4.2.1 The motor effect

Forces between magnets and currents

Definition

Motor effect

The force that acts on a current-carrying conductor when it is placed across the field lines of a magnetic field.

Definition

Current-carrying conductor

A wire or other piece of conducting material that has an electric current flowing through it.

Definition

Magnetic field

The region around a magnet in which another magnet or a magnetic material experiences a force.

Definition

Interaction pair

The two forces produced by one interaction, one acting on each of the two interacting objects.

A current-carrying conductor is surrounded by its own magnetic field, made of circular field lines centred on the wire.
This field exists only while charge is flowing, so a wire with no current in it has no magnetic field of its own.
When the conductor is placed between the poles of a permanent magnet, the field of the wire and the field of the magnet overlap in the same region of space.
The two magnetic fields interact, and this interaction produces a force on the conductor.
This effect is called the motor effect.
The force is a non-contact force, because the magnet and the wire act on each other through their fields without touching.
The magnet feels a force as well, because a force always arises from an interaction between two objects.
The force on the magnet is equal in size to the force on the conductor and acts in the opposite direction, so the two forces form an interaction pair, as Newton's third law describes.
In most arrangements the magnet is held still and only the conductor is free to move, so the conductor is the object you see moving.
If the conductor were clamped in place and the magnet were free to move, the magnet would move instead, in the opposite direction.
Like every other force, the motor-effect force is measured in newtons ( $N\text{N}$ ).

The combined magnetic field

Definition

Magnetic field line

A line drawn to represent a magnetic field, whose arrow shows the direction of the force on a north pole placed at any point along it.

Between the flat, opposite poles of two magnets, the magnetic field lines run straight across from the north pole to the south pole, evenly spaced and parallel.
Around the wire, the field lines form concentric circles, and the direction of these circles depends on the direction of the current.
On one side of the wire, the circular field lines point the same way as the magnet's field lines, so the two fields add together and the combined field is stronger.
On the opposite side of the wire, the circular field lines point the opposite way to the magnet's field lines, so the two fields partly cancel and the combined field is weaker.
The combined field lines are therefore crowded together on the strong side of the wire and spread out on the weak side.
The conductor is pushed out of the region of stronger field towards the region of weaker field.
The crowded field lines behave like the stretched elastic of a catapult that flings the wire towards the weaker side, which is why this pattern is often called a catapult field.
The magnet is pushed the opposite way by the same interaction, which is the equal and opposite force on the magnet.

When the force acts

There must be a current in the conductor, because without a current the wire has no magnetic field to interact with the field of the magnet.
The conductor must lie across the magnetic field lines rather than along them.
The force is largest when the conductor is at right angles to the magnetic field lines.
The force becomes smaller as the conductor is turned towards the direction of the field lines, and it is zero when the conductor is parallel to them.
Reversing the direction of the current reverses the direction of the force on the conductor.
Reversing the direction of the magnetic field, by swapping over the poles of the magnet, also reverses the direction of the force.
Reversing both the current and the field at the same time leaves the force acting in its original direction.
Switching the current off removes the force straight away, so the conductor is no longer pushed.

Practical

Observing the force on a wire

Aim: to show that a magnet and a current-carrying conductor exert forces on one another, and to find how the direction of the force depends on the current and the magnetic field.
Apparatus: two parallel horizontal copper rails fixed about $cm5\ \text{cm}$ apart on a wooden base, a short bare copper rod that rests across the rails, two flat magnets on a U-shaped steel yoke, a low-voltage d.c. power supply, a switch, a variable resistor, an ammeter, connecting leads with crocodile clips, a top-pan balance and a clamp stand.
Method, the rolling rod:
- Rub the rails and the rod with fine emery paper until they shine, because a dull, oxidised surface stops the current passing between them.
- Clip a lead to the end of each rail and connect the rails in series with the switch, the variable resistor, the ammeter and the power supply, so the only path for the current from one rail to the other is through the loose copper rod.
- Slide the magnet yoke over the rails so the rod sits between the poles, with the north pole above and the south pole below, so the magnetic field passes vertically through the rod.
- Set the supply to about $V2\ \text{V}$ , close the switch for a moment, and adjust the variable resistor until the ammeter reads about $A2\ \text{A}$ .
- Close the switch for only a second or two, watch which way the rod rolls along the rails, and then open the switch.
- Swap the two leads at the power supply so the current flows through the rod the other way, close the switch again, and note the direction the rod rolls.
- Turn the yoke over so the poles are swapped, return the leads to their original positions, and repeat.
- Finally swap both the leads and the poles together and repeat once more.
Method, the force on the magnet:
- Stand the magnet yoke on the top-pan balance and press the tare button so the reading is zero.
- Clamp a stiff straight copper wire horizontally so it passes through the gap between the poles without touching the yoke or the balance.
- Connect the wire into the same circuit, close the switch and read the balance.
- Reverse the current and read the balance again.
Results:
- The rod starts to roll the moment the current flows and is no longer pushed once the switch opens.
- Reversing the current, or reversing the magnetic field, makes the rod roll the opposite way.
- Reversing both the current and the field makes the rod roll in its original direction.
- The balance shows a positive reading for one direction of current and a negative reading of the same size for the other, even though nothing touches the magnet, which shows that the wire exerts a force on the magnet.
Watch out: the rails and rod warm up quickly at large currents, so keep the switch closed only briefly, and make sure the rod lies straight across both rails so it does not jam.
Safety: use only a low-voltage supply, let the rod and rails cool before touching them, and handle the magnets carefully because they can snap onto the steel yoke and trap fingers.

Common Mistake

Do not say that the wire is attracted to one pole of the magnet, because the force acts across the field, at right angles to the line joining the poles.
Do not say that the current pushes on the magnet directly, because the current produces a magnetic field and it is the interaction of the two fields that produces the force.
Do not forget the force on the magnet, which is equal in size and opposite in direction to the force on the wire.

Exam technique

Explaining why a wire moves

A two-mark explanation needs a cause and an effect: the current in the wire produces a magnetic field, and this field interacts with the field of the magnet to produce a force.
Name both fields in your answer, because writing "the magnetic field" on its own does not tell the examiner which field you mean.
When a question asks about the forces on both objects, state that the force on the magnet is equal and opposite to the force on the wire.
When asked how to make the wire move the other way, give one change, such as reversing the current, because reversing the current and the field together leaves the force unchanged.

Self review

What is the motor effect?
Explain why a wire with no current in it feels no force between the poles of a magnet.
How does the force on the magnet compare with the force on the current-carrying wire?
At what angle to the field lines is the force on a wire largest, and when is it zero?
What happens to the force when both the current and the magnetic field are reversed?

4.2.2 Fleming's left-hand rule

Fleming's left-hand rule

Definition

Fleming's left-hand rule

A rule in which the thumb, first finger and second finger of the left hand, held at right angles to each other, show the directions of the force, the magnetic field and the conventional current.

Definition

Mutually perpendicular

Describes three directions that are each at right angles to the other two.

Definition

Conventional current

The direction of current shown on circuit diagrams, from the positive terminal of the supply, around the circuit, to the negative terminal.

The motor effect produces a force on a current-carrying conductor in a magnetic field, and Fleming's left-hand rule tells you which way that force acts.
The rule is used when the current and the magnetic field are at right angles to each other.
In that arrangement the force, the magnetic field and the current are mutually perpendicular, so each one is at right angles to the other two.
Hold your left hand with the thumb pointing up, the first finger pointing straight ahead and the second finger bent sideways, so the three point in directions at right angles.
The first finger points in the direction of the magnetic field, which runs from the north pole to the south pole.
The second finger points in the direction of the conventional current, which runs from the positive terminal of the supply to the negative terminal.
The thumb then points in the direction of the force on the conductor, which is the direction in which the conductor moves.
A memory aid links each digit to a letter: First finger for Field, seCond finger for Current, and thuMb for Motion.

Left hand held with the thumb, first finger and second finger at right angles: the first finger gives the magnetic field, the second finger gives the conventional current and the thumb gives the force on the conductor.

Using the rule step by step

Find the direction of the magnetic field on the diagram, remembering that field lines run from north to south.
Point your first finger in that direction and keep it fixed.
Turn your wrist about the first finger until your second finger points along the conventional current.
Read off the direction of your thumb, which gives the direction of the force on the conductor.
Diagrams often show a direction going into or out of the page with a cross or a dot.
A cross ( $⊗\otimes$ ) shows a direction into the page, like the feathered tail of an arrow flying away from you.
A dot ( $⊙\odot$ ) shows a direction out of the page, like the point of an arrow flying towards you.
Reversing the current reverses the force, because your second finger now points the other way and your thumb flips over.
Reversing the magnetic field also reverses the force, because your first finger now points the other way.
Reversing both the current and the field leaves the force acting in the same direction as before.
The rule gives only the direction of the force and tells you nothing about its size.

Conventional current and electron flow

Definition

Electron flow

The movement of electrons through a circuit from the negative terminal to the positive terminal, in the opposite direction to the conventional current.

A cell connected to a lamp, with one arrow showing conventional current leaving the positive terminal and a second arrow showing electrons travelling the opposite way, from the negative terminal.

In a metal wire the current is carried by electrons, which are negatively charged.
The electrons drift from the negative terminal of the supply, around the circuit, towards the positive terminal.
Conventional current was defined before electrons were discovered, as flowing from positive to negative, so it points the opposite way to the electron flow.
Fleming's left-hand rule always uses the direction of conventional current.
If a diagram shows the direction in which electrons move, point your second finger the opposite way to the electron flow.
The same rule finds the force on a beam of charged particles moving through a magnetic field, because a stream of moving charge is a current.
For positively charged particles the current is in the same direction as their motion, while for electrons the current is in the opposite direction to their motion.
A positive particle and an electron moving the same way through the same field are therefore pushed in opposite directions.

Example

Wire carrying current up the page

A straight wire runs up the page and carries a conventional current upwards.
The magnetic field between two poles points from left to right across the page.
Point the first finger of your left hand to the right, along the field.
Turn your hand so the second finger points up the page, along the current.
Your thumb now points into the page, so the force on the wire acts into the page.

Example

Electron flow shown on the diagram

A wire lies across a magnetic field that points into the page.
The diagram shows electrons moving along the wire from left to right.
The conventional current therefore flows from right to left.
Point your first finger into the page and your second finger to the left.
Your thumb points down the page, so the force on the wire acts downwards.

Common Mistake

Use your left hand, because the right hand gives the opposite answer every time.
Do not point your second finger along the electron flow, because the rule is built on conventional current.
Do not try to use the rule when the wire is parallel to the field, because there is then no force to find.

Exam technique

Stating a direction

Turn your hand to match the diagram on the paper rather than relying on the memory of a similar diagram, because examiners often rotate the set-up.
Give the direction in the terms the question uses, such as up the page, to the left or into the page.
When asked to show the force on a diagram, draw a single arrow that starts on the conductor and points in the direction of the force.
If the question reverses one quantity, apply the rule again from the start rather than guessing which way the force now acts.

Self review

Which finger of the left hand represents the magnetic field, and which represents the current?
What does a cross drawn in a circle mean on a diagram?
A wire carries conventional current to the right through a field pointing up the page; in which direction is the force?
How do you use the rule when a diagram shows the direction of electron flow?
What happens to the force on a wire if the current and the field are both reversed?

4.2.3 The force on a current-carrying conductor

The size of the force

Definition

Magnetic flux density

A measure of the strength of a magnetic field, measured in tesla (T).

Definition

Tesla

The unit of magnetic flux density, where a field of one tesla exerts a force of one newton on each metre of wire carrying a current of one ampere at right angles to the field.

Definition

Directly proportional

A relationship between two quantities in which multiplying one by a factor multiplies the other by the same factor.

The size of the force on a current-carrying conductor in a magnetic field depends on three quantities.
The magnetic flux density, $B$ , measures how strong the magnetic field is, and a stronger field gives a bigger force.
The current, $I$ , in the conductor sets the strength of the wire's own magnetic field, so a bigger current gives a bigger force.
The length, $L$ , of conductor inside the field matters as well, because every part of the wire in the field is pushed, so a longer length gives a bigger force.
When the conductor is at right angles to the field, the force is given by $F = B I L$ , or $density×current×length\text{force}=\text{magnetic flux density}\times\text{current}\times\text{length}$ .
In this equation $F$ is the force in newtons ( $N\text{N}$ ), $B$ is the magnetic flux density in tesla ( $T\text{T}$ ), $I$ is the current in amperes ( $A\text{A}$ ) and $L$ is the length in metres ( $m\text{m}$ ).
The force is directly proportional to each of the three quantities, so doubling any one of them while the other two stay the same doubles the force.
The equation applies only when the conductor is perpendicular to the magnetic field, which is the arrangement that gives the largest force.
A conductor lying parallel to the field feels no force at all, whatever the values of $B$ , $I$ and $L$ .
The length $L$ is only the part of the conductor that lies inside the magnetic field, not the whole length of wire in the circuit.
The equation gives the size of the force, while Fleming's left-hand rule gives its direction.

Rearranging and converting units

The equation can be rearranged to make any one of the four quantities the subject.
To find the magnetic flux density, divide the force by the current multiplied by the length: $B=FILB=\dfrac{F}{IL}$ .
To find the current, use $I=FBLI=\dfrac{F}{BL}$ .
To find the length of conductor in the field, use $L=FBIL=\dfrac{F}{BI}$ .
The rearranged form $B=FILB=\dfrac{F}{IL}$ shows what one tesla means: a field of $T1\ \text{T}$ exerts a force of $N1\ \text{N}$ on each $m1\ \text{m}$ of wire carrying $A1\ \text{A}$ at right angles to it.
One tesla is a very strong field, and the magnets used in school laboratories produce fields well below $T1\ \text{T}$ , so the forces measured in class are small fractions of a newton.
Lengths given in centimetres or millimetres must be converted to metres before substituting, so $m5.0\ \text{cm}=0.050\ \text{m}$ and $m25\ \text{mm}=0.025\ \text{m}$ .
Currents given in milliamperes must be converted to amperes, so $A250\ \text{mA}=0.25\ \text{A}$ .
Flux densities given in millitesla must be converted to tesla, so $T40\ \text{mT}=0.040\ \text{T}$ .
Very small forces are often written in standard form, so $N0.0045\ \text{N}$ is written as $N4.5\times10^{-3}\ \text{N}$ .

