle; the commutator, connected to the supply through the brushes, reverses the current every half turn; the forces reverse; the coil keeps turning the same way.

Worked examples

Example 1. A coil lies horizontally between the poles of a magnet, with the N pole on the left and the S pole on the right. You look at it end-on, along the axle. In the left-hand long side the current flows away from you. In the right-hand long side it flows towards you. Determine the direction of the force on each side, and the direction in which the coil starts to turn as you see it.

Show the working

Find the field direction first. Field lines run from N to S, so the field points from left to right.
Left-hand side: point your first finger to the right (field) and your second finger away from you (current). Your thumb points down. So the force on the left-hand side is downwards.
Right-hand side: the field is the same, but the current is reversed, flowing towards you. Reversing the current reverses the force, so the force on the right-hand side is upwards. You do not need the rule a second time.
Now picture the coil as a horizontal bar on a pivot. The left end is pushed down and the right end is pushed up. A bar whose right end rises turns anticlockwise.

Answer: The force on the left-hand side is downwards and the force on the right-hand side is upwards, so the coil starts to turn anticlockwise as you see it.

Example 2. A d.c. motor is changed in each of these ways, one at a time: (i) the current is increased; (ii) the coil is rewound with more turns, with the current kept the same; (iii) the two connections to the supply are swapped; (iv) the connections are swapped and the magnet is also turned round so that its poles swap. Predict the effect of each change on the motor.

Show the working

Sort the changes first. Two of them change the size of the force on each wire, and two change its direction.
(i) A larger current gives a larger force on each long side, so the turning effect increases and the motor turns faster. The direction is unchanged.
(ii) More turns at the same current means more wires, each with its own pair of forces. The forces add, so the turning effect increases and the motor turns faster. The direction is unchanged.
(iii) Swapping the connections reverses the current in the coil. That reverses the force on each side, so the motor turns the opposite way. The size of the turning effect does not change.
(iv) Reversing the current alone would reverse the rotation. Reversing the field alone would also reverse it. Doing both reverses the forces twice, so they end up in their original directions.

Answer: (i) Larger turning effect, turns faster, same direction. (ii) Larger turning effect, turns faster, same direction. (iii) Turns in the opposite direction. (iv) Turns in the original direction, because the two reversals cancel.

Example 3. A student mounts a coil on an axle between two magnets and joins the ends of the coil to a d.c. supply with two ordinary fixed wires. The coil turns, overshoots the vertical position, swings back and forth, and comes to rest with its plane at right angles to the field. Explain why it comes to rest there, and describe how a split-ring commutator and brushes would keep it turning.

Show the working

Explain why it stops first. At the vertical position the forces on the long sides act through the axle, so the turning effect is zero.
Once a side passes vertical, it is on the other side of the axle. It still carries current the same way, so it is still pushed the same way, and on this side of the axle that push turns the coil back. So the coil swings back towards vertical and settles there.
Now describe the fix. Replace the fixed wires with a split-ring commutator on the axle: a ring in two halves, insulated from each other, each half joined to one end of the coil. It rotates with the coil.
Two fixed brushes press on the commutator and connect it to the supply. They make a sliding contact, so the current reaches the coil as it rotates.
Every half turn, as the coil passes vertical, the halves of the ring swap brushes. That reverses the current in the coil.
With the current reversed, the force on each side reverses. The turning effect is now in the original direction, so the coil keeps rotating the same way. Its momentum carries it through the vertical position, where the turning effect is zero.

Answer: Past vertical, each side is still pushed the same way but is now on the other side of the axle, so the turning effect reverses and pulls the coil back to vertical, where the turning effect is zero. A split-ring commutator, which rotates with the coil and is connected to the supply through fixed brushes, reverses the current in the coil every half turn. The forces on the sides then reverse, so the turning effect stays in one direction and the coil keeps rotating.

Example 4. The coil of a d.c. motor turns at $1500$ revolutions per minute. Calculate how many times per second the commutator reverses the current in the coil, and the time between one reversal and the next.

