• Lumidisk Feynman's Disk Paradox pixel-art physics T-shirt, white tee, front view

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Lumidisk

Feynman's Disk Paradox Physics

\[\oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Lumidisk sits perfectly still, charged studs on her disc and a coil humming at her core. Switch the coil off, and she spins.

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.

L_field = ε₀ ∫ r × (E × B) dV

  • L_field: angular momentum of the fields (kg·m²/s)
  • ε₀: vacuum permittivity (F/m)
  • r: position from the axle (m)
  • E: electric field (V/m)
  • B: magnetic field (T)
  • dV: volume element (m³)

The puzzle and its answer are in The Feynman Lectures on Physics, Vol. II, sections 17-4 and 27-6.

On the shirt

Front: Lumidisk, the pixel-art character, with the title. Back: Faraday's law, the field momentum density, field angular momentum and the conservation line, with the Lectures reference. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

För skalans skull: storlek M är 20 in (50.80 cm) bred, ungefär 3.143 × 1034 Plancklängder (lP = 1.616255 × 10−35 m). Ett måttband går snabbare.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

I sit perfectly still: charged studs on my disc, a coil humming at my core. Switch the coil off, and I spin.

Feynman set this puzzle in his Lectures. Nothing is moving, so where can the spin come from? As the current dies, the falling magnetic flux induces an electric field around the coil, ∮E·dl = −dΦ_B/dt, and that field pushes my charges round. Yet angular momentum is supposed to be conserved.

The answer is that it was there all along, stored in the static fields. Crossed electric and magnetic fields carry momentum density ε₀E × B, and so angular momentum. When the magnetic field dies, that angular momentum passes to the disc. Disc plus field, the total is conserved.

Equations

\[\oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}\]
\[\mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \frac{\mathbf{S}}{c^{2}}\]
\[\mathbf{L}_{\text{field}} = \varepsilon_0 \int \mathbf{r}\times(\mathbf{E}\times\mathbf{B})\,dV\]
SymbolMeaningUnit
\(\mathbf{E}\) electric field \(\mathrm{V/m}\)
\(\mathbf{B}\) magnetic field \(\mathrm{T}\)
\(d\mathbf{l}\) line element of a closed loop \(\mathrm{m}\)
\(\Phi_B\) magnetic flux through the loop \(\mathrm{Wb}\)
\(t\) time \(\mathrm{s}\)
\(\mathbf{g}\) momentum density of the field \(\mathrm{kg·m⁻²·s⁻¹}\)
\(\mathbf{S}\) Poynting vector, E × B / μ₀ \(\mathrm{W/m²}\)
\(\varepsilon_0\) vacuum permittivity, 8.854 187 8188 × 10⁻¹² F/m \(\mathrm{F/m}\)
\(c\) speed of light, 299 792 458 m/s (exact) \(\mathrm{m/s}\)
\(\mathbf{r}\) position from the axle \(\mathrm{m}\)
\(dV\) volume element \(\mathrm{m³}\)
\(\mathbf{L}_{\text{field}}\) angular momentum stored in the fields \(\mathrm{kg·m²/s}\)

Sources

  1. Feynman, Leighton & Sands, The Feynman Lectures on Physics, Vol. II, §17-4 (the paradox)
  2. Feynman, Leighton & Sands, The Feynman Lectures on Physics, Vol. II, §27-6 (field momentum)
  3. NIST CODATA: vacuum electric permittivity

Field notes

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