A motor with two wrong explanations
The Mill Light Alone Does Not Explain
A Crookes radiometer spins silver-side-leading, opposite the direction pure light pressure predicts. Equal-pressure cancellation corrects only the equilibrium broad-face story. Out of equilibrium, area, edge and shear contributions have geometry- and regime-dependent weights.
Layer one · the wrong-way proofAsk light which way
A black surface absorbs a light ray's momentum. A perfect mirror sends the ray back, so its momentum change is twice as large. Set the vane and lamp below. The page recomputes all three forces from F = IA/c.
The silver face gets the larger push, so it trails behind.
The black face recedes from the lamp. The observed sign is opposite.
Calculating the sign and scale.
Calculating the disclosed spin-up comparison.
That is already fatal twice over. The ideal light-pressure torque points the wrong way, and its force is tiny beside even a deliberately gentle model of the force needed to spin up a rotor. The comparison is not an observation of a particular instrument. It is a replaceable mechanics model, surfaced in the check below so that no canned weakness ratio has to be trusted.
Layer two · the second correctionHotter is faster, but not harder
The quick replacement story says that molecules rebound faster from the hot black face and push it harder. Stop at that sentence and the explanation is still wrong. In a steady dilute gas, the warmer side is also less dense. The normal momentum flux over a broad, uniform face is pressure, p = nkT. Turn up the temperature split and watch the factors cancel.
Calculating density and molecular speed.
Reference density and molecular speed.
Calculating the broad-face momentum flux.
Now let the molecules find the edge
At an edge, gas can travel between cold and hot surface regions through a temperature gradient. The broad-face cancellation no longer tells the whole story. Rarefied-gas theory separates area, edge and shear contributions, with their balance depending on geometry and Knudsen number. This instrument shows the pressure scaling, not a universal force law for every commercial bulb.
The curve joins two published asymptotic behaviours: force proportional to √Kn in the small-Kn limit and to 1/Kn in the large-Kn limit. Its amplitude coefficient and the use of vane width as the characteristic length are free model choices. They are not fitted measurements.
What survives both correctionsThe motor lives in a narrow country
Too much gas
When the mean free path is much shorter than the vane scale, collisions restore near-continuum behaviour and viscous drag is strong. The rarefied edge imbalance becomes ineffective.
Too little gas
When the mean free path is enormous, the available molecular flux falls with pressure. There are too few impacts to maintain the torque. In a sufficiently good vacuum, the mill stops.
Between those limits, molecules sample both temperatures before ordinary collisions erase the gradient. Reynolds connected radiometer motion to thermal transpiration. Einstein later estimated a force that survives near a vane edge. Modern experiments and kinetic simulations make the honest story less tidy: area forces, edge thermal stresses and shear can all matter, and which dominates changes with pressure and construction.
The check
These rows are recomputed in this browser from the same equations that drive both instruments. The independent Node verifier starts again from constants and assertions.
| quantity | live result |
|---|---|
| Exact speed of light | calculating |
| Absorbing-face force | calculating |
| Reflecting-face force | calculating |
| Reflecting / absorbing ratio | calculating |
| Light-only direction | calculating |
| Modelled force per vane for spin-up | calculating |
| Spin-up demand / light difference | calculating |
| Broad-face hot / cold momentum flux | calculating |
| Pressure-model peak | calculating |
| Illustrative edge-force estimate | calculating |
Model assumptions, uncertainties and free choices
- Measured or defined: the speed of light and Boltzmann constant are exact SI defining constants. The silver-leading direction is a documented observation, not a measurement made by this page.
- Ideal optics: the light bench assumes normal incidence, equal illumination, one perfectly absorbing face and one perfectly reflecting face. Real black paint reflects some light and real metal absorbs some. Those departures change magnitude, not the ideal sign comparison shown here.
- Spin-up comparison: the rotor is modelled as four aluminium square vanes, each with the live area, at a fixed arm radius and foil thickness. The target speed and spin-up time are choices. Bearing friction and gas drag are omitted, so the displayed demand is not an observed radiometer force.
- Broad-face cancellation: the equality uses a steady, locally equilibrated ideal gas at equal pressure on both broad faces. A real radiometer is non-equilibrium precisely near gradients and boundaries.
- Pressure curve: the mean-free-path conversion assumes air represented by a single hard-sphere molecular diameter of 3.7e−10 m. Changing gas species changes that diameter, the mean free path, Knudsen number and horizontal peak location. Mean temperature, characteristic length, interpolation shape and amplitude coefficient are also model inputs. The curve reproduces the correct vanishing limits and published asymptotic scalings. Its peak location and force amplitude are illustrative, not a prediction for an identified bulb.
- Gas-surface accommodation: the illustrative force law has no separate accommodation coefficient, so its free amplitude coefficient implicitly holds that surface dependence fixed. Real accommodation can alter the force scale and the relative weights of area, edge and shear contributions.
- Mechanism: thermal transpiration and edge effects are real. The literature does not support one geometry-independent claim that a single edge term always dominates. Area and shear terms can be important in other Knudsen regimes.
Machine check: node research/the-mill-light-couldnt-turn/verify-the-mill-light-couldnt-turn.mjs
Sources and honest apparatus
- OpenStax, University Physics, volume 2, section 16.4. Derives I/c for a perfect absorber and 2I/c for a perfect reflector at normal incidence.
- National Institute of Standards and Technology, The International System of Units, SI Brochure, section 2. Gives the exact defining values used for c and k.
- Osborne Reynolds, “On Certain Dimensional Properties of Matter in the Gaseous State,” Philosophical Transactions of the Royal Society of London, volume 170, 1879, pages 727 to 845, doi:10.1098/rstl.1879.0078. The foundational thermal-transpiration treatment and radiometer discussion.
- Albert Einstein, “Zur Theorie der Radiometerkräfte,” Zeitschrift für Physik, volume 27, 1924, pages 1 to 6, doi:10.1007/BF01328006. A calculation of the residual force associated with the temperature transition near vane edges. The page does not present its simple curve as Einstein's formula.
- Andrew Ketsdever, Natalia Gimelshein, Sergey Gimelshein and Nathaniel Selden, “Radiometric phenomena: From the 19th to the 21st century,” Vacuum, volume 86, issue 11, 2012, pages 1644 to 1662, doi:10.1016/j.vacuum.2012.02.006. Modern review of area, edge and shear mechanisms and their regime dependence.
- N. Selden, C. Ngalande, S. Gimelshein, E. P. Muntz, A. Alexeenko and A. Ketsdever, “Area and edge effects in radiometric forces,” Physical Review E, volume 79, 2009, article 041201, doi:10.1103/PhysRevE.79.041201. Experiments and kinetic simulations across a broad Knudsen range.
- Hassan Akhlaghi, Ehsan Roohi and Stefan Stefanov, “A comprehensive review on micro- and nano-scale gas flow effects: Slip-jump phenomena, Knudsen paradox, thermally-driven flows, and Knudsen pumps,” Physics Reports, volume 997, 2023, pages 1 to 60, doi:10.1016/j.physrep.2022.10.004. Source for the two rarefied-gas asymptotic force scalings joined by the illustrative pressure curve.
- University of Queensland Physics Museum, “Crookes' Radiometer (Modern)”. Documents silver-side-leading rotation and the opposite radiation-pressure prediction for a museum instrument.
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