Planck's law of black body radiation

In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Planck's law of black body radiation:

<math>I(\nu) =\frac{2h\nu^{3}}{c^2}\frac{1}{\exp\left(\frac{h\nu}{kT}\right)-1}<math>

where:

I(ν) is the amount of energy per unit time per unit surface area per unit solid angle per unit frequency. Units are e.g. [W m^-2 Hz^-1 sr^-1];

ν is the frequency

T is the temperature of the black body

h is Planck's constant,:

c is the speed of light

k is Boltzmann's constant.

The law is sometimes written in terms of the spectral energy density

<math>U(\nu)=\frac{8\pi h\nu^3 }{c^3}~\frac{1}{e^{h\nu/kT}-1}<math>

which has units of energy per unit volume per unit frequency.

Max Planck originally produced this law in 1900 (published in 1901) in an attempt to improve upon Wien's law which fit the experimental data at short wavelengths but deviated from it at long wavelengths. He found that the above function fit the data for all wavelengths remarkably well. In constructing a derivation of this law, he considered the possible ways of distributing electromagnetic energy over the different modes of charged oscillators in matter. Planck's law emerges if it is assumed that these oscillators have energy proportional to frequency

<math>E=h\nu\,<math>.

History

Contrary to popular opinion Planck did not quantize light. This is plain in his writing in his original 1901 paper and in the references in this paper to his earlier work. It is also plainly explained in his book "Theory of Heat Radiation" where he explains that his constant refers to Hertzian oscillators. The idea of quantization was developed by others into what we now know as quantum mechanics. The next step along this road was made by Albert Einstein, who, by studying the photoelectric effect proposed a model and equation whereby light was not only emitted but also absorbed in packets or photons. Then, in 1924, Satyendra Nath Bose developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck's law.

Nor did Planck derive his law in an attempt to resolve the "Ultraviolet catastrophe", the name given to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is applied to black body radiation. Planck did not consider the equipartion theorem to be universally valid, so he never noticed any sort of "catastrophe" – it was only discovered some five years later by Einstein, Lord Rayleigh, and Sir James Jeans.

From the Planck's law of black body radiation we can obtain the Stefan-Boltzmann law by integrating the energy density over all wavelengths.

External link and references

• Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901).
• Radiation of a Blackbody – interactive simulation to play with Planck's law
• Scienceworld entry on the Planck Law
• Kragh, Helge Max Planck: The reluctant revolutionary Physics World, December 2000