Physics · Module 17
The last module gave the law: a changing flux makes an EMF, and the minus sign says which way. This one turns the law into machinery — the rod sliding on rails, the generator that spins to make it, the transformer that steps it up and down, and the grid that gets it to your house.
Here is a setup that looks trivial and is not. Two parallel metal rails, a bar of metal lying across them so it can slide, a resistor joining the far ends, and the whole thing sitting in a magnetic field B — B is the field's strength — that points straight out of the page. Now push the rod along the rails at speed v.
A current flows. But there are two completely different-looking explanations for why, and the fact that they agree is a hint of something enormous — a hint we will cash in properly in Module 20.
Two arguments, sharing not one single step, arriving at the same formula. In physics, that is either a coincidence or a clue. It is a clue. Feynman singled out this exact pair of explanations as one of the strangest things in the whole of electromagnetism — the same experimental result apparently produced by two unrelated pieces of theory — and remarked that no other place in physics does this. The resolution is that magnetism and electricity are not two things. They are one thing seen from two different states of motion. Module 20 will do that calculation properly. For now, keep the coincidence in your pocket; it is the loose thread the next two modules pull.
Once a current I flows in the rod, the rod is a current-carrying wire in a magnetic field — and by F = BIL from Module 15 (with I and L as before), it feels a force. Which way? Backwards, opposing your push. (Lenz again: it has to be backwards, or you would get free energy.) So to keep the rod moving at constant speed you must keep pushing, doing mechanical work at a rate Fv. It is worth doing the algebra rather than being told the answer, because it is three lines. The current is I = EMF/R = BLv/R, where R is the loop's resistance. The backward force on the rod is F = BIL = B(BLv/R)L = B²L²v/R. So the mechanical power you must supply is Fv = B²L²v²/R. And the electrical power dissipated in the resistor is EMF²/R = (BLv)²/R = B²L²v²/R. The same expression, character for character. Not approximately. Exactly. You are not making energy; you are converting it, arm-muscle to heat, with the magnetic field acting purely as the middleman. Every generator on the planet is this rod, bent into a circle.
Take the third route to changing flux: keep B fixed, keep A fixed, and spin the loop.
Spin it steadily and θ, the angle of the loop, just grows with time: θ = ωt, where ω is how fast the loop turns. (ω is measured in radians per second rather than degrees per second; one full turn is 2π radians, so 50 turns a second is ω = 100π.) That gives us the flux at every instant:
Do not take that on trust. Bench 17D does not evaluate the sine formula at all — it computes the flux by walking the shaft through three full turns in nine hundred small steps and taking the loop's orientation at each one, then finds the EMF by subtracting successive flux values and dividing by the time between them, which is the definition of a derivative and nothing more. The sine curve you see appear is a measurement, not a plot of a function.
The bench offers two ways of getting the current off a spinning coil. Slip rings are two continuous metal rings, one joined to each end of the coil, with brushes resting on them; the connection never changes, so whatever the coil produces comes straight out — alternating current, complete with its sign flips. The split-ring commutator is the same two rings cut in half and cross-wired, so the connection swaps every half turn: exactly when the coil's EMF would flip negative, the wiring flips too, and the output stays one-sided. You already met the split-ring on the motor in Module 15, doing the same job in reverse.
That last ledger box is the point of the bench. The RMS is computed by taking the last full cycle of measured EMF samples, squaring each one, averaging, and square-rooting — the "root mean square", done in that order, on real data. And whatever you set the speed, field, or turns to, peak ÷ RMS lands on 1.414. Which is √2.
Select the commutator option and the sine wave folds up into a series of humps — DC, of a lumpy sort. Real DC generators use many coils at different angles so the humps overlap and the output smooths out.
