This article provides answers to the following questions, among others:

  • What dows the Stefan-Boltzmann law state?
  • What does Kirchhoff’s law of thermal radiation state?
  • Why are blackbodies ideal radiant emitters?
  • How can the Stefan-Boltzmann law be derived from the laws of thermodynamics?
  • What is the relationship between the radiation pressure and the energy density of a photon gas?
  • How was it possible to calculate the Stefan Boltzmann constant?

Introduction

In the article Blackbody radiation it has already been explained in detail why every object emits radiation above absolute zero. This radiation is also called thermal radiation. Thermal radiation is caused by the motions of atoms that emit electromagnetic waves, i.e. radiation.

Thermal radiation can not only be proven by the fact that it is able to heat other objects, as one could derive from the term thermal radiation. At sufficiently high temperatures, the radiated wavelength spectrum shifts into the visible range and can thus be observed directly by the human eye. The reddish annealing of a heated metal rod during forging is, for example, the result of such a visible thermal radiation.

Visible radiation of a glowing steel rod
Figure: Visible radiation of a glowing steel rod

The visible glow of the filament of a light bulb at over 3000 °C is also a typical example which can be traced back to the phenomenon of thermal radiation. More than 90 % of the energy is radiated in the non-visible infrared range and can therefore only be perceived as heat. The remaining part, however, is in the visible wavelength spectrum and can be directly observed as a yellowish glow (a small part is also radiated as ultraviolet light).

Visible radiation of a light bulb
Figure: Visible radiation of a light bulb

Radiant power

As a heat lamp or a glowing steel block clearly shows, emitting radiation is obviously associated with emitting energy. How much radiant energy \(\Delta Q\) an object emits per unit time \(\Delta t\), i.e. how high its radiant power \(\Phi\) is, depends mainly on the temperature \(T\), but also on the area of the surface \(A\) as well as on the radiative property of the body (the so-called emissivity \(\varepsilon\)).

\begin{align}
&\boxed{\Phi = \frac{\Delta Q}{\Delta t}}=\Phi(T,A,\varepsilon) \\[5px]
\end{align}

Note that the radiant power (also called radiant flux) does not only refer to the thermal energy in the infrared spectrum or to the radiated energy in the visible wavelength range, but to the energy related to the entire wavelength spectrum, i.e. to the entire energetic radiation!

If, for example, an incandescent light bulb is operated at low current, the temperature of the filament is correspondingly low. The light bulb does not only glow less but it also does not heat up as much. Overall, the radiant power is relatively low at low temperatures. With a large current, on the other hand, the filament heats up strongly and the temperature is correspondingly high. It then not only glows intensely yellow but also radiates infrared radiation to a high degree, which is clearly noticeable as heat. The higher the temperature of a body, the higher the radiant power is!

Thermal radiation of a light bulb at different temperatures
Figure: Thermal radiation of a light bulb at different temperatures

Note: The fact that the filament is weakly reddish at low temperature and bright yellow at high temperature is due to the wavelength spectrum emitted, which shifts to the yellowish range with increasing temperature (more information on this in the article Blackbody radiation).

In addition to the temperature, the area of the surface of the radiating body also influences the radiant flux. The larger the surface, the more atoms can vibrate and emit radiation. Note that radiation emitted by the atoms inside the body is directly reabsorbed by the surrounding atoms. Thus only the atoms on the surface are relevant for the radiation of the electromagnetic waves. If the surface is twice as large, the radiant power should therefore be twice as high.

More detailed studies by the physicists Josef Stefan and Ludwig Boltzmann at the end of the 19th century showed that the radiant power is actually directly proportional to the surface area of the emitting object. The influence of temperature on radiant power, on the other hand, is far greater. It increases with the fourth power of the absolute temperature. A doubling of the temperature from e.g. 1000 K to 2000 K thus increases the radiant power by a factor of 16! For an ideal thermal radiator, a so-called blackbody, the radiant power \(\Phi_{ideal}\) results as a function of temperature \(T\) and surface area \(A\) as follows:

\begin{align}
&\Phi_{ideal} \sim A \cdot T^4 \\[5px]
&\boxed{\Phi_{ideal} = \sigma \cdot A \cdot T^4} ~~~~~ \sigma = 5,670 \cdot 10^{-8} \frac{\text{W}}{\text{m²K}^4} \\[5px]
\end{align}

The Stefan-Boltzmann law states that the radiant power of an object in thermal equilibrium is proportional to the fourth power of temperature and directly proportional to its surface area!

