PlanetPhysics/Stefan Boltzamann Law

Stefan-Boltzmann law

The Stefan-Boltzmann law, also known as Stefan's law, states that the total [[../CosmologicalConstant/|energy]] radiated per unit surface area of a black body in unit time, P is directly proportional to the fourth [[../Power/|power]] of the black body's [[../Thermodynamics/|Thermodynamic]] [[../BoltzmannConstant/|temperature]] T (also called [[../ThermodynamicLaws/|absolute temperature]]):

$P(T)=ϵ\sigma T^4$

The irradiance P has dimensions of power density (energy per time per [[../PiecewiseLinear/|square]] distance), and the SI units of measure are joules per second per square meter, or equivalently, watts per square meter. The SI unit for absolute temperature T is the kelvin. e is the emissivity of the blackbody; if it is a perfect blackbody e = 1.

The constant of proportionality $\sigma$, called the Stefan-Boltzmann constant or Stefan's constant, is non-fundamental in the sense that it derives from other known constants of nature. The value of the constant is

$\sigma =\frac{2{\pi }^5{k}^4}{15c^2{h}^3}=5.670400\times 10^{-8}[{\text{<mrow data-mjx-texclass="ORD">W<mspace width="0.167em"></mspace>s</mrow>}}^{-1}{\text{m}}^{-2}{\text{K}}^{-4}].$

Thus at 100 K the energy [[../AbsoluteMagnitude/|flux]] density is 5.67 W/m2, at 1000 K 56.7 kW/m2, etc.

The law was discovered experimentally by Jo\v{z}ef Stefan (1835-1893) in 1879 and derived theoretically, using thermodynamics, by Ludwig Boltzmann (1844-1906) in 1884. Boltzmann treated a certain ideal [[../Heat/|heat]] engine with the light as a working matter instead of the gas. This law is the only [[../PrincipleOfCorrespondingStates/|physical law]] of nature named after a Slovene physicist. The law is valid only for ideal black [[../TrivialGroupoid/|objects]], the perfect radiators, called black bodies. Stefan published this law on March 20 in the article \"Uber die Beziehung zwischen der W\"armestrahlung und der Temperatur (On the relationship between thermal [[../Cyclotron/|radiation]] and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.

Derivation

The Stefan-Boltzmann law can be derived by integrating over all wavelengths the spectral intensity of a black body as given by [[../PlancksRadiationLaw/|Planck's radiation law]].

$P(T)=\underset{0}{\overset{\mathrm{\infty }}{\int }}I(\lambda ,T)d\lambda$

where $I(\lambda ,T)$ is the amount of energy emitted by a black body at temperature T per unit surface per unit time per unit [[../CoIntersections/|solid]] angle. The equation for $I$ comes From Planck's radiation law and is given as

$I(\lambda ,T)=\frac{2\pi c^{2}h}{{\lambda }^5}\frac{1}{e^{hc/\lambda kT}-1}$

which leaves us to integrate

$P(T)=2\pi c^{2}h\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{d\lambda }{{\lambda }^5(e^{hc/\lambda kT}-1)}$

using u substitution by setting

$u=\frac{hc}{\lambda kT}$

so that

$du=-\frac{hc}{{\lambda }^{2}kT}d\lambda$ $d\lambda =-\frac{{\lambda }^{2}kT}{hc}du$

with the limits of integration changing to

$\lambda ->\mathrm{\infty },u->0$ $\lambda ->0,u->\mathrm{\infty }$

substituting this into the integral and yields

$P(T)=-2\pi c^{2}h\underset{\mathrm{\infty }}{\overset{0}{\int }}\frac{{\lambda }^{2}kTdu}{{\lambda }^{5}hc(e^u-1)}$

simplifying a little and switching the limits of integration to get rid of the minus sign

$P(T)=2\pi ckT\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{du}{{\lambda }^3(e^u-1)}du$

making sure we convert all ${\lambda }^{\prime }s$ to $u^{\prime }s$, use

${\lambda }^3={\left(\frac{hc}{ukT}\right)}^3$

leaving us with

$P(T)=2\pi ckT{\left(\frac{kT}{hc}\right)}^3\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{u^{3}du}{(e^u-1)}$

The theory needed to analytically solve this integral is beyond this article. Even looking up this integral in a table takes a few moments because the solution, given in [3], is defined as

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{u^{n-1}dx}{e^u-1}=\mathrm{\Gamma }(n)\zeta (n)$

where $\mathrm{\Gamma }(n)$ is the [[../GammaFunction/|gamma function]] and $\zeta (n)$ is the [[../RiemannZetaFunction/|Riemann zeta function]].

The values of the gamma function are simple for integers

$\mathrm{\Gamma }(n)=(n-1)!$

so for the case of $n=4$

$\mathrm{\Gamma }(4)=(3)!=3\cdot 2\cdot 1=6$

The values of the Riemann zeta function are more involved, but for even integers, we can use the [[../Formula/|theorem]] given on PlanetMath,

which lets us get $\zeta (4)$ from

$\zeta (n)=\frac{2^{n-1}|B_n|{\pi }^n}{n!}$

We still need $B_n$, the Bernoulli number, for $n=4$, we can get this from [5], PlanetMath,

$B_4=-\frac{1}{30}$

so

$\zeta (n)=\frac{2^3{\pi }^n}{30\cdot 4!}=\frac{{\pi }^4}{90}$

Finally, the integral solution is

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{u^{n-1}dx}{e^u-1}=\mathrm{\Gamma }(n)\zeta (n)=6\cdot \frac{{\pi }^4}{90}=\frac{{\pi }^4}{15}$

gathering all the constants from the original integral, we are left with

$P(T)=2\pi ckT{\left(\frac{kT}{hc}\right)}^3\frac{{\pi }^4}{15}$

simplifying yields the Stefan-Boltzmann law

$P(T)=\frac{2{\pi }^5{k}^4}{15h^3{c}^2}{T}^4$

References

[1] National Institute of Standards and Technology

[2] Krane, K., "Modern Physics." Second Edition. New York, John Wiley \& Sons, 1996.

[3] Thornton, S., Rex, A., "Modern Physics For Scientists and Engineers." Second Edition. Fort Worth, Saunders College Publishing, 2000.

[4] alozano, "values of the Riemann zeta function in terms of Bernoulli numbers" PlanetMath

[5] archibal, "Bernoulli Number" PlanetMath

This entry is a derivative of the Stefan-Boltzmann law article from Wikipedia, the Free Encyclopedia. Authors of the orginial article include: Yurivict, Patrick, XJamRastafire , Metacomet and Icairns. History page of the original is here