Show the working

Work in seconds, because the question asks for reversals per second. There are $60$ seconds in a minute, so divide by $60$ .
$\frac{1500}{60} = 25 revolutions per second$
The current reverses every half turn, so twice per revolution.
$2 \times 25 = 50 reversals per second$
Fifty equal intervals fit into one second, so each interval is one fiftieth of a second.
$\frac{1}{50} = 0.020 s$
Check: $50$ intervals of $0.020$ s make $1.0$ s, which is right.

Answer: $50$ times per second, with $0.020$ s between one reversal and the next.

Where people go wrong

Mistake: Saying the commutator reverses the direction of rotation of the coil.

Instead: The commutator reverses the current in the coil every half turn, and that keeps the coil rotating in the same direction.

Without the reversal the turning effect would reverse past vertical and the coil would stop. The commutator stops the rotation from reversing.

Mistake: Thinking the brushes rotate with the coil.

Instead: The brushes are fixed. The commutator rotates with the coil, and its two halves slide past the brushes.

The whole point of the brushes is a sliding contact between a fixed supply and a rotating coil. If they rotated, the halves would never swap and the current would never reverse.

Mistake: Thinking the turning effect is the same at every angle, or largest when the coil is vertical.

Instead: The turning effect is largest when the plane of the coil is parallel to the field and zero when it is at right angles to the field.

The forces are always at $90 °$ to the field. When the coil is vertical they act through the axle, so their perpendicular distance from it, and their moment, is zero.

Mistake: Thinking the current reverses because the supply changes direction.

Instead: The d.c. supply never changes. Only the current in the coil reverses, because the commutator halves swap brushes.

This is why the motor runs on d.c. at all: the reversal is done by the commutator, not by the supply.

Mistake: Predicting that reversing both the current and the magnetic field reverses the motor.

Instead: Reversing both leaves the rotation in its original direction.

Each change on its own reverses the forces on the sides. Two reversals bring the forces back to where they started.

Check yourself

Why are the forces on the two long sides of the coil in opposite directions?
Answer
The current flows in opposite directions in the two sides, and both sides are in the same field, so the forces are opposite.
State the three changes that increase the turning effect on a current-carrying coil in a magnetic field.
Answer
More turns on the coil, a larger current, and a stronger magnetic field.
At what angle of the coil is the turning effect largest, and at what angle is it zero?
Answer
Largest when the plane of the coil is parallel to the field. Zero when the plane of the coil is at right angles to the field, because the forces then act through the axle.
A student swaps the connections to the supply and also turns the magnet round. Predict the direction in which the motor now turns.
Answer
The original direction. Each change alone reverses the forces, so the two together cancel.
What do the brushes do in a d.c. motor?
Answer
They are fixed contacts that press on the rotating commutator, making a sliding connection between the supply and the coil so that current reaches the coil while it turns.
How many times does the commutator reverse the current in the coil in one revolution, and at what position of the coil?
Answer
Twice, once every half turn, as the coil passes the position where its plane is at right angles to the field.
At the vertical position the turning effect is zero. Why does the coil not stop there?
Answer
It is already moving, so its momentum carries it past the vertical position. Just past it, the current has reversed and the turning effect is in the same direction again.
A motor coil turns at $40$ revolutions per second. How many times does the current in the coil reverse in one minute?
Answer
$2 \times 40 = 80$ reversals per second, so $80 \times 60 = 4800$ reversals in one minute.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A simple d.c. motor has a rectangular coil on an axle between the N and S poles of a magnet. The ends of the coil are joined to a split-ring commutator, and two brushes connect the commutator to a d.c. supply.

(a) State three changes that would increase the turning effect on the coil. 3 marks No calculator

Mark scheme

B1 more turns on the coil
B1 a larger current (in the coil); accept a larger p.d. / voltage of the supply
B1 a stronger magnet / stronger magnetic field; bigger magnet is insufficient

(b) The coil is horizontal, with its plane parallel to the magnetic field. Explain why the coil experiences a turning effect. 3 marks No calculator

Mark scheme

B1 the current flows in opposite directions in the two long sides of the coil
B1 so the forces on the two sides are in opposite directions (one up, one down)
B1 the forces act on opposite sides of the axle / do not act through the axle, so they turn the coil, owtte

(c) Describe how the split-ring commutator and the brushes keep the coil rotating in the same direction. 4 marks No calculator