The proportionality factor \(\sigma\) is called the Stefan-Boltzmann constant and is a universal constant, i.e. it does not depend on the material of the radiating object as long as the body absorbs all incident radiation and can therefore be regarded as a blackbody.

A black-body is an ideal thermal radiator that absorbs all incident radiation and therefore radiates at maximum power!

Why a black body is not only a perfect absorber of radiation but also a perfect emitter of radiation will be explained later.

If the radiant power \(\Phi\) of the black body is related to its surface area \(A\), then one also speaks of the so-called intensity \(I\) (surface power density). The intensity indicates the strength of the radiant power per unit area. The intensity of black-body radiation depends only on temperature:

\begin{align}
&I=\frac{\Phi}{A} = \frac{\sigma \cdot A \cdot T^4}{A} = \sigma \cdot T^4\\[5px]
&\boxed{I = \sigma \cdot T^4 } \\[5px]
\end{align}

Kirchhoff’s law of thermal radiation

In the following a blackbody is considered, which is irradiated by a heat lamp. By definition, the blackbody will absorb all incident radiation. The absorbed energy leads to an increase in temperature and the blackbody begins to emit more and more radiation. Finally, over time, a thermal equilibrium will be reached in which the temperature no longer rises. In thermodynamic equilibrium, the radiant energy emitted within a certain time (emitted radiant power \(\Phi_e\)) must therefore be the same as the absorbed radiant energy (absorbed radiant power \(\Phi_a\)):

\begin{align}
\label{kirch}
&\boxed{\Phi_a \overset{!}{=} \Phi_e} \\[5px]
\end{align}

Energy flow diagram of a blackbody
Figure: Energy flow diagram of a blackbody

This radiation equilibrium between emitted radiation and absorbed radiation basically applies to every body in thermal equilibrium, including non-ideal black bodies that do not radiate at maximum power. After all, a constant temperature and thus a thermal equilibrium will be reached for every object after a certain time, in which emission and absorption must take place to the same extent. This law is also called Kirchhoff’s law of radiation.

Energy flow diagram of a real body
Figure: Energy flow diagram of a real body

Kirchhoff’s law of radiation states that in thermal equilibrium of a radiating body, emission and absorption take place to the same extent!

In the same way as the emitted radiant power increases with the fourth power of the temperature, the absorbed radiant power must also increase with the fourth power of the temperature. Otherwise there would be no thermal equilibrium. The law with which an ideal blackbody emits radiation according to the Stefan-Boltzmann law must therefore also apply to the absorbed radiation in thermal equilibrium!

\begin{align}
&\boxed{\Phi_{a,ideal} = \Phi_{e,ideal} = \sigma \cdot A \cdot T^4} \\[5px]
\end{align}

Real bodies

This situation can now be transferred to real objects, which are not perfect black bodies. With real objects a certain part of the radiation is always reflected and not completely absorbed like with black bodies. The absorbed radiant power of a real body will therefore be a factor \(\alpha<1\) less than that of an ideal black body. This factor \(\alpha\), which describes the absorbed part of the incident radiation and thus indicates the absorbed radiant power in comparison to an ideal black body, is also referred to as absorptivity \(\alpha\).

\begin{align}
&\boxed{\Phi_{a,real} = \alpha \cdot \Phi_{a,ideal} } ~~~~~\alpha<1 \\[5px]
&\Phi_{a,real} = \alpha \cdot \sigma \cdot A \cdot T^4 \\[5px]
\end{align}

Absorptivity \(\alpha\) is the absorbed radiation portion of an incident radiation, i.e. the absorbed radiant energy of a real body compared to an ideal blackbody!

According to Kirchhoff’s law of radiation, the emitted radiant power in thermal equilibrium is just as high as the absorbed radiant power. A real body that has a lower absorption power than an ideal black body will therefore also have a lower emission power to the same extent!

