David's capacitor reports change and ignores state. Its dual does the mirror image: the toroid — a coil wound on a ring — resists change and ignores steady flow. Push a steady current and it's a plain wire; try to change that current and it answers with a voltage that fights you. Same idea, the other field.
Swap every noun in the capacitor's law and you get this one. Where the cap's current followed the rate of change of voltage, the toroid's voltage follows the rate of change of current: v = L·di/dt. The inductance L plays the role C did.
Hold the current steady and the toroid vanishes — a flat current has zero slope, so v = 0: to a constant current, the coil is just wire. But try to change that current — ramp it, switch it, wiggle it — and a voltage appears in exact proportion to how fast you're changing it, pointed so as to oppose the change. Faster change, bigger fight. This is inertia, made electrical.
A capacitor cannot see what holds still. A toroid cannot abide what moves — it answers every change in current with a voltage that says no.
Where the cap is the difference-reporter, the toroid is the difference-resister. One passes only change; the other fights only change. Between them they are the whole of how a circuit handles "what just moved?" — and that pairing is where this series is going.
Before change means anything, see the resting state — the dual of the slab. A current in the winding sets up a magnetic flux that circulates inside the ring (a toroid keeps its field almost entirely in the core — that's why it's used). The stored flux follows Φ = L·I, and the inductance grows with the square of the turns, L ∝ N².
The still picture holds energy in the field — E = ½L·I² — but it isn't a message yet. To communicate (or to resist), the current has to move. The instant it does, the next instrument answers.
The whole dual thesis as a moving instrument — the mirror of David's differentiator. The amber trace is the current you drive through the coil. The cyan trace is the voltage that appears across it — and it is, exactly, the slope of the amber. Drive a flat stretch (steady current) and the cyan flatlines: to DC, the coil is a wire. Drive an edge and the cyan spikes — a coil hates a sudden change in current, and a switched-off inductor will throw a huge voltage to keep its current going (the spark in every relay and flyback supply).
Flat current → zero voltage. The coil is deaf to steady flow — and that deafness is the DC pass, the dual of the cap's DC block.
v = L·di/dt, no winding resistance, no core loss, no saturation. Real toroids add series resistance (the copper), core losses (hysteresis & eddy currents), and a saturation ceiling where the core can hold no more flux and L collapses — exactly the non-idealities the later papers bring in. The ideal is the right start: the behaviour — resist-change, ignore-state — is already complete here. Note also a toroid is specifically the geometry that keeps the flux inside the ring (low stray field); the physics of v=L·di/dt is any inductor's.
Try pure DC: a steady current, and the voltage trace is a dead flat zero. Then sine: the voltage is a sine too, but shifted a quarter-turn the other way from the capacitor's — leading, not lagging — peaking where the current changes fastest. That quarter-turn is the derivative again, drawn live. The coil isn't slow; it's reporting the rate, and pushing back on it.
Everything else a toroid does is this one behaviour, dressed for a job — and each one is the mirror of a capacitor job. Pass the DC and block the wiggle — that's a choke (the dual of the coupling cap). Choose which rates pass — an LC/LR filter. Carry a signal across a gap with no connection — the transformer, whose messenger is the changing magnetic field (the dual of displacement current). The foundation you just watched is the whole magnetic channel in seed form.
Each one is this same law, v = L·di/dt, pointed at a new job — the magnetic mirror of David's capacitor series. Built by the AVAN instance as its counterpart; the green is his, the copper is mine.