PlanckRadiationLaw
PlanckRadiationLaw[temperature,λ]
returns the spectral radiance for the specified temperature and wavelength λ.
PlanckRadiationLaw[temperature,f]
returns the spectral radiance for the specified temperature and frequency f.
PlanckRadiationLaw[temperature,property]
returns the value of the property for the specified temperature.
PlanckRadiationLaw[temperature,{λ1,λ2}]
returns the integrated result of the spectral radiance over the wavelength range λ1 to λ2.
PlanckRadiationLaw[temperature,{f1,f2}]
returns the integrated result of the spectral radiance over the frequency range f1 to f2.
Details
- Inputs temperature, λ, and f should be Quantity objects.
- Properties include:
-
"Color" color of the peak wavelength "MaxFrequency" peak frequency "MaxWavelength" peak wavelength "MeanFrequency" average frequency "MeanWavelength" average wavelength "SpectralPlot" plot of spectral radiance versus wavelength - Spectral radiance is returned in SI units.
Examples
open all close allBasic Examples (2)
PlanckRadiationLaw[Quantity[10000., "Kelvins"], Quantity[500, "Nanometers"]]Determine spectral radiance by frequency:
PlanckRadiationLaw[Quantity[10000, "Kelvins"], Quantity[12.1, "Gigahertz"]]Examine the shape of spectral radiance at Quantity[100,"DegreesCelsius"]:
PlanckRadiationLaw[Quantity[5000, "DegreesCelsius"], "SpectralPlot"]PlanckRadiationLaw[Quantity[5000., "DegreesCelsius"], "MeanWavelength"]Discover the color of the peak wavelength:
PlanckRadiationLaw[Quantity[5000, "DegreesCelsius"], "Color"]Scope (3)
Explore all the properties of PlanckRadiationLaw:
PlanckRadiationLaw["Properties"]Find the peak wavelength for 6000 K and its color:
PlanckRadiationLaw[Quantity[6000., "Kelvins"], "MaxWavelength"]PlanckRadiationLaw[Quantity[6000, "Kelvins"], "Color"]Determine the peak frequency at 6000 K:
PlanckRadiationLaw[Quantity[6000., "Kelvins"], "MaxFrequency"]Find the integrated spectral radiance over wavelength or frequency:
PlanckRadiationLaw[Quantity[300, "DegreesCelsius"], {Quantity[5, "Micrometers"], Quantity[25.2, "Micrometers"]}]PlanckRadiationLaw[Quantity[300, "DegreesCelsius"], {Quantity[0.5 * 10 ^ 12, "Hertz"], Quantity[10 ^ 12, "Hertz"]}]Applications (5)
Calculate the maximum radiance as a function of wavelength:
{maxweavelength = PlanckRadiationLaw[Quantity[1000., "Kelvins"], "MaxWavelength"],
PlanckRadiationLaw[Quantity[1000, "Kelvins"], maxweavelength]}Calculate the maximum radiance as a function of frequency:
{maxfrequency = PlanckRadiationLaw[Quantity[1000., "Kelvins"], "MaxFrequency"],
PlanckRadiationLaw[Quantity[1000, "Kelvins"], maxfrequency]}Note that the peak values do not correspond to the same wavelength of light:
UnitConvert[Quantity["SpeedOfLight"] / maxfrequency, "Nanometers"]Examine how the spectral radiance varies as a function of frequency:
LogLogPlot[QuantityMagnitude[PlanckRadiationLaw[Quantity[1000, "Kelvins"], Quantity[x, "Hertz"]]], {x, 1, 10 ^ 16}, AxesLabel -> {Quantity[None, "Hertz"], Quantity[None, "Watts" / ("Hertz" * "Meters" ^ 2 * "Steradians")]}]Use the directional temperature, corrected for relativistic effects, to see how the peak for spectral radiance is shifted to longer wavelengths for an object moving at relativistic speeds:
Teff[t_, β_, θ_] := t * Sqrt[1 - β ^ 2] / (1 - β Cos[θ])PlanckRadiationLaw[Teff[Quantity[5778, "Kelvins"], 0.9, Quantity[30, "AngularDegrees"]], "SpectralPlot"]PlanckRadiationLaw[Teff[Quantity[5778, "Kelvins"], 0.99, Quantity[30, "AngularDegrees"]], "SpectralPlot"]Demonstrate Wien's displacement law, that the peak wavelength is inversely proportional to the temperature:
sol = FindMaximum[QuantityMagnitude[PlanckRadiationLaw[Quantity[1000, "Kelvins"], Quantity[x, "Nanometers"]]], {x, 500, 800}];
UnitConvert[Quantity[x /. Last[sol], "Nanometers"] * Quantity[1000, "Kelvins"], "SI"]sol = FindMaximum[QuantityMagnitude[PlanckRadiationLaw[Quantity[10000, "Kelvins"], Quantity[x, "Nanometers"]]], {x, 50, 80}];
UnitConvert[Quantity[x /. Last[sol], "Nanometers"] * Quantity[10000, "Kelvins"], "SI"]Find the radiant exitance by approximating the integral of Planck's law by integrating the dominant part of the spectrum and using Lambert's cosine law to derive the angular factor for a point on the black body's surface:
m = UnitConvert[Quantity[Pi, "Steradians"] * PlanckRadiationLaw[Quantity[1000, "Kelvins"], {Quantity[10 ^ -10, "Meters"], Quantity[0.1, "Meters"]}], "SIBase"]Divide by the fourth power of the temperature to find the Stefan–Boltzmann constant:
m / Quantity[1000, "Kelvins"] ^ 4UnitConvert[N[Quantity["StefanBoltzmannConstant"]], "SIBase"]Properties & Relations (1)
The formula used by PlanckRadiationLaw is the same as presented by FormulaData:
FormulaData[{"PlanckRadiationLaw", "Wavelength"}]FormulaData[{"PlanckRadiationLaw", "Wavelength"}, {"λ" -> Quantity[500, "Nanometers"], "T" -> Quantity[10000., "Kelvins"]}, UnitSystem -> "Metric"]UnitConvert[PlanckRadiationLaw[Quantity[10000., "Kelvins"], Quantity[500, "Nanometers"]] * Quantity[1, "Steradians"], "Pascals" / "Femtoseconds"]Neat Examples (2)
Compare Planck's radiation law to Wien's distribution law:
wien[t_, ν_] := UnitConvert[Quantity[2, "PlanckConstant" / "SpeedOfLight" ^ 2 / "Steradians"] * ν ^ 3 *
E ^ (-Quantity[1, "PlanckConstant" / "BoltzmannConstant"] * ν /
t), "Watts" / "Meters" ^ 2 / "Hertz" / "Steradians"]Plot[QuantityMagnitude@{wien[Quantity[7000, "Kelvins"], Quantity[10 ^ x, "Hertz"]], PlanckRadiationLaw[Quantity[7000, "Kelvins"], Quantity[10 ^ x, "Hertz"]]}, {x, 1, 16}, PlotRange -> All]Compare Wien's distribution law to the Rayleigh–Jeans law and Planck's radiation law:
wlist = Table[{10 ^ x, QuantityMagnitude@wien[Quantity[7000, "Kelvins"], Quantity[10 ^ x, "Hertz"]]}, {x, 13, 15, 0.1}];
rjlist = Table[{10 ^ x, QuantityMagnitude@UnitConvert[Quantity[2, "BoltzmannConstant" / "SpeedOfLight" ^ 2 / "Steradians"] * Quantity[10 ^ x, "Hertz"] ^ 2 * Quantity[7000, "Kelvins"], "Watts" / ("Hertz" * "Meters" ^ 2 * "Steradians")]}, {x, 13, 15, 0.1}];
plist = Table[{10 ^ x, QuantityMagnitude@PlanckRadiationLaw[Quantity[7000, "Kelvins"], Quantity[10 ^ x, "Hertz"]]}, {x, 13, 15, 0.1}];ListLogLogPlot[{wlist, rjlist, plist}, Joined -> True]In a micrometer-sized box, quantum effects cause the minimum frequency possible to be the following:
νmin = UnitConvert[Quantity[Sqrt[3] / 2 , "SpeedOfLight"] / Quantity[1., "Micrometers"], "Hertz"]Plot the energy density within this box, accounting for the finite size effect relative to a blackbody in an infinite cavity:
data = Table[{Quantity[10 ^ t, "Kelvins"], Quiet[PlanckRadiationLaw[Quantity[10 ^ t, "Kelvins"], {Quantity[10 ^ 19, "Hertz"], νmin}] / PlanckRadiationLaw[Quantity[10 ^ t, "Kelvins"], {Quantity[10 ^ 19, "Hertz"], Quantity[10 ^ -1, "Hertz"]}]]}, {t, 2, 4, 0.1}];ListPlot[data, Joined -> True, AxesLabel -> {Quantity[None, "Kelvins"], "ratio"}]Text
Wolfram Research (2014), PlanckRadiationLaw, Wolfram Language function, https://reference.wolfram.com/language/ref/PlanckRadiationLaw.html (updated 2016).
CMS
Wolfram Language. 2014. "PlanckRadiationLaw." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/PlanckRadiationLaw.html.
APA
Wolfram Language. (2014). PlanckRadiationLaw. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PlanckRadiationLaw.html