The emission power of a real body in thermal equilibrium is as much lower as the absorption power!

However, this also means that a body that absorbs to the maximum also emits to the maximum. A blackbody is therefore not only an ideal absorber of radiation but also a perfect emitter of radiation!

A blackbody is an ideal thermal radiator with the maximum possible radiant power!

In purely formal terms, the emission power of a real body can also be expressed with a factor which then indicates the ratio of the emitted radiation of the real body compared to an ideal black body! This factor is referred to as emissivity \(\varepsilon\).

\begin{align}
&\boxed{\Phi_{e,real} = \varepsilon \cdot \Phi_{e,ideal} } ~~~~~\varepsilon<1 \\[5px]
&\Phi_{e,real} = \varepsilon \cdot \sigma \cdot A \cdot T^4 \\[5px]
\end{align}

An emissivity of e.g. \(\varepsilon\)=0.9 means that the considered object has 90 % of the radiant power of an ideal thermal radiator (perfect black body). Many non-metallic objects have an emissivity of more than 90 % and can therefore be regarded as blackbodies in very good approximation with respect to their emitted radiation. The emissivity for an ideal blackbody is \(\varepsilon=1\).

The emissivity represents the ratio of the radiation actually emitted by a real body to that of an ideal thermal radiator, a perfect blackbody!

Although the absorptivity and the emissivity are formally differentiated, in thermal equilibrium this distinction is obsolete, since both quantities have the same value according to Kirchhoff’s law of radiation (\ref{kirch}):

\begin{align}
\require{cancel}
&\Phi_{e,real} \overset{!}{=} \Phi_{e,real} \\[5px]
&\alpha \cdot \bcancel{\sigma \cdot A \cdot T^4} = \varepsilon \cdot \bcancel{\sigma \cdot A \cdot T^4} \\[5px]
&\boxed{\alpha = \varepsilon} \\[5px]
\end{align}

In thermal equilibrium, the absorptivity is equal to the emissivity!

Kirchhoff's law of thermal radiation
Figure: Kirchhoff’s law of thermal radiation

Remark

Real bodies which have a constantly lower (spectral) emission power in the entire radiated wavelength spectrum compared to an ideal black body are also called gray bodies. Gray bodies therefore have an absorptivity or emissivity that is not dependent on the wavelength.

In some cases, however, a body will also absorb or emit radiation to different degrees depending on the wavelength. The absorptivity or emissivity then depends on the wavelength. Such bodies are called selective absorbers or selective radiators.

Thermodynamic derivation of the Stefan-Boltzmann Law

In the following sections, the Stefan-Boltzmann Law is to be derived with the aid of the laws of thermodynamics. In order to be able to do this, fundamental relationships must first be clarified.

Relationship between energy density and pressure

Kinetic theory of gases with classical particles

With the help of the kinetic theory of gases, the following relationship between the pressure of a gas \(p\) and the speed \(v\) of the particles contained therein was derived in the article Pressure and temperature:

\begin{align}
\label{p}
&\boxed{p = \frac{1}{3}\frac{N}{V}m\cdot \overline{v^2}} \\[5px]
\end{align}

Here \(N\) denotes the number of gas particles in a volume \(V\) and \(m\) refers to the mass of one particle (not the entire gas mass!). The speed refers to the root mean square speed.

In a classical view of ideal gases, in which the particle speeds are far below the speed of light, the mean kinetic energy of the particles can be described in very good approximation with \(\frac{1}{2}m\overline{v^2}\) (Note that this formula only applies to non-relativistic mechanics!) The following relationship between the pressure and the mean energy of a particle becomes obvious:

\begin{align}
&p = \frac{2}{3} \cdot \frac{N}{V} \cdot \overbrace{\frac{1}{2} m \overline{v^2}}^{\overline{W_{kin}}} = \frac{2}{3} \cdot \frac{N}{V} \cdot \overline{W_{kin}} = \frac{2}{3} \cdot \frac{\overbrace{N \cdot \overline{W_{kin}}}^{\text{internal energy } U}}{V} = \frac{2}{3}\cdot \overbrace{\frac{U}{V}}^{\text{energy densitiy } u_v} = \frac{2}{3}\cdot u_v \\[5px]
&\boxed{p = \frac{2}{3} u_v} ~~~~~\text{only valid for a classical gas}
\end{align}