Practical

Measuring the force on a wire

Aim: to investigate how the force on a current-carrying wire in a magnetic field depends on the current in the wire.
Apparatus: a top-pan balance reading to $g0.01\ \text{g}$ , two flat magnets on a U-shaped steel yoke, a stiff straight copper rod, two clamp stands with bosses and clamps, a low-voltage d.c. power supply, a variable resistor, an ammeter, a switch, connecting leads and a ruler.
Method:
- Place the magnets on the yoke with opposite poles facing each other, stand the yoke on the balance and press the tare button so the reading is zero.
- Clamp the copper rod horizontally so it passes through the gap between the poles at right angles to the field, without touching the magnets, the yoke or the balance pan.
- Measure the width of the magnet faces with the ruler, because this is the length of rod inside the field.
- Connect the rod in series with the ammeter, the variable resistor, the switch and the power supply.
- Close the switch, adjust the variable resistor until the ammeter reads $A0.50\ \text{A}$ , and record the balance reading once it has settled.
- Open the switch between readings so the rod does not heat up, and check that the balance returns to zero.
- Increase the current in steps of $A0.50\ \text{A}$ up to $A3.0\ \text{A}$ , recording the balance reading at each step.
- Repeat the whole set of readings twice more and calculate a mean balance reading for each current.
Processing:
- The balance reading is a change in mass caused by the force between the rod and the magnet, so convert it to kilograms and multiply by the gravitational field strength, $N/kgg=10\ \text{N/kg}$ , to find the force.
- Plot force on the $y$ -axis against current on the $x$ -axis and draw a line of best fit.
Variables: the independent variable is the current, the dependent variable is the force, and the control variables are the length of rod in the field, the magnets used and the right angle between the rod and the field.
Results:
- The points lie close to a straight line through the origin, which shows that the force is directly proportional to the current.
- The gradient of the line equals $B L$ , so dividing the gradient by the measured length gives the magnetic flux density between the poles.
- Reversing the current turns a positive balance reading into a negative reading of the same size, because the force on the magnet reverses with the current.
Watch out: a rod that touches the yoke or the balance pushes on it directly and spoils the reading, and a draught or a knock on the bench makes a sensitive balance drift, so tare it again before each set of readings.
Safety: the rod and leads become warm at the largest currents, so switch off between readings and let the rod cool before touching it.

Example

Force on a wire

A wire carries a current of $A2.5\ \text{A}$ at right angles to a magnetic field of flux density $T0.15\ \text{T}$ .
The length of wire inside the field is $cm8.0\ \text{cm}$ , which is $m0.080\ \text{m}$ .
Substitute the values into the equation:

F=BIL=0.15\times2.5\times0.080

F=0.030\ \text{N}

Example

Finding the flux density

A $cm5.0\ \text{cm}$ length of wire carrying $A3.2\ \text{A}$ at right angles to a magnetic field feels a force of $N0.024\ \text{N}$ .
Convert the length into metres: $m5.0\ \text{cm}=0.050\ \text{m}$ .
Rearrange the equation to make $B$ the subject and substitute:

B=\frac{F}{IL}=\frac{0.024}{3.2\times0.050}=\frac{0.024}{0.16}

B=0.15\ \text{T}

Example

Finding the current

A $mm25\ \text{mm}$ length of wire lies at right angles to a $mT60\ \text{mT}$ field and feels a force of $N9.0\times10^{-3}\ \text{N}$ .
Convert the units: $T60\ \text{mT}=0.060\ \text{T}$ and $m25\ \text{mm}=0.025\ \text{m}$ .
Rearrange the equation to make $I$ the subject and substitute:

I=\frac{F}{BL}=\frac{9.0\times10^{-3}}{0.060\times0.025}

I=6.0\ \text{A}

Common Mistake

Do not use the total length of wire in the circuit, because $L$ is only the length that lies inside the magnetic field.
Do not apply $F = B I L$ to a wire that is parallel to the field, because the force on it is zero.

Exam technique

Calculating the force on a conductor

The equation $F = B I L$ is one you select and apply rather than recall, so the marks come from choosing it, rearranging it and handling the units correctly.
Write the equation, then the substituted values, then the answer with its unit, because each stage can earn a mark even if a later step goes wrong.
Convert centimetres, millimetres, milliamperes and millitesla before substituting, since a missed conversion changes the answer by a factor of $100$ or $1000$ .
When a question asks what happens to the force if one quantity changes, use proportionality, so tripling the current triples the force and halving the length halves it.

Self review

State the equation that links force, magnetic flux density, current and length.
What is the unit of magnetic flux density?
Calculate the force on $m0.20\ \text{m}$ of wire carrying $A4.0\ \text{A}$ at right angles to a $T0.050\ \text{T}$ field.
What happens to the force if the current is halved and the length in the field is doubled?
What is the force on a wire that lies parallel to the magnetic field lines?

4.2.4 The electric motor

Forces on a coil

Definition

Electric motor

A device that uses the forces on a current-carrying coil in a magnetic field to produce continuous rotation.

Definition

Coil

A length of insulated wire wound into many loops, called turns, so that the effects of the current in each turn add together.

Definition

Moment

The turning effect of a force about a pivot, equal to the force multiplied by the perpendicular distance from the pivot to the line of action of the force.

An electric motor uses the motor effect to turn a shaft, transferring energy from the electrical supply to the kinetic energy store of the rotating parts.
A simple motor has a rectangular coil of wire placed between the poles of a magnet, mounted on an axle so that it is free to spin.
When a current flows, it travels along one long side of the coil and back along the opposite long side, so the current in the two sides is in opposite directions.
Both long sides lie across the magnetic field, so the motor effect produces a force on each of them.
Because the currents are opposite, Fleming's left-hand rule gives forces in opposite directions, with one side pushed up and the other side pushed down.
The two forces act on opposite sides of the axle, so each produces a moment in the same rotational direction, and together they make the coil turn.
The short ends of the coil either lie along the field or feel forces that act along the line of the axle, so they add nothing to the turning effect.
A coil with many turns has many wires along each side, so the forces on them add up and the turning effect is larger.

Keeping the coil turning

Definition

Split-ring commutator

A ring split into two halves that turns with a coil and reverses the connections between the coil and the external circuit every half turn.

The turning effect is greatest when the plane of the coil is parallel to the field lines, because the two forces then act furthest from the axle.
As the coil turns, the forces keep acting straight up and down, so the perpendicular distance from the axle to each force shrinks and the moment gets smaller.
When the plane of the coil is perpendicular to the field lines, both forces act straight through the axle, so the moment is zero.
The coil keeps moving past this position because of its momentum.
If the current kept flowing the same way around the coil, the forces would now turn it backwards, so the coil would rock to and fro and settle in the perpendicular position.
To keep the coil turning in one direction, the current in the coil must reverse every half turn, at the moment the coil passes the perpendicular position.
In a simple d.c. motor a split-ring commutator does this by swapping the connections between the coil and the supply every half turn.
After each swap, the side of the coil next to the north pole always carries current in the same direction, so it is always pushed the same way and the rotation continues.

Speed and direction of rotation

The coil turns faster when the forces on its sides are larger.
Increasing the current increases the force on each side of the coil.
Using a stronger magnet increases the magnetic flux density and so increases the force on each side.
Adding more turns to the coil increases the total force along each side.
Reversing the current through the motor reverses the forces and so reverses the direction of rotation.
Reversing the magnetic field, by swapping the poles, also reverses the direction of rotation.
Reversing both the current and the field together leaves the motor turning the same way as before.
Electric motors drive anything that needs electricity to produce rotation, including the wheels of electric cars, the drum of a washing machine, cordless drills and the fans in hairdryers.

Example

Direction of rotation

Looking along the axle, the coil lies flat between a north pole on the left and a south pole on the right, so the field points from left to right.
The side of the coil on the left carries conventional current out of the page, and the side on the right carries it into the page.
For the left side, point your first finger to the right and your second finger out of the page, and your thumb points up.
For the right side, your second finger points into the page, so your thumb points down.
The left side is pushed up and the right side is pushed down, so the coil turns clockwise as seen from this end.

Practical

Building a simple motor

Aim: to build a working motor from a coil, a magnet and a cell, and to see how the direction of the current and of the field affects the rotation.
Apparatus: about $m1\ \text{m}$ of enamelled copper wire, a $V1.5\ \text{V}$ D-size cell, two large metal paper clips, a strong flat magnet, sticky tape, fine sandpaper and a marker pen or similar cylinder to wind the coil on.
Method:
- Wind the wire tightly around the marker pen about $10$ to $15$ times to make a neat circular coil, leaving about $cm3\ \text{cm}$ of straight wire sticking out on opposite sides to act as the axle.
- Slide the coil off the pen and wrap each loose end twice around the turns to hold them together, checking that both straight ends lie on one line through the centre of the coil so it balances.
- On one end, sand off all of the enamel insulation all the way round.
- On the other end, hold the coil upright and sand the enamel off the top half of the wire only, leaving the bottom half insulated.
- Bend each paper clip into a support with a small loop at the top, and tape one to each terminal of the cell so the loops are level with each other.
- Stick the magnet to the side of the cell, directly below the space between the two loops.
- Rest the axle ends in the loops so the coil hangs just above the magnet, then give it a gentle flick to start it spinning.
- Turn the magnet over, and then turn the cell round, and note the direction of spin each time.
Results:
- The coil spins steadily once it is balanced and the bare ends make good contact with the loops.
- The half-stripped end acts as a simple switch, so current flows only during the half of each turn when the bare wire touches the loop.
- The coil is pushed only during the part of each turn when the force turns it the right way, and it coasts round on its momentum for the rest of the turn.
- Turning the magnet over or reversing the cell makes the coil spin the other way.
Watch out: a coil that only rocks back and forth usually has the wrong half of the enamel removed, and a lopsided coil wobbles and falls out of the loops.
Safety: the coil has a very low resistance, so the current is large and the wire and cell become hot within a minute, so lift the coil out when you are not watching it.

Common Mistake

Do not say that the coil turns because it is attracted to the magnet, since the turning comes from the motor-effect forces on its two sides.
Do not say that the forces on the two sides act in the same direction, because they act in opposite directions and that is what produces the turning effect.

Exam technique

Explaining how a motor turns

Build your explanation as a chain: current in the coil, a magnetic field around each side, interaction with the field of the magnet, forces on the two sides, and a turning effect on the coil.
Use the phrase opposite directions for both the currents and the forces, because this is the step most often missed.
When explaining why the motor keeps turning the same way, state that the current in the coil reverses every half turn.
For ways to make the motor faster, give changes that increase the force, such as a larger current, a stronger magnet or more turns on the coil.

Self review

Why do the forces on the two long sides of a motor coil act in opposite directions?
In which position of the coil is the turning effect zero?
Why must the current in the coil reverse every half turn?
Give three ways of making a motor spin faster.
State two ways of reversing the direction in which a motor turns.

4.2.5 Electromagnetic induction

Inducing a potential difference

Definition

Electromagnetic induction

The production of a potential difference across a conductor when the magnetic field around it changes.

Definition

Induced potential difference

A potential difference produced across the ends of a conductor by a change in the magnetic field around it.

Definition

Induced current

A current driven around a complete circuit by an induced potential difference.

When the magnetic field around a conductor changes, a potential difference is produced across the ends of the conductor.
This effect is called electromagnetic induction, and the potential difference it produces is an induced potential difference.
One way to change the field is to move a magnet towards or away from a coil of wire, so the field passing through the coil grows or shrinks.
Another way is to move a wire across the field lines of a magnet, which is often described as the wire cutting the field lines.
Only relative motion matters, so moving a magnet past a still coil has the same effect as moving the coil past a still magnet.
A wire moving along the field lines does not cut them, so no potential difference is induced across it.
If there is no change in the field, nothing is induced, so a magnet resting inside a coil induces no potential difference however strong the magnet is.
If the conductor is part of a complete circuit, the induced potential difference drives an induced current around the circuit.
If the circuit is broken, a potential difference still appears across the ends of the conductor, but no current flows.

Size and direction of the effect

The induced potential difference is larger when the magnetic field changes faster, so moving the magnet more quickly gives a larger reading.
A stronger magnet gives a larger induced potential difference for the same movement, because the field changes by a greater amount.
A coil with more turns gives a larger induced potential difference, because a potential difference is induced in every turn and these add together along the coil.
Reversing the direction of movement, such as pulling the magnet out instead of pushing it in, reverses the direction of the induced potential difference.
Reversing the pole that faces the coil, such as pushing in a south pole instead of a north pole, also reverses the induced potential difference.
Reversing both the movement and the pole together gives an induced potential difference in the original direction.

Opposing the change

An induced current flowing in a coil produces its own magnetic field, just like any other current in a coil.
The induced current always flows in the direction that makes its magnetic field oppose the change that produced it.
When a north pole is pushed towards one end of a coil, the induced current makes that end of the coil a north pole, which repels the approaching magnet.
When the north pole is pulled away again, the induced current reverses and that end of the coil becomes a south pole, which attracts the magnet and resists it leaving.
In both cases the coil pushes back against the movement, so you have to do work to move the magnet.
The work you do against this opposing force is the energy that the induced current carries around the circuit.
If the induced field helped the movement instead, the magnet would speed up on its own and energy would appear from nothing, which cannot happen.
If the circuit is broken, no current flows and there is no opposing field, so the magnet moves freely even though a potential difference is still induced.

Practical

Inducing a current with a magnet

Aim: to show that a changing magnetic field induces a potential difference and a current, and to find what affects the size and direction of the effect.
Apparatus: a coil of insulated wire with about $600$ turns, a second coil with about $300$ turns, two strong bar magnets, a sensitive centre-zero ammeter (galvanometer), connecting leads and a cardboard tube that the magnet can slide through.
Method:
- Connect the ends of the $600$ -turn coil directly to the centre-zero meter, with nothing else in the circuit, and place the meter about a metre away so the magnet cannot pull on its needle.
- Hold the bar magnet still with its north pole just outside one end of the coil, and check that the needle reads zero.
- Push the north pole into the coil at a steady speed, and note the direction in which the needle moves.
- Hold the magnet still inside the coil and watch the needle.
- Pull the magnet back out and note the direction in which the needle moves.
- Repeat with the south pole facing the coil.
- Hold the magnet still and move the coil over it instead.
- Stand the coil upright with the tube passing through it, and drop the magnet through the tube from heights of $cm10\ \text{cm}$ , $cm20\ \text{cm}$ and $cm40\ \text{cm}$ , recording the largest reading each time, so that the speed is controlled by the drop height.
- Repeat the drops with two magnets held together, and then with the $300$ -turn coil in place of the $600$ -turn coil.
Results:
- The needle moves only while the magnet or the coil is moving, and it returns to zero whenever both are still.
- Pushing the magnet in and pulling it out move the needle in opposite directions.
- Turning the magnet round to use the south pole reverses each of these readings.
- Moving the coil instead of the magnet gives the same readings, which shows that it is the relative movement that matters.
- A larger drop height, two magnets and the coil with more turns each give a larger reading.
Watch out: a coil with only a few turns gives a reading too small to see, and a magnet held near the meter moves its needle directly, so keep the two well apart.
Safety: strong magnets can snap together and trap fingers, and should be kept away from phones and bank cards.