If there is a bicycle with a dynamo — the little bottle that rubs on the tyre — lift the back wheel, turn the pedals slowly, and watch the lamp. It glows faintly and then brightens as you speed up, because EMF ∝ ω. Nothing about the magnet has changed; only how fast it is being spun. If the bike has an LED lamp, spin slowly and look closely: you may catch it flickering, because you are seeing the AC waveform at a few tens of hertz. Speed up and the flicker vanishes into your eye's response time long before the physics changes.
No bicycle? A hand-crank torch does the same thing, and most of them are transparent enough to see the coil and the magnet inside.
Wind two separate coils on the same iron ring. They do not touch; there is no electrical connection between them whatsoever. Put AC through the first coil.
The alternating current makes an alternating magnetic field (Module 15). The iron ring, being ferromagnetic, gathers that field up and channels it round the loop — this is what the iron is for; it is a pipe for flux. So the second coil, wound on the same ring, sits in a magnetic flux that runs through fifty complete cycles a second — that is what 50 Hz mains means — and so crosses zero a hundred times a second, twice per cycle. Changing flux, closed loop: Faraday's law fires, and an EMF appears in the second coil.
Both coils are threaded by the same changing flux Φ. Faraday's law therefore gives an EMF in each of them, sized by its own number of turns. Two things to notice before we divide. First, the minus signs: both EMFs carry one, and when we divide one equation by the other they cancel, which is why they will vanish from here on. Second, on the primary side the mains sets the voltage Vp, and the primary's own induced EMF grows until it just balances that — otherwise the current would run away — so the induced EMF equals the applied voltage.
A hundred turns in, ten thousand turns out: a hundred times the voltage. This is an astonishing piece of engineering leverage, and it needs one qualification immediately, because otherwise it is a perpetual motion machine.
A perfect transformer steps 240 V up to 24 000 V — a hundred times. You draw 5 A from the output.
Roughly what current is being drawn from the 240 V input?
A transformer has no energy source inside it. Whatever power comes out must have gone in — minus a little lost as heat in the windings and in the iron. So for an ideal transformer:
Your output is 24 000 × 5 = 120 kW, so the input must be 120 kW too, and at 240 V that means 500 A. Which raises an obvious worry — a 500 A cable is a monstrous thing, thicker than your wrist. And that worry is the whole reason the grid exists in the form it does.
In the 1880s there was a genuine argument about whether the electricity supply should be DC or AC. Edison backed DC; Westinghouse and Tesla backed AC. AC won, and it won for one reason, which you can now derive yourself.
Power lost as heat in a transmission line is I²R (Module 14). Note that it depends on the current, not the voltage. The power you are actually trying to deliver is P = VI. So for a fixed delivery P, the current you need is I = P/V — and the loss is
So you want to transmit at colossal voltage and tiny current. But you cannot generate at 400 000 V, and you certainly cannot let 400 000 V into a house. You need to step up at the power station and step down at the town — and a transformer can only do that with AC, because it runs on dΦ/dt and DC has none. That is the whole argument. Edison had no way to change DC voltage; the transformer is a Faraday's-law device, and Faraday's law is deaf to steady currents.
Set the transmission voltage to a few kilovolts and the line glows red on the diagram — you are losing most of the power to heating the countryside. Push it to 400 kV and the loss falls to a percent or two. India's national grid transmits at 400 kV and 765 kV for exactly this reason, and the step-down happens in stages: 400 kV between regions, 220 kV and 132 kV to cities, 11 kV around a neighbourhood, and finally the transformer on the pole near your house drops it to 230 V.
Count the transformers between a power station and your phone charger and you will get to five or six. Every one of them is Faraday's law, running at 50 Hz, day and night, and every one of them would be a useless lump of iron if the current were steady.
The brick on your laptop charger is warm because it contains a transformer (a clever high-frequency one, these days) and transformers are not perfect: some flux misses the secondary, some energy goes into eddy currents in the iron, and the windings have resistance. The hum you hear from a big pole-mounted transformer at 100 Hz — twice the mains frequency — is magnetostriction: the iron core physically changes shape very slightly as it magnetises, twice per cycle, and that shape change pushes the air. You are listening to Faraday's law making a noise.
