Leave a powerful magnet sitting inside a coil of wire for a hundred years and the coil will produce exactly zero volts. Slide the same magnet through in a tenth of a second and you get a real, measurable pulse. The magnetic field was never the point — the change was. That one sentence, written down by Faraday in 1831, is why every power station on Earth is a machine for turning something round, and why a magnet dropped down a copper pipe falls as though the pipe were full of syrup.
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The coil does not respond to the magnet. It responds to the rate at which magnetic flux through it is changing. Watch the blue curve (flux) and the amber curve (voltage): the voltage is the slope of the flux, nothing more. It peaks where the flux is changing fastest — which is not where the magnet is closest — and it is exactly zero at the moment the magnet sits dead centre, because that is where the flux is momentarily flat. Choose Hold it still and the amber line lies flat on zero for as long as you care to watch, even though the magnet is right inside the coil. Then take You move it and drag the slider: you are the source of every volt.
A metal rod slides along two rails through a magnetic field. The circuit's area is growing, so its flux is growing, so a voltage appears and a current flows — and that current, in that field, feels a force. Lenz's law says the force pushes back against the motion, and the two power figures below say why it must: the mechanical power your hand supplies and the electrical power the resistor turns into heat are the same number to every digit shown. Flip the sign and the force would drive the rod instead, so the rod would speed up, so the current would grow, so the force would grow — the pale curve below, climbing off the top of the chart, powering itself out of nothing forever. The minus sign is the universe declining to do that.
Turn the coil steadily in the field and the flux through it goes as cos θ, so the voltage goes as sin θ — a sine wave, out of nothing but rotation. Notice where the voltage peaks: not when the coil faces the field squarely and the flux is largest, but a quarter turn later, when the coil is edge-on and the flux is passing through zero. That is the whole idea of an alternator, and it is why mains electricity alternates at all. A two-pole machine turning at 3000 rpm gives 50 Hz; at 3600 rpm it gives 60 Hz, which is the only real difference between the two halves of the world's wall sockets.
Faraday's law does not care what makes the flux change. Wrap a second coil around the same iron core and the alternating flux from the first induces a voltage in the second, in proportion to the turns each of them has. Nothing passes between them but the field. An ideal transformer changes voltage and current in opposite directions and leaves the product alone, so it is not a source of power — it is a lever. The second chart shows what that lever is for: resistive loss in a cable goes as I², so sending a megawatt at 400 kV instead of 11 kV cuts the loss by a factor of about 1300, which is the entire reason the pylons outside your window carry the voltage they do.
Why the grid runs at absurd voltages. One megawatt, down a line with the resistance you choose. The current needed falls as you raise the voltage, and the loss falls as the square of it.
Drop a strong magnet down a plastic tube and it falls the way anything falls. Drop it down a copper tube of identical size and it drifts down at a slow, eerily constant speed, even though copper is not magnetic and never touches it. As the magnet moves, the flux through each ring of the tube wall changes, currents circle in the metal, and by Lenz's law those currents oppose the change — which means they pull upward. The faster it falls the harder they pull, so it settles at the speed where the pull exactly balances its weight. Nothing is stored: every joule of height it loses comes out as heat in the copper.
| tube material | conductivity (MS/m) | terminal speed | 1 m takes |
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The laws are exact and were checked against their closed forms before this page was written. The generator's peak output is NBAω — a 200-turn, 100 cm² rotor in 0.5 T at 3000 rpm gives 314.159 V peak and 222.144 V rms, and differentiating the flux numerically reproduces that to within 10−7 V. The transformer conserves power to machine precision on every setting. The sliding rod's mechanical and electrical powers agree exactly, and the copper-tube motion matches vt(1 − e−t/τ) to about 4×10−15 m/s. The flux of a magnet through the coil in step 1 uses the exact point-dipole result Φ = μ₀ m a²/2(a²+z²)3/2, and its integral over a complete pass comes out at zero, as the flux rule demands.
The simplifications are real, though. The transformer is ideal: no core loss, no leakage flux, no winding resistance, no magnetising current, and no phase shift at all — a real 240 V unit runs at 95–99% efficiency and its secondary voltage sags under load, which this one never does. Nothing on this page includes self-inductance, so no circuit here resists a change in its own current; add that and the sliding rod's current would lag, and the generator would show a phase angle between voltage and current. The falling magnet is modelled as a point dipole with a single lumped drag constant k, calibrated so that a 12 g magnet in a 2 mm copper wall reaches about 0.5 m/s — matching the classroom demonstration, where a metre of copper pipe takes roughly two seconds against 0.45 s of free fall. Deriving k honestly from the tube's geometry and conductivity is a genuinely hard problem, and the numbers here would not survive it unchanged. Treat the material comparison as the right ordering and the right order of magnitude, not as a measurement.