In the above derivation it was used that the product of the number of particles \(N\) and the mean kinetic energy of a particle \(\overline{W_{kin}}\) corresponds to the total energy contained in the gas, i.e. the so-called internal energy \(U\). The quotient of internal energy \(U\) and volume \(V\) can therefore be interpreted as (volumetric) energy density \(u_v\) (to avoid misunderstandings with the specific internal energy \(u\) as a mass-related quantity, a “v” is added to the energy density in the index).

Due to the fact that classical mechanics was used for the derivation under the assumption of the validity of \(W_{kin} = \frac{1}{2}mv^2\), the obtained relationship between pressure and energy density applies only to classical ideal gases.

Kinetic theory of gases with relativistic particles (photons)

Due to the wave-particle duality, electromagnetic radiation can also be imagined as a beam of photons, as has already been done for the derivation of the radiation pressure. In this way, the radiation of a blackbody can now be described mathematically. In the article Blackbody radiation the realization of such a blackbody has already been explained in detail. For this purpose, a small hole is drilled into an object leading into a cavity.

Realization of a black (hollow) body
Figure: Realization of a black (hollow) body

All incident radiation is absorbed by the cavity, which also emits radiation itself. In thermal equilibrium, the emitted and absorbed radiation energy is the same, so that in principle a gas of photons is formed in the cavity. Instead of massive particles, in this case we are dealing with massless, relativistic photons. Here, too, a pressure \(p\) can be found in the cavity volume \(V\) due to the moving photons, the so-called radiation pressure.

Illustration of the photon gas in a cavity acting as a blackbody
Figure: Illustration of the photon gas in a cavity acting as a blackbody
Animation: Photon gas in a cavity acting as a blackbody

Equation (\ref{p}) can now also be used, it only has to be interpreted against the background of theory of relativity. For photons, the speed of the particles \(v\) corresponds to the speed of light \(c\) and the particle mass \(m\) must be considered as the relativistic mass \(m_{rel}\). At this point it is not necessary to differentiate between the mean value of the speed squares \(\overline{c^2}\) and the speed square \(c^2\), since in this case there is no classical speed distribution but all photons are travelling with the same speed of light. Both values are therefore identical.

\begin{align}
\label{pp}
&p = \frac{1}{3}\frac{N}{V}\cdot m_{rel} \cdot c^2 \\[5px]
\end{align}

Note that although photons do not have a rest mass, they do have a relativistic mass! For example, Einstein showed with his famous equation on energy-mass equivalence that every thing with energy can be assigned a corresponding (relativistic) mass \(m_{rel}\):

\begin{align}
\label{e}
&\boxed{E =mc^2} ~~~~~\text{or}~~~~~ \boxed{W=m_{rel}\cdot c^2} \\[5px]
\end{align}

According to quantum mechanics, the energy of a photon results from the product of frequency \(f\) and Planck constant \(h\), so that the relativistic mass can be determined from this:

\begin{align}
&W = h\cdot f \\[5px]
\end{align}

The product of the (relativistic) mass \(m\) (=\(m_{rel}\)) and the square of the speed of light \(c^2\) contained in equation (\ref{pp}) corresponds to the energy of a photon \(W\) according to equation (\ref{e}). Thus the following relationship applies between the radiation pressure \(p\) of the photon gas and its energy density \(u_v\):

\begin{align}
&p = \frac{1}{3}\frac{N}{V}\cdot \overbrace{ m_{rel}\cdot c^2}^{W}=\frac{1}{3}\frac{N}{V}\cdot W =\frac{1}{3}\frac{\overbrace{N\cdot W}^{\text{“internal” energy } U}}{V} =\frac{1}{3} \cdot \overbrace{\frac{U}{V}}^{\text{energy density }u_v} = \frac{1}{3} u_v \\[5px]
\label{h}
&\boxed{p = \frac{1}{3} u_v } ~~~~~\text{only valid for a photon gas (homogeneous radiation)}
\end{align}

The product of photon energy and number of photons corresponds to the total (internal) energy contained in the photon gas, i.e. the radiant energy (photon gas energy). The quotient of radiant energy and cavity volume can in turn be understood as the energy density of the photon gas or the energy density of the radiation.

