13.1.1 Producing an electric current by induction

Inducing a potential difference

1. A conductor has a potential difference induced across its ends whenever the magnetic field passing through that conductor changes.
2. The change can be produced by relative movement between the conductor and a magnet, or by a magnetic field that grows or collapses in strength while everything stays still.
3. A stationary conductor sitting in a steady magnetic field has nothing induced across it, however strong that field is, because nothing about the field through it is changing.
4. The induced potential difference exists across the ends of the conductor whether or not any current flows, in the same way that a cell has a potential difference across its terminals before it is connected to anything.
5. For a straight wire, the field through the wire only changes if the wire cuts across the magnetic field lines, so moving the wire along the field lines induces nothing.

When an induced current flows

1. An induced current flows only when the conductor forms part of a complete circuit.
2. Connecting the ends of a coil to a sensitive ammeter completes the circuit, so the needle deflects while the magnet is moving and returns to zero when the movement stops.
3. Leaving the ends of the coil unconnected still gives an induced potential difference, but no charge can circulate, so no current is measured.
4. The size of the induced current follows $I=\frac{V}{R}$, so a coil made of thicker wire or a circuit with less resistance gives a larger current for the same induced potential difference.
5. The needle deflects in one direction while the magnet approaches and in the opposite direction while it recedes, which shows that the direction of the induced current depends on the direction of the change.

Ways of producing the change

1. Pushing a bar magnet into a coil of wire increases the field through the coil, and pulling it out decreases the field, so both movements induce a potential difference.
2. Holding the magnet still and moving the coil over it works exactly as well, because only the relative movement of the two matters.
3. Moving a straight wire downwards between the poles of a horseshoe magnet makes the wire cut across the field lines and induces a potential difference along its length.
4. Rotating a coil continuously inside a magnetic field keeps the field through the coil changing, which is how a generator produces a continuous output.
5. Switching a current on or off in a nearby coil makes that coil's magnetic field grow or collapse, and the changing field induces a potential difference in the second coil even though nothing moves.
6. Moving a magnet sideways past a coil, so that the field through the coil barely changes, induces very little.

Direction of the induced potential difference

1. Reversing the direction of movement reverses the induced potential difference, so pushing a magnet in and pulling it out give deflections on opposite sides of zero.
2. Reversing the poles of the magnet, so that the south pole leads instead of the north pole, also reverses the induced potential difference.
3. Reversing the movement and the poles together leaves the direction of the induced potential difference unchanged, because the two reversals cancel.
4. Moving the magnet in and out repeatedly produces a potential difference that repeatedly changes direction, which is an alternating potential difference.

Energy transfer during induction

1. The induced current in the coil is itself a current in a loop of wire, so it produces its own magnetic field.
2. This induced field always acts to oppose the change that produced it, so the coil repels an approaching pole and attracts a receding pole.
3. A force must therefore be applied to keep the magnet moving, and the work done against that opposing force is the source of the electrical energy.
4. Energy is transferred from the kinetic store of the moving magnet, through the electrical working of the circuit, and finally to the thermal store of the surroundings as the induced current heats the wires.
5. If the induced field helped the change rather than opposing it, the magnet would accelerate on its own and energy would appear from nowhere, which conservation of energy forbids.
6. This is why a bicycle dynamo makes the pedals harder to turn once its lamp is switched on, and why a generator supplying more current needs more driving power.

Reading an induction trace

1. A data logger connected across the coil records how the induced potential difference changes with time, giving a pulse rather than a steady value.
2. Each pulse begins at zero, rises to a peak while the field is changing fastest, and returns to zero when the movement stops.
3. A faster movement gives a pulse that is taller and narrower, because the same change in the field is squeezed into a shorter time.
4. A pulse below the zero line means the induced potential difference is in the opposite direction to one above the line, not that it is smaller.

13.1.2 Factors affecting the induced potential difference

Rate of change of the magnetic field

1. The size of the induced potential difference is set by the rate at which the magnetic field through the conductor changes, not by the size of the field itself.
2. A change completed in a shorter time gives a larger induced potential difference, and the same change spread over a longer time gives a smaller one.
3. Anything that makes the field through the coil change by a larger amount, or change in a shorter time, increases the induced potential difference.
4. The induced current grows in proportion to the induced potential difference for a fixed circuit resistance, so the same list of factors controls the current reading.
5. Each factor works by changing either how much the field through the coil changes or how quickly that change happens.

