Lumidisk: Feynman's Disk Paradox
Electromagnetism & Plasma arc, page 78 of 193. Back to The Pixelated Physics Universe
Who they are
Lumidisk sits perfectly still, charged studs on her disc and a coil humming at her core. Switch the coil off, and she spins.
Lumidisk is a character of the Electromagnetism & Plasma arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
A disc carries fixed charges and a coil at its centre carries a current. Nothing moves. Switch off the current and the falling magnetic flux induces an electric field that pushes the charges round, so the disc turns. The angular momentum was there all along, stored in the static electric and magnetic fields, and the total of disc plus field is conserved. The puzzle and its answer are in The Feynman Lectures on Physics, Vol. II, sections 17-4 and 27-6.
$$ \begin{gathered} \oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \\ \mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \frac{\mathbf{S}}{c^{2}} \\ \mathbf{L}_{\text{field}} = \varepsilon_0 \int \mathbf{r}\times(\mathbf{E}\times\mathbf{B})\,dV \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\mathbf{E}$ | electric field | V/m |
| $\mathbf{B}$ | magnetic field | T |
| $d\mathbf{l}$ | line element of a closed loop | m |
| $\Phi_B$ | magnetic flux through the loop | Wb |
| $t$ | time | s |
| $\mathbf{g}$ | momentum density of the field | kg·m⁻²·s⁻¹ |
| $\mathbf{S}$ | Poynting vector, E × B / μ₀ | W/m² |
| $\varepsilon_0$ | vacuum permittivity, 8.854 187 8188 × 10⁻¹² F/m | F/m |
| $c$ | speed of light, 299 792 458 m/s (exact) | m/s |
| $\mathbf{r}$ | position from the axle | m |
| $dV$ | volume element | m³ |
| $\mathbf{L}_{\text{field}}$ | angular momentum stored in the fields | kg·m²/s |
Sources
- Feynman, Leighton & Sands, The Feynman Lectures on Physics, Vol. II, §17-4 (the paradox)
- Feynman, Leighton & Sands, The Feynman Lectures on Physics, Vol. II, §27-6 (field momentum)
- NIST CODATA: vacuum electric permittivity
Crossovers
Lumidisk meets Secundar (Gauge Invariance). Lumidisk starts spinning when her coil switches off, which looks like angular momentum from nowhere. Secundar points to where it was hiding: in the electromagnetic field, whose angular momentum is gauge-invariant and real.
Lumidisk meets Homopole (Faraday Paradox). Homopole and Lumidisk are both spinning-disk puzzles in induction. Homopole's EMF depends on the disk moving through the field. Lumidisk's spin comes from a changing magnetic flux inducing an electric field. In both, total angular momentum and energy are conserved once the fields are counted.
Wear the lore
Concept keywords: Feynman's Disk Paradox, Lumidisk, Electromagnetism & Plasma, physics lore, Pixelated Physics