Physics · Module 16
Module 15 finished with a current making a magnetic field. This module runs the sentence backwards: a magnetic field, if it changes, makes a current. Almost every joule of electricity in your house arrived through that one sentence.
In the autumn of 1831 Michael Faraday wound a coil of wire, connected the two ends to a galvanometer — a needle that swings when current flows through it — and pushed a bar magnet into the coil.
The needle kicked.
He left the magnet sitting inside the coil. The needle went back to zero and stayed there. He pulled the magnet out. The needle kicked the other way.
That is it. That is the entire discovery, and it is worth stopping on how strange it is. There is no battery anywhere in this circuit. Nothing is burning, nothing is being consumed chemically, no charges are being separated by a cell. And yet current flows in the wire — real current, which will warm the wire, deflect a compass, run a bulb. The only thing that has happened is that a lump of iron moved.
Go further. The magnet does not touch the coil. It does not have to come within a mile of touching. There is a gap of air between magnet and wire, and across that gap of air, something reaches out and pushes the electrons in the copper along.
In Module 13 we gave that reaching-across-a-gap a name: a field. In Module 15 we found that a moving charge makes a magnetic field. What Faraday found is the other half of the relationship — and it is not the mirror image you might expect. A magnetic field sitting there does nothing to the electrons in a stationary wire. Only a changing magnetic field does anything at all.
Put a strong magnet flat on the table and lay a loop of wire over it. Nothing. No current, not a microamp, forever. Lift the loop one centimetre — current, for the moment you are lifting. Stop — nothing. This is the thing to hold on to through the next three sections, and it is what the first bench is built to make you feel.
Before we write anything down: the magnet is resting exactly at the centre of the coil, not moving. The coil has 600 turns and the magnet is neodymium — the small, startlingly strong kind used in headphones and hard drives.
Predict the galvanometer reading.
Nothing about the strength of the magnet or the number of turns can rescue a stationary magnet. A field that is merely present is invisible to the electrons in a wire that is also merely present. Only change does work here. Now we need a way to say precisely what is changing.
Three things to do with that bench before you read on.
One. You have just predicted this one. Park the magnet in the middle and confirm it: zero, however deep you push it in. The amount of field through the coil is irrelevant; only its rate of change shows up on the needle.
Two. Shove it through slowly, then shove it through fast. Same magnet, same coil, same total change in flux — but the fast shove gives a much bigger kick. Double the speed and you double the EMF — "roughly" only because it is hard to shove at exactly twice the speed by hand, not because the law is approximate.
Three. Watch the sign as the magnet passes the middle. Going in, the flux is rising, and the EMF has one sign. Coming out the far side, the flux is falling, and the EMF flips. The needle swings one way then the other within a single shove, and the crossing point is exactly the moment the magnet is centred — the moment the flux is at its maximum and therefore, briefly, not changing at all.
That third observation is the fingerprint of a derivative. If you have met calculus, you have just watched a function pass through a maximum and its slope pass through zero. If you have not, you have just watched something more useful: a quantity whose steepness, not its size, is what matters.
We need one new quantity, and it is genuinely new — you have not met it in this course before, though you have met everything it is built from.
Here is the problem it solves. A magnetic field has a strength and a direction at every point in space — call it B. A loop of wire is a flat thing sitting somewhere in that space. We want a single number that says how much of the field is going through the loop. That number is called the magnetic flux, symbol Φ (Greek phi).
There is an older and rather beautiful way to picture Φ, and it is the way Faraday himself thought, because he was not a mathematician and did not think in cosines. He imagined the field as a bundle of lines, drawn closer together where the field is stronger — the picture we drew in Module 13 for electric fields and reused in Module 15 for magnets. Then:
The flux through a loop is proportional to the number of field lines that pass through the hoop. Faraday's picture — and every property of Φ is visible in it
One caution, because this picture has a limit like the bucket did. Field lines are not real objects you could count if you had a good enough microscope; how many to draw is a choice made by whoever draws the diagram. Draw twice as many and every count doubles. So line-counting tells you truthfully how Φ changes when you tilt, widen or move the loop — but only Φ = BA cos θ gives you the answer in webers.
Tilt the hoop and fewer lines thread it. Turn it edge-on and none do. Make the hoop bigger and it catches more. Move it into a stronger region, where the lines are packed tighter, and it catches more. Every property of Φ = BA cos θ is visible in the line-counting picture without a single symbol. The next bench does the counting for you — literally: it draws the field lines, then checks one by one which of them actually go through the loop, and adds them up. It uses the formula for one thing only — to say what a single line is worth. Everything after that is counting: which lines cross the hoop, and which way round they go. Then it shows the formula's full answer alongside, so you can watch the count reproduce the cosine on its own.
Push θ past 90° and watch the sign flip. This is not bookkeeping fussiness; it is going to matter enormously in the generator, where a loop spins all the way round and the flux swings smoothly from +BA to −BA and back, twice per revolution.
Why does nature care about this particular combination — field times area times cosine — rather than about the field at, say, the centre of the loop? Because a wire loop is not a point. Every bit of the loop is somewhere different, feeling a slightly different field. Flux is the honest way to summarise the whole loop's situation in one number. And it turns out that this particular summary is the one that controls what the loop does. That is not an accident of definition: it is a real theorem, called Stokes' theorem, which you will meet properly at university. Any other summary you might invent — average field, field at the centre, maximum field — fails to predict the experiment. Φ works.
Now we can say what the first bench was showing you, in one line.
