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| {{See also |Black body|Black body radiation|Planck's law|Thermal radiation}}
| | I am 19 years old and my name is Mittie Thielen. I life in Cavenago D'adda (Italy).<br><br>Also visit my webpage ... [http://www.crestwoodroofingcompany.com Crestwood Roofing] |
| [[File:Stefan Boltzmann 001.svg|thumb|Graph of a function of total emitted energy of a black body <math>j^{\star}</math> proportional to its thermodynamic temperature <math>T\,</math>. In blue is a total energy according to the [[Wien approximation]], <math> j^{\star}_{W} = j^{\star} / \zeta(4) \approx 0.924 \, \sigma T^{4} \!\, </math>]]
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| The '''Stefan–Boltzmann law''', also known as '''Stefan's law''', describes the power radiated from a [[black body]] in terms of its [[thermodynamic temperature|temperature]]. Specifically, the Stefan–Boltzmann law states that the total [[energy]] radiated per unit [[area|surface area]] of a [[black body]] across [[Black_body_radiation#Spectrum|all wavelengths]] per unit [[time]] (also known as the black-body ''[[radiant exitance]]'' or ''emissive power''), <math> j^{\star}</math>, is directly [[Proportionality (mathematics)|proportional]] to the fourth power of the black body's [[thermodynamic temperature]] ''T'':
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| :<math> j^{\star} = \sigma T^{4}.</math>
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| The [[constant of proportionality]] σ, called the [[Stefan–Boltzmann constant]] or '''Stefan's constant''', derives from other known [[constants of nature]]. The value of the constant is
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| :<math>
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| \sigma=\frac{2\pi^5 k^4}{15c^2h^3}= 5.670 400 \times 10^{-8}\, \mathrm{J\, s^{-1}m^{-2}K^{-4}},
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| </math>
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| where k is the [[Boltzmann constant]], h is [[Planck's constant]], and c is [[speed of light|the speed of light in a vacuum]]. Thus at 100 K the energy flux is 5.67 W/m<sup>2</sup>, at 1000 K 56,700 W/m<sup>2</sup>, etc. The [[radiance]] (watts per square metre per [[steradian]]) is given by
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| :<math> L = \frac{j^{\star}}\pi = \frac\sigma\pi T^{4}.</math>
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| 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]], <math>\varepsilon < 1</math>:
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| :<math> j^{\star} = \varepsilon\sigma T^{4}.</math>
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| The irradiance <math> j^{\star}</math> has dimensions of energy flux (energy per time per area), and the [[SI]] units of measure are [[joule]]s per second per square metre, or equivalently, [[watt]]s per square metre. The SI unit for absolute temperature ''T'' is the [[kelvin]]. ''<math>\varepsilon</math>'' is the [[emissivity]] of the grey body; if it is a perfect blackbody, <math>\varepsilon=1</math>. In the still more general (and realistic) case, the emissivity depends on the wavelength, <math>\varepsilon=\varepsilon(\lambda)</math>.
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| To find the total [[Power (physics)|power]] radiated from an object, multiply by its surface area, <math>A</math>:
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| :<math> P= A j^{\star} = A \varepsilon\sigma T^{4}.</math>
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| [[Metamaterials]] may be designed to exceed the Stefan–Boltzmann law.<ref>[http://arxiv.org/pdf/1109.5444 "Beyond Stefan-Boltzmann Law: Thermal Hyper-Conductivity."] 26 September 2011.</ref>
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| ==History==
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| The law was deduced by [[Joseph Stefan|Jožef Stefan]] (1835–1893) in 1879 on the basis of experimental measurements made by [[John Tyndall]] and was derived from theoretical considerations, using [[thermodynamics]], by [[Ludwig Boltzmann]] (1844–1906) in 1884. Boltzmann considered a certain ideal [[heat engine]] with [[light]] as a working matter instead of gas. The law is highly accurate only for ideal black objects, the perfect radiators, called [[black body|black bodies]]; it works as a good approximation for most "grey" bodies. Stefan published this law 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.