Example

Magnet falling towards a coil

A bar magnet falls with its south pole downwards towards the top of a vertical coil connected to a meter.
As the south pole approaches, the magnetic field through the coil increases, so a current is induced in the coil.
The induced field opposes the approach, so the top of the coil becomes a south pole.
The two south poles repel, so there is an upward force on the falling magnet and it speeds up less than it would in free fall.
If the ends of the coil are disconnected, no current flows, there is no opposing field, and the magnet falls freely.

Common Mistake

Do not say that a magnet held still inside a coil induces a current, because the magnetic field must be changing.
Do not say that the induced field helps the magnet along, because it always opposes the change that produced it.
Do not say that a broken circuit has no induced potential difference, because the potential difference is still induced and only the current is missing.

Exam technique

Describing induction

Use the word induced whenever you describe a potential difference or current produced by a changing magnetic field.
Link the cause and the effect in one sentence: a change in the magnetic field around the conductor induces a potential difference across its ends.
For ways to increase the reading, give specific changes, such as moving the magnet faster, using a stronger magnet or using a coil with more turns.
When asked why a magnet is harder to push into a coil in a complete circuit, explain that the induced current produces a magnetic field that opposes the movement.

Self review

What is electromagnetic induction?
Why does a magnet held still inside a coil induce no potential difference?
State three ways of increasing the induced potential difference.
What is needed for an induced potential difference to drive a current?
Explain why it is harder to push a magnet into a coil when the coil is part of a complete circuit.

4.2.6 Alternators and dynamos

Generating with a rotating coil

Definition

Generator

A device that induces a potential difference by rotating a coil and a magnet relative to each other.

A generator uses electromagnetic induction to produce a potential difference from rotation.
In a simple generator a coil of wire is turned between the poles of a magnet, so its sides move across the magnetic field lines.
The magnetic field through the coil keeps changing as it turns, so a potential difference is induced across the ends of the coil.
The induced potential difference is largest when the plane of the coil is parallel to the field lines, because the sides are then moving straight across the lines and cutting them at the fastest rate.
The induced potential difference is zero when the plane of the coil is perpendicular to the field lines, because for that instant the sides are moving along the lines and not cutting them.
After every half turn, each side of the coil has moved to the other pole and is crossing the field in the opposite direction.
The potential difference induced in the coil therefore reverses every half turn, so the rotating coil always produces an alternating potential difference.
What reaches the external circuit depends on how the ends of the coil are connected to it, and this connection is the difference between an alternator and a dynamo.

The alternator

Definition

Alternator

A generator that uses slip rings to deliver an alternating current to the external circuit.

Definition

Alternating current

A current that repeatedly reverses its direction of flow around a circuit.

Definition

Slip rings

Two complete metal rings that turn with a coil and keep each end of the coil connected to the same side of the external circuit.

Definition

Brushes

Fixed carbon contacts that press against rotating rings to connect a turning coil to the external circuit.

An alternator connects each end of the coil to its own slip ring, a complete metal ring that turns with the axle.
Fixed carbon brushes press against the rings, so each end of the coil stays connected to the same side of the external circuit throughout the rotation.
The reversal of potential difference in the coil every half turn is passed straight on, so the output is an alternating potential difference that drives an alternating current.
A graph of the output against time is a smooth wave that rises to a positive peak, falls through zero to a negative peak of the same size, and rises back to zero.
One complete cycle of this wave is produced by one complete rotation of the coil.
Many alternators keep the coil still and spin a magnet or electromagnet inside it instead, which produces the same changing field without the output passing through moving contacts.
Power stations use very large alternators of this kind to generate the mains supply, which in the UK alternates at $Hz50\ \text{Hz}$ .

The dynamo

Definition

Dynamo

A generator that uses a split-ring commutator to deliver a direct current to the external circuit.

Definition

Direct current

A current in which charge flows around a circuit in one direction only.

Definition

Split-ring commutator

A ring split into two halves that turns with a coil and reverses the connections between the coil and the external circuit every half turn.

A dynamo connects the ends of the coil to a split-ring commutator, a single ring cut into two halves that turns with the axle.
The brushes slip from one half of the ring to the other at the moment the coil is perpendicular to the field, when the induced potential difference is zero.
This is exactly the moment the potential difference in the coil reverses, so the connections to the external circuit are reversed at the same instant.
The two reversals cancel out, so the output always acts in the same direction and drives a direct current.
A graph of the output against time is a row of humps that all lie on the positive side of the time axis, with two humps for every rotation of the coil.
The current from a dynamo flows in one direction but is not steady, because its size still rises and falls as the coil turns.
A dynamo has the same construction as a simple d.c. motor, but it is turned by an outside force instead of being driven by a current.

Changing the output

Turning the coil faster makes the field through it change more quickly, so the peak potential difference increases.
Turning it faster also means each rotation takes less time, so the cycles are closer together on the graph and the frequency of an alternator's output increases.
A stronger magnet or a coil with more turns increases the peak potential difference without changing the frequency.
When the generator supplies a current, the current in the coil produces a magnetic field that opposes the rotation, so the coil becomes harder to turn.
This is why a hand-turned generator becomes noticeably stiffer when a lamp is connected and lit, because more work has to be done to keep it turning.

Practical

Displaying a generator's output

Aim: to display the output of a simple generator as a trace, and to see how it depends on the connections to the coil and on the speed of rotation.
Apparatus: a demonstration generator with a coil on an axle between two magnets that can be fitted with either slip rings or a split-ring commutator, a hand crank, an oscilloscope or a voltage sensor connected to a data logger, connecting leads and a small filament lamp in a holder.
Method:
- Fit the slip rings, connect the brushes to the oscilloscope input, and set the time base so that a few cycles fit across the screen.
- Turn the crank slowly and steadily, and sketch or save the trace.
- Turn the crank at about twice the speed, and record the new trace on the same scales.
- Replace the slip rings with the split-ring commutator, and repeat both speeds.
- Connect the lamp across the brushes and turn the crank fast enough to light it, noticing how hard the crank is to turn compared with when the lamp is disconnected.
Results:
- With slip rings, the trace is a wave that swings above and below zero, showing an alternating output.
- Faster turning gives taller peaks and squeezes more cycles across the screen.
- With the commutator, every hump of the trace lies on the same side of zero, showing a direct output whose size still rises and falls.
- The crank becomes harder to turn when the lamp is lit, because the current in the coil produces a field that opposes the rotation.
Watch out: turn the crank at as steady a rate as possible, and set the volts-per-division control so the trace fills most of the screen without running off the top.
Safety: keep fingers and loose clothing clear of the turning coil, and keep the magnets away from the oscilloscope screen.

Example

Reading an alternator trace

An alternator trace shows one complete cycle every $s0.02\ \text{s}$ .
One cycle is produced by one rotation, so the coil turns once every $s0.02\ \text{s}$ .
The number of rotations each second is:

\frac{1}{0.02\ \text{s}}=50\ \text{rotations per second}

The frequency of the alternating output is therefore $Hz50\ \text{Hz}$ .
Turning the coil twice as fast makes each cycle last $s0.01\ \text{s}$ , raises the frequency to $Hz100\ \text{Hz}$ and makes the peaks taller.

Common Mistake

Do not say that a dynamo gives a steady direct current, because its output flows one way but its size still rises and falls.
Do not say that the output is largest when the coil is perpendicular to the field, because the sides are then moving along the field lines and the output is zero.
Do not mix up the two connections: slip rings give a.c. from an alternator, and a split-ring commutator gives d.c. from a dynamo.

Exam technique

Comparing alternators and dynamos

When comparing the two, focus on the connection: slip rings keep the same connections so the output alternates, while a commutator reverses the connections every half turn so the output stays in one direction.
When sketching a trace for faster turning, show both changes, taller peaks and a shorter time for each cycle.
Explain the shape of a trace in terms of the coil's position, with the largest output when the sides cut the field lines fastest and zero output when they move along the lines.

Self review

What does an alternator produce, and which part makes this possible?
What does a dynamo produce, and which part makes this possible?
In which position of the coil is the induced potential difference zero?
State two effects of turning an alternator's coil faster.
Why is a generator harder to turn when it is lighting a lamp?

4.2.7 How transformers work

Parts of a transformer

Definition

Transformer

A device in which an alternating current in one coil induces an alternating potential difference in a second coil wound on the same iron core.

Definition

Primary coil

The input coil of a transformer, which is connected to the alternating supply.

Definition

Secondary coil

The output coil of a transformer, across which a potential difference is induced.

Definition

Iron core

The soft iron frame that passes through both coils of a transformer and carries the changing magnetic field from one coil to the other.

A transformer is made of two separate coils of insulated wire wound on the same iron core.
The primary coil is the input coil, and it is connected to an alternating supply.
The secondary coil is the output coil, and it is connected to the device or circuit that the transformer supplies.
The wire of each coil is insulated, so current cannot jump from one turn to the next, and there is no electrical connection between the two coils.
The core is made of soft iron, which is magnetised very easily and loses its magnetism as soon as the magnetising current falls, so its field follows every change in the current.
The core forms a closed loop that passes through both coils, so it guides almost all of the magnetic field made by the primary coil through the secondary coil.

How the coils are linked

Definition

Alternating current

A current that repeatedly reverses its direction of flow around a circuit.

Definition

Electromagnetic induction

The production of a potential difference across a conductor when the magnetic field around it changes.

An alternating current in the primary coil is constantly changing in size and reversing in direction.
This current turns the primary coil into an electromagnet whose magnetic field also changes in size and reverses in direction.
The iron core is magnetised by this field, so an alternating magnetic field is set up in the core.
The core carries this changing magnetic field through the secondary coil.
A magnetic field that changes around a conductor induces a potential difference across it, so an alternating potential difference is induced across the secondary coil by electromagnetic induction.
If the secondary coil is part of a complete circuit, this induced potential difference drives an alternating current in the secondary circuit.
The induced current alternates at the same frequency as the current in the primary coil, because the field in the core changes at that frequency.
Energy passes from the primary circuit to the secondary circuit through the changing magnetic field in the core, not through any current flowing between the coils.
In this way an alternating current in one circuit induces a current in a completely separate circuit.

Why direct current does not work

Definition

Direct current

A current in which charge flows around a circuit in one direction only.

A direct current in the primary coil produces a magnetic field that is steady once the current has settled.
A steady field in the core does not change, so no potential difference is induced across the secondary coil.
A transformer connected to a d.c. supply therefore gives no output while the current is steady.
A brief potential difference is induced in the secondary only at the instant the d.c. supply is switched on or off, because the field in the core changes at those moments.
This is why transformers are always used with alternating supplies, such as the UK mains supply, which alternates at $Hz50\ \text{Hz}$ .

Practical

Linking two circuits

Aim: to show that an alternating current in one coil induces a current in a separate coil, and that the effect needs a changing magnetic field.
Apparatus: two coils with the same number of turns, a U-shaped soft-iron core with a straight iron bar (the yoke) that closes it, a clamp, a low-voltage a.c. supply, a low-voltage d.c. supply, a switch, a small filament lamp in a holder, a sensitive centre-zero meter and connecting leads.
Method:
- Slide one coil onto each limb of the U-shaped core, and clamp the yoke across the top so the iron forms a closed loop through both coils.
- Connect the first coil to the a.c. supply through the switch, and connect the second coil to the lamp only, so no wire links the two circuits.
- Set the supply to a low potential difference, close the switch and look at the lamp.
- Lift the yoke off the core and look at the lamp again, then slide out the core altogether and switch off quickly.
- Rebuild the closed core, swap the a.c. supply for the d.c. supply, and replace the lamp with the centre-zero meter.
- Watch the meter as the switch is closed, while it stays closed, and as it is opened again.
Results:
- The lamp lights even though no wire connects the two circuits.
- Removing the yoke makes the lamp dimmer, and removing the core makes it go out or almost out, because much less of the changing field reaches the second coil.
- With the d.c. supply, the needle flicks one way at the moment the switch is closed, returns to zero while the current is steady, and flicks the other way when the switch is opened.
Watch out: without the core, the current in the first coil rises sharply and the coil heats up, so take that observation quickly and switch off.
Safety: use only a low-voltage supply and never connect the coils to the mains, let the coils cool before handling them, and keep fingers clear when the yoke snaps onto the core.

Common Mistake

Do not say that current flows through the iron core from one coil to the other, because the core carries a changing magnetic field and the two coils are electrically separate.
Do not say that a transformer works on direct current, because a steady current gives a steady field and induces nothing in the secondary coil.

Exam technique

Explaining how a transformer works

Set your answer out as a chain in this order: alternating current in the primary coil, alternating magnetic field, field carried by the iron core, changing field through the secondary coil, alternating potential difference induced, and alternating current if the secondary circuit is complete.
Describe the field as changing or alternating every time you mention it, because a field that is not described as changing earns no credit.
If asked why the core is made of iron, say that iron is easily magnetised and carries the magnetic field from the primary coil to the secondary coil.

Self review

Name the three main parts of a transformer.
Why must the supply to the primary coil be alternating?
What links the primary coil to the secondary coil?
Why is the core of a transformer made of soft iron?
What is induced in the secondary coil, and what is needed for it to drive a current?

4.2.8 Transformer turns ratio

Turns and potential difference

Definition

Turns ratio

The number of turns on the primary coil of a transformer compared with the number of turns on its secondary coil.

Each coil of a transformer is made of a number of turns, which are complete loops of wire around the iron core.
The iron core carries the same changing magnetic field through every turn of both coils.
Every turn therefore has the same potential difference induced across it, whichever coil it belongs to.
The turns of a coil are joined one after another, like cells connected in series, so the potential differences across the individual turns add up.
The total potential difference across a coil is therefore proportional to the number of turns it has.
In the primary coil, the supply potential difference is shared equally between its turns, which sets the potential difference per turn for the whole transformer.
The ratio of the potential differences across the two coils is equal to the ratio of their numbers of turns, which is called the turns ratio.
A secondary coil with twice as many turns as the primary coil has twice the potential difference across it.
A secondary coil with half as many turns as the primary coil has half the potential difference across it.
This simple link holds for an ideal transformer, in which all of the magnetic field from the primary coil passes through the secondary coil and no energy is wasted.

Step-up and step-down transformers

Definition

Step-up transformer

A transformer with more turns on its secondary coil than on its primary coil, so its output potential difference is greater than its input potential difference.

Definition

Step-down transformer

A transformer with fewer turns on its secondary coil than on its primary coil, so its output potential difference is smaller than its input potential difference.

A step-up transformer has more turns on its secondary coil than on its primary coil.
The output potential difference of a step-up transformer is therefore greater than its input potential difference.
A step-down transformer has fewer turns on its secondary coil than on its primary coil.
The output potential difference of a step-down transformer is therefore smaller than its input potential difference.
A transformer with the same number of turns on both coils gives an output potential difference equal to its input.
The transformer inside a microwave oven is a step-up transformer, raising the $V230\ \text{V}$ mains to a few thousand volts.
The transformer inside a doorbell supply is a step-down transformer, lowering the $V230\ \text{V}$ mains to a few volts.

