How a field pushes a moving charge: F = q(E + v × B). The electric part shoves along E and does work. The magnetic part is turned 90° from the motion — it steers the charge in a circle and never adds a single joule. Down the center the state goes in, the engine computes the force live, and the proven invariants come out. The blue team builds it; the red team tries to make the magnetic force do work.
source H. A. Lorentz, Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern (Leiden: Brill, 1895) — the force on a moving charge. No stable canonical DOI; scan at archive.org/details/versucheinerthe00loregoog. AMBER Rendered, not quoted.
The force on a charge q in fields E, B moving at velocity v is one vector with two halves:
electric qE — parallel to E, along the motion it can push or brake, so it changes speed and energy. magnetic q(v×B) — a cross product, so it is perpendicular to both v and B. Perpendicular to v means zero work: it bends the path but leaves the speed alone.
Live decomposition for the panel's current state:
| vector | x | y | z |
|---|
Two neighbours give you where fields come from: the-ampere-law makes B from a current; the-faraday-law makes an EMF from a changing flux.
Lorentz supplies the missing half — what the field then does to a charge. F = q(E + v×B) closes the loop: fields act on charges, charges move, moving charges are currents that remake the fields. The magnetic term being workless is why a static B can confine a plasma forever without feeding it energy. Each sphere is the next one's premise.
The blue team's live re-check: over many random states, confirm v·(v×B) = 0 (no work), that the E×B drift zeroes the net force, and that a simulated orbit keeps its speed. If red tampers, this badge is where it shows.
A single charge is fully described by five inputs. Feed them to the panel and the force is fixed:
| symbol | is | kind |
|---|---|---|
| q | charge (sign & size) | scalar |
| m | mass | scalar |
| v | velocity | vector |
| E | electric field | vector |
| B | magnetic field | vector |
The only nonlinearity is the cross product v×B — a right-hand-rule turn that makes the magnetic force depend on direction of travel, not just position. That is what you feed the engine below.
B points into the page. The dot is the charge; the cyan arrow is v, the amber arrow is F. Watch F stay perpendicular to v — it turns the charge without ever speeding it up.
Change any control — the force, radius, cyclotron frequency and work-rate are computed from F = q(E + v×B) on the spot, never looked up.
What the machine proves, live and to tolerance:
1 the magnetic force is workless — v·(v×B) = 0 exactly. 2 with v⊥B a charge circles at radius r = mv/(qB), obeying q v B = m v²/r and ω = qB/m. 3 v ∥ B gives zero magnetic force. 4 crossed E,B produce a charge-independent drift vd = (E×B)/B² that cancels the net force.
The blue witness (left) confirms these live; the red team (right) tries to make the magnetic force do work.
It is also frame-dependent in its split: what looks purely magnetic in one frame is partly electric in another. The total four-force is the invariant; "the magnetic force does no work" is a true statement about this frame, not a frame-free law.
"Magnetic fields do work on moving charges." Cut. q(v×B) is perpendicular to v, so v·F = 0 — zero work, always. A magnet that seems to do work (a motor) is spending energy through the electric field driving the current.
"F = qv×B is the Lorentz force." Cut. That is only the magnetic half. Drop qE and the charge can never gain or lose energy — no acceleration, no drift start, nothing.
"The magnetic force curves the path, so it must add energy." Kept, corrected. It redirects momentum without changing its magnitude: the speed, and the kinetic energy, stay fixed while the direction turns.
"Lorentz derived this from relativity." Cut. 1895 predates Special Relativity (1905). Einstein built on Lorentz's transformations — not the reverse.
The red team's move: swap the cross product v×B for the dot product v·B (a scalar, laid back along v). The magnetic force is no longer perpendicular — and it starts doing work.
Replace the cross with a dot and the magnetic force gains a component along v. The witness (window 7) recomputes v·F, finds it nonzero, and turns red. Nothing is faked; the attack is real and it is caught.