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| == Examples ==
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| === Temperature of the Sun ===
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| With his law Stefan also determined the temperature of the [[Sun]]'s surface. He learned from the data of [[Charles Soret]] (1854–1904) that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella (a thin plate). A round lamella was placed at such a distance from the measuring device that it would be seen at the same angle as the Sun. Soret estimated the temperature of the lamella to be approximately 1900 [[Celsius|°C]] to 2000 °C. Stefan surmised that ⅓ of the energy flux from the Sun is absorbed by the [[Earth's atmosphere]], so he took for the correct Sun's energy flux a value 3/2 times greater, namely 29 × 3/2 = 43.5.
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| Precise measurements of atmospheric [[Absorption (electromagnetic radiation)|absorption]] were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.57<sup>4</sup> = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K (the modern value is 5778 K<ref>http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html</ref>). This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as 13,000,000 °C were claimed. The lower value of 1800 °C was determined by [[Claude Servais Mathias Pouillet]] (1790–1868) in 1838 using the [[Dulong-Petit law]]. Pouillet also took just half the value of the Sun's correct energy flux.
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| === Temperature of stars ===
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| The temperature of [[star]]s other than the Sun can be approximated using a similar means by treating the emitted energy as a [[black body]] radiation.<ref name="luminosity">{{cite web | url = http://outreach.atnf.csiro.au/education/senior/astrophysics/photometry_luminosity.html | title = Luminosity of Stars | publisher = Australian Telescope Outreach and Education | accessdate = 2006-08-13 }}</ref> So:
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| : <math>L = 4 \pi R^2 \sigma T_{e}^4 </math>
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| where '''L''' is the [[luminosity]], '''σ''' is the [[Stefan–Boltzmann constant]], '''R''' is the stellar radius and '''T''' is the [[effective temperature]]. This same formula can be used to compute the approximate radius of a main sequence star relative to the sun:
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| : <math>\frac{R}{R_\odot} \approx \left ( \frac{T_\odot}{T} \right )^{2} \cdot \sqrt{\frac{L}{L_\odot}}</math>
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| where <math>R_\odot</math>, is the [[solar radius]], and so forth.
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| With the Stefan–Boltzmann law, [[astronomer]]s can easily infer the radii of stars. The law is also met in the [[Black hole thermodynamics|thermodynamics]] of [[black hole]]s in so-called [[Hawking radiation]].
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| === Temperature of the Earth ===
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| Similarly we can calculate the [[effective temperature]] of the Earth ''T''<sub>E</sub> by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation. The amount of power, E<sub>S</sub>, emitted by the Sun is given by:
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| :<math>
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| E_S = 4\pi r_S^2 \sigma T_S^4
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| </math>
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| At Earth, this energy is passing through a sphere with a radius of ''a''<sub>0</sub>, the distance between the Earth and the Sun, and the energy passing through each square metre of the sphere is given by
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| :<math>
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| E_{a_0} = \frac{E_S}{4\pi a_0^2}
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| </math>
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| The Earth has a radius of r<sub>E</sub>, and therefore has a cross-section of <math>\pi r_E^2</math>. The amount of solar power absorbed by the Earth is thus given by:
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| :<math>
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| E_{abs} = \pi r_E^2 \times E_{a_0}
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| :</math>
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| The amount of energy emitted must equal the amount of energy absorbed, and so:
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| :<math>
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| \begin{align}
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| 4\pi r_E^2 \sigma T_E^4 &= \pi r_E^2 \times E_{a_0} \\
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| &= \pi r_E^2 \times \frac{4\pi r_S^2\sigma T_S^4}{4\pi a_0^2} \\
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| \end{align}
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| </math>
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| T<sub>E</sub> can then be found:
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| :<math>
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| \begin{align}
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| T_E^4 &= \frac{r_S^2 T_S^4}{4 a_0^2} \\
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| T_E &= T_S \times \sqrt\frac{r_S}{2 a_0} \\
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| & = 5780 \; {\rm K} \times \sqrt{696 \times 10^{6} \; {\rm m} \over 2 \times 149.598 \times 10^{9} \; {\rm m} } \\
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| & \approx 279 \; {\rm K}
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| \end{align}
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| </math>
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| where ''T''<sub>S</sub> is the temperature of the Sun, ''r''<sub>S</sub> the radius of the Sun, and ''a''<sub>0</sub> is the distance between the Earth and the Sun. This gives an effective temperature of 6°C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.