Mark scheme

B1 the commutator rotates with the coil while the brushes are fixed and press on it, connecting the coil to the supply
B1 every half turn / when the plane of the coil is at right angles to the field, the halves of the commutator swap brushes
B1 so the current in the coil reverses
B1 so the forces on the sides reverse and the turning effect stays in the same direction

(d) The coil makes $1200$ revolutions per minute. Calculate the number of times per second that the current in the coil reverses. 2 marks

Mark scheme

C1 20 revolutions per second seen, or 2 reversals per revolution used
A1 40 per second

Question 2. A d.c. motor is running. At one instant the plane of its coil is at right angles to the magnetic field.

(a) State the size of the turning effect on the coil at this instant. 1 mark No calculator

Mark scheme

B1 zero

(b) Explain why the coil does not stop at this position, and why it continues to turn in the same direction afterwards. 2 marks No calculator

Mark scheme

B1 the coil is already moving / its momentum / its inertia carries it past this position
B1 the commutator then reverses the current in the coil, so the turning effect is in the same direction as before

(c) Predict the effect on the motor of swapping the two connections to the d.c. supply. 1 mark No calculator

Mark scheme

B1 the coil rotates in the opposite direction

(d) The connections are swapped and the magnet is also turned round so that its poles swap. Predict the direction in which the coil now rotates compared with the original direction. Give a reason. 2 marks No calculator

Mark scheme

B1 the same direction as originally
B1 each change on its own reverses the forces on the sides, so the two reversals cancel, owtte

How this is examined

How the split-ring commutator and brushes keep the coil turning in one direction: a Describe or Explain of the motor's operation, which may come with a diagram of a coil, magnet, commutator and brushes Four separate points: the commutator rotates with the coil while the fixed brushes press on it and connect it to the supply; every half turn (when the plane of the coil is at right angles to the field) the halves swap brushes; so the current in the coil reverses; so the forces on the sides reverse and the turning effect stays in the same direction. "It changes the direction" without saying of what does not earn the reversal point.
Why a coil carrying a current in a magnetic field turns at all: an Explain about a coil between two poles, with the current direction marked The current in the two long sides flows in opposite directions; so the forces on them are in opposite directions (one up, one down); the forces act on opposite sides of the axle, so together they produce a turning effect.
The three ways to increase the turning effect: a State asking for two or three changes that make the motor turn faster or with a larger turning effect One point each: more turns on the coil; a larger current; a stronger magnet or magnetic field. Write "stronger", not "bigger". Under the list rule a wrong extra answer counts towards the number asked for, so give only the changes on the syllabus list.
Where the turning effect is largest and zero, and why the coil passes the zero point: a State or Explain about the coil at a given angle, such as the vertical position Largest when the plane of the coil is parallel to the field; zero when it is at right angles to the field, because the forces act through the axle. The coil keeps going past vertical because its momentum carries it on.
Predicting the direction of rotation after a change: a Predict asking what happens when the supply connections are swapped, the magnet is turned round, or both One reversal (current or field) reverses the rotation. Both together leave it unchanged, and the reason is that each reversal reverses the forces, so two cancel.
Counting how often the current in the coil reverses: an application of the half-turn reversal, not a separate syllabus requirement: a Calculate giving the speed of the coil in revolutions per minute or per second Two reversals per revolution, and conversion between minutes and seconds where the units differ. The final value needs "per second" or "per minute" with it.

Words to know

Coil: A loop, or many loops, of insulated wire. In a motor it sits between the poles of a magnet on an axle.
Turning effect: The tendency of forces to rotate an object about a pivot. On a motor coil it comes from the opposite forces on the two long sides.
Plane of the coil: The flat surface the coil's loops lie in. The turning effect is largest when it is parallel to the field and zero when it is at right angles to it.
Split-ring commutator: A ring in two halves, insulated from each other, fixed to the axle and joined to the ends of the coil. It reverses the current in the coil every half turn.
Brushes: Fixed contacts, often carbon, that press on the commutator and connect the rotating coil to the supply.
Fleming's left-hand rule: First finger along the field, second finger along the current, thumb gives the direction of the force (motion).

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