D. Fun with polarizers
In one respect, Einstein's mathematical analysis (like Rayleigh's earlier one) proves quite accurate, in a way that's easy to demonstrate. This has to do with how the sky's scattered light is polarized.
Try looking at a patch of clear sky through one lens of a pair of polarizing sunglasses while you rotate the lens. You'll notice that the sky looks brighter as you look through the lens in some positions, and darker when the lens is in other positions. If the sun is not far from the patch of sky you're looking at, you'll find that the sky looks brightest when the sun is to the left or right of the lens, and darkest when the sun is "above the top" or "below the bottom" of the lens. Why is this?
Any kind of wave-whether sound wave, water wave, light wave-is associated with one obvious direction: its direction of travel. But some waves also possesses another significant direction perpendicular to that one. When a rope is held at one end and shaken from side to side, a wave will travel down the length of the rope. While the wave travels along the rope in the direction of its length, each point along the rope also has a side-to-side motion perpendicular to this.
Light waves also have a "side-to-side" direction. As we've noted, a light wave is a wave in an electromagnetic force field. Whenever a light wave passes by an electrically charged object, it tends to accelerate that object along the same line in which the light wave's electric force field vibrates. This direction is always out of the light wave's path, perpendicular to the wave's own direction of travel.
In particular, a scattered light wave's electric field is perpendicular to its direction of travel. It turns out that a scattered wave will be most intense if its direction of travel is also perpendicular to the electric field of the original wave. But if the scattered wave's direction is not perpendicular to the original wave's electric field, the scattered wave will be less intense-its own electric field will be weaker. Indeed, a "scattered wave" traveling parallel to the original wave's electric field would have zero strength-that is, the original wave can never be scattered into the direction of its own electric field.
The side-to-side polarization of light waves makes polarizing sunglasses useful. The electric fields of light waves scattered from horizontal surfaces mostly vibrate horizontally themselves. Since large horizontal surfaces are so common outdoors (bodies of water, roads, vehicles), most outdoor glare is light waves whose electric fields vibrate horizontally. Polarizing sunglasses filter out waves of glare more than other light waves because their lenses contain molecules whose electric charges are freer to move horizontally than vertically. The horizontal electric fields that dominate reflected glare accelerate these electric charges so that they also vibrate horizontally, and thus absorb energy from the glare, making it less intense. The same electric charges aren't so free to respond to light waves whose electric fields are vertical, so they absorb much less energy from them, and more of the vertically-oriented light waves pass through.
Polarizing sunglasses act this way, of course, when they're used in their normal orientation. If you turn them sideways and look at horizontal surfaces, the glare won't be nearly as filtered and the surfaces will look much brighter.
The sky, however, is not so full of glare or horizontal surfaces. According to what Einstein and Rayleigh both found, the electric fields of scattered light are mostly perpendicular to the direction the original light waves are traveling, away from the sun. Thus if you view a patch of clear daylight sky through polarizing sunglasses, it will look its brightest if the sun is to the left or right of the sunglasses. Turn the lenses 90o from that direction, and the same patch of sky will look its darkest. The sky will have intermediate brightness if viewed through the lenses at any angle in between. Different regions of the sky will look bright or dark as you turn the polarizing lenses in different directions.
You can do more than just check Einstein's formula this way. The same phenomenon provides an entertaining way to locate the sun during the daytime if the sun is below the horizon or behind a cloud. If there is a clear patch of sky to look at, try looking at it through a polarizing lens turned in different directions, and find which direction makes that patch as dark as possible. The sun will then be somewhere along a line parallel to the lens' "vertical" direction through that patch of sky. Find a second patch of clear sky and do the same thing again to find a second line to the sun. The two lines-actually great circles across the sky-will cross at two opposite points, one above the horizon and one below. The sun will be at one of those points. The other point will be directly above the spot on earth that's facing away from the sun, where the local solar time is midnight.
Next article: The Momentum of Light
References, Links, and Comments:
"Theorie der Opaleszenz von homogenen Flüssigkeiten und
Flüssigkeitsgemischen in der Nähe des kritischen Zustandes" ("The Theory
of the Opalescence of Homogeneous Fluids and Liquid Mixtures near the Critical
State") by Albert Einstein. Available
in the original German, with annotations in English, in The Collected Papers of Albert Einstein, Volume 3:
The Swiss Years: Writings,
1909-1911, edited by Martin J.
Klein, A. J. Kox, Jürgen Renn, and Robert Schulmann.
A translation is available in this volume's English translation
supplement by Anna Beck (translator) and Don Howard (consultant).