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| The Earth has an [[albedo]] of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of [[effective temperature]], which is what we are calculating). This approximation reduces the temperature by a factor of 0.7<sup>1/4</sup>, giving 255 K (−18 °C).<ref name= "IPCC4_ch01">[http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-chapter1.pdf Intergovernmental Panel on Climate Change Fourth Assessment Report. Chapter 1: Historical overview of climate change science] page 97</ref><ref>[http://eesc.columbia.edu/courses/ees/climate/lectures/radiation/ Solar Radiation and the Earth's Energy Balance<!-- Bot generated title -->]</ref>
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| However, long-wave radiation from the surface of the earth is partially absorbed and re-radiated back down by [[greenhouse gases]], namely [[water vapor]], [[carbon dioxide]] and [[methane]].<ref>P. K. Das, ''[http://www.ias.ac.in/resonance/Mar1996/pdf/Mar1996p54-65.pdf The Earth's Changing Climate]'', Resonance. Vol. 1. No. 3. pp. 54-65, 1996</ref><ref name="Cole">{{cite book | author=Cole, George H. A.; Woolfson, Michael M.
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| | title=Planetary Science: The Science of Planets Around Stars (1st ed.)
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| | publisher=Institute of Physics Publishing | year=2002 | isbn=0-7503-0815-X | pages = 36–37, 380–382 | url = http://books.google.com/books?id=Bgsy66mJ5mYC&pg=RA3-PA382&dq=black-body+emissivity+greenhouse+intitle:Planetary-Science+inauthor:cole }}</ref> Since the emissivity with greenhouse effect (weighted more in the longer wavelengths where the Earth radiates) is reduced more than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation) is reduced, the equilibrium temperature is higher than the simple black-body calculation estimates. As a result, the Earth's actual average surface temperature is about 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.
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| ==Derivation==
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| === Thermodynamic derivation of the energy density ===
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| <ref>http://www.pha.jhu.edu/~kknizhni/StatMech/Derivation_of_Stefan_Boltzmann_Law.pdf</ref>
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| The fact that the energy density of the box containing radiation is proportional to <math>T^{4}</math> can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure <math>p</math> is related to the internal energy density <math>u</math>:
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| <math> p = \frac{u}{3}</math>.
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| From the [[fundamental thermodynamic relation]]
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| <math> dU = T dS - p dV </math>,
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| we obtain the following expression, after dividing by <math> dV </math> and fixing <math> T </math> :
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| <math> \left(\frac{\partial U}{\partial V}\right)_{T} = T \left(\frac{\partial S}{\partial V}\right)_{T} - p = T \left(\frac{\partial p}{\partial T}\right)_{V} - p </math>.
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| The last equality comes from the following [[Maxwell relations|Maxwell relation]]:
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| <math> \left(\frac{\partial S}{\partial V}\right)_{T} = \left(\frac{\partial p}{\partial T}\right)_{V} </math>.
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| From the definition of energy density it follows that
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| <math> u = \frac{U}{V} </math>
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| and
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| <math> \left(\frac{\partial U}{\partial V}\right)_{T} = u </math>.
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| Now, the equality
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| <math> \left(\frac{\partial U}{\partial V}\right)_{T} = T \left(\frac{\partial p}{\partial T}\right)_{V} - p </math>,
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| after substitution of <math> \left(\frac{\partial U}{\partial V}\right)_{T}</math> and <math> p </math> for the corresponding expressions, can be written as
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| <math> u = \frac{T}{3} \left(\frac{\partial u}{\partial T}\right)_{V} - \frac{u}{3} </math>.