Practical

Investigating the turns on a transformer

Aim: to find how the output potential difference of a transformer depends on the number of turns on its secondary coil.
Apparatus: a U-shaped soft-iron core with a yoke and clamp, a reel of insulated wire, a low-voltage a.c. supply, two a.c. voltmeters (or multimeters set to a.c. volts), connecting leads and a switch.
Method:
- Wind $50$ turns of insulated wire neatly onto one limb of the core to make the primary coil, and connect it to the a.c. supply through the switch, with a voltmeter across it.
- Wind $10$ turns of wire onto the other limb, all in the same direction, to make the secondary coil, and connect the second voltmeter across its ends.
- Clamp the yoke firmly across the top of the core so the iron forms a closed loop.
- Close the switch, set the supply so the primary voltmeter reads $V2.0\ \text{V}$ , and record the secondary potential difference.
- Open the switch, add $10$ more turns to the secondary coil, and repeat, keeping the primary potential difference at $V2.0\ \text{V}$ .
- Continue in steps of $10$ turns up to $100$ turns.
- Plot the secondary potential difference against the number of secondary turns, and draw a line of best fit.
Variables: the independent variable is the number of turns on the secondary coil, the dependent variable is the secondary potential difference, and the control variables are the primary potential difference, the number of turns on the primary coil, the core and the supply frequency.
Results:
- The secondary potential difference rises in equal steps as turns are added, giving a straight line through the origin, so the output is directly proportional to the number of secondary turns.
- With fewer than $50$ secondary turns the output is smaller than the input, as in a step-down transformer, and with more than $50$ it is larger, as in a step-up transformer.
- The measured outputs are slightly smaller than the turns ratio predicts, because some of the magnetic field leaks out of the core and some energy is transferred to the thermal store of the core and coils.
Watch out: turns wound in the wrong direction cancel part of the induced potential difference, and a loose yoke lets field leak out, so keep the winding direction the same and clamp the yoke tightly.
Safety: use only a low-voltage a.c. supply, open the switch before changing the number of turns, and let the core cool if it becomes warm.

Common Mistake

Do not decide whether a transformer steps up or down from the total number of turns, because what matters is whether the secondary coil has more or fewer turns than the primary coil.
Do not reverse the names: a step-up transformer increases the potential difference and has more turns on its secondary coil.

Exam technique

Explaining the turns ratio

When explaining why the output depends on the turns, include the two key steps: the same changing magnetic field passes through every turn, and the potential differences across the turns add up along each coil.
To identify a step-up or step-down transformer, compare the secondary turns with the primary turns, and state the comparison in your answer.
If a question gives the potential differences instead of the turns, use the same link in reverse, so a larger output potential difference means more turns on the secondary coil.

Self review

What does the turns ratio of a transformer compare?
Explain why a coil with more turns has a larger potential difference induced across it.
How can you tell from the numbers of turns whether a transformer is step-up or step-down?
A secondary coil has three times as many turns as the primary coil; how does the output potential difference compare with the input?
Why is the measured output of a real transformer slightly less than the turns ratio predicts?

4.2.9 Potential differences and turns in transformers

The transformer equation

Definition

Potential difference

The energy transferred per unit charge as the charge passes between two points in a circuit.

Definition

Turns ratio

The number of turns on the primary coil of a transformer compared with the number of turns on its secondary coil.

The potential differences across the two coils of a transformer are linked to their numbers of turns by $VpVs=NpNs\dfrac{V_p}{V_s}=\dfrac{N_p}{N_s}$ .
In words, $coil\dfrac{\text{potential difference across primary coil}}{\text{potential difference across secondary coil}}=\dfrac{\text{number of turns in primary coil}}{\text{number of turns in secondary coil}}$ .
$V_p$ is the potential difference across the primary coil and $V_s$ is the potential difference across the secondary coil, both measured in volts ( $V\text{V}$ ).
$N_p$ is the number of turns on the primary coil and $N_s$ is the number of turns on the secondary coil.
Numbers of turns are pure numbers with no unit, so the turns ratio $NpNs\dfrac{N_p}{N_s}$ has no unit either.
Both potential differences must be in the same unit, so convert kilovolts to volts, or volts to kilovolts, before substituting.
Turns ratios are often written in the form primary : secondary, so a ratio of $1 : 20$ means the secondary coil has $20$ times as many turns as the primary coil.
The equation assumes an ideal transformer, which is the assumption made in every calculation of this kind.

Rearranging the equation

The equation can be written the other way up, as $VsVp=NsNp\dfrac{V_s}{V_p}=\dfrac{N_s}{N_p}$ , which is often easier when the secondary is the unknown.
Whichever way up you write it, both primary quantities must be on the top or both must be on the bottom.
To find the secondary potential difference, use $Vs=Vp×NsNpV_s=V_p\times\dfrac{N_s}{N_p}$ .
To find the primary potential difference, use $Vp=Vs×NpNsV_p=V_s\times\dfrac{N_p}{N_s}$ .
To find the number of turns on the secondary coil, use $Ns=Np×VsVpN_s=N_p\times\dfrac{V_s}{V_p}$ .
To find the number of turns on the primary coil, use $Np=Ns×VpVsN_p=N_s\times\dfrac{V_p}{V_s}$ .
A quick check catches most mistakes: if the secondary coil has more turns than the primary coil, the secondary potential difference must be the larger of the two.

Example

Finding the output potential difference

A transformer has $500$ turns on its primary coil and $25$ turns on its secondary coil.
The primary coil is connected to the $V230\ \text{V}$ mains.
Write the equation with the secondary quantities on top:

\frac{V_s}{V_p}=\frac{N_s}{N_p}

Rearrange and substitute:

V_s=230\times\frac{25}{500}

V_s=11.5\ \text{V}

The secondary coil has fewer turns, so a smaller output is expected, which matches the answer.

Example

Finding the number of turns

A step-up transformer changes $V12\ \text{V}$ into $kV3.0\ \text{kV}$ , and its primary coil has $40$ turns.
Convert the output into volts: $V3.0\ \text{kV}=3000\ \text{V}$ .
Rearrange to make $N_s$ the subject and substitute:

N_s=N_p\times\frac{V_s}{V_p}=40\times\frac{3000}{12}

N_s=10\,000\ \text{turns}

Example

Using a turns ratio

A transformer has a turns ratio of $20 : 1$ , primary to secondary, and gives an output of $V6.0\ \text{V}$ .
The primary coil has $20$ times as many turns as the secondary coil, so its potential difference is $20$ times larger:

\frac{V_p}{V_s}=\frac{N_p}{N_s}=\frac{20}{1}

V_p=20\times6.0=120\ \text{V}

Common Mistake

Do not write the equation with $V_p$ on top on one side and $N_s$ on top on the other, because the primary quantities must be on the same level of both fractions.
Do not mix kilovolts and volts in the same calculation, because the answer will be wrong by a factor of $1000$ .

Exam technique

Transformer calculations

This equation is one you select and apply rather than recall, so the marks come from rearranging it correctly and putting each value in the right place.
Label your values $V_p$ , $V_s$ , $N_p$ and $N_s$ before substituting, because swapping the primary and secondary values is the most common way to lose the marks.
Give a number of turns as a whole number with no unit, and give a potential difference with the unit $V\text{V}$ .
Check the size of your answer against the type of transformer before moving on.

Self review

Write down the equation that links the potential differences and the numbers of turns in a transformer.
A transformer has $100$ turns on its primary coil and $2000$ turns on its secondary coil, and the input is $V12\ \text{V}$ ; calculate the output potential difference.
Rearrange the equation to make $N_s$ the subject.
Is a transformer with a turns ratio of $1 : 5$ , primary to secondary, a step-up or a step-down transformer?
Why must both potential differences be in the same unit before you substitute them?

4.2.10 Microphones, loudspeakers and headphones

Sound as pressure variations

Definition

Sound wave

A longitudinal wave that travels through a medium as a series of compressions and rarefactions, which are regions of higher and lower pressure.

A sound wave travels through air as a series of compressions, where the air pressure is slightly higher than normal, and rarefactions, where it is slightly lower.
At any one point, the air pressure rises and falls over and over again as the wave passes, and these changes are the pressure variations of the sound.
The frequency of the pressure variations sets the pitch of the sound, and the size of the variations sets its loudness.
A microphone turns these pressure variations into matching variations in an electric current, and a loudspeaker turns variations in current back into pressure variations.

How a microphone works

Definition

Microphone

A device that converts the pressure variations of a sound wave into matching variations in an electric current.

Definition

Diaphragm

A thin, flexible sheet in a microphone or headphone that vibrates in step with a sound wave.

A dynamic microphone contains a thin, flexible diaphragm attached to a small, light coil of wire.
The coil sits in the narrow gap around a permanent magnet, so it is surrounded by a strong magnetic field.
When a sound wave reaches the microphone, the pressure variations push and pull on the diaphragm, so the diaphragm vibrates back and forth.
The coil is fixed to the diaphragm, so it vibrates with it and moves back and forth through the magnetic field.
As the coil moves, its wire cuts across the magnetic field lines, so a potential difference is induced across the ends of the coil by electromagnetic induction.
Each time the coil changes direction, the induced potential difference reverses, so the output is an alternating potential difference.
When the microphone is part of a complete circuit, this potential difference drives an alternating current whose variations match the pressure variations of the sound.
A higher-pitched sound makes the coil vibrate more times each second, so the current alternates at a higher frequency.
A louder sound makes the coil vibrate with a larger amplitude and move faster through the field, so the induced current is larger.
The current from a microphone is very small, so it is passed to an amplifier, which makes it larger without changing its pattern.

Loudspeakers and headphones

Definition

Loudspeaker

A device that uses the motor effect to convert a varying electric current into vibrations of a cone that produce sound.

Definition

Headphones

A pair of small loudspeakers worn on or in the ears, each using the motor effect to turn a varying current into sound.

A loudspeaker does the reverse job of a microphone, turning variations in current into pressure variations in the air.
It has a coil attached to a light paper or plastic cone, and the coil sits in the field of a permanent magnet.
An amplifier sends an alternating current through the coil, with the same pattern of variations as the original sound.
The current in the coil lies across the magnetic field, so the motor effect produces a force on the coil.
Each time the current reverses, the force on the coil reverses, so the coil and the cone are pushed out and pulled in again and again.
As the cone moves outwards it squeezes the air in front of it into a compression, and as it moves back it leaves a rarefaction.
The vibrating cone therefore sends out a sound wave whose pressure variations match the variations in the current.
The frequency of the current sets the frequency of the vibrations, and so sets the pitch of the sound.
A larger current produces a larger force, a larger amplitude of vibration and a louder sound.
Headphones contain a tiny loudspeaker for each ear that works in exactly the same way, with a small coil and magnet driving a thin diaphragm close to the ear.
A microphone relies on electromagnetic induction, while a loudspeaker and headphones rely on the motor effect, even though all three use a coil and a magnet.

Practical

Using a loudspeaker both ways

Aim: to show that an alternating current in a loudspeaker produces sound by the motor effect, and that sound reaching a loudspeaker induces an alternating potential difference.
Apparatus: a small loudspeaker, a signal generator, an oscilloscope (or a voltage sensor and data logger), connecting leads, a few grains of dry rice and a tuning fork.
Method, loudspeaker as a speaker:
- Lay the loudspeaker on its back with the cone facing upwards, and sprinkle a few grains of rice onto the cone.
- Connect the loudspeaker to the signal generator, set the frequency to about $Hz5\ \text{Hz}$ and the output to a low amplitude, and watch the cone and the rice.
- Raise the frequency slowly to a few hundred hertz and listen to the note, then increase the amplitude and listen again.
Method, loudspeaker as a microphone:
- Disconnect the signal generator and connect the loudspeaker straight to the oscilloscope input instead.
- Strike the tuning fork and hold it close to the cone, then hum a low note and a high note, and a quiet note and a loud note, close to the cone, watching the trace each time.
Results:
- At low frequency the cone moves visibly out and in, and the rice bounces in time with the signal.
- A higher frequency gives a higher note, and a larger amplitude gives a louder sound and makes the rice jump higher.
- When the loudspeaker is used as a microphone, the oscilloscope shows an alternating trace whenever sound reaches the cone, with more waves across the screen for a higher note and taller waves for a louder one.
- The same coil and magnet use the motor effect in one direction and electromagnetic induction in the other.
Watch out: the potential difference produced by a loudspeaker used as a microphone is tiny, so set the oscilloscope to its most sensitive range.
Safety: keep the signal generator output low, because large amplitudes can damage the loudspeaker cone and very loud sounds can damage your hearing.

Common Mistake

Do not say that a microphone works by the motor effect, because the moving coil in a microphone has a potential difference induced in it, while the loudspeaker coil has a force acting on it.
Do not say that the magnet in a loudspeaker vibrates, because the magnet is fixed and it is the coil and cone that move.

Exam technique

Describing the conversions

For a microphone, link every stage in order: pressure variations, vibrating diaphragm, coil moving through the magnetic field, induced potential difference, varying current.
For a loudspeaker, run the chain the other way: varying current in the coil, force from the magnetic field, vibrating coil and cone, pressure variations in the air.
Name the physics at the key step, writing electromagnetic induction for a microphone and motor effect for a loudspeaker.
When asked how the current changes for a louder or higher sound, link loudness to the size of the current and pitch to its frequency.

Self review

What makes the diaphragm of a microphone vibrate?
Explain why a potential difference is induced across the coil of a dynamic microphone.
Which effect makes the cone of a loudspeaker move?
How does the current in a loudspeaker change to make a louder sound?
How are headphones similar to a loudspeaker?

4.2.1 The motor effect

Forces between magnets and currents

Definition

Motor effect

The force that acts on a current-carrying conductor when it is placed across the field lines of a magnetic field.

Definition

Current-carrying conductor

A wire or other piece of conducting material that has an electric current flowing through it.

Definition

Magnetic field

The region around a magnet in which another magnet or a magnetic material experiences a force.

Definition

Interaction pair

The two forces produced by one interaction, one acting on each of the two interacting objects.

A current-carrying conductor is surrounded by its own magnetic field, made of circular field lines centred on the wire.
This field exists only while charge is flowing, so a wire with no current in it has no magnetic field of its own.
When the conductor is placed between the poles of a permanent magnet, the field of the wire and the field of the magnet overlap in the same region of space.
The two magnetic fields interact, and this interaction produces a force on the conductor.
This effect is called the motor effect.
The force is a non-contact force, because the magnet and the wire act on each other through their fields without touching.
The magnet feels a force as well, because a force always arises from an interaction between two objects.
The force on the magnet is equal in size to the force on the conductor and acts in the opposite direction, so the two forces form an interaction pair, as Newton's third law describes.
In most arrangements the magnet is held still and only the conductor is free to move, so the conductor is the object you see moving.
If the conductor were clamped in place and the magnet were free to move, the magnet would move instead, in the opposite direction.
Like every other force, the motor-effect force is measured in newtons ( $N\text{N}$ ).

