Stefan–Boltzmann law

The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time j ⋆ {displaystyle j^{star }} (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T: The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time j ⋆ {displaystyle j^{star }} (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T: The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. The value of the constant is where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. The radiance (watts per square metre per steradian) is given by A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, ε < 1 {displaystyle varepsilon <1} : The radiant emittance j ⋆ {displaystyle j^{star }} has dimensions of energy flux (energy per time per area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. ε {displaystyle varepsilon } is the emissivity of the grey body; if it is a perfect blackbody, ε = 1 {displaystyle varepsilon =1} . In the still more general (and realistic) case, the emissivity depends on the wavelength, ε = ε ( λ ) {displaystyle varepsilon =varepsilon (lambda )} . To find the total power radiated from an object, multiply by its surface area, A {displaystyle A} : Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law. In 1864, John Tyndall presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament. The proportionality to the fourth power of the absolute temperature was deduced by Josef Stefan (1835–1893) in 1879 on the basis of Tyndall's experimental measurements, in the article Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (On the relationship between thermal radiation and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.A derivation of the law from theoretical considerations was presented by Ludwig Boltzmann (1844–1906) in 1884, drawing upon the work of Adolfo Bartoli. Bartoli in 1876 had derived the existence of radiation pressure from the principles of thermodynamics. Following Bartoli, Boltzmann considered an ideal heat engine using electromagnetic radiation instead of an ideal gas as working matter. The law was almost immediately experimentally verified. Heinrich Weber in 1888 pointed out deviations at higher temperatures, but perfect accuracy within measurement uncertainties was confirmed up to temperatures of 1535 K by 1897.The law, including the theoretical prediction of the Stefan–Boltzmann constant as a function of the speed of light, the Boltzmann constant and Planck's constant, is a direct consequence of Planck's law as formulated in 1900.