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| Since the partial derivative <math> \left(\frac{\partial u}{\partial T}\right)_{V} </math> can be expressed as a relationship between only <math> u </math> and <math> T </math> (if one isolates it on one side of the equality), the partial derivative can be replaced by the ordinary derivative. After separating the differentials the equality becomes
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| <math> \frac{du}{4u} = \frac{dT}{T} </math>,
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| which leads immediately to <math> u = A T^4 </math>, with <math> A </math> as some constant of integration.
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| === Stefan–Boltzmann's law in ''n''-dimensional space ===
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| It can be shown that the radiation pressure in <math>n</math>-dimensional space is given by
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| <math>P=\frac{u}{n}</math>{{citation needed|date=August 2012}}
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| So in <math>n</math>-dimensional space,
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| <math>T dS= (n+1)P dV + n V dP\,</math>
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| So,
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| :<math>\frac{1}{P}\frac{dP}{dT}=\frac{(n+1)}{T}</math>
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| yielding
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| :<math>P \propto T^{n+1} </math>
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| or
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| :<math>u \propto T^{n+1} </math>
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| implying
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| :<math>\frac{dQ}{dt} \propto T^{n+1} </math>
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| The same result is obtained as the integral over frequency of Planck's law for <math>n</math>-dimensional space, albeit with a different value for the Stefan-Boltzmann constant at each dimension. In general the constant is
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| :<math>
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| \sigma=\frac{1}{p(n)} \frac{\pi^{\frac{n}{2}}}{\Gamma(1+\frac{n}{2})} \frac{1}{c^{n-1}} \frac{n(n-1)}{h^{n}} k^{(n+1)} \Gamma(n+1) \zeta(n+1)</math> {{citation needed|date=August 2012}}
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| where <math>\zeta(x)</math> is Riemann's zeta function and <math>p(n)</math> is a certain function of <math>n</math>, with <math>p(3)=4</math>.
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| === Derivation from Planck's law ===
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| The law can be derived by considering a small flat [[black body]] surface radiating out into a half-sphere. This derivation uses [[spherical coordinates]], with ''φ'' as the zenith angle and ''θ'' as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where ''φ'' = <sup>π</sup>/<sub>2</sub>.
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| The intensity of the light emitted from the blackbody surface is given by [[Planck's law]] :
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| ::<math>I(\nu,T) =\frac{2 h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{kT}}-1}.</math>
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| :where
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| :*<math>I(\nu,T)\,</math> is the amount of [[energy]] per unit [[surface area]] per unit [[time]] per unit [[solid angle]] emitted at a frequency <math>\nu \,</math> by a black body at temperature ''T''.
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| :*<math>h \,</math> is [[Planck's constant]]
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| :*<math>c \,</math> is the [[speed of light]], and
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| :*<math>k \,</math> is [[Boltzmann's constant]].
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| The quantity <math>I(\nu,T) ~A ~d\nu ~d\Omega</math> is the [[Power (physics)|power]] radiated by a surface of area A through a [[solid angle]] ''dΩ'' in the frequency range between ''ν'' and ''ν'' + ''dν''.
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| The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,
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| ::<math>\frac{P}{A} = \int_0^\infty I(\nu,T) d\nu \int d\Omega \,</math>
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| To derive the Stefan–Boltzmann law, we must integrate ''Ω'' over the half-sphere and integrate ''ν'' from 0 to ∞. Furthermore, because black bodies are ''Lambertian'' (i.e. they obey [[Lambert's cosine law]]), the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle ''φ'', and in spherical coordinates, ''dΩ'' = sin(''φ'') ''dφ dθ''.