The combined magnetic field

Definition

Magnetic field line

A line drawn to represent a magnetic field, whose arrow shows the direction of the force on a north pole placed at any point along it.

Between the flat, opposite poles of two magnets, the magnetic field lines run straight across from the north pole to the south pole, evenly spaced and parallel.
Around the wire, the field lines form concentric circles, and the direction of these circles depends on the direction of the current.
On one side of the wire, the circular field lines point the same way as the magnet's field lines, so the two fields add together and the combined field is stronger.
On the opposite side of the wire, the circular field lines point the opposite way to the magnet's field lines, so the two fields partly cancel and the combined field is weaker.
The combined field lines are therefore crowded together on the strong side of the wire and spread out on the weak side.
The conductor is pushed out of the region of stronger field towards the region of weaker field.
The crowded field lines behave like the stretched elastic of a catapult that flings the wire towards the weaker side, which is why this pattern is often called a catapult field.
The magnet is pushed the opposite way by the same interaction, which is the equal and opposite force on the magnet.

When the force acts

There must be a current in the conductor, because without a current the wire has no magnetic field to interact with the field of the magnet.
The conductor must lie across the magnetic field lines rather than along them.
The force is largest when the conductor is at right angles to the magnetic field lines.
The force becomes smaller as the conductor is turned towards the direction of the field lines, and it is zero when the conductor is parallel to them.
Reversing the direction of the current reverses the direction of the force on the conductor.
Reversing the direction of the magnetic field, by swapping over the poles of the magnet, also reverses the direction of the force.
Reversing both the current and the field at the same time leaves the force acting in its original direction.
Switching the current off removes the force straight away, so the conductor is no longer pushed.

Practical

Observing the force on a wire

Aim: to show that a magnet and a current-carrying conductor exert forces on one another, and to find how the direction of the force depends on the current and the magnetic field.
Apparatus: two parallel horizontal copper rails fixed about $cm5\ \text{cm}$ apart on a wooden base, a short bare copper rod that rests across the rails, two flat magnets on a U-shaped steel yoke, a low-voltage d.c. power supply, a switch, a variable resistor, an ammeter, connecting leads with crocodile clips, a top-pan balance and a clamp stand.
Method, the rolling rod:
- Rub the rails and the rod with fine emery paper until they shine, because a dull, oxidised surface stops the current passing between them.
- Clip a lead to the end of each rail and connect the rails in series with the switch, the variable resistor, the ammeter and the power supply, so the only path for the current from one rail to the other is through the loose copper rod.
- Slide the magnet yoke over the rails so the rod sits between the poles, with the north pole above and the south pole below, so the magnetic field passes vertically through the rod.
- Set the supply to about $V2\ \text{V}$ , close the switch for a moment, and adjust the variable resistor until the ammeter reads about $A2\ \text{A}$ .
- Close the switch for only a second or two, watch which way the rod rolls along the rails, and then open the switch.
- Swap the two leads at the power supply so the current flows through the rod the other way, close the switch again, and note the direction the rod rolls.
- Turn the yoke over so the poles are swapped, return the leads to their original positions, and repeat.
- Finally swap both the leads and the poles together and repeat once more.
Method, the force on the magnet:
- Stand the magnet yoke on the top-pan balance and press the tare button so the reading is zero.
- Clamp a stiff straight copper wire horizontally so it passes through the gap between the poles without touching the yoke or the balance.
- Connect the wire into the same circuit, close the switch and read the balance.
- Reverse the current and read the balance again.
Results:
- The rod starts to roll the moment the current flows and is no longer pushed once the switch opens.
- Reversing the current, or reversing the magnetic field, makes the rod roll the opposite way.
- Reversing both the current and the field makes the rod roll in its original direction.
- The balance shows a positive reading for one direction of current and a negative reading of the same size for the other, even though nothing touches the magnet, which shows that the wire exerts a force on the magnet.
Watch out: the rails and rod warm up quickly at large currents, so keep the switch closed only briefly, and make sure the rod lies straight across both rails so it does not jam.
Safety: use only a low-voltage supply, let the rod and rails cool before touching them, and handle the magnets carefully because they can snap onto the steel yoke and trap fingers.

Common Mistake

Do not say that the wire is attracted to one pole of the magnet, because the force acts across the field, at right angles to the line joining the poles.
Do not say that the current pushes on the magnet directly, because the current produces a magnetic field and it is the interaction of the two fields that produces the force.
Do not forget the force on the magnet, which is equal in size and opposite in direction to the force on the wire.

Exam technique

Explaining why a wire moves

A two-mark explanation needs a cause and an effect: the current in the wire produces a magnetic field, and this field interacts with the field of the magnet to produce a force.
Name both fields in your answer, because writing "the magnetic field" on its own does not tell the examiner which field you mean.
When a question asks about the forces on both objects, state that the force on the magnet is equal and opposite to the force on the wire.
When asked how to make the wire move the other way, give one change, such as reversing the current, because reversing the current and the field together leaves the force unchanged.

Self review

What is the motor effect?
Explain why a wire with no current in it feels no force between the poles of a magnet.
How does the force on the magnet compare with the force on the current-carrying wire?
At what angle to the field lines is the force on a wire largest, and when is it zero?
What happens to the force when both the current and the magnetic field are reversed?

4.2.2 Fleming's left-hand rule

Fleming's left-hand rule

Definition

Fleming's left-hand rule

A rule in which the thumb, first finger and second finger of the left hand, held at right angles to each other, show the directions of the force, the magnetic field and the conventional current.

Definition

Mutually perpendicular

Describes three directions that are each at right angles to the other two.

Definition

Conventional current

The direction of current shown on circuit diagrams, from the positive terminal of the supply, around the circuit, to the negative terminal.

The motor effect produces a force on a current-carrying conductor in a magnetic field, and Fleming's left-hand rule tells you which way that force acts.
The rule is used when the current and the magnetic field are at right angles to each other.
In that arrangement the force, the magnetic field and the current are mutually perpendicular, so each one is at right angles to the other two.
Hold your left hand with the thumb pointing up, the first finger pointing straight ahead and the second finger bent sideways, so the three point in directions at right angles.
The first finger points in the direction of the magnetic field, which runs from the north pole to the south pole.
The second finger points in the direction of the conventional current, which runs from the positive terminal of the supply to the negative terminal.
The thumb then points in the direction of the force on the conductor, which is the direction in which the conductor moves.
A memory aid links each digit to a letter: First finger for Field, seCond finger for Current, and thuMb for Motion.

Using the rule step by step

Find the direction of the magnetic field on the diagram, remembering that field lines run from north to south.
Point your first finger in that direction and keep it fixed.
Turn your wrist about the first finger until your second finger points along the conventional current.
Read off the direction of your thumb, which gives the direction of the force on the conductor.
Diagrams often show a direction going into or out of the page with a cross or a dot.
A cross ( $⊗\otimes$ ) shows a direction into the page, like the feathered tail of an arrow flying away from you.
A dot ( $⊙\odot$ ) shows a direction out of the page, like the point of an arrow flying towards you.
Reversing the current reverses the force, because your second finger now points the other way and your thumb flips over.
Reversing the magnetic field also reverses the force, because your first finger now points the other way.
Reversing both the current and the field leaves the force acting in the same direction as before.
The rule gives only the direction of the force and tells you nothing about its size.

Conventional current and electron flow

Definition

Electron flow

The movement of electrons through a circuit from the negative terminal to the positive terminal, in the opposite direction to the conventional current.

A cell connected to a lamp, with one arrow showing conventional current leaving the positive terminal and a second arrow showing electrons travelling the opposite way, from the negative terminal.

In a metal wire the current is carried by electrons, which are negatively charged.
The electrons drift from the negative terminal of the supply, around the circuit, towards the positive terminal.
Conventional current was defined before electrons were discovered, as flowing from positive to negative, so it points the opposite way to the electron flow.
Fleming's left-hand rule always uses the direction of conventional current.
If a diagram shows the direction in which electrons move, point your second finger the opposite way to the electron flow.
The same rule finds the force on a beam of charged particles moving through a magnetic field, because a stream of moving charge is a current.
For positively charged particles the current is in the same direction as their motion, while for electrons the current is in the opposite direction to their motion.
A positive particle and an electron moving the same way through the same field are therefore pushed in opposite directions.

Example

Wire carrying current up the page

A straight wire runs up the page and carries a conventional current upwards.
The magnetic field between two poles points from left to right across the page.
Point the first finger of your left hand to the right, along the field.
Turn your hand so the second finger points up the page, along the current.
Your thumb now points into the page, so the force on the wire acts into the page.

Example

Electron flow shown on the diagram

A wire lies across a magnetic field that points into the page.
The diagram shows electrons moving along the wire from left to right.
The conventional current therefore flows from right to left.
Point your first finger into the page and your second finger to the left.
Your thumb points down the page, so the force on the wire acts downwards.

Common Mistake

Use your left hand, because the right hand gives the opposite answer every time.
Do not point your second finger along the electron flow, because the rule is built on conventional current.
Do not try to use the rule when the wire is parallel to the field, because there is then no force to find.

Exam technique

Stating a direction

Turn your hand to match the diagram on the paper rather than relying on the memory of a similar diagram, because examiners often rotate the set-up.
Give the direction in the terms the question uses, such as up the page, to the left or into the page.
When asked to show the force on a diagram, draw a single arrow that starts on the conductor and points in the direction of the force.
If the question reverses one quantity, apply the rule again from the start rather than guessing which way the force now acts.

Self review

Which finger of the left hand represents the magnetic field, and which represents the current?
What does a cross drawn in a circle mean on a diagram?
A wire carries conventional current to the right through a field pointing up the page; in which direction is the force?
How do you use the rule when a diagram shows the direction of electron flow?
What happens to the force on a wire if the current and the field are both reversed?

4.2.3 The force on a current-carrying conductor

The size of the force

Definition

Magnetic flux density

A measure of the strength of a magnetic field, measured in tesla (T).

Definition

Tesla

The unit of magnetic flux density, where a field of one tesla exerts a force of one newton on each metre of wire carrying a current of one ampere at right angles to the field.

Definition

Directly proportional

A relationship between two quantities in which multiplying one by a factor multiplies the other by the same factor.

The size of the force on a current-carrying conductor in a magnetic field depends on three quantities.
The magnetic flux density, $B$ , measures how strong the magnetic field is, and a stronger field gives a bigger force.
The current, $I$ , in the conductor sets the strength of the wire's own magnetic field, so a bigger current gives a bigger force.
The length, $L$ , of conductor inside the field matters as well, because every part of the wire in the field is pushed, so a longer length gives a bigger force.
When the conductor is at right angles to the field, the force is given by $F = B I L$ , or $density×current×length\text{force}=\text{magnetic flux density}\times\text{current}\times\text{length}$ .
In this equation $F$ is the force in newtons ( $N\text{N}$ ), $B$ is the magnetic flux density in tesla ( $T\text{T}$ ), $I$ is the current in amperes ( $A\text{A}$ ) and $L$ is the length in metres ( $m\text{m}$ ).
The force is directly proportional to each of the three quantities, so doubling any one of them while the other two stay the same doubles the force.
The equation applies only when the conductor is perpendicular to the magnetic field, which is the arrangement that gives the largest force.
A conductor lying parallel to the field feels no force at all, whatever the values of $B$ , $I$ and $L$ .
The length $L$ is only the part of the conductor that lies inside the magnetic field, not the whole length of wire in the circuit.
The equation gives the size of the force, while Fleming's left-hand rule gives its direction.

Rearranging and converting units

The equation can be rearranged to make any one of the four quantities the subject.
To find the magnetic flux density, divide the force by the current multiplied by the length: $B=FILB=\dfrac{F}{IL}$ .
To find the current, use $I=FBLI=\dfrac{F}{BL}$ .
To find the length of conductor in the field, use $L=FBIL=\dfrac{F}{BI}$ .
The rearranged form $B=FILB=\dfrac{F}{IL}$ shows what one tesla means: a field of $T1\ \text{T}$ exerts a force of $N1\ \text{N}$ on each $m1\ \text{m}$ of wire carrying $A1\ \text{A}$ at right angles to it.
One tesla is a very strong field, and the magnets used in school laboratories produce fields well below $T1\ \text{T}$ , so the forces measured in class are small fractions of a newton.
Lengths given in centimetres or millimetres must be converted to metres before substituting, so $m5.0\ \text{cm}=0.050\ \text{m}$ and $m25\ \text{mm}=0.025\ \text{m}$ .
Currents given in milliamperes must be converted to amperes, so $A250\ \text{mA}=0.25\ \text{A}$ .
Flux densities given in millitesla must be converted to tesla, so $T40\ \text{mT}=0.040\ \text{T}$ .
Very small forces are often written in standard form, so $N0.0045\ \text{N}$ is written as $N4.5\times10^{-3}\ \text{N}$ .

Practical

Measuring the force on a wire

Aim: to investigate how the force on a current-carrying wire in a magnetic field depends on the current in the wire.
Apparatus: a top-pan balance reading to $g0.01\ \text{g}$ , two flat magnets on a U-shaped steel yoke, a stiff straight copper rod, two clamp stands with bosses and clamps, a low-voltage d.c. power supply, a variable resistor, an ammeter, a switch, connecting leads and a ruler.
Method:
- Place the magnets on the yoke with opposite poles facing each other, stand the yoke on the balance and press the tare button so the reading is zero.
- Clamp the copper rod horizontally so it passes through the gap between the poles at right angles to the field, without touching the magnets, the yoke or the balance pan.
- Measure the width of the magnet faces with the ruler, because this is the length of rod inside the field.
- Connect the rod in series with the ammeter, the variable resistor, the switch and the power supply.
- Close the switch, adjust the variable resistor until the ammeter reads $A0.50\ \text{A}$ , and record the balance reading once it has settled.
- Open the switch between readings so the rod does not heat up, and check that the balance returns to zero.
- Increase the current in steps of $A0.50\ \text{A}$ up to $A3.0\ \text{A}$ , recording the balance reading at each step.
- Repeat the whole set of readings twice more and calculate a mean balance reading for each current.
Processing:
- The balance reading is a change in mass caused by the force between the rod and the magnet, so convert it to kilograms and multiply by the gravitational field strength, $N/kgg=10\ \text{N/kg}$ , to find the force.
- Plot force on the $y$ -axis against current on the $x$ -axis and draw a line of best fit.
Variables: the independent variable is the current, the dependent variable is the force, and the control variables are the length of rod in the field, the magnets used and the right angle between the rod and the field.
Results:
- The points lie close to a straight line through the origin, which shows that the force is directly proportional to the current.
- The gradient of the line equals $B L$ , so dividing the gradient by the measured length gives the magnetic flux density between the poles.
- Reversing the current turns a positive balance reading into a negative reading of the same size, because the force on the magnet reverses with the current.
Watch out: a rod that touches the yoke or the balance pushes on it directly and spoils the reading, and a draught or a knock on the bench makes a sensitive balance drift, so tare it again before each set of readings.
Safety: the rod and leads become warm at the largest currents, so switch off between readings and let the rod cool before touching it.