The relationship between pressure and energy density of a classical gas thus differs by a factor of 2 from that of a relativistic photon gas.

Derivation from radiation pressure

The relationship between radiation pressure and energy density for a homogeneous photon gas can also be derived from the radiation pressure of a directed beam. In the article Radiation pressure it was shown that the radiation pressure \(p_{beam}\), which a directed beam exerts on an object at complete reflection, corresponds to twice the value of the energy density:

\begin{align}
\label{s}
&\boxed{p_{beam} =2 \cdot u_v} ~~~~~\text{only valid for complete reflection of directed beams} \\[5px]
\end{align}

This equation does not contradict equation (\ref{h})! Because equation (\ref{s}) is only valid for directed radiation, i.e. when all photons move in the same direction and hit a boundary surface. In the present case of cavity radiation, however, it is a matter of radiation that is permanently reflected, absorbed and emitted inside, i.e. the photons all move in completely random directions; it is actually a photon gas.

In order to make a completely random photon motion out of a directed photon beam, all photons contained in it would suddenly have to move in different directions. Assuming a homogeneous statistical distribution, one sixth of the photons would move upwards, one sixth downwards, one sixth to the left and one sixth to the right, and one sixth to the rear, and finally one sixth of the photons would move forwards towards the boundary surface under consideration, on which a pressure would be exerted.

Radiation pressure of a photon beam and radiation pressure of a photon gas
Figure: Radiation pressure of a photon beam and radiation pressure of a photon gas

At the same energy density (photon density), the pressure of a homogeneous photon gas is thus one sixth lower than that of a directed photon beam, since only one sixth of the photons effectively collide with the boundary surface. Thus, for a homogeneous photon gas, one obtains the same relationship:

\begin{align}
& p = \frac{1}{6} \cdot p_{beam} = \frac{1}{6} \cdot 2 \cdot u_v= \frac{1}{3} \cdot u_v \\[5px]
\end{align}

One could now argue at this point that the cavity radiation is not based on a complete reflection and thus equation (\ref{s}) is not valid at all. Rather, a blackbody or a cavity radiation is a complete absorption of the radiation. This is correct, but in thermal equilibrium exactly the same number of photons are absorbed as emitted. This is where the thermodynamic equilibrium comes into being.

Kinematically speaking, a reflection is nothing more than an absorption (“impact”) with subsequent emission (“rebound”). Only because absorption and emission do not take place at identical times in the case of cavity radiation, they are essentially identical to reflection, in which both processes in principle take place immediately one after the other. For this reason, a completely reflective behavior must be assumed for cavity radiation in thermodynamic equilibrium!

Relationship between a change in volume and the change in photon gas energy

As the experimental study of blackbody radiation has already shown, the energy of the radiation emitted depends only on the temperature. The energy density is therefore only a function of the temperature. At constant temperature and thus constant energy density \(u_v\), an increase of the cavity volume by \(\text{d}V\) means a corresponding increase of the photon gas energy \(\text{d}U\):

\begin{align}
& U(T,V) = u_v(T) \cdot V \\[5px]
& \frac{\text{d}U(T,V)}{\text{d}V} =\frac{\text{d}(u_v(T) \cdot V )}{\text{d}V} = \overbrace{\underbrace{\frac{\partial{u_v(T)}}{\partial{V}}}_{=0} \cdot V +\underbrace{\frac{\partial{V} }{\partial{V}}}_{=1} \cdot u_v(T)}^{\text{product rule}} = u_v(T) = u_v \\[5px]
\label{t}
&\boxed{\left(\frac{\partial{U}}{\partial{V}}\right)_T = u_v = 3p} \\[5px]
\end{align}

Note that the (partial) derivation of the energy density with respect to the volume equals to zero, since the energy density is not a function of the volume. With respect to the variable \(V\), this is the derivation of a constant. In the last step it was used that the energy density \(u_v\) corresponds to three times the value of the pressure \(p\) [see equation (\ref{h})].

