Black-body radiation refers to the electromagnetic radiation emitted by a perfect black body, which is an idealized physical object that absorbs all incident radiation, regardless of frequency or angle. When a black body is in thermal equilibrium, it emits radiation in a characteristic spectrum that depends solely on its temperature. Key characteristics of black-body radiation include: 1. **Spectrum**: A black body emits radiation across a continuous spectrum of wavelengths, with the intensity peaking at a particular wavelength that is determined by its temperature.
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Black-body radiation is the electromagnetic radiation in thermal equilibrium with an ideal blackbody. It can be described as a photon gas: a cavity whose walls are all at the same temperature contains radiation with the universal Planck's law spectrum. A small opening in such a cavity approximates a blackbody emitter because incoming radiation undergoes repeated absorption inside. The radiation's equilibrium spectrum depends on temperature, rather than the wall material. [ref]
https://openstax.org/books/university-physics-volume-3/pages/6-1-blackbody-radiation. OpenStax, University Physics 3, §6.1, blackbody cavity and emission laws
There are three useful quantities to keep distinct: directional spectral radiance or , total energy density in the cavity, and outward radiant exitance from its opening. Planck's law supplies the first; integrals over frequency and directions supply the others. For isotropy, every direction has the same , so the full solid angle givesHere is the reduced Planck constant, and the radiation constant is . The explicit Boltzmann constant makes this an SI formula; adopting natural units with recovers .
For a short integral outline, substitute and useThe expansion is a geometric series with nonnegative terms; repeated integration by parts evaluates each integral, and is a value of the Riemann zeta function. This gives the coefficient above.
The photon gas also exerts radiation pressure, . The factor comes from the angular average of the squared directional component of photon momentum, . The thermodynamic derivation of blackbody radiation pressure gives the same result. In contrast, outgoing power involves one projected-direction factor over an outward hemisphere, yielding radiant exitanceThis is the Stefan–Boltzmann law with unit emissivity; is the Stefan-Boltzmann constant. The factor follows from Lambert's cosine law, rather than the full-sphere factor used for energy density.
For example, an ideal cavity has , radiation pressure , and outward radiant exitance . A stipulated cavity has one-sixteenth each value because all three scale as . These describe stored energy, mechanical pressure, and outgoing power per area, respectively. In a closed equilibrium cavity, equal opposing radiation balances the directional power flow even though and are nonzero.
The usefulness of black-body radiation is this common reference: Planck's law, Wien's displacement law, and the Stefan–Boltzmann law consistently connect spectral measurements, temperature, and total emission. The three illustrative curves in Planck's law are ideal equilibrium spectra; applying them to a material requires considering its emissivity and whether thermal equilibrium is a suitable approximation.