Example

Force on a wire

A wire carries a current of $A2.5\ \text{A}$ at right angles to a magnetic field of flux density $T0.15\ \text{T}$ .
The length of wire inside the field is $cm8.0\ \text{cm}$ , which is $m0.080\ \text{m}$ .
Substitute the values into the equation:

F=BIL=0.15\times2.5\times0.080

F=0.030\ \text{N}

Example

Finding the flux density

A $cm5.0\ \text{cm}$ length of wire carrying $A3.2\ \text{A}$ at right angles to a magnetic field feels a force of $N0.024\ \text{N}$ .
Convert the length into metres: $m5.0\ \text{cm}=0.050\ \text{m}$ .
Rearrange the equation to make $B$ the subject and substitute:

B=\frac{F}{IL}=\frac{0.024}{3.2\times0.050}=\frac{0.024}{0.16}

B=0.15\ \text{T}

Example

Finding the current

A $mm25\ \text{mm}$ length of wire lies at right angles to a $mT60\ \text{mT}$ field and feels a force of $N9.0\times10^{-3}\ \text{N}$ .
Convert the units: $T60\ \text{mT}=0.060\ \text{T}$ and $m25\ \text{mm}=0.025\ \text{m}$ .
Rearrange the equation to make $I$ the subject and substitute:

I=\frac{F}{BL}=\frac{9.0\times10^{-3}}{0.060\times0.025}

I=6.0\ \text{A}

Common Mistake

Do not use the total length of wire in the circuit, because $L$ is only the length that lies inside the magnetic field.
Do not apply $F = B I L$ to a wire that is parallel to the field, because the force on it is zero.

Exam technique

Calculating the force on a conductor

The equation $F = B I L$ is one you select and apply rather than recall, so the marks come from choosing it, rearranging it and handling the units correctly.
Write the equation, then the substituted values, then the answer with its unit, because each stage can earn a mark even if a later step goes wrong.
Convert centimetres, millimetres, milliamperes and millitesla before substituting, since a missed conversion changes the answer by a factor of $100$ or $1000$ .
When a question asks what happens to the force if one quantity changes, use proportionality, so tripling the current triples the force and halving the length halves it.

Self review

State the equation that links force, magnetic flux density, current and length.
What is the unit of magnetic flux density?
Calculate the force on $m0.20\ \text{m}$ of wire carrying $A4.0\ \text{A}$ at right angles to a $T0.050\ \text{T}$ field.
What happens to the force if the current is halved and the length in the field is doubled?
What is the force on a wire that lies parallel to the magnetic field lines?

4.2.4 The electric motor

Forces on a coil

Definition

Electric motor

A device that uses the forces on a current-carrying coil in a magnetic field to produce continuous rotation.

Definition

Coil

A length of insulated wire wound into many loops, called turns, so that the effects of the current in each turn add together.

Definition

Moment

The turning effect of a force about a pivot, equal to the force multiplied by the perpendicular distance from the pivot to the line of action of the force.

An electric motor uses the motor effect to turn a shaft, transferring energy from the electrical supply to the kinetic energy store of the rotating parts.
A simple motor has a rectangular coil of wire placed between the poles of a magnet, mounted on an axle so that it is free to spin.
When a current flows, it travels along one long side of the coil and back along the opposite long side, so the current in the two sides is in opposite directions.
Both long sides lie across the magnetic field, so the motor effect produces a force on each of them.
Because the currents are opposite, Fleming's left-hand rule gives forces in opposite directions, with one side pushed up and the other side pushed down.
The two forces act on opposite sides of the axle, so each produces a moment in the same rotational direction, and together they make the coil turn.
The short ends of the coil either lie along the field or feel forces that act along the line of the axle, so they add nothing to the turning effect.
A coil with many turns has many wires along each side, so the forces on them add up and the turning effect is larger.

Keeping the coil turning

Definition

Split-ring commutator

A ring split into two halves that turns with a coil and reverses the connections between the coil and the external circuit every half turn.

The turning effect is greatest when the plane of the coil is parallel to the field lines, because the two forces then act furthest from the axle.
As the coil turns, the forces keep acting straight up and down, so the perpendicular distance from the axle to each force shrinks and the moment gets smaller.
When the plane of the coil is perpendicular to the field lines, both forces act straight through the axle, so the moment is zero.
The coil keeps moving past this position because of its momentum.
If the current kept flowing the same way around the coil, the forces would now turn it backwards, so the coil would rock to and fro and settle in the perpendicular position.
To keep the coil turning in one direction, the current in the coil must reverse every half turn, at the moment the coil passes the perpendicular position.
In a simple d.c. motor a split-ring commutator does this by swapping the connections between the coil and the supply every half turn.
After each swap, the side of the coil next to the north pole always carries current in the same direction, so it is always pushed the same way and the rotation continues.

Speed and direction of rotation

The coil turns faster when the forces on its sides are larger.
Increasing the current increases the force on each side of the coil.
Using a stronger magnet increases the magnetic flux density and so increases the force on each side.
Adding more turns to the coil increases the total force along each side.
Reversing the current through the motor reverses the forces and so reverses the direction of rotation.
Reversing the magnetic field, by swapping the poles, also reverses the direction of rotation.
Reversing both the current and the field together leaves the motor turning the same way as before.
Electric motors drive anything that needs electricity to produce rotation, including the wheels of electric cars, the drum of a washing machine, cordless drills and the fans in hairdryers.

Example

Direction of rotation

Looking along the axle, the coil lies flat between a north pole on the left and a south pole on the right, so the field points from left to right.
The side of the coil on the left carries conventional current out of the page, and the side on the right carries it into the page.
For the left side, point your first finger to the right and your second finger out of the page, and your thumb points up.
For the right side, your second finger points into the page, so your thumb points down.
The left side is pushed up and the right side is pushed down, so the coil turns clockwise as seen from this end.

Practical

Building a simple motor

Aim: to build a working motor from a coil, a magnet and a cell, and to see how the direction of the current and of the field affects the rotation.
Apparatus: about $m1\ \text{m}$ of enamelled copper wire, a $V1.5\ \text{V}$ D-size cell, two large metal paper clips, a strong flat magnet, sticky tape, fine sandpaper and a marker pen or similar cylinder to wind the coil on.
Method:
- Wind the wire tightly around the marker pen about $10$ to $15$ times to make a neat circular coil, leaving about $cm3\ \text{cm}$ of straight wire sticking out on opposite sides to act as the axle.
- Slide the coil off the pen and wrap each loose end twice around the turns to hold them together, checking that both straight ends lie on one line through the centre of the coil so it balances.
- On one end, sand off all of the enamel insulation all the way round.
- On the other end, hold the coil upright and sand the enamel off the top half of the wire only, leaving the bottom half insulated.
- Bend each paper clip into a support with a small loop at the top, and tape one to each terminal of the cell so the loops are level with each other.
- Stick the magnet to the side of the cell, directly below the space between the two loops.
- Rest the axle ends in the loops so the coil hangs just above the magnet, then give it a gentle flick to start it spinning.
- Turn the magnet over, and then turn the cell round, and note the direction of spin each time.
Results:
- The coil spins steadily once it is balanced and the bare ends make good contact with the loops.
- The half-stripped end acts as a simple switch, so current flows only during the half of each turn when the bare wire touches the loop.
- The coil is pushed only during the part of each turn when the force turns it the right way, and it coasts round on its momentum for the rest of the turn.
- Turning the magnet over or reversing the cell makes the coil spin the other way.
Watch out: a coil that only rocks back and forth usually has the wrong half of the enamel removed, and a lopsided coil wobbles and falls out of the loops.
Safety: the coil has a very low resistance, so the current is large and the wire and cell become hot within a minute, so lift the coil out when you are not watching it.

Common Mistake

Do not say that the coil turns because it is attracted to the magnet, since the turning comes from the motor-effect forces on its two sides.
Do not say that the forces on the two sides act in the same direction, because they act in opposite directions and that is what produces the turning effect.

Exam technique

Explaining how a motor turns

Build your explanation as a chain: current in the coil, a magnetic field around each side, interaction with the field of the magnet, forces on the two sides, and a turning effect on the coil.
Use the phrase opposite directions for both the currents and the forces, because this is the step most often missed.
When explaining why the motor keeps turning the same way, state that the current in the coil reverses every half turn.
For ways to make the motor faster, give changes that increase the force, such as a larger current, a stronger magnet or more turns on the coil.

Self review

Why do the forces on the two long sides of a motor coil act in opposite directions?
In which position of the coil is the turning effect zero?
Why must the current in the coil reverse every half turn?
Give three ways of making a motor spin faster.
State two ways of reversing the direction in which a motor turns.

4.2.5 Electromagnetic induction

Inducing a potential difference

Definition

Electromagnetic induction

The production of a potential difference across a conductor when the magnetic field around it changes.

Definition

Induced potential difference

A potential difference produced across the ends of a conductor by a change in the magnetic field around it.

Definition

Induced current

A current driven around a complete circuit by an induced potential difference.

When the magnetic field around a conductor changes, a potential difference is produced across the ends of the conductor.
This effect is called electromagnetic induction, and the potential difference it produces is an induced potential difference.
One way to change the field is to move a magnet towards or away from a coil of wire, so the field passing through the coil grows or shrinks.
Another way is to move a wire across the field lines of a magnet, which is often described as the wire cutting the field lines.
Only relative motion matters, so moving a magnet past a still coil has the same effect as moving the coil past a still magnet.
A wire moving along the field lines does not cut them, so no potential difference is induced across it.
If there is no change in the field, nothing is induced, so a magnet resting inside a coil induces no potential difference however strong the magnet is.
If the conductor is part of a complete circuit, the induced potential difference drives an induced current around the circuit.
If the circuit is broken, a potential difference still appears across the ends of the conductor, but no current flows.

Size and direction of the effect

The induced potential difference is larger when the magnetic field changes faster, so moving the magnet more quickly gives a larger reading.
A stronger magnet gives a larger induced potential difference for the same movement, because the field changes by a greater amount.
A coil with more turns gives a larger induced potential difference, because a potential difference is induced in every turn and these add together along the coil.
Reversing the direction of movement, such as pulling the magnet out instead of pushing it in, reverses the direction of the induced potential difference.
Reversing the pole that faces the coil, such as pushing in a south pole instead of a north pole, also reverses the induced potential difference.
Reversing both the movement and the pole together gives an induced potential difference in the original direction.

Opposing the change

An induced current flowing in a coil produces its own magnetic field, just like any other current in a coil.
The induced current always flows in the direction that makes its magnetic field oppose the change that produced it.
When a north pole is pushed towards one end of a coil, the induced current makes that end of the coil a north pole, which repels the approaching magnet.
When the north pole is pulled away again, the induced current reverses and that end of the coil becomes a south pole, which attracts the magnet and resists it leaving.
In both cases the coil pushes back against the movement, so you have to do work to move the magnet.
The work you do against this opposing force is the energy that the induced current carries around the circuit.
If the induced field helped the movement instead, the magnet would speed up on its own and energy would appear from nothing, which cannot happen.
If the circuit is broken, no current flows and there is no opposing field, so the magnet moves freely even though a potential difference is still induced.

Practical

Inducing a current with a magnet

Aim: to show that a changing magnetic field induces a potential difference and a current, and to find what affects the size and direction of the effect.
Apparatus: a coil of insulated wire with about $600$ turns, a second coil with about $300$ turns, two strong bar magnets, a sensitive centre-zero ammeter (galvanometer), connecting leads and a cardboard tube that the magnet can slide through.
Method:
- Connect the ends of the $600$ -turn coil directly to the centre-zero meter, with nothing else in the circuit, and place the meter about a metre away so the magnet cannot pull on its needle.
- Hold the bar magnet still with its north pole just outside one end of the coil, and check that the needle reads zero.
- Push the north pole into the coil at a steady speed, and note the direction in which the needle moves.
- Hold the magnet still inside the coil and watch the needle.
- Pull the magnet back out and note the direction in which the needle moves.
- Repeat with the south pole facing the coil.
- Hold the magnet still and move the coil over it instead.
- Stand the coil upright with the tube passing through it, and drop the magnet through the tube from heights of $cm10\ \text{cm}$ , $cm20\ \text{cm}$ and $cm40\ \text{cm}$ , recording the largest reading each time, so that the speed is controlled by the drop height.
- Repeat the drops with two magnets held together, and then with the $300$ -turn coil in place of the $600$ -turn coil.
Results:
- The needle moves only while the magnet or the coil is moving, and it returns to zero whenever both are still.
- Pushing the magnet in and pulling it out move the needle in opposite directions.
- Turning the magnet round to use the south pole reverses each of these readings.
- Moving the coil instead of the magnet gives the same readings, which shows that it is the relative movement that matters.
- A larger drop height, two magnets and the coil with more turns each give a larger reading.
Watch out: a coil with only a few turns gives a reading too small to see, and a magnet held near the meter moves its needle directly, so keep the two well apart.
Safety: strong magnets can snap together and trap fingers, and should be kept away from phones and bank cards.

Example

Magnet falling towards a coil

A bar magnet falls with its south pole downwards towards the top of a vertical coil connected to a meter.
As the south pole approaches, the magnetic field through the coil increases, so a current is induced in the coil.
The induced field opposes the approach, so the top of the coil becomes a south pole.
The two south poles repel, so there is an upward force on the falling magnet and it speeds up less than it would in free fall.
If the ends of the coil are disconnected, no current flows, there is no opposing field, and the magnet falls freely.

Common Mistake

Do not say that a magnet held still inside a coil induces a current, because the magnetic field must be changing.
Do not say that the induced field helps the magnet along, because it always opposes the change that produced it.
Do not say that a broken circuit has no induced potential difference, because the potential difference is still induced and only the current is missing.

Exam technique

Describing induction

Use the word induced whenever you describe a potential difference or current produced by a changing magnetic field.
Link the cause and the effect in one sentence: a change in the magnetic field around the conductor induces a potential difference across its ends.
For ways to increase the reading, give specific changes, such as moving the magnet faster, using a stronger magnet or using a coil with more turns.
When asked why a magnet is harder to push into a coil in a complete circuit, explain that the induced current produces a magnetic field that opposes the movement.