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| :: <math>
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| \begin{align}
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| \frac{P}{A} & = \int_0^\infty I(\nu,T) \, d\nu \int_0^{2\pi} \, d\theta \int_0^{\pi/2} \cos \phi \sin \phi \, d\phi \\
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| & = \pi \int_0^\infty I(\nu,T) \, d\nu
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| \end{align}
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| </math>
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| Then we plug in for ''I'':
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| :: <math>\frac{P}{A} = \frac{2 \pi h}{c^2} \int_0^\infty \frac{\nu^3}{ e^{\frac{h\nu}{kT}}-1} d\nu \,</math>
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| To do this integral, do a substitution,
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| ::<math> u = \frac{h \nu}{k T} \,</math>
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| <!-- extra space for legibility between two lines of "displayed" [[TeX]] -->
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| ::<math> du = \frac{h}{k T} \, d\nu </math>
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| which gives:
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| : <math>\frac{P}{A} = \frac{2 \pi h }{c^2} \left(\frac{k T}{h} \right)^4 \int_0^\infty \frac{u^3}{ e^u - 1} \, du.</math>
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| The integral on the right can be done in a number of ways (one is included in this article's appendix) – its answer is <math> \frac{\pi^4}{15} </math>, giving the result that, for a perfect blackbody surface:
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| : <math>j^\star = \sigma T^4 ~, ~~ \sigma = \frac{2 \pi^5 k^4 }{15 c^2 h^3} = \frac{\pi^2 k^4}{60 \hbar^3 c^2}. </math>
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| Finally, this proof started out only considering a small flat surface. However, any [[differentiable]] surface can be approximated by a bunch of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface—so this law holds for all [[convex set|convex]] blackbodies, too, so long as the surface has the same temperature throughout. The law extends to radiation from non-convex bodies by using the fact that the [[convex hull]] of a black body radiates as though it were itself a black body.
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| ===Appendix===
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| In one of the above derivations, the following integral appeared:
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| :<math>J=\int_0^\infty \frac{x^{3}}{\exp\left(x\right)-1} \, dx = \Gamma(4)\,\mathrm{Li}_4(1) = 6\,\mathrm{Li}_4(1) = 6 \zeta(4)</math>
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| where <math>\mathrm{Li}_s(z)</math> is the [[Polylogarithm#Integral_representations|polylogarithm]] function and <math>\zeta(z)</math> is the [[Riemann zeta function]]. If the polylogarithm function and the Riemann zeta function are not available for calculation, there are a number of ways to do this integration; a simple one is given in the appendix of the [[Planck's law#Appendix|Planck's law]] article. This appendix does the integral by [[contour integration]]. Consider the function:
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| :<math>f(k) = \int_0^\infty \frac{\sin\left(kx\right)}{\exp\left(x\right)-1} \, dx. </math>
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| Using the [[Taylor expansion]] of the sine function, it should be evident that the coefficient of the ''k''<sup>3</sup> term would be exactly -''J''/6.
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| By expanding both sides in powers of <math>k</math>, we see that <math>J</math> is minus 6 times the coefficient of <math>k^3</math> of the series expansion of <math>f(k)</math>. So, if we can find a closed form for ''f''(''k''), its [[Taylor expansion]] will give J.
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| In turn, sin(x) is the imaginary part of e<sup>ix</sup>, so we can restate this as:
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| :<math>
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| f(k)=\lim_{\varepsilon\rightarrow 0}~\text{Im}~\int_\varepsilon^\infty \frac{\exp\left(ikx\right)}{\exp\left(x\right)-1} \, dx.
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| </math>
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| To evaluate the integral in this equation we consider the contour integral:
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| :<math>
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| \oint_{C(\varepsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1} \, dz
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| </math>
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| where <math>C(\varepsilon,R)</math> is the contour from <math>\varepsilon</math> to <math>R</math>, then to <math>R+2\pi i</math>, then to <math>\varepsilon+2\pi i</math>, then we go to the point <math>2\pi i - \varepsilon i</math>, avoiding the pole at <math>2\pi i</math> by taking a clockwise quarter circle with radius <math>\varepsilon</math> and center <math>2\pi i</math>. From there we go to <math>\varepsilon i</math>, and finally we return to <math>\varepsilon</math>, avoiding the pole at zero by taking a clockwise quarter circle with radius <math>\varepsilon</math> and center zero.