Einstein found that the same basic analysis would apply to two different kinds of material: homogeneous fluids like the air, and liquid mixtures made of more than one kind of molecule. Light waves are scattered when they encounter overall density variations in a homogeneous fluid, or changes in the proportions of each kind of molecule in a liquid mixture.
The Rise of the New
Physics (in two volumes) by Abraham D'Abro
Volume 1's section on "Fluctuations", pp. 415-417, describes critical opalescence in terms of Smoluchowski's theory of fluid-density variations.
The significance of irregularities in a wave's path over one wavelength affects how much those irregularities will disturb a wave's motion. This is mentioned on page 279 and is a recurring theme in D'Abro's book as he discusses many kinds of wave phenomena in physics. The concept is useful for understanding critical opalescence: a fluid in its critical state has density fluctuations of many sizes, which means they will scatter waves of many different lengths.
Opalescence is exemplified by (and named for) the milky appearance of common opal. The play of colors seen in other types of opal is also called opalescence, but this is a different phenomenon. ("Opal", [exit federal site] in Wikipedia.) [exit federal site]
The daytime sky's color is treated in the following references:
is the sky blue?" [exit federal site] by Matt McIrvin.
"In these pages I explain why the sky is blue, given the structure of air molecules, the known behavior of electromagnetic fields and potentials, and the known behavior of the human visual system. Not everything is known about these subjects (the human visual system is by far the most mysterious of the three!) But I can confidently say that these are the subjects that people need to learn more about, in order to understand more thoroughly why the sky is blue." Avoids equations; concentrates on the physical processes involved.
the sky is blue (on a clear day with an unpolluted Earth atmosphere)" [exit federal site] by
11-page paper, some mathematical detail. Deals with the light from the sun, its interaction with the atmosphere, and color perception by the human eye.
"A Blue Sky History" by Pedro
Lilienfeld, Optics & Photonics News,
June 2004, pp. 32-39.
Goes further back into history than the present article, but ends with Lord Rayleigh's 19th-century contribution. For a history confined to understanding the sky's color, Rayleigh's work makes a reasonable stopping point, since the work of Smoluchowski, Einstein, and Zimm also dealt with the effects of interference phenomena, which are important in dense fluids but less so for the atmosphere.
Hyperphysics [exit federal site]
by C. R. Nave: "Blue
Sky". [exit federal site]
Brief, illustrated explanation of the sky's color in terms of the Rayleigh light scattering by molecules discussed above. The web page also describes the scattering of light waves by larger particles suspended in the atmosphere.
"Human color vision and the
unsaturated blue color of the daytime sky" by Glenn S. Smith, American
Journal of Physics, July 2005, pp. 590-597.
The preceding articles emphasize how air molecules scatter more high-frequency light than low-frequency light. This alone would explain why the sky isn't mostly a low-frequency , long-wavelength color like red, but not why it isn't violet, a color associated with even higher-frequency and shorter-length waves than blue. The reason has to do with the human retina's having three different kinds of color sensors (cones), each of which responds differently to the same frequencies. This article picks up where Bussink's article leaves off, showing how the cones respond to the different frequencies of skylight the same as they do to unsaturated blue rather than to violet or some other color.
The polarization of light waves is a subject well represented on the World Wide Web.
Hyperphysics [exit federal site]
by C. R. Nave:
"Polarization by Scattering". [exit federal site] A schematic illustration of how light becomes polarized when scattered by molecules.
"Polaroid Material", [exit federal site] "Polaroid Sunglasses". [exit federal site] How the sunglasses and the material their lenses are made from work.
"Polarization". [exit federal site] A hyperlinked concept map about polarized light, how to produce and analyze it, and its different types.
Several references are available on the broader subject of atmospheric optics, including the following:
Hyperphysics [exit federal site]
by C. R. Nave: "Atmospheric
Optics Concepts". [exit federal site]
A hyperlinked concept map of many aspects of the sky's appearance, including phenomena related to light-wave scattering, rainbows, coronas, and mirages.
Light and Color in the Outdoors (also published as The
Nature of Light and Color in the Open Air; in the original Dutch, De
Natuurkunde van het Vrije Veld) by M.G.J. Minnaert.
A classic book about 278 different outdoor optical phenomena.
"Atmospheric Optics" by Craig
38-page article in The Optics Encyclopedia.
Out of the Blue: A 24-hour
Skywatcher's Guide by John Naylor.
Deals with both daytime and nighttime phenomena. First chapter includes sections entitled "Why is the sky blue?" and "Polarised light from the sky".