Self review

What is electromagnetic induction?
Why does a magnet held still inside a coil induce no potential difference?
State three ways of increasing the induced potential difference.
What is needed for an induced potential difference to drive a current?
Explain why it is harder to push a magnet into a coil when the coil is part of a complete circuit.

4.2.6 Alternators and dynamos

Generating with a rotating coil

Definition

Generator

A device that induces a potential difference by rotating a coil and a magnet relative to each other.

A generator uses electromagnetic induction to produce a potential difference from rotation.
In a simple generator a coil of wire is turned between the poles of a magnet, so its sides move across the magnetic field lines.
The magnetic field through the coil keeps changing as it turns, so a potential difference is induced across the ends of the coil.
The induced potential difference is largest when the plane of the coil is parallel to the field lines, because the sides are then moving straight across the lines and cutting them at the fastest rate.
The induced potential difference is zero when the plane of the coil is perpendicular to the field lines, because for that instant the sides are moving along the lines and not cutting them.
After every half turn, each side of the coil has moved to the other pole and is crossing the field in the opposite direction.
The potential difference induced in the coil therefore reverses every half turn, so the rotating coil always produces an alternating potential difference.
What reaches the external circuit depends on how the ends of the coil are connected to it, and this connection is the difference between an alternator and a dynamo.

The alternator

Definition

Alternator

A generator that uses slip rings to deliver an alternating current to the external circuit.

Definition

Alternating current

A current that repeatedly reverses its direction of flow around a circuit.

Definition

Slip rings

Two complete metal rings that turn with a coil and keep each end of the coil connected to the same side of the external circuit.

Definition

Brushes

Fixed carbon contacts that press against rotating rings to connect a turning coil to the external circuit.

An alternator connects each end of the coil to its own slip ring, a complete metal ring that turns with the axle.
Fixed carbon brushes press against the rings, so each end of the coil stays connected to the same side of the external circuit throughout the rotation.
The reversal of potential difference in the coil every half turn is passed straight on, so the output is an alternating potential difference that drives an alternating current.
A graph of the output against time is a smooth wave that rises to a positive peak, falls through zero to a negative peak of the same size, and rises back to zero.
One complete cycle of this wave is produced by one complete rotation of the coil.
Many alternators keep the coil still and spin a magnet or electromagnet inside it instead, which produces the same changing field without the output passing through moving contacts.
Power stations use very large alternators of this kind to generate the mains supply, which in the UK alternates at $Hz50\ \text{Hz}$ .

The dynamo

Definition

Dynamo

A generator that uses a split-ring commutator to deliver a direct current to the external circuit.

Definition

Direct current

A current in which charge flows around a circuit in one direction only.

Definition

Split-ring commutator

A ring split into two halves that turns with a coil and reverses the connections between the coil and the external circuit every half turn.

A dynamo connects the ends of the coil to a split-ring commutator, a single ring cut into two halves that turns with the axle.
The brushes slip from one half of the ring to the other at the moment the coil is perpendicular to the field, when the induced potential difference is zero.
This is exactly the moment the potential difference in the coil reverses, so the connections to the external circuit are reversed at the same instant.
The two reversals cancel out, so the output always acts in the same direction and drives a direct current.
A graph of the output against time is a row of humps that all lie on the positive side of the time axis, with two humps for every rotation of the coil.
The current from a dynamo flows in one direction but is not steady, because its size still rises and falls as the coil turns.
A dynamo has the same construction as a simple d.c. motor, but it is turned by an outside force instead of being driven by a current.

Changing the output

Turning the coil faster makes the field through it change more quickly, so the peak potential difference increases.
Turning it faster also means each rotation takes less time, so the cycles are closer together on the graph and the frequency of an alternator's output increases.
A stronger magnet or a coil with more turns increases the peak potential difference without changing the frequency.
When the generator supplies a current, the current in the coil produces a magnetic field that opposes the rotation, so the coil becomes harder to turn.
This is why a hand-turned generator becomes noticeably stiffer when a lamp is connected and lit, because more work has to be done to keep it turning.

Practical

Displaying a generator's output

Aim: to display the output of a simple generator as a trace, and to see how it depends on the connections to the coil and on the speed of rotation.
Apparatus: a demonstration generator with a coil on an axle between two magnets that can be fitted with either slip rings or a split-ring commutator, a hand crank, an oscilloscope or a voltage sensor connected to a data logger, connecting leads and a small filament lamp in a holder.
Method:
- Fit the slip rings, connect the brushes to the oscilloscope input, and set the time base so that a few cycles fit across the screen.
- Turn the crank slowly and steadily, and sketch or save the trace.
- Turn the crank at about twice the speed, and record the new trace on the same scales.
- Replace the slip rings with the split-ring commutator, and repeat both speeds.
- Connect the lamp across the brushes and turn the crank fast enough to light it, noticing how hard the crank is to turn compared with when the lamp is disconnected.
Results:
- With slip rings, the trace is a wave that swings above and below zero, showing an alternating output.
- Faster turning gives taller peaks and squeezes more cycles across the screen.
- With the commutator, every hump of the trace lies on the same side of zero, showing a direct output whose size still rises and falls.
- The crank becomes harder to turn when the lamp is lit, because the current in the coil produces a field that opposes the rotation.
Watch out: turn the crank at as steady a rate as possible, and set the volts-per-division control so the trace fills most of the screen without running off the top.
Safety: keep fingers and loose clothing clear of the turning coil, and keep the magnets away from the oscilloscope screen.

Example

Reading an alternator trace

An alternator trace shows one complete cycle every $s0.02\ \text{s}$ .
One cycle is produced by one rotation, so the coil turns once every $s0.02\ \text{s}$ .
The number of rotations each second is:

\frac{1}{0.02\ \text{s}}=50\ \text{rotations per second}

The frequency of the alternating output is therefore $Hz50\ \text{Hz}$ .
Turning the coil twice as fast makes each cycle last $s0.01\ \text{s}$ , raises the frequency to $Hz100\ \text{Hz}$ and makes the peaks taller.

Common Mistake

Do not say that a dynamo gives a steady direct current, because its output flows one way but its size still rises and falls.
Do not say that the output is largest when the coil is perpendicular to the field, because the sides are then moving along the field lines and the output is zero.
Do not mix up the two connections: slip rings give a.c. from an alternator, and a split-ring commutator gives d.c. from a dynamo.

Exam technique

Comparing alternators and dynamos

When comparing the two, focus on the connection: slip rings keep the same connections so the output alternates, while a commutator reverses the connections every half turn so the output stays in one direction.
When sketching a trace for faster turning, show both changes, taller peaks and a shorter time for each cycle.
Explain the shape of a trace in terms of the coil's position, with the largest output when the sides cut the field lines fastest and zero output when they move along the lines.

Self review

What does an alternator produce, and which part makes this possible?
What does a dynamo produce, and which part makes this possible?
In which position of the coil is the induced potential difference zero?
State two effects of turning an alternator's coil faster.
Why is a generator harder to turn when it is lighting a lamp?

4.2.7 How transformers work

Parts of a transformer

Definition

Transformer

A device in which an alternating current in one coil induces an alternating potential difference in a second coil wound on the same iron core.

Definition

Primary coil

The input coil of a transformer, which is connected to the alternating supply.

Definition

Secondary coil

The output coil of a transformer, across which a potential difference is induced.

Definition

Iron core

The soft iron frame that passes through both coils of a transformer and carries the changing magnetic field from one coil to the other.

A transformer is made of two separate coils of insulated wire wound on the same iron core.
The primary coil is the input coil, and it is connected to an alternating supply.
The secondary coil is the output coil, and it is connected to the device or circuit that the transformer supplies.
The wire of each coil is insulated, so current cannot jump from one turn to the next, and there is no electrical connection between the two coils.
The core is made of soft iron, which is magnetised very easily and loses its magnetism as soon as the magnetising current falls, so its field follows every change in the current.
The core forms a closed loop that passes through both coils, so it guides almost all of the magnetic field made by the primary coil through the secondary coil.

How the coils are linked

Definition

Alternating current

A current that repeatedly reverses its direction of flow around a circuit.

Definition

Electromagnetic induction

The production of a potential difference across a conductor when the magnetic field around it changes.

An alternating current in the primary coil is constantly changing in size and reversing in direction.
This current turns the primary coil into an electromagnet whose magnetic field also changes in size and reverses in direction.
The iron core is magnetised by this field, so an alternating magnetic field is set up in the core.
The core carries this changing magnetic field through the secondary coil.
A magnetic field that changes around a conductor induces a potential difference across it, so an alternating potential difference is induced across the secondary coil by electromagnetic induction.
If the secondary coil is part of a complete circuit, this induced potential difference drives an alternating current in the secondary circuit.
The induced current alternates at the same frequency as the current in the primary coil, because the field in the core changes at that frequency.
Energy passes from the primary circuit to the secondary circuit through the changing magnetic field in the core, not through any current flowing between the coils.
In this way an alternating current in one circuit induces a current in a completely separate circuit.

Why direct current does not work

Definition

Direct current

A current in which charge flows around a circuit in one direction only.

A direct current in the primary coil produces a magnetic field that is steady once the current has settled.
A steady field in the core does not change, so no potential difference is induced across the secondary coil.
A transformer connected to a d.c. supply therefore gives no output while the current is steady.
A brief potential difference is induced in the secondary only at the instant the d.c. supply is switched on or off, because the field in the core changes at those moments.
This is why transformers are always used with alternating supplies, such as the UK mains supply, which alternates at $Hz50\ \text{Hz}$ .

Practical

Linking two circuits

Aim: to show that an alternating current in one coil induces a current in a separate coil, and that the effect needs a changing magnetic field.
Apparatus: two coils with the same number of turns, a U-shaped soft-iron core with a straight iron bar (the yoke) that closes it, a clamp, a low-voltage a.c. supply, a low-voltage d.c. supply, a switch, a small filament lamp in a holder, a sensitive centre-zero meter and connecting leads.
Method:
- Slide one coil onto each limb of the U-shaped core, and clamp the yoke across the top so the iron forms a closed loop through both coils.
- Connect the first coil to the a.c. supply through the switch, and connect the second coil to the lamp only, so no wire links the two circuits.
- Set the supply to a low potential difference, close the switch and look at the lamp.
- Lift the yoke off the core and look at the lamp again, then slide out the core altogether and switch off quickly.
- Rebuild the closed core, swap the a.c. supply for the d.c. supply, and replace the lamp with the centre-zero meter.
- Watch the meter as the switch is closed, while it stays closed, and as it is opened again.
Results:
- The lamp lights even though no wire connects the two circuits.
- Removing the yoke makes the lamp dimmer, and removing the core makes it go out or almost out, because much less of the changing field reaches the second coil.
- With the d.c. supply, the needle flicks one way at the moment the switch is closed, returns to zero while the current is steady, and flicks the other way when the switch is opened.
Watch out: without the core, the current in the first coil rises sharply and the coil heats up, so take that observation quickly and switch off.
Safety: use only a low-voltage supply and never connect the coils to the mains, let the coils cool before handling them, and keep fingers clear when the yoke snaps onto the core.

Common Mistake

Do not say that current flows through the iron core from one coil to the other, because the core carries a changing magnetic field and the two coils are electrically separate.
Do not say that a transformer works on direct current, because a steady current gives a steady field and induces nothing in the secondary coil.

Exam technique

Explaining how a transformer works

Set your answer out as a chain in this order: alternating current in the primary coil, alternating magnetic field, field carried by the iron core, changing field through the secondary coil, alternating potential difference induced, and alternating current if the secondary circuit is complete.
Describe the field as changing or alternating every time you mention it, because a field that is not described as changing earns no credit.
If asked why the core is made of iron, say that iron is easily magnetised and carries the magnetic field from the primary coil to the secondary coil.

Self review

Name the three main parts of a transformer.
Why must the supply to the primary coil be alternating?
What links the primary coil to the secondary coil?
Why is the core of a transformer made of soft iron?
What is induced in the secondary coil, and what is needed for it to drive a current?

4.2.8 Transformer turns ratio

Turns and potential difference

Definition

Turns ratio

The number of turns on the primary coil of a transformer compared with the number of turns on its secondary coil.

Each coil of a transformer is made of a number of turns, which are complete loops of wire around the iron core.
The iron core carries the same changing magnetic field through every turn of both coils.
Every turn therefore has the same potential difference induced across it, whichever coil it belongs to.
The turns of a coil are joined one after another, like cells connected in series, so the potential differences across the individual turns add up.
The total potential difference across a coil is therefore proportional to the number of turns it has.
In the primary coil, the supply potential difference is shared equally between its turns, which sets the potential difference per turn for the whole transformer.
The ratio of the potential differences across the two coils is equal to the ratio of their numbers of turns, which is called the turns ratio.
A secondary coil with twice as many turns as the primary coil has twice the potential difference across it.
A secondary coil with half as many turns as the primary coil has half the potential difference across it.
This simple link holds for an ideal transformer, in which all of the magnetic field from the primary coil passes through the secondary coil and no energy is wasted.

Step-up and step-down transformers

Definition

Step-up transformer

A transformer with more turns on its secondary coil than on its primary coil, so its output potential difference is greater than its input potential difference.

Definition

Step-down transformer

A transformer with fewer turns on its secondary coil than on its primary coil, so its output potential difference is smaller than its input potential difference.

A step-up transformer has more turns on its secondary coil than on its primary coil.
The output potential difference of a step-up transformer is therefore greater than its input potential difference.
A step-down transformer has fewer turns on its secondary coil than on its primary coil.
The output potential difference of a step-down transformer is therefore smaller than its input potential difference.
A transformer with the same number of turns on both coils gives an output potential difference equal to its input.
The transformer inside a microwave oven is a step-up transformer, raising the $V230\ \text{V}$ mains to a few thousand volts.
The transformer inside a doorbell supply is a step-down transformer, lowering the $V230\ \text{V}$ mains to a few volts.

