How does a light-mill work?
In 1873, while investigating infrared radiation and the element thallium,
the eminent Victorian experimenter Sir William Crookes developed a special
kind of radiometer, an instrument for measuring radiant energy of heat
and light. Crookes's
Radiometer is today marketed as a conversation piece called a light-mill
or solar engine. It consists of four vanes each of which is blackened on one
side and silvered on the other. These are attached to the arms of a rotor
which is balanced on a vertical support in such a way that it can turn with
very little friction. The mechanism is encased inside a clear glass bulb
which has been pumped out to a high, but not perfect, vacuum.
When sunlight falls on the light-mill the vanes turn with the black
surfaces apparently being pushed away by the light. Crookes at first
believed this demonstrated that light radiation pressure on
the black vanes was turning it round just like water in a water mill.
His paper reporting the device was refereed by James Clerk Maxwell
who accepted the explanation Crookes gave. It seems that
Maxwell was delighted to see a demonstration of the effect of radiation
pressure as predicted by his theory of electromagnetism.
But there is a problem with this explanation. Light falling on the
black side should be absorbed, while light falling on the silver
side of the vanes should be reflected. The net result is that there
is twice as much radiation pressure on the metal side as on the black.
In that case the mill is turning the wrong way.
When this was realised other explanations for the radiometer
effect were sought and some of
the ones that people came up with are still mistakenly quoted as the
correct one. It was clear that the black side would absorb heat from
infrared radiation more than the silver side. This would cause the
rarefied gas to be heated on the black side. The obvious explanation
in that case, is that the pressure of the gas on the darker size
increases with it's temperature creating a higher force on that
side of the vane. This force would push the rotor round. Maxwell
analysed this theory carefully presumably being wary about making a
second mistake. He discovered that in fact the warmer gas would
simply expand in such a way that there would be no net force from
this effect, just a steady flow of heat across the vanes. So it is
wrong, but even the Encyclopaedia Britanica gives this false
explanation today. As a variation on this theme, it is sometimes said that the motion
of the hot molecules on the black side of the vane provide the push.
Again this is not correct and could only work if the mean free path
between molecular collisions were as large as the container, but in
fact it is typically less than a millimetre.
To understand why these common explanations are wrong think
first of a simpler set-up in which a tube of gas is kept hot at one end
and cool at the other. If the gas behaves according to the ideal gas
laws with isotropic pressure, it will settle into a steady state with a
temperature gradient along the tube. The pressure will be the same
throughout otherwise net forces would disturb the gas. The density would
vary inversely to temperature along the tube. There will be a flow of
heat from the hot end to the other but the force on both ends will be
the same because the pressure is equal.
Any mechanism you might conjecture which would give a stronger force on
the hot end than on the cool end with no tangential forces along the
length of the tube cannot be correct since otherwise there would be a
net force on the tube with no opposite reaction. The radiometer is a
little more complex but the same principle should apply. No net force
can be generated by normal forces on the faces of the vanes because
pressure would quickly equalise to a steady state with just a flow of
heat through the gas.
Another blind alley was the theory that the heat vaporised gases
dissolved in the black coating which then leaked out. This outgassing
would propel the vanes round. Actually, such an effect does exist
but it is not the real explanation as can be demonstrated by
cooling the radiometer. It is found that the rotor then turns the other
way. Furthermore, if the gas is pumped out to make a much higher vacuum,
the vanes stop turning. This suggests that the rarefied gas is involved in
the effect. For similar reasons, the theory that the rotation
is propelled by electrons dislodged by the photoelectric effect is also
ruled out. One last incorrect explanation which is sometimes given is
that the heating sets up convection currents with a horizontal component
that turns the vanes. Sorry, wrong again. The effect cannot be
explained this way.
The correct solution to the problem was provided qualitatively by
Osborne Reynolds, better remembered for the "Reynolds number".
Early in 1879 Reynolds submitted a paper to the Royal Society in which
he considered what he called "thermal transpiration", and
also discussed the theory of the radiometer.
