1 Stefan–Boltzmann Law

The Stefan–Boltzmann law, also known as Stefan’s law, states that the total radiant power emitted per unit surface area of a body is proportional to the fourth power of its absolute temperature. For a gray body,

where P is the radiant exitance in watts per square meter, 𝜖 is the emissivity, and T is the absolute temperature in kelvins. For an ideal blackbody, 𝜖 = 1.

The constant of proportionality σ, called the Stefan–Boltzmann constant, can be written in terms of other physical constants as

Thus, for an ideal blackbody at 100 K,

while at 1000 K,

The law was discovered experimentally by Jožef Stefan (1835–1893) in 1879 and derived theoretically, using Thermodynamics, by Ludwig Boltzmann (1844–1906) in 1884. Stefan published the result in the article Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (“On the relationship between thermal radiation and temperature”).

2 Derivation

The Stefan–Boltzmann law can be derived by integrating Planck’s blackbody spectrum over all wavelengths. The spectral radiant exitance is

The total radiant exitance is therefore

or

Use the substitution

Then

As λ →∞, u → 0, and as λ → 0, u →∞. Substitution gives

The standard integral

gives, for n = 4,

and

Hence

Therefore,

so

3 References

[1] National Institute of Standards and Technology

[2] K. Krane, Modern Physics, 2nd ed., John Wiley & Sons, New York (1996).

[3] S. Thornton and A. Rex, Modern Physics for Scientists and Engineers, 2nd ed., Saunders College Publishing, Fort Worth (2000).

[4] alozano, “Values of the Riemann zeta function in terms of Bernoulli numbers,” PlanetMath

[5] archibal, “Bernoulli Number,” PlanetMath

This entry is a derivative of the Stefan–Boltzmann law article from Wikipedia, the Free Encyclopedia. Authors of the original article include Yurivict, Patrick, XJamRastafire, Metacomet, and Icairns. The history page of the original is here.