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| [[File:Contplanck.png|thumb|200px|Integration contour]]
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| Because there are no poles in the integration contour we have:
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| :<math>
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| \oint_{C(\varepsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1} \, dz = 0.
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| </math>
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| We now take the limit <math>R\rightarrow\infty</math>. In this limit the contribution from the segment from <math>R</math> to <math>R+2\pi i</math> tends to zero. Taking together the integrations over the segments from <math>\varepsilon</math> to <math>R</math> and from <math>R+2\pi i</math> to <math>\varepsilon+2\pi i</math> and using the fact that the integrations over clockwise quarter circles withradius <math>\varepsilon</math> about [[simple pole]]s are given up to order <math>\varepsilon</math> by minus <math>\textstyle \frac{i \pi}{2}</math> times the residues at the poles we find:
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| :<math>
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| \left[1-\exp\left(-2\pi k\right) \right]\int_\varepsilon^\infty \frac{\exp\left(ikx\right)}{\exp\left(x\right)-1} \, dx = i \int_\varepsilon^{2\pi-\varepsilon} \frac{\exp\left(-ky\right)}{\exp\left(iy\right)-1} \, dy + i\frac{\pi}{2}\left[1 + \exp \left(-2\pi k\right)\right] + \mathcal{O} \left(\varepsilon\right) \qquad \text{ (1)}
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| </math>
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| The left hand side is the sum of the integral from <math>\varepsilon</math> to <math>R</math> and from <math>R+2 \pi i</math> to <math>2 \pi i + \varepsilon</math>. We can rewrite the integrand of the integral on the r.h.s. as follows:
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| :<math>
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| \frac{1}{\exp\left(iy\right)-1} = \frac{\exp\left(-i\frac{y}{2}\right)}{\exp \left(i \frac{y}{2}\right) - \exp\left(-i\frac{y}{2}\right)} = \frac{1}{2i} \frac{\exp\left(-i\frac{y}{2}\right)}{\sin\left(\frac{y}{2}\right)}
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| </math>
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| If we now take the imaginary part of both sides of Eq. (1) and take the limit <math>\varepsilon\rightarrow 0</math> we find:
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| :<math>f(k) = -\frac{1}{2k} + \frac{\pi}{2}\coth\left(\pi k\right)
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| </math>
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| after using the relation:
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| :<math> \coth\left(x\right) = \frac{1+\exp\left( -2x\right)}{1 - \exp\left( -2x \right)}.</math>
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| Using that the series expansion of <math>\coth(x)</math> is given by:
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| :<math>
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| \coth(x)= \frac{1}{x}+\frac{1}{3}x-\frac{1}{45}x^{3} + \cdots
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| </math>
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| we see that the coefficient of <math>k^3</math> of the series expansion of <math>f(k)</math> is <math>\textstyle -\frac{\pi^4}{90}</math>. This then implies that <math>\textstyle J = \frac{\pi^4}{15} </math> and the result
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| :<math>j^\star = \frac{2\pi^5 k^4}{15 h^3 c^2} T^4 </math>
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| follows.
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| == See also ==
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| *[[Wien's displacement law]]
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| *[[Rayleigh–Jeans law]]
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| *[[Radiance]]
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| *[[Climate model#Zero-dimensional models|Zero-dimensional models]]
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| *[[Black body]]
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| *[[Sakuma–Hattori equation]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Stefan, J.: ''Über die Beziehung zwischen der Wärmestrahlung und der Temperatur'', in: ''Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften'', Bd. 79 (Wien 1879), S. 391-428.
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| * Boltzmann, L.: ''Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie'', in: ''Annalen der Physik und Chemie'', Bd. 22 (1884), S. 291-294
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| {{DEFAULTSORT:Stefan-Boltzmann law}}
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| [[Category:Thermodynamics]]
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| [[Category:Power laws]]
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| [[Category:Heat transfer]]
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| {{Link FA|sl}}
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