From Planck’s Law to Stefan–Boltzmann

The Stefan–Boltzmann law, also called Stefan’s law, defines the total energy emitted per unit surface area by a black body in thermal equilibrium. To determine the total radiated power from an object, multiply the radiant exitance by its surface area A. The law expresses radiant exitance M as a function of temperature. The radiant exitance with a superscript circle denotes quantities specific to a black body.

[math]M(T)\ =\ \varepsilon \ M^{\circ }(T)=\ \varepsilon \ \sigma \ T^4\ [/math]

The constant of proportionality σis called the Stefan–Boltzmann constant [math]\sigma[/math] and is derived from fundamental physical constants. 

[math]\sigma=\frac{2\pi^{5}.k_B^4}{15c^{2}.h^{3}}=5670.10^{-8}W \cdot m^{-2}.K^{-4}[/math]

In the general case, the Stefan–Boltzmann law for radiant exitance takes the form of the emissivity of the surface-emitting the radiation. For a grey body, the power [math]P\ [/math] of the emitted radiation for an area [math]A\ [/math] is adjusted by the emissivity ε, which accounts for the material’s ability to emit infrared radiation, 

[math]P\left(T\right)=AM\left(T\right)=A.\epsilon.\sigma.T^{4}[/math]

The law can be derived by integrating Planck’s law over the complete spectrum and half sphere, considering Lambert’s cosine law, where [math]B_{\lambda }(\lambda ,T)[/math] is the radiance spectrum from Planck’s law, [math]d\Omega \ =\sin (\theta )\ d\theta \ d\phi [/math] is the solid angle element in spherical polar coordinates, [math]\theta [/math] is the angle between the surface normal and the radiation direction, [math]\phi [/math] is the angle of direction in the half-space, and the term [math]\cos (\theta )[/math] accounts for Lambert’s cosine law. 

[math]M\left(\lambda,T\right)=\int_{0}^{\infty}\int_{}^{} B_{\lambda}\left(\lambda,T\right)\cos\left(\sigma\right)d\Omega\ d\lambda[/math]

Since [math]B_{\lambda }(\lambda ,T)[/math] is independent of direction, the angular integral over the hemisphere simplifies to as the integration of the trigonometric terms results in a simple factor [math]\pi [/math].

[math]M°\left(\lambda,T\right)=\int_{0}^{\infty} B_{\lambda}\left(\lambda,T\right)d\lambda\int_{0}^{\pi/2} \cos\left(\theta\right)\sin\left(\theta\right)d\left(\theta\right)\int_{0}^{2\pi} d\phi=\int_{0}^{\infty} B_{\lambda}\left(\lambda,T\right)d\lambda.\frac{1}{2}.2\pi[/math]

The integration of the spectral radiance [math]B_{\lambda }(\lambda ,T)[/math] requires the substitution of the exponential component [math]x=\frac{hc}{\lambda \ k_B\ T}[/math] so that [math]d\lambda =-\frac{h}{\lambda ^2\ k_B\ T}\ dx\ [/math] and under consideration of the Bose-Einstein integral, the equation simplifies to the following form.  

[math]M°\left(\lambda,T\right)=\pi\int_{0}^{\infty} B_{\lambda}\left(\lambda,T\right)d\lambda=2\pi.hc^{2}\left(\frac{k_{B}T}{hc}\right)^{4}\int_{0}^{\infty} \frac{x^{3}}{e^{x}-1}dx=2\pi.hc^{2}\frac{k_{B}T}{hc}\frac{\pi^{4}}{15}=\frac{2\pi^{5}k_B^4}{15c^{2}h^{3}}T^{4}[/math]

The last term represents the already introduced Stefan–Boltzmann constant and proves that the emitted by a black body is proportional to the fourth power of its temperature. Figure 1 illustrates the relationship between the emitted spectrum and the emitted power.