# Black-body radiation **Black-body radiation** is the [[Thermal_radiation|thermal radiation]] emitted by an idealised body that absorbs every wavelength falling on it and is in thermal equilibrium at [[Temperature|temperature]] T. Its defining property is that the spectrum depends on T and on nothing else — not on material, shape or history. That single fact turns a glowing object into a thermometer, and the attempt to derive the spectrum classically failed so completely that it forced the quantum. In the microsim below the reader has one control, T on a logarithmic scale from 3 K to 30,000 K, and four things move together: the curve of Planck's law, B_lambda = (2·h·c²/lambda⁵)/(exp(h·c/(lambda·k_B·T)) − 1); its peak, sliding as Wien's law lambda_max = b/T with b = 2,900 μm·K; the area beneath it, growing as sigma·T⁴ with sigma = 5.67×10⁻⁸ W m⁻² K⁻⁴; and a colour swatch running from dull red through white to blue as the peak crosses the visible band.[^rain-ch12][^manual10] Three presets pin the scale — the [[Sun]] at 5,780 K peaking at 0.50 μm, the [[Earth]] at 287 K peaking at 10.1 μm, and the [[Cosmic_microwave_background|cosmic microwave background]] at 2.725 K — and a dashed Rayleigh–Jeans line is drawn beside the Planck curve, so the classical prediction can be watched climbing without limit toward short wavelengths while the real curve turns over and dies. On the Physics flagship this article serves Part II — Core theories at the section *Black-body radiation and the spectrum* (row P33), between interference and the [[Schrödinger_equation|Schrödinger equation]]: it is where the [[Electromagnetic_spectrum|electromagnetic spectrum]] stops being continuous in its energy content. It carries the cluster's only Planck-curve microsim; the Energy flagship's [[Thermal_radiation|thermal radiation]] page places this same sim rather than building a second one. ## Theory A hot opaque body radiates, and a cold one absorbs. The theory of black-body radiation is the statement that at equilibrium these two processes are tied so tightly that the emitted spectrum carries no information about the emitter beyond its temperature. That is what makes remote thermometry possible: a pyrometer aimed at a furnace, a radiometer aimed at a cloud top and a telescope aimed at a star run the same inference. ### Spectrum The spectral radiance of a black body rises steeply from zero at short wavelengths, passes through a single maximum, and falls away as a long tail. Both ends have simple limits. Where the photon energy is small compared with the thermal energy, h·c/lambda ≪ k_B·T, the exponential can be expanded and Planck's law collapses to the classical Rayleigh–Jeans form B_lambda → 2·c·k_B·T/lambda⁴, which has no maximum and diverges as lambda → 0. Where the photon energy is large, h·c/lambda ≫ k_B·T, the −1 becomes negligible and the law reduces to Wien's exponential tail, B_lambda ≈ (2·h·c²/lambda⁵)·exp(−h·c/(lambda·k_B·T)). The real spectrum interpolates between them, and the crossover sits at the peak. The Sun's curve is the familiar case. At 5,780 K the peak is at 0.50 μm, in the middle of the visible band, and numerical integration of Planck's law over 380–750 nm gives 43.8 percent of the radiated power in the visible (derived) — matching the figure of about 44 percent quoted for sunlight, with roughly 7 percent in the ultraviolet.[^geog] Of the sunlight arriving at the top of the atmosphere only about 53 percent reaches the ground, 31 percent direct and 22 percent diffuse.[^rain-ch12] The Earth's own curve, at 287 K, peaks at 10.1 μm, twenty times longer, and the two spectra barely overlap — which is why surface [[Earth's_energy_budget|energy budget]] accounting can be run as shortwave in and longwave out. ### Black body A black body is defined by its absorption: it reflects and transmits nothing, so its emissivity is 1 at every wavelength. Nothing is perfectly black, but a cavity with a small hole is an excellent approximation, since radiation entering the hole is absorbed after enough internal reflections that almost none returns. Real surfaces carry an emissivity epsilon between 0 and 1 multiplying the emitted flux. | surface | emissivity | |---|---| | Sun, snow and ice | ≈0.99 | | water | 0.98–0.99 | | vegetation | 0.95–0.98 | | soil | 0.86–0.96, rising with water content | These values are measured in the thermal infrared and are typical for the land-surface work the source treats.