# The law of Stefan-Boltzmann: definition, formula and conclusion

The Stefan-Boltzmann law is associated with thermal phenomena and radiation processes in physics. According to this law, the emitter, which represents the emission of energy in the form of electromagnetic radiation, proportional to the fourth degree of absolute temperature, per second per unit area of its surface.

## Concept of black body

Before describing the law of radiation of Stefan-Boltzmann, one should look into the question of what constitutes a black body. A black body is a theoretical object that is able to absorb absolutely all the electromagnetic energy that falls on it. That is, electromagnetic radiation does not pass through the black body and is not reflected from it. The black body should not be confused with the dark matter in space, since the black body is capable of radiating electromagnetic energy. The concept of black body was introduced into physics to simplify the study of the radiation processes of real bodies.The term “black body” itself was introduced by Gustav Kirchhoff in 1862.

## Body radiation

Every real body radiates energy in the form of electromagnetic waves into the surrounding space. In this case, in accordance with the law of Stefan-Boltzmann, this radiation will be the more intense, the higher the body temperature. If the body has a low temperature, for example, an ambient temperature, then the energy it emits is small and most of it is emitted in the form of long electromagnetic waves (infrared radiation). An increase in body temperature leads not only to an increase in the amount of radiated energy, but also to a shift of the emission spectrum to higher frequencies. That is why the color of the body changes when it is heated. The amount of energy that a body emits, heated to a specific temperature in a certain narrow frequency range, is described by Planck’s law.

The amount and spectrum of the radiated electromagnetic energy depends not only on the temperature of the body, but also on the nature of the radiating surface. So, the matte or black surface has a greater emissivity than the bright or shiny.This means that the amount of energy that a red-hot carbon fiber emits is greater than, for example, a platinum filament heated to the same temperature. Kirchhoff's law states that if a body radiates energy well, then it will absorb it well. Thus, black bodies are good absorbers of electromagnetic radiation.

## Real objects, similar in their characteristics to the black body

The emissivity and absorption properties of a black body are an idealized case, but in nature there are objects that, according to these characteristics, can be considered as a black body in the first approximation.

The simplest object, which in its ability to absorb visible light is close to a black body, is an insulated container that has a small hole in its body. Through this hole, a beam of light enters the cavity of the object and experiences multiple reflections from the inner walls of the container. With each reflection, a part of the beam energy is absorbed, and this process continues until all the energy is absorbed.

Another object that almost completely absorbs the light falling on it is an alloy of nickel and phosphorus.This alloy was obtained in 1980 by Hindus and Americans, and in 1990 it was perfected by Japanese scientists. This alloy reflects only 0.16% of the light energy incident on it, which is 25 times less than the equivalent value for the black paint itself.

A real example of a radiator in space, which in its properties is close to the emissivity of a black body, are stars of galaxies.

## Black body radiation energy

In accordance with the definition of the Stefan-Boltzmann law, black body radiation energy from a surface of 1 m2in one second is determined by the formula:

E = σ (Tuh)4,

where tuhis the effective radiation temperature, that is, the absolute temperature of the body surface, σ is the Stefan-Boltzmann constant, equal to 5.67 · 10-8W / (m2·TO4).

The closer the radiative characteristics of real bodies to the properties of a black body, the closer will be the energy calculated by the specified formula to the radiated energy of real bodies.

## Radiation energy of real bodies

The formula of the law of Stefan-Boltzmann for the radiation of real bodies is:

E = εσ (Tuh)4,

where ε is the emissivity coefficient of a real body, which lies within 0 <ε <1. This coefficient is not a constant, but depends on the absolute temperature, frequency of electromagnetic radiation and surface properties of a real body.

## The story of the discovery of the law of Stefan-Boltzmann

This law was discovered in 1879 by the Austrian physicist Joseph Stefan on the basis of experimental measurements. The experiments themselves were carried out by the Irish physicist John Tyndall. In 1884, as a result of theoretical studies using thermodynamics, Ludwig Boltzmann also came to this law of black body radiation. In his reasoning, Boltzmann considered some ideal engine in which the source of energy was light.

Stefan published the experimentally obtained law in an article entitled “On the relationship between radiation and absolute temperature” in one of the brochures of the Academy of Sciences of Vienna.

## The mathematical derivation of the radiation law formula

The derivation of the formula of the Stefan-Boltzmann law is quite simple, for this you only need to integrate energy over all frequencies, which is determined by Planck’s law for black body radiation. As a result of this integration, it can be shown that the Stefan-Boltzmann constant is defined through other fundamental physical constants:

σ = 2pi5k4/ (15c2h3),

here pi = 3.14 (pi), k = 1.38 · 10–23J / C (Boltzmann constant), c = 3 · 108m / s (the speed of light in a vacuum), h = 6.63 · 10-34J · s (Planck constant).

As a result of calculations, we obtain that σ = 5.67 · 10-8W / (m2·TO4), which exactly corresponds to the experimentally determined value.

## An example of using the law of Stefan-Boltzmann: the temperature of the surface of the sun

Using independently open law, Stefan determined the temperature of the surface of our star - the Sun. For this, he used the data of Charles Soret, according to which the density of the flow of solar energy is 29 times greater than the density of electromagnetic radiation of a heated metal plate. The scientist placed the plate from the electromagnetic flow detector at the same angle from which the Sun can be seen from the Earth. As a result, Soret estimated the plate temperature at 1900-2000 ° C. Stephen, in turn, also took into account the atmospheric absorption of solar radiation on Earth, suggesting that the actual energy flow from the Sun is 43.5 times greater than that from a heated plate. Note that accurate measurements of the atmospheric absorption of solar energy were carried out in a series of experiments from 1888 to 1904.

Further, according to the law of Stefan-Boltzmann, you can easily showthat the surface temperature of the Sun must be 2.57 times the temperature of the metal plate (to obtain this figure, you must take the fourth degree root of the ratio of the energy fluxes of the Sun's radiation and the plate). Thus, Stefan obtained that the surface temperature of our star is 5713 K (the current value is 5780 K).

The obtained value of the surface temperature of the Sun was the most accurate in the XIX century. Prior to the work of Stefan, other scientists obtained both too low temperatures for the surface of the Sun (1,800 ° C) and too high values of it (13,000,000 ° C).