Since a black body. merely emits and does not transmit or reflect radiation, and since emission depends merely on temperature, the from a black body is the same whether it is in an enclosure at constant tem perature or not. Hence the radiation given out by a black body is always normal. For this reason normal radiation is often called black body radiation.
Composition of Black Body Radiation.— Consider the radiation in an enclosure where the absolute temperature is T. We have seen that the character of this radiation is a function of the temperature only. If then u(•) dX is the amount of heat energy per unit volume which exists in wave trains of which the wave lengths lie between X and X + dl', the quantity u(X) will be a function of X and T only. The determination of this function is one of the most important problems in the theory of radia tion.
The first formula for u(X) was obtained by Jeans and Lord Rayleigh. Jeans' method was to start from the well-known result in statistical mechanics, that, in a dynamical system of a large number of degrees of freedom the energy on the average is distributed equally among the various degrees of freedom, so that each has an amount of energy RT, where R is an ab solute constant. By counting the number of ways in which wave lengths between a and + dz can occur in stationary vibrations in a fixed enclosure and giving to each two de grees of freedom, corresponding to the two plans of polarization, Jeans found that . 8r.RT • This is known as Rayleigh's law of radia tion. For large values of 1t it represents the facts fairly well. But for small values of X it disagrees entirely with experiment and leads to the rather questionable conclusion that the total energy per unit volume • is infinite.
A formula that does not have these de fects and that seems entirely in accord with experimental values wa3 first obtained ' by Planck. His result is Iltrch a6 where c is the velocity of light, h and R constants. , To obtain this formula, Planck assumed that molecular systems. do not emit radiant energy continuously but in chunks or Radiation of frequency v is .only emitted in multiples of hv, where Is is a uni versal constant. The simplest proof of the formula is a modification of that given by Jeans. Consider a system of stationary electro magnetic waves in a rectangular enclosure.
The number of ways per unit volume in which such waves with frequency between v and v ? dv can be sat up is aartodv cs as in Jeans' proof. If now energy occurs only in multiples of E = ha, and is distributed as is Gibb's canonical ensemble, the average energy. for a wave of particular frequency is 2e EEG e 0 Tet +2 e e Kt 1+ e te RT Hence uv dv 8revit 8rh todv • ( RT — 1) I 11 hv By using the relations — dethis gives the formula above.
This ((quantum theory» that radiant energy is emitted only in multiples of hu has been the subject of very general discussion. There are some who still doubt its validity. Poincare bas shown, however, that any law of radiation. in which the total energy in unit volume is finite requires discontinuities similar to those involved in the hypothesis of quanta. N. Bohr has given a •suggestion as to the physical sig nificance of quanta. He assumes that atoms consist of positive nuclei with negative electrons revolving like planets around them and that there is a series of stable orbits in which each electron can revolve continuously. If dis turbed it will either return to its orbit and give out no energy or it will change to another orbit and, in so doing, give off energy equal to the difference of its energies in the two orbits. In this way he caleulated the spectrum of hydrogen, the calculation as given by Millikan, agreeing with measurement tea less than one part • in a 'thousand: .This and other equally striking results indicate that the quantum hypothesis has a substantial basis of truth.
Deductions from Planck's Law.— In a constant temperature enclosure, radiant energy travels with the velocity • of light, spreading out equally in all directions. The rate at which it crosses a given area in a given direction is then proportional to the density of radiation In the enclosure. If E(a) da is the portion cross. big unit area in unit time canned by wave trains with wave length between 1 and 1 + da, by Planck's law ite 1 ) where A and C2 are constants. This is also the radiation emitted by a black body at temperature T whether it is in a region of constant temperature or not.
. The total radiation of all wave lengths across unit area ef..ihe surface of. a black body is • . . - CV (I