[^rain-ch12] A surface of constant emissivity is called grey; for a grey body the flux is scaled by epsilon but the shape of the spectrum, and therefore the peak, is unchanged. ### Additional explanations Two points repay care. First, "spectral radiance" is a density, and the shape of a density depends on the variable it is taken against. The curve per unit wavelength and the curve per unit frequency are not the same function, and their peaks do not correspond: lambda_max·nu_max is not c. The sim plots per unit wavelength throughout and says so, because the two conventions give visibly different-looking peaks for the same body. Second, the universality of the spectrum is a consequence of the second law rather than of any model of matter. If two bodies at the same temperature inside a cavity emitted differently at some wavelength, a filter between them would move energy from one to the other with no other change, which is forbidden. That argument fixes the spectrum without saying what it is; deriving the function itself took another forty years and a new constant. ## Equations Three equations do the work: one exact law and two of its consequences, each found empirically decades before the law that contains them. That is why the two consequences are still quoted as laws in their own right — Wien's for a colour temperature, Stefan–Boltzmann's for an energy budget. The microsim shows all three at once, since one control moves the curve, its peak and its area together. ### Planck's law of black-body radiation Planck's law gives the spectral radiance of a black body per unit wavelength: B_lambda(lambda, T) = (2·h·c²/lambda⁵) / (exp(h·c/(lambda·k_B·T)) − 1), with h the [[Planck_constant|Planck constant]], 6.626×10⁻³⁴ J·s, c the [[Speed_of_light|speed of light]] and k_B the Boltzmann constant.[^murphy] Integrating it over all wavelengths and over the hemisphere gives the Stefan–Boltzmann law; differentiating it and setting the result to zero gives Wien's displacement law. Both appear on the page below as limits, and both were known before 1900 as separate empirical facts. This is the equation the microsim computes. The single control is T, logarithmic from 3 K to 30,000 K. The readouts are lambda_max, the flux epsilon·sigma·T⁴ on a logarithmic bar because T⁴ spans many decades over that range, the peak [[Photon|photon]] energy from E in electronvolts = 1.24/lambda in micrometres,[^murphy] and the fraction of the power in the visible band. The colour swatch is produced from a baked 1,024-entry lookup table converting the curve to sRGB; that swatch is ILLUSTRATIVE, a rendering convention for what an eye adapted to the scene would report, not a measured quantity. So is the dashed Rayleigh–Jeans overlay in one respect worth stating: it is drawn from the classical formula 2·c·k_B·T/lambda⁴, which is not in the source books and is included to show what Planck's quantum removed.[^manual10] The sources themselves are honest about a gap here. The Portal Books that supply the constants and the two limit laws print only Wien's and Stefan–Boltzmann's equations; the Planck curve is supplied from outside them, and the sub-manual for this cluster records that explicitly.[^manual10] Worked numbers from the books are used to test it: 2,900/5,780 = 0.502 μm and 2,900/287 = 10.1 μm for the two presets,[^rain-ch12] and visible light from 0.4 to 0.7 μm carrying 5.0 to 2.8×10⁻¹⁹ J, or 3.1 to 1.8 eV per photon.[^murphy] ### Wien's displacement law The peak wavelength is inversely proportional to temperature: lambda_max = b/T, b = 2,900 μm·K.[^rain-ch12] The rounded constant is the one the source prints; the recommended value is 2,897.77 μm·K, and the sub-manual flags the rounding as a pitfall.[^manual10][^codata] The law is why a heated poker glows red before orange, why a star's colour is a temperature measurement, and why an object at room temperature is invisible in the dark but obvious to an infrared camera. Two derived checks: a source peaking at 555 nm, the wavelength of maximum photopic sensitivity, is at 5,225 K, and the 2.725 K background peaks at 1.06 mm, in the microwave (derived).[^rain-ch12] Emissivity does not enter: epsilon scales the whole curve and therefore moves the flux, not the peak.