By "thermal transpiration" Reynolds meant the flow
of gas through porous plates caused by a temperature difference on
the two sides of the plates. If the gas is initially at the same
pressure on the two sides, there is a flow of gas from the colder to
the hotter side, resulting in a higher pressure on the hotter side
if the plates cannot move. Equilibrium is reached when the ratio
of pressures on either side is the square root of the ratio of
absolute temperatures. This is a counterintuitive effect due to
tangential forces between the gas molecules and the sides of the
narrow pores in the plates. The effect of these thermomolecular forces
is very similar to the thermomechanical effects of superfluid liquid helium.
The liquid, which lacks all viscosity, will climb the sides of its container
towards a warmer region. If a thin capillary is dipped into the
superfluid it flows up the tube at such speed that a fountain effect
is produced at the other end.
The vanes of a radiometer are not porous. To explain the radiometer,
therefore, one must focus attention not on the faces of the vanes,
but on their edges. The faster molecules from the warmer side strike
the edges obliquely and impart a higher force than the colder molecules.
Again these are the same thermomolecular forces which are responsible for
thermal transpiration. The effect is also known as thermal creep since it
causes gases to creep along a surface where there is a temperature gradient.
The net movement of the vane due to the tangential forces around the edges
is away from the
warmer gas and towards the cooler gas with the gas passing round the edge
in the opposite direction. The behaviour is just as if there were a
greater force on the blackened side of the vane (which as Maxwell
showed is not the case), but the explanation must be in terms of what
happens not at the faces of the vanes but near their edges.
Maxwell refereed Reynolds's paper, and so became aware of Reynolds's
suggestion. Maxwell at once made a detailed mathematical analysis of
the problem, and submitted his paper, "On stresses in rarefied gases
arising from inequalities of temperature", for publication in the
Philosophical Transactions; it appeared in 1879, shortly before his
death. The paper gave due credit to Reynolds's suggestion that the
effect is at the edges of the vanes, but criticised Reynolds's
mathematical treatment. Reynolds's paper had not yet appeared (it was
published in 1881), and Reynolds was incensed by the fact that
Maxwell's paper had not only appeared first, but had criticised his
unpublished work! Reynolds wanted his protest to be published by the
Royal Society, but after Maxwell's death this was thought to be
By the way. It is possible to measure radiation
pressure using a more refined apparatus. To make it work you have to use a
much better vacuum, suspend the vanes from fine fibers and
coat the vanes with an inert glass to prevent out-gassing. When
you succeed the vanes are deflected the other way as predicted by Maxwell.
The experiment is very difficult but was first done successfully
in 1901 by Pyotr Lebedev and also by Eenest Nichols and Gordon Hull.
Original papers by Maxwell and Reynolds:
"On stresses in rarefied gases arising from inequalities of
temperature" James Clerk Maxwell, Royal Society Phil. Trans. (1879)
"On Certain dimensional properties of matter in the gaseous state"
Osborne Reynolds, Royal Society Phil. Trans., Pt. 2, (1879)
Original papers on detection of radiation pressure:
P.N. Lebedev, Ann Phys. (Leipzig) 6:433 (1901)
E.F. Nichols and G.F. Hull, Phys Rev 13:307 (1901)
Historical details are taken from these sources:
"The genius of James Clerk Maxwell"
by Keith J. Laidler" in Phys 13 news of the University of Waterloo
Department of Physics
"The Kind of Motion that we Call Heat" S.G. Brush
Other useful articles about the radiometer:
"Crookes' Radiometer and Otheoscope" Norman Heckenberg,
Bulletin of the Scientific Instrument Society, 50, 40-42 (1996)
"Concerning the Action of the Crookes Radiometer"
Gorden F. Hull, American J. Phys., 16, 185-186 (1948)
"The Radiometer and How it Does Not Work"
Arther E. Woodruff, The Physics Teacher 6, 358-363 (1968)
General text books:
"Light", R.W. Ditchburn, Blackie & Son (1954)
"Kinetic Theory of Gasses", Kennard, McGrawHill (1938)
LightMill image and animation by Torsten Hiddessen
Thanks to Norman Heckenberg and Bob Ehrlich for useful comments.