The five controlling factors

1. Speed of relative movement:
   1. Moving the magnet or the coil faster completes the same change in the field in less time, so the rate of change is greater.
   2. Doubling the speed roughly doubles the peak induced potential difference for the same magnet and coil.
2. Number of turns on the coil:
   1. Each turn of wire has a potential difference induced across it, and the turns are joined in series, so their contributions add together.
   2. A coil of $200$ turns gives about twice the induced potential difference of an otherwise identical coil of $100$ turns.
3. Strength of the magnetic field:
   1. A stronger magnet, or two magnets used together, produces a larger change in the field for the same movement.
   2. Slotting a soft iron core into the coil concentrates the field lines through the coil and increases the induced potential difference sharply.
4. Area of the coil:
   1. A coil of larger cross-sectional area has more of the magnetic field passing through it, so the same movement changes the field through it by more.
5. Orientation of the movement:
   1. Moving a wire at right angles to the field lines makes it cut through the greatest number of them and gives the largest induced potential difference.
   2. Moving a wire along the field lines makes it cut none of them, so nothing is induced.

Factors that make no difference

1. The resistance of the external circuit changes the induced current but not the induced potential difference.
2. The material of the wire changes the resistance of the coil, so it affects the current rather than the induced potential difference.
3. The length of time for which the magnet is left inside the coil makes no difference once it has stopped moving, because the reading has already fallen to zero.
4. The absolute position of the magnet matters only through how much the field through the coil changes as it moves.

Reading the graphs and traces

1. A straight line through the origin on a graph of peak induced potential difference against speed shows that the two quantities are directly proportional.
2. A straight line through the origin on a graph of peak induced potential difference against number of turns shows the same proportional relationship for the turns.
3. Doubling the speed makes the pulse on an oscilloscope twice as tall and half as wide, because the same overall change in the field happens in half the time.
4. The area under a pulse of induced potential difference against time barely changes when only the speed changes, since the total change in the field is the same.
5. Reading a value off a straight line through the origin is a matter of scaling, so a coil of triple the turns at double the speed gives six times the original peak.

13.1.3 Alternators and dynamos

Generating with a rotating coil

1. A generator turns a coil of wire inside a magnetic field so that the field through the coil changes continuously and a potential difference is induced without pause.
2. As the coil rotates, the two long sides of the coil sweep through the field lines, and the rate at which they cut the field lines varies smoothly through each turn.
3. When the plane of the coil lies parallel to the field lines, the sides are moving straight across the field lines, so they cut them fastest and the induced potential difference is at its peak.
4. When the plane of the coil lies at right angles to the field lines, the sides are moving momentarily along the field lines, so they cut none of them and the induced potential difference is zero.
5. The coil is driven by an external source of energy, such as a steam turbine in a power station, a bicycle wheel or a hand crank, and energy is transferred from the kinetic store of the rotating coil to the electrical circuit.
6. Both types of generator can be built the other way round, with the magnet rotating inside a fixed coil, because only the relative movement matters.

The alternator and slip rings

1. An alternator connects each end of the rotating coil to its own slip ring, and each slip ring is pressed against a fixed carbon brush.
2. Each end of the coil therefore stays connected to the same brush and the same side of the external circuit for the whole rotation.
3. Every half turn the two sides of the coil swap places in the field, so the induced potential difference reverses direction, and the external circuit sees that reversal.
4. The output over one rotation rises from zero to a positive peak, falls back through zero, reaches an equal negative peak and returns to zero, tracing a smooth curve.
5. One complete rotation of the coil gives one complete cycle of the output, so the frequency of the output equals the number of rotations each second.
6. The alternators in UK power stations turn at a rate that gives mains electricity a frequency of $50Hz$, so the output completes $50$ cycles every second.

The dynamo and the commutator

1. A dynamo connects the coil to a split-ring commutator, which is a single ring cut into two halves that each press against one of the fixed brushes.
2. At the moment when the induced potential difference in the coil would reverse, the two halves of the commutator swap brushes.
3. The connections to the external circuit reverse at the same instant as the potential difference in the coil reverses, so the two reversals cancel and the output keeps one direction.
4. The output over one rotation is two positive humps, each rising from zero to a peak and falling back to zero, because the negative half of each cycle has been flipped over.
5. The output is direct because it never crosses the zero line, even though its size changes constantly, so it is described as a pulsing direct potential difference.
6. The commutator is the only difference in principle between the two machines, so an alternator becomes a dynamo when its slip rings are replaced by a split ring.

