Stefan–Boltzmann law

The Stefan - Boltzmann law is a law of physics that indicates the thermally radiated power of an ideal black body as a function of its temperature. It is named after the physicists Josef Stefan and Ludwig Boltzmann.

Stefan- Boltzmann law

Every body whose temperature is above absolute zero, is thermal radiation from its surroundings. A blackbody is an idealized body can fully absorb all radiation incident on it ( absorption coefficient = 1). According to the Kirchhoff's law of radiation, therefore, be achieved emissivity ε is 1, and sends the maximum possible at the relevant temperature thermal performance. The Stefan -Boltzmann law states that radiation power emits a black body of the area and the absolute temperature. It is

With the Stefan- Boltzmann constant. The radiation power of a black body is proportional to the fourth power of its absolute temperature: Doubling the temperature causes the radiated power increases by a factor of 16. This law is often called the " Boltzmann- T- high - four - law."

The value of these constants of nature is, according to current measurement accuracy

1 - and 2-dimensional case,

The above-mentioned Act applies to three -dimensional body, ie all length dimensions are in the range, that is, are very large relative to the wavelength considered. If a body dimension is very small compared to the wavelength, ie, if true, it is a two -dimensional body ( surface ). And if two body dimensions are very small compared to the wavelength, so it is a one -dimensional body ( rod). In these cases, the wave can not propagate in the body in three dimensions, and thus the total internal energy is smaller. It is

And

The Riemann zeta function and is also referred to as apery constant. The radiated energy of a black body is generally proportional to the so -th power of its absolute temperature, the dimension of the body respectively.

Derivation of the thermodynamics

The Stefan - Boltzmann law was discovered experimentally in 1879 by Josef Stefan. Boltzmann introduced in 1884, this law of radiation from the laws of thermodynamics and the classical Maxwell's electrodynamics from. Starting from one of the basic thermodynamic equations

Can be found in compliance with the integrability condition the expression

S = entropy, U = internal energy, V = volume of the closed system, p = pressure, T = temperature

(Explanation of symbols see thermodynamics)

Maxwell showed in 1873 that the radiation pressure as

Write leaves. is in this case the energy density of the electromagnetic radiation. Adolfo Bartoli could justify the existence of a radiation pressure thermodynamically further in 1876 by ​​stating that in case of non- existence of the second law of thermodynamics would be violated. The prefactor 1/3 follows but only for electro-dynamic considerations.

Substituting this expression for in the previous relationship and takes into account that the total energy can be described as writing in a volume, it follows by integration

Or for the total energy

However, the constant of integration is initially undetermined. You had to go through experiments, such as those of Joseph Stefan, can be determined. The fact that this is a composite of other fundamental constants size, only appeared in quantum mechanics. In 1900, ie 21 years after the Stefan- Boltzmann law, Max Planck discovered named after him Planck's radiation law, from the Stefan- Boltzmann law follows simply by integrating over all directions and wavelengths. The Planck radiation law was also the first time return the Stefan- Boltzmann constant on fundamental constants of nature with the introduction of the quantum of action.

In older literature, the size is also referred to as the Stefan- Boltzman constant. The guided by the CODATA under this name constants is, the so-called radiation constant, but over

In relationship. Expressed in figures:

Derivation from quantum mechanics

The derivation starts from the spectral radiance of a blackbody and integrates both the entire half-space, the radiating element of the surface considered, and all the frequencies:

According to the Lambert 's law, taking into account the cosine factor the fact that it is transmitted in any of the projection of this direction perpendicular to the surface occurs due to the given angle, and direction as the effective beam area. The term is an element of solid angle.

Since the black body basically a diffuse reflector and its spectral radiance is therefore independent of direction, gives the integral performed on the half-space, the value. For the integration over the frequencies

Observed. Integrating the emittance obtained even over the radiating surface, we obtain the Stefan- Boltzmann law in the form given above.

For the 1 - and 2-dimensional case, there are two other integral to solve. The following applies:

Thus it follows for

And it follows

These integrals are solved for example by skillful transformation or by using the theory of functions.

Non - Blackbody

The Stefan - Boltzmann law applies in the above form only for blackbody radiators. If a non - black body given that radiates directionally independent (so-called Lambert radiator ) and its emissivity for all frequencies has the same value (so-called gray body ), then

The radiation emitted by this performance. Here, the emissivity of the weighted average emissivity over all wavelengths, and the weighting function is the blackbody energy distribution. scatters depending on the material from 0.012 to 0.98. If the emissivity is wavelength dependent, so the radiation distribution changes not only due to the change of the Planck distribution. This additional temperature dependence of the total radiation power is not strictly proportionate to the fourth power of absolute temperature.

For a spotlight, in which the direction of independence or the independence of the emission frequency is not given, the integral has to be calculated individually on the basis of the relevant laws to determine the hemispheric total emission level ε (T). Many bodies differ only slightly from the ideal Lambertian; if the emissivity in the frequency range in which the body gives off a significant portion of its radiation power varies only slightly, the Stefan- Boltzmann law can be at least approximately apply.

Example

Outside the earth's atmosphere in sun -earth distance receives an aligned to the sun surface and an irradiation intensity of S = 1.367 kW / m² ( solar constant ). Determine the temperature of the sun's surface, assuming that the sun in sufficient approximation is a black body. The solar radius is R = 6.963 · 108 m, the average distance between Earth and Sun is D = 1.496 · 1011 m.

The output from the solar surface radiation power P passes through a concentric spherical shell around the sun set with the radius D of the irradiance S, ie totals P = 4πD ² · S = 3.845 · 1026 W ( luminosity of the sun ). According to the Stefan- Boltzmann law, the temperature of the radiating surface

The thus determined temperature of the sun's surface is called effective temperature. It is the temperature that would have an equally large black body to deliver the same radiant power as the sun.