CYCLOTRON
T = 2πm/qB — no speed in it: five radii, one period, audited to 1e-15. The spiral obeys √k to the bit.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Two hollow D-shaped electrodes, a magnetic field through both, and a voltage that flips across the gap. Lawrence’s 1930 bet: if the orbital timing never needs adjusting, one fixed oscillator can accelerate a particle forever. The question is why the timing would never need adjusting.
The period
T equals two pi m over q B — no speed anywhere in it. Faster particles run bigger circles, and the two effects cancel exactly: every half-turn takes the same time whether the particle is crawling or tearing along. One frequency stays in step forever. That single cancellation is the entire machine.
The ledger
Each gap crossing deposits exactly q V of kinetic energy — an integer ledger. And since the radius grows as the square root of the energy, the spiral obeys r-k over r-one equals root k, audited to the bit. But watch the right panel: gamma is already creeping, and the period is drifting away from the oscillator. The classical cyclotron carries its own death certificate.
Twenty-five crossings
The sweep lays down endpoint-joined half-circles one crossing at a time. Each new arc begins exactly at the previous gap endpoint and its center shifts as the radius grows, producing one continuous orbit rather than disconnected concentric semicircles.
Audit
Audited: the period by formula and the period by circumference-over-speed agree at five different radii within one part in ten to the fifteen; the square-root spiral law is bitwise at k equals four, nine, sixteen, and twenty-five; the proton defaults lock at 65.5945 nanoseconds and 15.2452 megahertz; and the relativistic drift is quantified through massenergy’s own gamma — 0.106579 percent after twenty gaps.