Understanding this phenomenon theoretically and correctly predicting the emitted
spectrum was one of the most important topics of physics in the late nineteenth century.
For discussing thermal radiation, the concept of the blackbody is very useful. A
blackbody, which does not exist in nature, absorbs all the radiation that is incident on it,
regardless of wavelength and angle of incidence. Its reflectivity therefore is 0. Of course,
since it will also emit light according to its equilibrium temperature, it does not need to
appear black to the eye.
Two approximations for the blackbody spectrum were presented around the turn of
the century: First, in 1896, Wilhelm Wien empirically derived the following expression for
the spectral blackbody radiance:
where λ and T are the wavelength and the temperature, respectively. While this
approximation gives good results for short wavelengths, it fails to predict the emitted
spectrum at long wavelengths, thus in the infrared.
Secondly, in 1900 and in a more complete version in 1905, Lord Rayleigh and James
Jeans, derived the equation
where k B ≈ 1.381 × 10
−23 J/K is the Boltzmann constant. The derivation of this equation
was based on electrodynamic arguments. While
is in good agreement to measured
values at long wavelengths, it diverges to infinity for short wavelength. Further, the
radiant emittance, which is obtained via integration over all wavelengths, diverges towards
infinity. This so-called ultraviolet catastrophe demonstrates that Rayleigh and Jeans did
not succeed in developing a model that could adequately describe thermal radiation.
In 1900, Max Planck found an equation that interpolates between the Wien
approximation and the Rayleigh–Jeans law,
where c ≈ 2.998 × 10
8
m/s is the speed of light in vacuo and h ≈ 6.626 × 10
−34 m
2 kg/s,
which is now known as the Planck constant. Via Eq. (5.7) we find the Planck law
expressed as a function of the frequency ν,
It is remarkable to see that the Planck law contains three fundamental constants, c, k B , and
spectrum was one of the most important topics of physics in the late nineteenth century.
For discussing thermal radiation, the concept of the blackbody is very useful. A
blackbody, which does not exist in nature, absorbs all the radiation that is incident on it,
regardless of wavelength and angle of incidence. Its reflectivity therefore is 0. Of course,
since it will also emit light according to its equilibrium temperature, it does not need to
appear black to the eye.
Two approximations for the blackbody spectrum were presented around the turn of
the century: First, in 1896, Wilhelm Wien empirically derived the following expression for
the spectral blackbody radiance:
where λ and T are the wavelength and the temperature, respectively. While this
approximation gives good results for short wavelengths, it fails to predict the emitted
spectrum at long wavelengths, thus in the infrared.
Secondly, in 1900 and in a more complete version in 1905, Lord Rayleigh and James
Jeans, derived the equation
where k B ≈ 1.381 × 10
−23 J/K is the Boltzmann constant. The derivation of this equation
was based on electrodynamic arguments. While
is in good agreement to measured
values at long wavelengths, it diverges to infinity for short wavelength. Further, the
radiant emittance, which is obtained via integration over all wavelengths, diverges towards
infinity. This so-called ultraviolet catastrophe demonstrates that Rayleigh and Jeans did
not succeed in developing a model that could adequately describe thermal radiation.
In 1900, Max Planck found an equation that interpolates between the Wien
approximation and the Rayleigh–Jeans law,
where c ≈ 2.998 × 10
8
m/s is the speed of light in vacuo and h ≈ 6.626 × 10
−34 m
2 kg/s,
which is now known as the Planck constant. Via Eq. (5.7) we find the Planck law
expressed as a function of the frequency ν,
It is remarkable to see that the Planck law contains three fundamental constants, c, k B , and