There’s no standard-level content in this topic. Induction is an additional higher-level treatment of what happens when conductors, magnetic fields and motion are combined.

Flux: how much magnetic field links an area

Magnetic flux density is a vector field quantity that measures the magnetic field passing normally through unit area; it is measured in tesla, where $1\ \mathrm{T} = 1\ \mathrm{Wb\,m}^{-2}$. In earlier work, you may simply have called this magnetic field strength. Here, “flux density” is the more useful wording because induction depends on how much field links an area.

Magnetic flux is a scalar quantity that measures the component of a magnetic field passing through a specified area. For a uniform field,

Don’t use the angle between the field and the plane of the coil unless the question has defined it that way. The formula uses the angle to the normal. If the field goes straight through the face of the coil, $\theta = 0$ and the flux is maximum. If the field lies along the plane of the coil, $\theta = 90^\circ$ and the flux is zero.

Field lines are a model, not little strings in space

A useful picture is to draw magnetic field lines closer together where the field is stronger. More lines through the same drawn area suggests a larger magnetic flux density; more total lines through the area suggests a larger magnetic flux. Treat this as a representation, not a literal count of real lines. It’s a good Nature of Science example: the field-line model is powerful because it compresses direction and strength into a visual language, but it is still a model.

The unit weber is tied directly to induction. A change of flux of one weber in one second corresponds to an induced emf of one volt in a single turn. That statement will make more sense once Faraday’s law is in place.

What induction actually says

Electromagnetic induction is the process where a changing magnetic flux linkage produces an emf in a conductor. Induced emf is the energy-per-charge quantity produced by induction, measured in volts; if there is a complete conducting circuit, it can drive a current.

A standard demonstration uses a coil connected to a centre-zero galvanometer, with a bar magnet moved towards or away from the coil. The important observation is not “there is a magnet nearby”. It is “the flux through the coil is changing”. You can do that by moving the magnet, moving the coil, changing the field strength, or rotating the coil.

Faraday’s law

In a coil with many turns, every turn adds to the total flux linkage. Magnetic flux linkage is the product of the number of turns and the magnetic flux through each turn. Faraday’s law is

The induced emf depends on how fast the flux linkage changes. Change the same flux slowly and you get a small emf; change it quickly and the emf is larger. On a graph of flux linkage against time, the induced emf is the negative gradient of the graph. A flat section means no induced emf.

The negative sign is doing real work. It represents Lenz’s law: the induced emf acts in the direction that opposes the change producing it. We will discuss that energy argument properly in D.4.4.

Ways to produce an induced emf

The syllabus expects you to recognize these examples:

• a time-varying magnetic field, such as a stationary coil near a field that is being switched on, switched off or varied;
• a coil rotating in a uniform magnetic field, the basis of an ac generator;
• relative motion between a conductor and a magnetic field, such as a magnet oscillating near a coil or a coil moving into or out of a magnetic-field region.

For calculations, the course restricts you to straight conductors moving at right angles to fields, and rectangular coils moving in and out of fields or rotating in fields. That restriction is deliberate: the focus is the physics of changing flux, not awkward three-dimensional geometry.

Self-induction, only qualitatively

Self-induction is an induction effect where a changing current in a conductor changes the magnetic flux linked with that same conductor, producing an emf that opposes the current change. Coils show this effect strongly because they link much of their own magnetic field. When the current is increasing, the self-induced emf acts against the increase. When the current is decreasing, it acts to maintain the current. With a steady current, there is no changing flux, so there is no self-induced emf.

No quantitative treatment of inductance, and no resistance–inductor circuit analysis, is required here.

Rate of change as a recurring physics idea

Faraday’s law is another rate-of-change law. Physics uses the same pattern again and again: velocity is the rate of change of displacement, acceleration is the rate of change of velocity, power is the rate of energy transfer, and radioactive activity is the rate at which nuclei decay. On graphs, “rate of change” usually means “gradient”, so Faraday’s law is a graph-reading idea as well as an equation.

Charge separation in a moving conductor

A metal rod has mobile electrons. When the rod moves through a magnetic field, it carries those charges with it, and the charges experience a magnetic force. Electrons are pushed towards one end of the rod, leaving the other end short of electrons. This charge separation produces an emf across the rod.

If the rod is isolated, the separated charge quickly sets up an electric field inside the rod, stopping any further movement of charge. An induced emf is still present, but there is no continuous current. For a continuous induced current, the conductor needs a complete path.

The motional-emf equation

For a straight conductor moving at right angles to a uniform magnetic field,

This equation is one of the main quantitative results in the topic. It works neatly for a straight rod cutting a uniform field at 90°. If the conductor does not move at right angles to the field, only the perpendicular component of its motion contributes, but syllabus calculations stay with the right-angle case.

Rectangular coils moving into and out of fields

As a rectangular coil enters a uniform magnetic-field region, its flux linkage increases, so an emf is induced. Once the whole coil is fully inside a uniform field and keeps moving without changing orientation, the flux linkage stays constant and the net induced emf is zero. Students often say “it is cutting field lines, so there must be an emf”; for the whole loop, the emfs on opposite sides cancel when the loop is entirely within the same uniform field.

