Stefan-Boltzmann constant
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What is the Stefan-Boltzmann Constant?
The Stefan-Boltzmann constant is used mostly in equations involving the Stefan Boltzmann law, which states that the energy radiated per unit of surface area of a black body per unit time is proportional to the fourth power of the thermodynamic temperature of that body. One could apply this law to calculate the temperature of the sun, for example. In its standard use, the Stefan-Boltzmann constant uses the symbol σ, and has units of W/(m^{2}.K^{4}).
Its value is 5.670400x10^{-8}.
One can also derive the Stefan-Boltzmann constant both by experiment and from the standard Boltzmann constant. Mathematically, this amounts to the following:
σ = (2.π^{5}.k_{b}^{4}) / (15.h^{3}.c_{o}^{2})
where k_{b} is the Boltzmann constant (1.3x10^{-23} J/K), h is the Planck constant (6.26x10^{-34} J.s), and c_{o} is the speed of light (3x10^{8} m/s).
The Stefan-Boltzmann law uses this constant in the following way, relating the radiant flux of a black body to temperature with this formula:
j^{*} = σT^{4}
where j^{*} is the radiant flux of a perfect black body and T is the thermodynamic temperature of that black body. j^{*} has energy flux units (watts/m^{2}). In most cases, as is
also observed in nature, the black body radiator will not be perfect and j^{*} will depend on an emissivity value, ε, such that:
j^{*} = ε.σT^{4}
ε has a value between zero and one where an emissivity of one is that of a perfect black body. In most cases, the emissivity is some function of the wavelength of the radiation emitted (ε = ε(λ)).
We can also find the total power of energy emitted from a black body with the Stefan-Boltzmann law, using the constant σ in doing so. In order to find the power
P, the surface area, A, of the black body radiator must be taken into
consideration:
P = A.j^{*} = A.ε.σT^{4}
One application of this equation is to calculate the temperature of stars from measurements of their luminosity. The equation used to do this takes surface area of the star in question
which has a known radius, r. The following equation relates the luminosity to
this surface area and temperature:
L = 4πr^{2}.ε.σT_{e}^{4}
where L is the luminosity of the star and T_{e} is the effective temperature,
which is the temperature of a black body that would radiate the same
amount of electromagnetic energy (i.e. the equivalent luminosity per unit area). This is typically done when we do not have an emissivity function ε(λ).