# The Boltzmann constant is always constant

The **Stefan Boltzmann Law** is a physical law that specifies the power radiated thermally by a black body as a function of its temperature. It is named after the physicists Josef Stefan and Ludwig Boltzmann.

### Stefan Boltzmann Law

Every body with a temperature above absolute zero emits thermal radiation. A black body is an idealized body that can completely absorb all radiation hitting it (degree of absorption = 1). According to the kirchhoff radiation law, its emissivity also reaches*ε* the value 1, and it sends out the maximum possible thermal power at the relevant temperature. The Stefan-Boltzmann law indicates which radiation power *P.* a black body of the plane *A.* and the absolute temperature*T* emitted. It is

with the Stefan-Boltzmann constant σ. The radiant power of a black body is proportional to the fourth power of its absolute temperature: a doubling of the temperature causes the radiated power to increase by a factor of 16.

The Stefan-Boltzmann constant is a natural constant and its numerical value is according to CODATA 2000

- .

Are there *k* the Boltzmann constant (not to be confused with σ), *H* Planck's quantum of action and *c* the speed of light.

### Derivation

The derivation is based on the spectral radiance of a black body and integrates this over the entire half-space into which the surface element under consideration radiates, as well as over all frequencies.

- .

The cosine factor takes into account the fact that in the case of radiation in any direction through the angles β and given direction only the projection perpendicular to this direction the area d*A.* occurs as an effective jet area. The term is a solid angle element.

Since the black body is basically a diffuse radiator and its spectral radiance is therefore direction-independent, the integral, carried out over the half-space, results in the value π. For the integration over the frequencies it should be noted that

- .

One integrates the specific charisma obtained in this way *M.*^{O}(*T*) over the radiating surface, one obtains the Stefan-Boltzmann law in the form given above.

### Non-black bodies

The Stefan-Boltzmann law in the above form only applies to black bodies. Is there a non-black body that radiates in a direction-independent manner (so-called Lambert radiator) and its emissivity has the same value for all frequencies (so-called gray body), then is the radiant power emitted by this

- Gray Lambert radiator.

Here is the emissivity the weighted averaged emissivity over all wavelengths and the weighting function is the blackbody energy distribution. If the emissivity is wavelength-dependent, the radiation distribution changes not only because of the change in the Planck distribution. Through this *additional* As a function of temperature, the total radiation power is no longer strictly proportional to the fourth power of the absolute temperature.

For a radiator in which the directional independence or the frequency independence of the emission is not given, the determination of the *hemispherical total emissivity*ε (*T*) the integral can be calculated individually on the basis of the relevant laws. Many bodies deviate only slightly from the ideal Lambert radiator; if the emissivity varies only slightly in the frequency range in which the body emits a noticeable proportion of its radiation power, the Stefan-Boltzmann law can be applied at least approximately.

### History

The Stefan-Boltzmann law was discovered experimentally by Josef Stefan in 1879 and theoretically derived from the classical electromagnetic theory of radiation by Ludwig Boltzmann in 1884 through thermodynamic considerations. In 1900, 21 years after the Stefan-Boltzmann law, Max Planck discovered Planck's law of radiation, named after him, from which the Stefan-Boltzmann law follows simply through integration over all directions and wavelengths. Planck's law of radiation could with the introduction of the quantum of action *H* also trace the Stefan-Boltzmann constant back to fundamental natural constants for the first time.

### example

Outside the earth's atmosphere, a surface facing the sun receives an irradiance of *S.* = 1.367 kW / m^{2} (Solar constant). Determine the temperature *T* the surface of the sun under the assumption that the sun is a black body in a sufficient approximation. The solar radius is *R.* = 6,963·10^{8}m, which is the mean distance between the earth and the sun *D.* = 1,496·10^{11} m.

The radiant power emitted by the sun's surface *P.* penetrates a spherical shell of the radius, which is placed concentrically around the sun *D.* with the irradiance *S.*, is therefore in total *P = 4πD ^{2}· S* = 3,845·10

^{26}W (luminosity of the sun). According to the Stefan-Boltzmann law, the temperature of the radiating surface is

The so determined temperature of the sun's surface is called *Effective temperature*. It is the temperature that a black body of the same size would have to have in order to emit the same radiation power as the sun.

### See also

The Stefan-Boltzmann law makes a statement about the total radiation power emitted by a black body at all frequencies. The division into individual frequencies or wavelengths is described by Planck's law of radiation. Wien's law of displacement connects the temperature of a black body with the most strongly emitted wavelength.

### literature

- Stefan, J .:
*About the relationship between thermal radiation and temperature*, in:*Session reports of the mathematical and scientific class of the Imperial Academy of Sciences*, Vol. 79 (Vienna 1879), pp. 391-428. - Boltzmann, L .:
*Derivation of Stefan's law regarding the dependence of thermal radiation on temperature from electromagnetic light theory*, in:*Annals of Physics and Chemistry*, Vol. 22 (1884), pp. 291-294

Categories: Thermodynamics | Atomic physics

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