[^manual10] ### Stefan–Boltzmann law The total flux radiated from a surface is J = epsilon·sigma·T⁴, sigma = 5.67×10⁻⁸ W m⁻² K⁻⁴,[^rain-ch12] so doubling the temperature multiplies the output by sixteen. The chain of derived numbers the sim uses runs: the Sun at 5,780 K with epsilon = 0.990 emits 6.27×10⁷ W/m²; over a radius of 6.96×10⁸ m that is a total output of 3.81×10²⁶ W; spread over a sphere of radius 1.50×10¹¹ m it arrives as 1,349 W/m² at the Earth's distance (derived from the source's own problem set).[^rain-ch12] The Earth at 287 K emits 385 W/m² by the same law (derived). [[Ludwig_Boltzmann|Boltzmann]] derived the fourth power in 1884 from thermodynamics applied to radiation pressure, four years after Josef Stefan had fitted it to measurements.[^stefan1879][^boltzmann1884] ## Applications The law is used wherever a temperature has to be inferred from radiation, or a radiative loss budgeted. The four cases below span thirteen orders of magnitude in flux, and each is worked from the same two constants. Three are measurements read backwards — body temperature from an infrared image, a planet's balance from its albedo, the moment the universe turned transparent — and the fourth is a design failure the arithmetic predicts. ### Human-body emission Skin at about 307 K with emissivity 0.98 over roughly 2 m² radiates about 990 W and absorbs about 820 W back from surroundings at 293 K, a net loss near 170 W (derived). That exceeds the roughly 100 W a resting adult produces, which is the arithmetic behind clothing: the effective radiating area is well below the skin area, because limbs face the torso and fabric intervenes. The peak is at 9.4 μm (derived), the band thermal cameras are built for and the reason they see people through darkness but not through glass. ### Temperature relation between a planet and its star A planet in radiative balance absorbs sunlight over its disc and radiates over its whole sphere, so its effective temperature is T = [S·(1 − a)/(4·sigma)]^(1/4), with S the incident flux and a the [[Albedo|albedo]]. The source-derived solar constant of 1,349 W/m² and an albedo of 0.30 give 254 K for the Earth (derived), against a measured mean surface temperature near 288 K. The 34 K difference is the [[Greenhouse_effect|greenhouse effect]]: the atmosphere is largely transparent at the Sun's 0.5 μm and largely opaque at the Earth's 10 μm, so outgoing longwave radiation is intercepted and partly returned. ### Cosmology The cosmic microwave background is the most nearly perfect black body ever measured, with a temperature of 2.7255 ± 0.0006 K.[^fixsen2009] Its peak is at 1.06 mm and its total flux 3.1×10⁻⁶ W/m² (derived). The spectrum is thermal because the early universe was dense enough to be opaque, so matter and radiation reached equilibrium, and it stayed thermal through the [[Expansion_of_the_universe|expansion]] because stretching every wavelength by one factor turns a black-body spectrum into another at lower temperature: at the redshift of last scattering, about 1,100, the same radiation was at roughly 3,000 K (derived), the temperature at which hydrogen recombines and the universe becomes transparent. ### Light bulb An incandescent filament is a black body used as a lamp, and the spectrum is the reason the design is inefficient. Tungsten at 2,800 K peaks at 1.04 μm, in the near infrared, and only 8.2 percent of its radiated power falls between 380 and 750 nm (derived). Pushing it to 3,000 K raises that to 10.9 percent (derived) at the cost of evaporation and a shorter life. No filament temperature available in a sealed bulb puts most of the output in the visible, which is why thermal lamps were displaced by [[Light-emitting_diode|light-emitting diodes]], whose emission comes from a band gap and is not a Planck curve. ## History The subject begins as a question about equilibrium and ends by breaking classical physics. After Kirchhoff's theorem defined the problem in 1860, Josef Stefan found the fourth-power law empirically in 1879 and Boltzmann derived it in 1884.[^stefan1879][^boltzmann1884] Wilhelm Wien obtained the displacement law in 1893 and an exponential fit that worked at short wavelengths but failed at long ones.[^wien1893] Lord Rayleigh's 1900 classical calculation worked at long wavelengths and diverged at short ones.