When the coil leaves the field, the flux linkage decreases, and the induced emf changes direction. Its magnitude depends on the rate of change of flux linkage: faster motion, more turns, a stronger field, or a larger coil side cutting the boundary all increase the emf.

A familiar example is an aircraft flying through Earth’s magnetic field. Its wingspan can act like a moving conductor, so a small emf is induced between the wingtips when the motion cuts the vertical component of the field.

Lenz’s law: oppose the change, not necessarily the motion

Lenz’s law is a direction rule: an induced emf acts in the direction that opposes the change in magnetic flux linkage that produced it.

That wording is worth being careful with. When a north pole approaches a coil, the coil’s near face becomes a north pole, so it repels the magnet and opposes the increasing flux. When the north pole moves away, the near face becomes a south pole, so it attracts the magnet and opposes the decreasing flux. Either way, the induced effect resists the change.

Why this is conservation of energy

Imagine the induced current helped the change instead. A magnet approaching a coil would be pulled in faster, causing a larger change of flux, which would create a larger current and pull it faster again. Kinetic energy and electrical energy would appear without any external energy source. Lenz’s law stops that impossible result.

In a moving-rod generator, closing the circuit lets the induced current in the rod experience a magnetic force opposite to the rod’s motion. The external agent pulling the rod has to do work against this force. That mechanical work becomes electrical energy in the circuit. With the circuit open, an emf can still exist, but there is no continuous current, so there is no magnetic braking force associated with an induced current.

A bicycle dynamo shows the same idea. It turns easily when the lamp circuit is open, but it is harder to turn when the lamp is lit. The extra effort is not a mysterious loss; it is the mechanical input required for electrical output.

Links to earlier field ideas

The force on charges in the moving conductor comes from the same magnetic-force idea that can make a charged particle move in a circular path: a force perpendicular to velocity changes the direction of motion. By measuring the radius of a charged particle’s path in a magnetic field, physicists can find the particle’s charge-to-mass character, which is why field methods are so useful for identifying particles. In induction, the same family of ideas comes back as Lenz’s law: fields exert forces, forces transfer energy, and energy is conserved.

There is a shared representation point across electric and magnetic fields as well. Field lines and flux diagrams help us picture invisible field properties, but the diagrams are still models. Good physicists use the model, and also know its limits.

Why a rotating coil gives alternating emf

An alternating current generator uses electromagnetic induction to produce an emf that reverses direction periodically. It needs a coil, a magnetic field, relative rotation between the coil and field, and connections so charge can flow through an external circuit.

As the coil turns, the angle between the coil’s normal and the field keeps changing. The magnetic flux linkage changes with it. When the flux linkage is at its maximum, its gradient is zero, so the induced emf is zero. When the flux linkage passes through zero, it is changing fastest, so the emf has its maximum magnitude.

For a rectangular coil rotating at constant angular speed in a uniform magnetic field, the induced emf is sinusoidal:

So the peak emf is proportional to $N$, $B$, $A$ and $\omega$. The rotation rate can also be written in frequency form:

A practical ac generator may spin the coil in a fixed magnetic field, or spin the magnetic field around a fixed coil. The same physics applies: the flux linkage must change. In a rotating-coil design, slip rings and brushes keep contact with the external circuit without twisting the wires.

Generators, power and industrialization

Induction made large-scale electrical generation possible. Mechanical energy from falling water, steam turbines, wind turbines or engines can be transferred to electrical energy by rotating coils or magnetic fields. This was one of the major physics-to-engineering steps behind industrial electrification: energy could be generated centrally, carried by cables, and used far from the original energy source.

The efficiency of electricity generation depends strongly on the source and on the sequence of energy transfers before the generator. A hydroelectric station has a different loss chain from a fossil-fuel or nuclear station, where thermal processes and turbines come before the generator stage. In every case, the generator cannot deliver electrical energy unless mechanical energy is supplied to it, and Lenz’s law explains why a loaded generator is harder to turn.

Engineers can increase generator output without changing the supply frequency by increasing the number of turns, using a stronger magnetic field, or increasing the coil area. Spinning the generator faster does increase the emf, but it also changes the frequency of the alternating supply, which is usually not acceptable for a grid.

Faster rotation changes two things

Spin the same coil faster in the same magnetic field, and the flux linkage still reaches the same maximum and minimum values. The difference is timing: it gets from one to the other in less time. By Faraday’s law, a faster rate of change gives a larger peak induced emf.

Since $\varepsilon = NBAA\omega \sin(\omega t)$, increasing the rotation frequency increases $\omega$, so the peak emf increases in direct proportion. If the rotation frequency is doubled, the peak emf doubles too, as long as $N$, $B$ and $A$ stay unchanged.

Second effect

There’s another effect: the alternating emf repeats more times each second. A faster generator therefore has a larger amplitude and a higher output frequency. On an emf–time graph, the wave becomes taller and more squeezed horizontally. On an emf–angle graph, the angular pattern still repeats over the same angles, but the amplitude is larger because the coil moves through those angles more quickly.

That’s why increasing a generator’s power output is not as simple as “turn it faster”. In many electrical systems the supply frequency has to remain fixed, so output is usually increased by improving the magnetic field, coil area, number of turns, or mechanical input at the required rotational speed.