01An honest mystery
Every wall socket on Earth traces back to this trick. In 1831 Michael Faraday wrapped a coil of wire, connected it to a sensitive meter, and moved a magnet nearby. Current flowed — but only while something was changing. A magnet parked inside the coil, however strong, does nothing at all.
That's the strange part. A battery pushes current because chemistry keeps its terminals charged. Here there is no battery — yet charges in the wire clearly feel a push. The push appears when the magnet moves and vanishes when it stops. Whatever creates it cares nothing for how strong the magnetic field is — only for how fast it is changing.
To make that precise we need one visual idea. Picture the magnet's influence as threads sprouting from one pole, arcing through space, and returning to the other — the field lines you see when iron filings arrange themselves. Now look at the coil and simply count the threads passing through it. That count (each thread weighted by field strength) is called the magnetic flux, Φ. Faraday's discovery, in one sentence: the wire feels a push proportional to how fast the thread-count through it is changing.
02The live experiment
Try this, in order: ① Drag the magnet slowly toward the coil and watch three things at once: the thread-count graph (Φ) rising, the push graph (EMF) showing a bump, the needle leaning. ② Stop moving — with the magnet deep inside the coil — and watch everything die to zero. Strong field, no change, no current. ③ Now yank the magnet through fast: the Φ curve is the same shape but steeper, so the EMF spike is taller. Same journey, more violence. ④ Flip the magnet: everything mirrors. ⑤ Turn on auto-sweep, sit back, and watch the machinery breathe — that, minus the hand, is a power-station generator.
Magnet & coil · dipole flux, Faraday's law, and a galvanometer with real needle dynamics
The magnet is modelled as a magnetic dipole; the flux through each turn uses the exact on-axis dipole formula, and the EMF is its true time-derivative under your hand's motion. The needle obeys its own differential equation (a damped torsion pendulum driven by the current), which is why it swings and settles like a real meter. The bookkeeping check integrates the EMF over time and compares it against −N·ΔΦ — Faraday's law says the two must match to numerical precision, no matter how erratically you drag.
03How to think about it
The rule has a personality, and it's a stubborn one: the circuit acts to oppose whatever you're doing to its flux.
Look at the arrows the simulation draws on the coil. Push the north pole in — flux through the coil grows — and the induced current circulates so that the coil becomes a weak electromagnet facing the intruder with its own north pole, pushing back. Pull the magnet away and the current reverses: now the coil makes a south pole, tugging the magnet back like a reluctant goodbye. This contrariness is Lenz's law, and it isn't spite — it's energy conservation wearing a personality. If the coil ever helped the change instead of opposing it, the magnet would be sucked in faster, making more current, sucking it in faster still — a free-energy avalanche. Nature doesn't do free.
The opposition is also why generating electricity is honest work. Drag the magnet with auto-sweep and imagine doing it by hand forever: every watt of electrical output shows up as a real mechanical drag on your arm. A power station is exactly this exhibit scaled up — steam, water, or wind forcing magnets past coils against that magnetic reluctance, megawatt by megawatt. The electricity lighting your screen right now was pushed through this very mechanism, against this very drag, a few hundred kilometres from where you sit.
The wrong picture — and where it breaks
Wrong idea #1: "The magnetic field pushes the current along." Then a strong stationary magnet should drive a strong steady current — and the experiment kills that idea in one move. Park the magnet dead-centre in the coil, where its field is at maximum, and the needle sits at exactly zero. The field itself pushes nothing. Only its change does. (What actually pushes the charges is an electric field that a changing magnetic field conjures into existence — a fact strange enough that it forms half of Maxwell's equations and, ultimately, the explanation of light.)
Wrong idea #2: "Moving it faster just makes the same current happen sooner." Faster motion makes a genuinely bigger push — steeper flux change, taller EMF spike. What stays the same is the spike's area: the bookkeeping chip shows ∫EMF·dt depends only on where the magnet started and ended, not on how fast it travelled. Speed buys current; the total charge shoved through the meter is fixed by geography alone. Slow drag: long low hum. Fast yank: short sharp shove. Same area under the curve — watch it.
Wrong idea #3: "You need the magnet to move." Only the relative change matters — move the coil instead and everything works identically (Einstein opened his 1905 relativity paper with exactly this observation). More broadly, anything that changes the thread-count works: spin the coil (that's an alternator), deform it, or change a neighbouring current (that's a transformer — induction with no motion at all).
04The law behind it
Faraday's law of induction
The electromotive force (the push, in volts) equals the rate of change of magnetic flux through the circuit — once per turn, so N turns multiply it. The minus sign is Lenz's stubbornness.
Flux is the thread-count made precise: Φ = ∫B·dA, the magnetic field integrated over the loop's area. In the simulation the dipole's on-axis flux through one turn of radius R at distance s is
— which is why the Φ graph swells as the magnet approaches and saturates inside the coil: past a point, no new threads are gained, and the EMF fades even while the magnet still moves.
Faraday's law runs civilization: generators and alternators (flux changed by rotation), transformers (flux changed by other currents), induction cooktops and wireless chargers (eddy currents in your pan and phone), electric-guitar pickups, credit-card readers, and the regenerative braking that recharges an electric car — all one mechanism: change the thread-count, get a push.
About this exhibit: flux uses the exact on-axis dipole formula per turn; EMF is its analytic derivative under the magnet's measured velocity; the galvanometer needle integrates a damped torsion-oscillator ODE; the running ∫EMF·dt vs −N·ΔΦ comparison is a live self-check of Faraday's law. Field lines are integrated streamlines of the dipole field. Hero image is an AI-generated illustration (Nano Banana via the KIE API). Single offline file — view source for the numerics.