Supplement 2.3: The Stefan-Boltzmann Law (2/2)

Derivation from electrodynamics and thermodynamics

In theoretical electrodynamics it is shown (keyword: energy-momentum tensor of the electromagnetic field), that electromagnetic waves generate a radiation pressure that acts as a force on an illuminated surface. This has already been proven experimentally. This pressure is directly related to the intensity of the illumination on the surface, and hence the Poynting vector $⟨ S ⟩$ as calculated in Supplement 1.4:

p = ⟨ S ⟩ / c

where $c$ is the speed of light. Also, in Supplement 1.4 it was shown that the Poynting vector can be derived by multiplying the energy density $U$ with $c$ , which gives:

⟨ S ⟩ = c U

Radiation pressure and energy density are therefore closely linked. Now, we consider electromagnetic waves in thermal equilibrium in an evacuated cacity. Under these conditions, the radiation pressure inside the vacuum can be characterised:

p = \frac{1}{3} U

Diese Beziehung ist die Zustandsgleichung der Hohlraumstrahlung.

This relationship is the equation of state of cavity radiation.

The energy density depends on the temperature, but not on the volume of the vacuum; an isothermal change in volume has no influence on the cavity radiation. This also applies to the radiation pressure. In this respect, the equation of state of cavity radiation differs from the equation of state of an ideal gas $p V = R T$ , where conditions depend on volume as well as pressure and temperature; $R$ is the universal gas constant. That being said, the equation of state of cavity radiation does support the photon model of light with the concept of a photon gas in the vacuum; although in contrast to the particles of the ideal gas, the number of photons may change due to absorption and emission.

According to $E = U V$ , the total energy $E$ in the vacuum depends on the energy density and volume. It corresponds to the internal energy in the vacuum, whose infinitesimal change can be derived with $U$ and $V$ :

d E = V d U + U d V

According to the first law of thermodynamics, the internal energy increases with supplied heat $δ q$ and reduces by doing work through increasing the volume $p d V$ :

d E = δ q - p d V

Equating the two relationships for the internal energy, replacing the pressure with the energy density according to the equation of the cavity radiation mentioned above and rewriting gives:

δ q = \frac{4}{3} U d V + V d U

The entropy in the cavity changes as a result of a change in heat according to

d S = \frac{δ q}{T} = \frac{4}{3} \frac{U}{T} d V + \frac{V}{T} \frac{\partial U}{\partial T} d T

On the other hand, the following applies to entropy when the volume and temperature change:

d S = \frac{\partial S}{\partial V} d V + \frac{\partial S}{\partial T} d T

Since the entropy is a continuous and differentiable quantity (and therefore a state function), the derivatives of the coefficients are equal:

\frac{\partial}{\partial T} \frac{\partial S}{\partial V} = \frac{\partial}{\partial V} \frac{\partial S}{\partial T}

\frac{\partial^{2} S}{\partial T \partial V} = \frac{\partial^{2} S}{\partial V \partial T}

i.e., the sequence of the derivations is irrelevant. So here:

\frac{4}{3} \frac{\partial}{\partial T} \frac{U}{T} = \frac{\partial}{\partial V} (\frac{V}{T} \frac{\partial U}{\partial T})

Further deriving gives:

\frac{4}{3} (\frac{\partial U}{\partial T} - \frac{U}{T^{2}}) = \frac{1}{T} \frac{\partial U}{\partial T}

After shortening and rewriting, you get the following equation:

\frac{1}{U} d U = \frac{4}{T} d T

Integrating the left and right sides within the respective boundaries of $U$ and $T$ , we arrive at the Stefan-Boltzmann law:

U = const . \cdot T^{4}

The coefficient on the right-hand side resulting from the integration cannot be determined by classical physics; as shown on the previous page, this requires Planck's law of radiation and thus the quantum theory.

The basics of the differentials and derivatives of functions of multiple variables used on this page can be found in Supplement 2.6.

Information about thermodynamic variables and functions as well as the meaning of the symbols " $d$ " and " $δ$ " can be found in Supplement 2.7.

Understanding Spectra from the Earth

Supplement 2.3: The Stefan-Boltzmann Law (2/2)

Derivation from electrodynamics and thermodynamics