The fact that the photon gas energy increases at a constant energy density to the same extent as the volume also becomes clear, because a constant energy density also means a constant photon density. If one increases the volume of the cavity, then with constant photon density there are now more photons in it (the cavity then emits more photons until the thermal equilibrium is restored). Since each photon is associated with a certain energy, the total energy contained in the volume has also increased.

First law of thermodynamics

Since the photon gas is considered analogous to a classical ideal gas, the first law of thermodynamics can also be applied at this point. In differential form, it is represented as follows (in which \(S\) denotes the entropy):

\begin{align}
&\boxed{\text{d}U = T \cdot \text{d}S – p \cdot \text{d}V } ~~~~~\text{First law of thermodynamics} \\[5px]
\end{align}

If this equation is divided by \(\text{d}V\), the following relationship is obtained at constant temperature:

\begin{align}
\label{ss}
& \left(\frac{\partial{U}}{\partial{V}}\right)_T = T \left(\frac{\partial{S}}{\partial{V}}\right)_T – p \\[5px]
\end{align}

Without going deeper into the derivation of the so-called Maxwell relations of thermodynamics, these provide another important relationship at this point: A change in entropy and a change in volume at constant temperature are in the same relationship as a change in pressure and a change in temperature at constant volume. Mathematically this is expressed as follows:

\begin{align}
&\boxed{\left(\frac{\partial{S}}{\partial{V}}\right)_T = \left(\frac{\partial{p}}{\partial{T}}\right)_V} ~~~~~\text{Maxwell relation} \\[5px]
\end{align}

This Maxwell relation can now be used in equation (\ref{ss}):

\begin{align}
\label{x}
& \left(\frac{\partial{U}}{\partial{V}}\right)_T = T \left(\frac{\partial{p}}{\partial{T}}\right)_V – p \\[5px]
\end{align}

According to equation (\ref{t}) the term \(\left(\frac{\partial{U}}{\partial{V}}\right)_T\) can be replaced by the energy density \(u_v\). In addition, the pressure \(p\) according to the equation (\ref{h}) corresponds to just one third of the energy density:

\begin{align}
u_v &= T \left(\frac{\partial{\frac{1}{3}u_v}}{\partial{T}}\right)_V – \frac{1}{3} u_v \\[5px]
u_v &= \frac{1}{3} T \left(\frac{\partial{u_v}}{\partial{T}}\right)_V – \frac{1}{3} u_v \\[5px]
3 u_v &= T \left(\frac{\partial{u_v}}{\partial{T}}\right)_V – u_v \\[5px]
\end{align}

Since the energy density is only a function of temperature, the partial derivative with respect to the temperature can be expressed as an ordinary derivative. After separation of variables (Fourier method) we obtain:

\begin{align}
3 u_v &= T \frac{\text{d}u_v}{\text{d}T} – u_v \\[5px]
4 u_v &= T \frac{\text{d}u_v}{\text{d}T} \\[5px]
\frac{\text{d}u_v}{u_v} &=4 \frac{\text{d}T}{T} \\[5px]
\end{align}

Both sides of the equation can now be integrated, whereby constants of integration must be taken into account when calculating the primitive function, which are summarized in the constant \(a\) (in this case, \(a\) generally stands for a constant, not for a special value!):

\begin{align}
\int \frac{\text{d}u_v}{u_v} &= \int 4 \frac{\text{d}T}{T} \\[5px]
\ln{u_v} &= 4 \cdot \ln{T} + a\\[5px]
e^{\ln{u_v}} &= e^{4 \cdot \ln{T}+a} \\[5px]
e^{\ln{u_v}} &= e^{4 \cdot \ln{T}} \cdot e^{a} \\[5px]
e^{\ln{u_v}} &= \left(e^{\ln{T}}\right)^4 \cdot a \\[5px]
&\underline{u_v = a \cdot T^4} \\[5px]
\end{align}

The energy density \(u_v\) is thus proportional to the fourth power of the absolute temperature \(T\). A volume \(V\) filled with photons (e.g. the volume of the considered cavity), which are emitted by a blackbody at the temperature \(T\), contains the following total energy \(U\):

\begin{align}
&U = u_v \cdot V = a \cdot T^4 \cdot V \\[5px]
\label{st}
&\boxed{U = a \cdot T^4 \cdot V} \\[5px]
\end{align}

The constant \(a\) could now be determined experimentally. Usually, however, it is not the energy of a photon gas that is of interest but the power with which the body emits the photons.