Read it slowly, because it is doing several jobs at once.
dΦ/dt means rate of change. If you have not done calculus, read it as "how many webers the flux changes by, per second". If the flux goes from 0 to 0.02 Wb in one-tenth of a second, dΦ/dt is 0.2 Wb/s, and a single-turn loop gets 0.2 V. Nothing more mysterious than that. A weber per second is a volt — that is what the equation says, and it is worth pausing to notice that a unit built entirely out of magnetism turns out to be the unit of electrical push.
N multiplies. Wind the wire round twenty times and each turn is its own little loop, each getting its own EMF, all of them in series like twenty cells in a row. Twenty times the voltage. This is why a real generator or transformer is a coil and not a single hoop: turns are free voltage.
EMF, not voltage. We met this distinction in Module 14, with a battery's internal resistance. EMF is the energy given per coulomb pushed round — the cause. If the loop is a closed circuit, that EMF drives a current — call it I — equal to EMF / R, where R is the loop's resistance, and the terminal voltage you would measure is a little less. If the loop is broken — a coil with two loose ends — the EMF is still there, driving nothing, waiting.
It does not care why the flux changed. This is the deep part. There are three quite different ways to change Φ = BA cos θ:
| Change what | How you do it | What it becomes |
|---|---|---|
| Change B | Move a magnet nearer; switch on a nearby electromagnet; use an alternating current in a neighbouring coil | Bench 16A · the transformer |
| Change A | Slide one side of the circuit so the enclosed area grows; stretch or squash the loop | Motional EMF, the rod on rails |
| Change θ | Rotate the loop in a steady field | Every generator on Earth |
Faraday's law lumps all three together and treats them identically. Whichever route you take, the loop responds to the same number: how fast Φ is changing. That is a large claim, and it is one of the reasons this law is considered fundamental rather than a rule of thumb.
The black stripe on an old credit card, or a hotel key card, is a strip of tiny magnets. Swiping it drags those magnets past a coil at speed. Each little magnet's field sweeping past the coil is a change of flux, and each change of flux is a pulse of EMF — one bit of data. The reason you have to swipe rather than hold the card against the reader is Faraday's law: hold it still and dΦ/dt is zero, and the reader sees nothing at all.
We now have to account for that minus sign, and it is not a bookkeeping convention. It is conservation of energy, wearing a disguise.
Here is the argument, and I want to run it as a proof by contradiction — assume the opposite and watch the universe break.
Notice what Lenz's law does not say. It does not say the induced current opposes the field. It says it opposes the change in flux. If the flux is already large but shrinking, the induced current works to prop it up — flowing in the same sense as the field that is dying, trying to keep it alive. "Opposes the change" is the whole rule, and getting into the habit of asking "which way is Φ moving?" rather than "which way does B point?" will save you a great deal of trouble.
You drop a strong magnet down a vertical copper pipe. Copper is not magnetic — a magnet will not stick to it, and a copper pipe will not attract a paperclip.
What happens as the magnet falls through?
Copper is not ferromagnetic — meaning it cannot be turned into a magnet itself, and a magnet will not stick to it, unlike iron or steel. That is quite true, and it is why the answer surprises people. But copper is an outstanding conductor, and that is what matters here. As the magnet falls, the flux through each ring-shaped slice of pipe wall changes, and a current is induced going right round that slice. That current makes each slice into a temporary magnet, and by Lenz's law every one of them is oriented to oppose the magnet's motion: the slices below push up, the slices above pull back. The magnet slows until the magnetic drag exactly balances its weight, then drifts down at a nearly constant speed, sometimes taking several seconds to fall a metre. Bench 16C runs the whole thing step by step so you can watch the terminal speed appear on its own; the pipe wall is modelled as a stack of discrete conducting rings, calibrated once so the drag reads out cleanly rather than to match a specific real pipe to the millimetre.
Look at the energy panel at the bottom of that bench while it runs. The gravitational energy the magnet gives up is accounted for to the last millijoule. Some of it goes into kinetic energy, speeding the magnet up. The rest comes out as heat in the pipe. And once the magnet has stopped speeding up — once it has reached what is called its terminal speed, where drag balances weight — the kinetic energy stops growing, so from then on essentially all of the gravitational energy pours straight into heat. Nothing is lost and nothing is invented. This is the same ledger we built in Module 5, and it is still balancing, ten modules later, in a situation Module 5 could not have described.
And it is worth being clear about the mechanism of the heating: the induced currents in the pipe wall are real currents in a real resistance, so they dissipate I²R exactly as in Module 14. A copper pipe used this way gets measurably warm. The magnet's fall is being paid, in cash, into the pipe.
Those currents swirling round in a solid block of metal have a name: eddy currents. Modern roller coasters stop with them — the car carries a fin of copper or aluminium that passes between two rows of powerful magnets at the end of the ride. No contact, no pads, nothing to wear out, and the braking force grows automatically with speed, which is exactly what you want. The same trick stops trains, saws, and the drum of a treadmill. The catch, and it is a real one: eddy braking cannot hold you still. At zero speed there is no flux change and therefore no force at all, so every eddy brake needs a friction brake to finish the job.
| Idea | What it says |
|---|---|
| Flux Φ | Φ = BA cos θ. How much field the loop catches. Webers. Faraday's picture: the number of field lines through the hoop. |
| Faraday's law | EMF = −N dΦ/dt. Only a changing flux counts. A field that is strong but unchanging through a fixed loop does nothing; a steady field can still give an EMF if the loop moves or grows, because then Φ changes. |
| Three routes | Change B, change A, or change θ. The law cannot tell which you used. |
| Lenz's law | The minus sign. Induced effects oppose the change that made them. It is conservation of energy in disguise. |
The people who explain this best, in their own words. Nothing here is a summary of a summary.