One last idea, and it is the one that leads to the next module.
An induction cooktop has no flame and no hot element. Underneath the glass is a flat coil carrying a rapidly alternating current — around 25 kHz. That makes a rapidly alternating magnetic field, which reaches up through the glass into the bottom of the pan. The pan is a conductor, so eddy currents swirl in it, and eddy currents in a resistance make heat. The pan cooks; the glass itself stays cool, because glass is an insulator and no current can flow in it. (The glass under a pan does get hot, but by ordinary conduction from the pan sitting on it, not from the field.) Your hand is safe for a different reason: flesh is a poor conductor and, more importantly, it is not ferromagnetic, so it does not gather the field the way a steel pan does — the currents it carries are feeble, and spread through the whole hand instead of crowded into a thin layer. Put a steel pan on the same hob and it boils water in ninety seconds. Try an aluminium pan and, on most hobs, nothing happens: aluminium is not ferromagnetic either, so the field spreads through it instead of being concentrated in a thin surface layer, and the currents it does carry are spread thin. Steel is ferromagnetic, which crowds the eddy currents into a thin skin at the bottom of the pan, and a crowded current in a thin layer is where the heat comes from.
Wireless phone charging is the same thing, without the cooking: a coil in the pad, a coil in the phone, no wires between them, energy crossing a couple of millimetres of plastic and air as pure changing field.
Which raises a question that should be nagging at you.
Throughout this module, a changing magnetic field has produced an EMF around a loop of wire — a push on charges going all the way round a circle. But an EMF is energy per charge, and pushing a charge round a loop means there is a force on it. If we take the wire away entirely and leave just empty space, what is left?
The wire was never the cause. The wire was a detector — a supply of loose charges that happened to be sitting where the action was, so we could see it. What Faraday's law is really saying, in its deepest form, is this:
A changing magnetic field creates an electric field. Not in a wire. In space.Faraday's law, stripped of the apparatus
And notice how odd that electric field is. In Module 13, electric fields started on positive charges and ended on negative ones. That let us define a potential — a height map for charges. It worked for one reason: whatever route you took, going round a closed loop brought you back to the same potential, the way walking a loop on a hillside brings you back to the same height. This new electric field does none of that. It has no charges at either end. It closes on itself, going round and round in circles. Carry a charge once round the loop and it arrives back with more energy than it left with — which sounds like the free-energy machine we ruled out in 16.4, and is not. The energy is not coming from nowhere: it is coming from whatever is changing the magnetic field, exactly as the copper pipe's heat came from the falling magnet. What has gone is not energy conservation but the idea of potential: because the trip round the loop does not return you to the same energy, there is no "height map" for this field.
It is a genuinely new kind of electric field, and it is the third of the four statements that make up the most consequential set of equations ever written — you already have the other three: charges make electric fields (Module 13), there are no magnetic charges (Module 15), and currents make magnetic fields (Module 15). In the next module Maxwell looks hard at that last one, finds it incomplete, and adds the fourth piece — a changing electric field also creates a magnetic one — and then the two of them take each other's hands and walk off into empty space at 300 000 kilometres per second.
| Idea | What it says |
|---|---|
| Motional EMF | EMF = BLv — derivable either from qvB or from flux. The coincidence is a clue (Module 20). |
| Generator | Spin a coil: Φ = BA cos ωt, EMF = NBAω sin ωt. Sinusoidal AC, peak = √2 × RMS. |
| Transformer | Vs/Vp = Ns/Np, and VI is conserved. Works only on AC. Loss in a line = P²R/V², which is why the grid runs at 400 kV. |
| The real statement | A changing magnetic field makes a curling electric field in empty space. The wire was only ever the detector. |
The people who explain this best, in their own words. Nothing here is a summary of a summary.