Changing the output of a generator

1. Turning the coil faster raises the peak potential difference, because the coil sides cut the field lines more quickly.
2. Turning the coil faster also raises the frequency, because each rotation takes less time, so the peaks on a trace move closer together.
3. Adding more turns to the coil, using stronger magnets or using a coil of larger area raises the peak potential difference while leaving the frequency unchanged.
4. Placing the coil on a soft iron armature concentrates the field lines through the coil and raises the peak value further.
5. Drawing more current from a generator makes it harder to turn, because the induced current in the coil produces a field that opposes the rotation.

Reading a generator output trace

1. Identify the machine first: a trace that crosses the zero line and goes negative is from an alternator, and a trace of humps all on one side of zero is from a dynamo.
2. Read the peak potential difference by counting the vertical divisions from the zero line to the top of the trace and multiplying by the volts per division.
3. Read the period by counting the horizontal divisions for one complete cycle and multiplying by the time per division.
4. Convert the period into the rate of rotation using $f=\frac{1}{T}$, remembering that an alternator completes one cycle per rotation.
5. A dynamo trace shows two humps per rotation, so its period must be measured over one complete pair of humps rather than one hump.

13.1.4 Microphones and loudspeakers

Microphones and the generator effect

1. A moving-coil microphone has a thin flexible diaphragm fixed to a light coil of wire, and the coil sits in the field of a permanent magnet.
2. A sound wave arriving at the diaphragm is a series of compressions, where the air pressure is above normal, and rarefactions, where it is below normal.
3. The changing air pressure pushes the diaphragm inwards during a compression and lets it spring outwards during a rarefaction, so the diaphragm vibrates at the frequency of the sound.
4. The coil is attached to the diaphragm, so the coil vibrates with it and moves back and forth inside the magnetic field.
5. The moving coil changes the magnetic field passing through itself, so a potential difference is induced across the coil by the generator effect.
6. The coil moves one way during a compression and the opposite way during a rarefaction, so the induced potential difference reverses each half cycle and the output is an alternating signal.
7. A louder sound carries a larger pressure variation, which pushes the diaphragm further and faster, so the coil moves faster and the amplitude of the induced potential difference is larger.
8. A higher-pitched sound vibrates the diaphragm more times each second, so the frequency of the electrical signal is higher.
9. The signal therefore has the same frequency as the sound wave and an amplitude that follows its loudness, which is what makes it an accurate electrical copy of the sound.

Loudspeakers and the motor effect

1. A loudspeaker has a coil wound around one pole of a permanent magnet, and the coil is fixed to a stiff paper or plastic cone.
2. An alternating current from an amplifier is passed through the coil, so the coil becomes a current-carrying conductor sitting in a magnetic field.
3. A current-carrying conductor in a magnetic field experiences a force, so the coil is pushed along the axis of the magnet.
4. The direction of the force reverses whenever the direction of the current reverses, so the coil is pushed outwards and then pulled inwards, once for each cycle of the current.
5. The cone moves with the coil and pushes on the air in front of it, creating a compression as it moves out and a rarefaction as it moves back, and these travel outwards as a sound wave.
6. A larger current amplitude produces a larger force, a larger cone movement and therefore a louder sound.
7. A higher current frequency makes the cone vibrate more times each second, giving a higher-pitched sound of the same frequency as the current.
8. Fleming's left-hand rule gives the direction of the force on the coil at any instant from the directions of the current and the field.

Comparing the two devices

1. Effect used: a microphone works by the generator effect, in which movement induces a potential difference, and a loudspeaker works by the motor effect, in which a current produces a force.
2. Direction of energy transfer: a microphone transfers energy from a sound wave to an electrical circuit, and a loudspeaker transfers energy from an electrical circuit to a sound wave.
3. Input and output: the sound wave is the input to a microphone and the output from a loudspeaker.
4. Shared parts: both devices contain a coil, a permanent magnet and a movable surface, so the two are close to being the same device operated in opposite directions.
5. Signal matching: in both devices the frequency of the electrical signal equals the frequency of the sound, and the amplitude of the signal corresponds to the loudness.
6. A small loudspeaker will in fact act as a rough microphone if sound is directed at its cone, because the moving coil then induces a potential difference.