Practical

Investigating the turns on a transformer

Aim: to find how the output potential difference of a transformer depends on the number of turns on its secondary coil.
Apparatus: a U-shaped soft-iron core with a yoke and clamp, a reel of insulated wire, a low-voltage a.c. supply, two a.c. voltmeters (or multimeters set to a.c. volts), connecting leads and a switch.
Method:
- Wind $50$ turns of insulated wire neatly onto one limb of the core to make the primary coil, and connect it to the a.c. supply through the switch, with a voltmeter across it.
- Wind $10$ turns of wire onto the other limb, all in the same direction, to make the secondary coil, and connect the second voltmeter across its ends.
- Clamp the yoke firmly across the top of the core so the iron forms a closed loop.
- Close the switch, set the supply so the primary voltmeter reads $V2.0\ \text{V}$ , and record the secondary potential difference.
- Open the switch, add $10$ more turns to the secondary coil, and repeat, keeping the primary potential difference at $V2.0\ \text{V}$ .
- Continue in steps of $10$ turns up to $100$ turns.
- Plot the secondary potential difference against the number of secondary turns, and draw a line of best fit.
Variables: the independent variable is the number of turns on the secondary coil, the dependent variable is the secondary potential difference, and the control variables are the primary potential difference, the number of turns on the primary coil, the core and the supply frequency.
Results:
- The secondary potential difference rises in equal steps as turns are added, giving a straight line through the origin, so the output is directly proportional to the number of secondary turns.
- With fewer than $50$ secondary turns the output is smaller than the input, as in a step-down transformer, and with more than $50$ it is larger, as in a step-up transformer.
- The measured outputs are slightly smaller than the turns ratio predicts, because some of the magnetic field leaks out of the core and some energy is transferred to the thermal store of the core and coils.
Watch out: turns wound in the wrong direction cancel part of the induced potential difference, and a loose yoke lets field leak out, so keep the winding direction the same and clamp the yoke tightly.
Safety: use only a low-voltage a.c. supply, open the switch before changing the number of turns, and let the core cool if it becomes warm.

Common Mistake

Do not decide whether a transformer steps up or down from the total number of turns, because what matters is whether the secondary coil has more or fewer turns than the primary coil.
Do not reverse the names: a step-up transformer increases the potential difference and has more turns on its secondary coil.

Exam technique

Explaining the turns ratio

When explaining why the output depends on the turns, include the two key steps: the same changing magnetic field passes through every turn, and the potential differences across the turns add up along each coil.
To identify a step-up or step-down transformer, compare the secondary turns with the primary turns, and state the comparison in your answer.
If a question gives the potential differences instead of the turns, use the same link in reverse, so a larger output potential difference means more turns on the secondary coil.

Self review

What does the turns ratio of a transformer compare?
Explain why a coil with more turns has a larger potential difference induced across it.
How can you tell from the numbers of turns whether a transformer is step-up or step-down?
A secondary coil has three times as many turns as the primary coil; how does the output potential difference compare with the input?
Why is the measured output of a real transformer slightly less than the turns ratio predicts?

4.2.9 Potential differences and turns in transformers

The transformer equation

Definition

Potential difference

The energy transferred per unit charge as the charge passes between two points in a circuit.

Definition

Turns ratio

The number of turns on the primary coil of a transformer compared with the number of turns on its secondary coil.

The potential differences across the two coils of a transformer are linked to their numbers of turns by $VpVs=NpNs\dfrac{V_p}{V_s}=\dfrac{N_p}{N_s}$ .
In words, $coil\dfrac{\text{potential difference across primary coil}}{\text{potential difference across secondary coil}}=\dfrac{\text{number of turns in primary coil}}{\text{number of turns in secondary coil}}$ .
$V_p$ is the potential difference across the primary coil and $V_s$ is the potential difference across the secondary coil, both measured in volts ( $V\text{V}$ ).
$N_p$ is the number of turns on the primary coil and $N_s$ is the number of turns on the secondary coil.
Numbers of turns are pure numbers with no unit, so the turns ratio $NpNs\dfrac{N_p}{N_s}$ has no unit either.
Both potential differences must be in the same unit, so convert kilovolts to volts, or volts to kilovolts, before substituting.
Turns ratios are often written in the form primary : secondary, so a ratio of $1 : 20$ means the secondary coil has $20$ times as many turns as the primary coil.
The equation assumes an ideal transformer, which is the assumption made in every calculation of this kind.

Rearranging the equation

The equation can be written the other way up, as $VsVp=NsNp\dfrac{V_s}{V_p}=\dfrac{N_s}{N_p}$ , which is often easier when the secondary is the unknown.
Whichever way up you write it, both primary quantities must be on the top or both must be on the bottom.
To find the secondary potential difference, use $Vs=Vp×NsNpV_s=V_p\times\dfrac{N_s}{N_p}$ .
To find the primary potential difference, use $Vp=Vs×NpNsV_p=V_s\times\dfrac{N_p}{N_s}$ .
To find the number of turns on the secondary coil, use $Ns=Np×VsVpN_s=N_p\times\dfrac{V_s}{V_p}$ .
To find the number of turns on the primary coil, use $Np=Ns×VpVsN_p=N_s\times\dfrac{V_p}{V_s}$ .
A quick check catches most mistakes: if the secondary coil has more turns than the primary coil, the secondary potential difference must be the larger of the two.

Example

Finding the output potential difference

A transformer has $500$ turns on its primary coil and $25$ turns on its secondary coil.
The primary coil is connected to the $V230\ \text{V}$ mains.
Write the equation with the secondary quantities on top:

\frac{V_s}{V_p}=\frac{N_s}{N_p}

Rearrange and substitute:

V_s=230\times\frac{25}{500}

V_s=11.5\ \text{V}

The secondary coil has fewer turns, so a smaller output is expected, which matches the answer.

Example

Finding the number of turns

A step-up transformer changes $V12\ \text{V}$ into $kV3.0\ \text{kV}$ , and its primary coil has $40$ turns.
Convert the output into volts: $V3.0\ \text{kV}=3000\ \text{V}$ .
Rearrange to make $N_s$ the subject and substitute:

N_s=N_p\times\frac{V_s}{V_p}=40\times\frac{3000}{12}

N_s=10\,000\ \text{turns}

Example

Using a turns ratio

A transformer has a turns ratio of $20 : 1$ , primary to secondary, and gives an output of $V6.0\ \text{V}$ .
The primary coil has $20$ times as many turns as the secondary coil, so its potential difference is $20$ times larger:

\frac{V_p}{V_s}=\frac{N_p}{N_s}=\frac{20}{1}

V_p=20\times6.0=120\ \text{V}

Common Mistake

Do not write the equation with $V_p$ on top on one side and $N_s$ on top on the other, because the primary quantities must be on the same level of both fractions.
Do not mix kilovolts and volts in the same calculation, because the answer will be wrong by a factor of $1000$ .

Exam technique

Transformer calculations

This equation is one you select and apply rather than recall, so the marks come from rearranging it correctly and putting each value in the right place.
Label your values $V_p$ , $V_s$ , $N_p$ and $N_s$ before substituting, because swapping the primary and secondary values is the most common way to lose the marks.
Give a number of turns as a whole number with no unit, and give a potential difference with the unit $V\text{V}$ .
Check the size of your answer against the type of transformer before moving on.

Self review

Write down the equation that links the potential differences and the numbers of turns in a transformer.
A transformer has $100$ turns on its primary coil and $2000$ turns on its secondary coil, and the input is $V12\ \text{V}$ ; calculate the output potential difference.
Rearrange the equation to make $N_s$ the subject.
Is a transformer with a turns ratio of $1 : 5$ , primary to secondary, a step-up or a step-down transformer?
Why must both potential differences be in the same unit before you substitute them?

4.2.10 Microphones, loudspeakers and headphones

Sound as pressure variations

Definition

Sound wave

A longitudinal wave that travels through a medium as a series of compressions and rarefactions, which are regions of higher and lower pressure.

A sound wave travels through air as a series of compressions, where the air pressure is slightly higher than normal, and rarefactions, where it is slightly lower.
At any one point, the air pressure rises and falls over and over again as the wave passes, and these changes are the pressure variations of the sound.
The frequency of the pressure variations sets the pitch of the sound, and the size of the variations sets its loudness.
A microphone turns these pressure variations into matching variations in an electric current, and a loudspeaker turns variations in current back into pressure variations.

How a microphone works

Definition

Microphone

A device that converts the pressure variations of a sound wave into matching variations in an electric current.

Definition

Diaphragm

A thin, flexible sheet in a microphone or headphone that vibrates in step with a sound wave.

A dynamic microphone contains a thin, flexible diaphragm attached to a small, light coil of wire.
The coil sits in the narrow gap around a permanent magnet, so it is surrounded by a strong magnetic field.
When a sound wave reaches the microphone, the pressure variations push and pull on the diaphragm, so the diaphragm vibrates back and forth.
The coil is fixed to the diaphragm, so it vibrates with it and moves back and forth through the magnetic field.
As the coil moves, its wire cuts across the magnetic field lines, so a potential difference is induced across the ends of the coil by electromagnetic induction.
Each time the coil changes direction, the induced potential difference reverses, so the output is an alternating potential difference.
When the microphone is part of a complete circuit, this potential difference drives an alternating current whose variations match the pressure variations of the sound.
A higher-pitched sound makes the coil vibrate more times each second, so the current alternates at a higher frequency.
A louder sound makes the coil vibrate with a larger amplitude and move faster through the field, so the induced current is larger.
The current from a microphone is very small, so it is passed to an amplifier, which makes it larger without changing its pattern.

Loudspeakers and headphones

Definition

Loudspeaker

A device that uses the motor effect to convert a varying electric current into vibrations of a cone that produce sound.

Definition

Headphones

A pair of small loudspeakers worn on or in the ears, each using the motor effect to turn a varying current into sound.

A loudspeaker does the reverse job of a microphone, turning variations in current into pressure variations in the air.
It has a coil attached to a light paper or plastic cone, and the coil sits in the field of a permanent magnet.
An amplifier sends an alternating current through the coil, with the same pattern of variations as the original sound.
The current in the coil lies across the magnetic field, so the motor effect produces a force on the coil.
Each time the current reverses, the force on the coil reverses, so the coil and the cone are pushed out and pulled in again and again.
As the cone moves outwards it squeezes the air in front of it into a compression, and as it moves back it leaves a rarefaction.
The vibrating cone therefore sends out a sound wave whose pressure variations match the variations in the current.
The frequency of the current sets the frequency of the vibrations, and so sets the pitch of the sound.
A larger current produces a larger force, a larger amplitude of vibration and a louder sound.
Headphones contain a tiny loudspeaker for each ear that works in exactly the same way, with a small coil and magnet driving a thin diaphragm close to the ear.
A microphone relies on electromagnetic induction, while a loudspeaker and headphones rely on the motor effect, even though all three use a coil and a magnet.

Practical

Using a loudspeaker both ways

Aim: to show that an alternating current in a loudspeaker produces sound by the motor effect, and that sound reaching a loudspeaker induces an alternating potential difference.
Apparatus: a small loudspeaker, a signal generator, an oscilloscope (or a voltage sensor and data logger), connecting leads, a few grains of dry rice and a tuning fork.
Method, loudspeaker as a speaker:
- Lay the loudspeaker on its back with the cone facing upwards, and sprinkle a few grains of rice onto the cone.
- Connect the loudspeaker to the signal generator, set the frequency to about $Hz5\ \text{Hz}$ and the output to a low amplitude, and watch the cone and the rice.
- Raise the frequency slowly to a few hundred hertz and listen to the note, then increase the amplitude and listen again.
Method, loudspeaker as a microphone:
- Disconnect the signal generator and connect the loudspeaker straight to the oscilloscope input instead.
- Strike the tuning fork and hold it close to the cone, then hum a low note and a high note, and a quiet note and a loud note, close to the cone, watching the trace each time.
Results:
- At low frequency the cone moves visibly out and in, and the rice bounces in time with the signal.
- A higher frequency gives a higher note, and a larger amplitude gives a louder sound and makes the rice jump higher.
- When the loudspeaker is used as a microphone, the oscilloscope shows an alternating trace whenever sound reaches the cone, with more waves across the screen for a higher note and taller waves for a louder one.
- The same coil and magnet use the motor effect in one direction and electromagnetic induction in the other.
Watch out: the potential difference produced by a loudspeaker used as a microphone is tiny, so set the oscilloscope to its most sensitive range.
Safety: keep the signal generator output low, because large amplitudes can damage the loudspeaker cone and very loud sounds can damage your hearing.

Common Mistake

Do not say that a microphone works by the motor effect, because the moving coil in a microphone has a potential difference induced in it, while the loudspeaker coil has a force acting on it.
Do not say that the magnet in a loudspeaker vibrates, because the magnet is fixed and it is the coil and cone that move.

Exam technique

Describing the conversions

For a microphone, link every stage in order: pressure variations, vibrating diaphragm, coil moving through the magnetic field, induced potential difference, varying current.
For a loudspeaker, run the chain the other way: varying current in the coil, force from the magnetic field, vibrating coil and cone, pressure variations in the air.
Name the physics at the key step, writing electromagnetic induction for a microphone and motor effect for a loudspeaker.
When asked how the current changes for a louder or higher sound, link loudness to the size of the current and pitch to its frequency.

Self review

What makes the diaphragm of a microphone vibrate?
Explain why a potential difference is induced across the coil of a dynamic microphone.
Which effect makes the cone of a loudspeaker move?
How does the current in a loudspeaker change to make a louder sound?
How are headphones similar to a loudspeaker?

4.2 Uses of magnetism

4.2.1 The motor effect

Forces between magnets and currents

The combined magnetic field

When the force acts

Observing the force on a wire

Explaining why a wire moves

4.2.2 Fleming's left-hand rule

Fleming's left-hand rule

Using the rule step by step

Conventional current and electron flow

Wire carrying current up the page

Electron flow shown on the diagram

Stating a direction

4.2.3 The force on a current-carrying conductor

The size of the force

Rearranging and converting units

Measuring the force on a wire

Force on a wire

Finding the flux density

Finding the current

Calculating the force on a conductor

4.2.4 The electric motor

Forces on a coil

Keeping the coil turning

Speed and direction of rotation

Direction of rotation

Building a simple motor

Explaining how a motor turns

4.2.5 Electromagnetic induction

Inducing a potential difference

Size and direction of the effect

Opposing the change

Inducing a current with a magnet

Magnet falling towards a coil

Describing induction

4.2.6 Alternators and dynamos

Generating with a rotating coil

The alternator

The dynamo

Changing the output

Displaying a generator's output

Reading an alternator trace

Comparing alternators and dynamos

4.2.7 How transformers work

Parts of a transformer

How the coils are linked

Why direct current does not work

Linking two circuits

Explaining how a transformer works

4.2.8 Transformer turns ratio

Turns and potential difference

Step-up and step-down transformers

Investigating the turns on a transformer

Explaining the turns ratio

4.2.9 Potential differences and turns in transformers

The transformer equation

Rearranging the equation

Finding the output potential difference

Finding the number of turns

Using a turns ratio

Transformer calculations

4.2.10 Microphones, loudspeakers and headphones

Sound as pressure variations

How a microphone works

Loudspeakers and headphones

Using a loudspeaker both ways

Describing the conversions

Review 4.2 Uses of magnetism by teaching Genie

Recap your knowledge with an interactive lesson

Put it into practice with exam-style questions

Remember key concepts with flashcards

4.2 Uses of magnetism Revision Guide

4.2 Uses of magnetism

4.2.1 The motor effect

Forces between magnets and currents

The combined magnetic field

When the force acts

Observing the force on a wire

Explaining why a wire moves