[^rayleigh1900] [[Max_Planck|Max Planck]] presented an interpolation between the two in October 1900 and, in December, a derivation requiring that the oscillators exchange energy only in elements of size h·nu.[^planck1901] The step was taken as a mathematical device; [[Albert_Einstein|Einstein]]'s 1905 treatment of the [[Photoelectric_effect|photoelectric effect]] made it physical by attributing the quantisation to the radiation itself. Paul Ehrenfest named the classical divergence the ultraviolet catastrophe in 1911, after it had already been repaired. ### Balfour Stewart In 1858 Balfour Stewart compared the emission of polished plates of different materials with that of a lamp-black surface at the same temperature, and found that a good absorber is a good emitter, wavelength by wavelength.[^stewart1858] The result anticipated Kirchhoff's law, and the priority was contested at the time; Stewart's version was experimental and stated for the materials he measured rather than proved as a general theorem. ### Gustav Kirchhoff Gustav Kirchhoff proved in 1859–60 that the ratio of emissive power to absorptivity is the same universal function of wavelength and temperature for every body, and introduced the term "black body" for the case where absorptivity is 1.[^kirchhoff1860] The theorem converts a question about materials into a question about a single function, and Kirchhoff explicitly posed finding that function as a task for physics. It took forty years, a new constant of nature, and the abandonment of the [[Equipartition_theorem|equipartition theorem]] that Rayleigh's calculation had assumed. ## Doppler effect A black body viewed by a moving observer is still a black body, at a shifted temperature: every wavelength is multiplied by the same Doppler factor, and a Planck curve scaled in wavelength is another Planck curve. To first order in v/c the observed temperature varies over the sky as T'/T ≈ 1 + (v/c)·cos(theta), producing a dipole pattern rather than a distortion of the spectral shape. The cosmic microwave background shows exactly this: a dipole of about 3.36 mK on a 2.725 K sky corresponds to v/c = 1.2×10⁻³, or roughly 370 km/s for the motion of the Solar System relative to the radiation (derived).[^fixsen2009] The same reasoning applies to cosmological redshift in the [[Hubble's_law|expanding universe]], where the common factor is 1 + z, and to the thermal emission of any fast-moving source, whose colour temperature then reports its velocity as well as its heat. ## See also - [[Planck's_law]] - [[Wien's_displacement_law]] - [[Stefan–Boltzmann_law]] - [[Thermal_radiation]] - [[Cosmic_microwave_background]] - [[Electromagnetic_spectrum]] - [[Emissivity]] - [[Photoelectric_effect]] ## References [^rain-ch12]: Ochsner, Tyson (2019). *Rain or Shine: An Introduction to Soil Physical Properties and Processes*. Chapter 12, "Surface Energy Balance and Evapotranspiration", pp. 275–296 (Wien's law with b = 2,900 μm·K at p. 278; the Stefan–Boltzmann law with σ = 5.67×10⁻⁸ W m⁻² K⁻⁴, the 5,780 K and 287 K peaks and the emissivity values at p. 279; the 53 percent surface fraction at p. 280; the unworked problem set from which the 5,225 K, 6.27×10⁷ W/m², 3.81×10²⁶ W and 1,349 W/m² figures are computed at p. 283). Portal Book 119. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/rain-or-shine [^murphy]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 5, "Energy and Fossil Fuels", p. 99 (photon energy E = hν = hc/λ with h = 6.626×10⁻³⁴ J·s; E in eV = 1.24/λ in μm; visible light at 0.4–0.7 μm carrying 5.0–2.8×10⁻¹⁹ J, or 3.1–1.8 eV). Portal Book 097. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^geog]: Patrich, Jeremy (2020). *Physical Geography*, version 1, pp. 69 and 71 (about 44 percent of sunlight in the visible and about 7 percent in the ultraviolet; the atmospheric fate of UVA, UVB and UVC). Portal Book 125. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/physical-geography [^manual10]: Portal Books research note, sub-manual 10 "Earth, Soil and Energy", §5.1 "Blackbody emission: Wien, Stefan–Boltzmann and the photon": the equation images were lost in extraction and the two laws were supplied as standard forms against the surviving symbols; Planck's law itself is external to Portal Books 119 and 097 and is flagged as such; the constant 2,900 μm·K is recorded as a rounding of 2,898; emissivity is noted to scale the flux and not λ_max; the flux is to be drawn on a logarithmic axis because T⁴ spans five decades over the control range (pages to pin against the source PDFs). [^codata]: National Institute of Standards and Technology. "CODATA Internationally recommended values of the Fundamental Physical Constants" (Wien wavelength displacement law constant; Stefan–Boltzmann constant; Planck constant; Boltzmann constant). NIST Reference on Constants, Units and Uncertainty. https://physics.nist.gov/cuu/Constants/ [^up3-ch6]: Sanny, Jeff; Ling, Samuel; et al. (2016). *University Physics Volume 3*. OpenStax. Chapter 6, "Photons and Matter Waves", pp. 241–294 (black-body radiation, Planck's hypothesis, the photoelectric effect and the photon; page to pin). Portal Book 079. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^stewart1858]: Stewart, Balfour (1858). "An account of some experiments on radiant heat, involving an extension of Prévost's theory of exchanges." *Transactions of the Royal Society of Edinburgh*, volume 22. [^kirchhoff1860]: Kirchhoff, Gustav (1860). "Über das Verhältniß zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht." *Annalen der Physik und Chemie*, volume 109. [^stefan1879]: Stefan, Josef (1879). "Über die Beziehung zwischen der Wärmestrahlung und der Temperatur." *Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften*, volume 79. [^boltzmann1884]: Boltzmann, Ludwig (1884). "Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie." *Annalen der Physik und Chemie*, volume 258. [^wien1893]: Wien, Wilhelm (1893). "Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie." *Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin*. [^rayleigh1900]: Rayleigh, Lord (1900). "Remarks upon the law of complete radiation." *Philosophical Magazine*, Series 5, volume 49. [^planck1901]: Planck, Max (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum." *Annalen der Physik*, 309 (3): 553–563. [^fixsen2009]: Fixsen, Dale J. (2009). "The temperature of the cosmic microwave background." *The Astrophysical Journal*, 707 (2): 916–920. ### Bibliography - Ochsner (2019), *Rain or Shine*, Chapter 12 — the page-cited source for Wien, Stefan–Boltzmann and the emissivity table. Portal Book 119. - Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5 — the photon-energy relations. Portal Book 097. - Sanny, Ling et al. (2016), *University Physics Volume 3*, Chapter 6 — the introductory account of Planck's hypothesis. Portal Book 079. - Primary papers: Stewart (1858), Kirchhoff (1860), Stefan (1879), Boltzmann (1884), Wien (1893), Rayleigh (1900), Planck (1901) and Fixsen (2009), all cited above. ## Further reading - Sanny, Ling et al. (2016). *University Physics Volume 3*. OpenStax. Chapter 6, "Photons and Matter Waves", pp. 241–294 — the introductory treatment of black-body radiation and the quantum hypothesis, at the level of this page.[^up3-ch6] - Patrich (2020). *Physical Geography*, version 1 — the atmospheric fate of the solar spectrum, band by band. Portal Book 125. - Likharev, Konstantin (2013). *Part QM: Quantum Mechanics*. Chapter 1, pp. 5–30 — the quantum of radiation in its later, operator form. Portal Book 047. ## External links - [Rain or Shine](https://open.umn.edu/opentextbooks/textbooks/rain-or-shine), Open Textbook Library record — Portal Book 119, Chapter 12 - [Energy and Human Ambitions on a Finite Planet](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet), Open Textbook Library record — Portal Book 097 - [Fundamental Physical Constants](https://physics.nist.gov/cuu/Constants/), NIST — the Wien displacement and Stefan–Boltzmann constants <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Black-body_radiation.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Black-body radiation* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Black-body_radiation.html" data-title="Black-body radiation"></div> *Built from `MICROSIM_GUIDE/specs/sims/Black-body_radiation.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Black-body_radiation) : [Wikitube](https://en.wikitube.io/wiki/Black-body_radiation) · pinned revision [1372973026](https://en.wikipedia.org/w/index.php?oldid=1372973026) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]], [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P33 · sim pending (matter/Black-body_radiation).*