Radiant power

In the following, a blackbody with a surface area \(A\) is considered, which is in thermodynamic equilibrium at the temperature \(T\). This black body emits photons that move away from the surface at the speed of light \(c\) (in the case of a hollow body into the cavity and then through the hole into the environment, or in the case of a solid body directly into the environment). Within the infinitesimal time \(\text{d}t\) these photons cover a certain distance \(\text{d}s\):

\begin{align}
& \text{d}s = c \cdot \text{d}t \\[5px]
\end{align}

Derivation of the radiant power of a blackbody
Figure: Derivation of the radiant power of a blackbody

Since the photons are emitted from the surface area \(A\), they occupy a certain shell volume around the blackbody within the time \(\text{d}t\). Due to the infinitesimal distance, the volume \(\text{d}V\) of this emitted photon shell results from the product of base area \(A\) and height \(\text{d}s\):

\begin{align}
& \text{d}V = A \cdot \text{d}s = A \cdot c \cdot \text{d}t \\[5px]
\end{align}

At a given temperature \(T\) the energy \(\text{d}U\) contained in the emitted photon volume \(\text{d}V\) is given by equation (\ref{st}).

\begin{align}
&\text{d}U = a \cdot T^4 \cdot \text{d}V= a \cdot T^4 \cdot A \cdot c \cdot \text{d}t \\[5px]
\end{align}

Within the time \(\text{d}t\) the blackbody radiates the energy \(\text{d}U\). From this the radiant power \(\Phi\) can be determined as the emitted radiant energy per unit time:

\begin{align}
\require{cancel}
&\Phi =\frac{\text{d}U}{\text{d}t} = \frac{a \cdot T^4 \cdot A \cdot c \cdot \bcancel{\text{d}t} }{\bcancel{\text{d}t}} =\underbrace{a\cdot c}_{\sigma} \cdot A \cdot T^4 \\[5px]
\end{align}

The constant \(a\) and the constant speed of light \(c\) can be combined to a new constant, the so-called Stefan-Boltzmann constant \(\sigma\) (not to be confused with the Boltzmann constant \(k_B\)!). Thus the radiant power of a black body in thermodynamic equilibrium at a given temperature \(T\) results from the following formula:

\begin{align}
\label{bb}
&\boxed{\Phi = \sigma \cdot A \cdot T^4} ~~~~~ \sigma = 5,670 \cdot 10^{-8} \frac{\text{W}}{\text{m²K}^4} \\[5px]
\end{align}

This law was experimentally derived by the physicist Josef Stefan and later mathematically derived by Ludwig Boltzmann. This law is therefore called the Stefan-Boltzmann Law.

If the radiant power \(\Phi\) at this point is related to the surface area \(A\) of the blackbody, then the intensity \(I\) is obtained:

\begin{align}
&I=\frac{\Phi}{A} = \frac{\sigma \cdot A \cdot T^4}{A} = \sigma \cdot T^4\\[5px]
&\boxed{I = \sigma \cdot T^4 } \\[5px]
\end{align}

The Stefan-Boltzmann constant \(\sigma\), which was initially determined empirically by experiments, is a fundamental physical constant that could actually only be derived from other fundamental constants by quantum mechanics:

\begin{align}
&\boxed{\sigma = \frac{2 \pi^5k_B^4}{15h^3c^2} } \\[5px]
\end{align}

As already explained at the beginning of this article, Blackbodies only exist in the ideal. In reality, therefore, bodies do not radiate with the intensity of a black body but with lower power. This is expressed by the emissivity \(\varepsilon<1\):

\begin{align}
&I=\varepsilon \cdot \sigma \cdot T^4\\[5px]
\end{align}