|I. Introductory Thoughts||Refer to HRW Section 19-11|
All objects give off radiation, i.e., electromagnetic waves, because of their temperature; we refer to
this as thermal radiation. As well they absorb such radiation from their surroundings.
If a body is hotter than its surroundings it emits more radiation than it absorbs, and tends to cool;
if a body is cooler than its surroundings it absorbs more radiation than it emits, and tends to warm.
Usually it will eventually come to thermal equilibrium with its surroundings: a condition in which its
rates of absorbtion and emission of radiation are equal.
The spectrum of the thermal radiation from a hot solid body is continuous; that is, it emits electromagnetic radiation in a continuous range of wavelengths (or frequencies), over a rather large band of allowed wavelengths. It is usual for the amount of radiation at any given frequency to be different than at other frequencies, depending of course on the body, and the atoms that constitute it. We refer to the set of all these frequencies as its spectrum and are often interested not only in the range of frequencies but also the amount of radiation in some small frequency interval. In particular it is interesting to look at the portion of the spectrum where ``a majority" of the radiation is emitted by the body, i.e., the ``predominant'' part of its radiation spectrum. It seems reasonable that this would have something to do with what the body is made of and also what is its temperature. Most bodies with which we normally come into contact have the predominant portion of their radiation spectrum in the infrared, or far-infrared, portion of the spectrum, so that it is not visible to us. However, as the temperature of a body is increased more radiation is emitted, and also the dominant portion of its spectrum shifts so that its radiation begins to be visible to humans. We usually could describe this by saying that, as some object is heated more and more, the predominant color of the hot object shifts from dull red through bright yellow-orange to bluish, ``white heat."
As already mentioned, the radiation depend not only on the temperature but also the composition of the object. For example, at 2000 K a polished, flat tungsten surface emits radiation at a rate of 23.5 Watts/cm2; for molybdenum, however, the corresponding rate is only 19.2 Watts/cm2. In each of these cases the rate increases somewhat if the surface is roughened, instead of being polished. Other differences appear if we measure the distribution in wavelengths of the emitted radiation. This dependence on materials makes the problem quite complicated. Hopefully it reminds us of some of the difficulties one has with describing gases of atoms. Gases of different sorts of atoms have different properties; the more elementary parts of their behavior could, however, be understood rather well by introducing, instead, the idea of an ideal gas, which was fairly easy to study, and could then be used as a beginning to understand more of the details of more complicated gases. This ideal gas was really an idealization, or abstraction, of the basic ideas surrounding the behavior of gases, which was fairly straight-forward to study because of their simplicity. Likewise, in considering ``the radiation problem,'' it is useful to introduce the concept of ``an ideal radiator." For such an object we assume that the spectrum of the emitted thermal radiation depends only on the temperature of the radiator and not on the material or the nature of the surface.
In the real world, a gas which behaves not very differently than an ideal gas can be obtained by considering a gas of monatomic atoms, such as helium, neon, etc., at temperatures that are not too high. Likewise, in the real world, one may obtain a very good approximation to an ideal radiator by considering the radiation emitted through a small hole of a hollow body, heated to some temperature. The radiation is emitted by the interior walls of the cavity, and becomes quite mixed before it manages to find the small hole to escape; as a result of this it depends very little on the actual material of the body, or its surface. We therefore refer to this radiation as . We can also consider this situation in reverse, and consider the radiation that is absorbed--rather than emitted--by this hole. As it is simply a hole--with a dark hollow on the far side--it absorbs all radiation that strikes it, independent, again, of its material. Therefore, often such radiation is referred to, instead, as . We will now spend the rest of this material trying, therefore, to understand the details of the spectrum of black-body radiation, and its dependence upon temperature. Such studies were made near the beginning of the twentieth century, and shortly thereafter.
II. The Stefan-Boltzmann Law
The total radiated power per unit area of a cavity aperture, summed over all wavelengths, is called its . It depends on the temperature T, has units of Watt/m2, and is given by a simple formula, given below, which involves , the Stefan-Boltzmann constant, which is a universal constant, independent of material, surface, or temperature. We note, however, that ordinary hot objects always radiate less efficiently than do cavity radiators; therefore, we associate with them an additional, positive quantity, , referred to as the emissivity, which is always less than 1, and depends on the material, the surface, and (usually) the temperature as well. Although it is called the emissivity, it also is used to describe the absorption of radiation by that material as well:
III. The Spectral Radiancy
While the total radiant intensity, or intensity per unit area is a very interesting quantity, much more interesting is the various proportions of that intensity emitted (or absorbed) in a particular band of wavelengths. Therefore we introduce the spectral radiancy, , which tells us how the intensity of the cavity radiation varies with wavelength, for a given temperature.
We also notice that the curves begin at zero intensity at zero wavelength, near gamma rays and X-rays,
then rise to a single maximum
value, and then fall, rather more slowly, to zero as the wavelength increases toward very large
values, such as radio waves.
These curves were measured in the nineteenth century, and many people endeavored to understand them. An early effort in this direction was made by Wilhelm Wien (Germany, 1864-1928) who was able to show that the wavelength at which the maximum of the curve occurred was simply inversely proportional to the temperature. He was awarded the 1911 Nobel prize for his research in this area; the Wien displacement law says, very simply, that
IV. Planck's Law for the Spectral Radiancy
At this point, Max Planck (German, 1858-1947) made an inspired interpolation between the two formulas that turned out to fit the data at all wavelengths. In October of 1900 he published the following guess, and then managed, two months later, to create a derivation of his equation, which included the speed of light, c, and a brand-new constant, h, which we now refer to as :
Planck himself was never able to create an argument he would believe for ``why" this was true; however,
it fit the data exactly, so that it must in fact be true. It is this beginning that created what
today we call ``quantum physics." It is worth noting that the very, very small size of Planck's
constant h causes this difference between two adjacent values to usually be very, very small.
In particular, since
it is this very small size that allowed the Rayleigh-Jeans
theory to fit the data very well for large values of ,
where hf is indeed very small
relative to the energies involved.
This text adapted (slightly) from Halliday, Resnick, and Krane (Ch. 49).