# Stefan Boltzmann law

**the Stefan Boltzmann law** is a physical law, which indicates the achievement radiated thermally from a black body as a function of its temperature.

## To table of contents |

## Stefan Boltzmann law

each body,its temperature over the absolute zero lies, sends radiant heat . A black body is an idealized body, which can absorb all radiation meeting it completely (absorption factor = 1). To the kirchhoffschen radiation law reached therefore also its emissivity the value 1 and it sends thoseat the temperature concerned maximally possible thermal achievement out. The Stefan Boltzmann law indicates, which radiating power <math> P< /math> a black body of the surface <math> A< /math> and the absolute temperature <math> T< /math> emitted. It is

<math> P = \ sigma \ cdot A \ cdot T^4< /math> |

with the Stefan Boltzmann constants <math> \ sigma< /math>. The radiating power of a black one Body is thus proportional to the fourth power of his absolute temperature: a doubling of the temperature causes that the eradiated power rises around the factor 16.

The Stefan Boltzmann constant is a natural constant and its numerical value amounts to in accordance with CODATA 2000

Are *k* (not with <math> \ sigma< /math> ) Boltzmann constant which can be confounded, *h* the Planck quantum of action and *C* the speed of light.

## derivation

for derivation proceed one from the spectral radiance of a black body and integrateit both over the entire semi-infinite space, into which the regarded two dimensional element radiates, and over all frequencies, around the specific radiant emittance <math> M^o (T)< /math> to receive:

- <math> M^o (T) = \ int_ {\ nu=0} ^ {\ infty} \ int_ {semi-infinite space} \ frac {2 h \ nu^ {3}} {c^2} \ frac {1} {e^ {\ left (\ frac {h \ nu} {kT} \ right)}- 1} \, \ cos (\ beta) \, \ sin (\ beta) \ mathrm {D} \ beta \ mathrm {D} \ varphi \, \ mathrm {D} \ nu </math>.

The cosine factor considers thereby thatCircumstance that with radiation into any through <math> \ beta< /math> and <math> \ varphi< /math> given direction only the projection standing perpendicularly on this direction <math> \ cos (\ beta) \ mathrm {D} A< /math> the surface <math> \ mathrm {D} A< /math> as effective jet surface appears. The term <math> \ sin (\ beta) \ mathrm {D} \ beta \ mathrm {D} \ varphi< /math> is a solid angle element.

There the black body in principle a vague emitterand its spectral radiance therefore is direction-independent, has the integral over the semi-infinite space the value <math> \ pi< /math>. For the integration over the frequencies it is to be noted that

- < math> \ int_ {0} ^ {\ infty} \ frac {x^3} {e^ {x} - 1} \, \ mathrm {D} x = \ frac {\ pi^4} {15}< /math>.

Integrates one in such a way received specific radiant emittance <math> M^o (T)< /math> still over the radiating surface,one receives the Stefan Boltzmann law in the form indicated above.

## non--black bodies

the Stefan Boltzmann law applies only to black bodies. A non-black body is given, that direction-independent radiates (Lambert emitter so mentioned) and its emissivity <math> \ varepsilon (T)< /math> for all frequencies has the same value (so mentioned Grey body), then is radiating power delivered of this

If the emissivity is temperature-dependent, then the entire radiating power *is * no longer strictly proportional to the fourth power of the absolute temperature because of this additional temperature dependence.

For an emitter, with which the direction-independentness and/or the frequency-independentness are not given to the emission, /math must for the determination* of the hemispherical*< entire eating ion degree> math \ epsilon (T<)> the integral individually using the regularities concerned to be computed. Many bodies deviate only little from the ideal Lambert emitter; if the emissivity inthe frequency range, in which the body delivers a noticeable portion of its radiating power, varied, the Stefan Boltzmann law can only little be used at least to approach.

## historical

the Stefan Boltzmann law was experimentally discovered in the year 1879 by Josef Stefan and 1884 of Ludwig Boltzmann theoretically by thermodynamicConsiderations from the classical electromagnetic theory of the radiation deduced. In the year 1900, thus 21 years after the Stefan Boltzmann law, discovered Max Planck the Planck radiation law designated after it, from which the Stefan Boltzmann law follows simply by integration over all directions and wavelengths. The Planck radiation lawh /math could with the introduction <>of the quantum of action< math> attribute the Stefan Boltzmann constant also for the first time to fundamental natural constants.

## example

outside of the terrestrial atmosphere receives a surface an irradiancy aligned to the sun from *S* = 1,376 W/m^{ 2} (solar constant). One determines the temperature <math> T< /math> thatSun surface on the assumption that the sun is in sufficient approximation a black body. The sun radius amounts to* R* = 6,963·10^{ 8} m, the middle distance between earth and sun is *D* = 1,496·10^{ 11} M.

The radiating power P delivered by *the sun surface* penetrates one concentricallyaround the sun put socket pad of the radius* D* with the irradiancy *S*, amounts to thus altogether *P=4πD ^{ 2}·S* = 3,870·10

^{ 26}W (luminosity of the sun). After the Stefan Boltzmann law the temperature of the radiating surface amounts to

- < math> T = \ sqrt [4] {\ frac {P} {\ sigma A}} = \ sqrt [4] {\ frac {P} {\ sigma \, 4 \ piR^2}} = \ sqrt [4] {\ frac {3 {,} 870 \ cdot 10^ {26}} {5 {,} 670 \ cdot 10^ {- 8} \, \ cdot \, 6 {,} 093 \ 10^ cdot {18}}} \; \ mathrm {K} = 5785 \, \ mathrm {K}< /math>

In such a way determined temperature of the sun surface (a more exact value is 5777 K) is called *effective temperature*. It is the temperature, which a equivalent large black body haswould have, around the same radiating power to deliver as the sun.

## see also

the Stefan Boltzmann law makes a statement about altogether the radiating power delivered by a black body on all frequencies. The allocation on individual frequencies and/or. Wavelengths one describes by the Planck radiation law.

## Literature

- Stefan, J.:
*Over the relationship between the radiant heat and the temperature*, in:*Minutes of the meeting of the mathematical-scientific Classe of the imperial academy of the sciences*, Bd. 79 (Vienna 1879), S. 391-428. - Boltzmann, L.:
*Derivative of the Stefan' law, concerning the dependence of the radiant heat from the temperaturethe electromagnetischen light theory*, in:*Annals of physics and chemistry*, Bd. 22 (1884), S. 291-294