h, which are among the most important constants in physics.
Figure 5.3 shows the spectrum of a blackbody with a 6,000 K temperature and the
Wien approximation and the Rayleigh–Jeans law. We indeed see that the Wien
approximation fits well at short wavelengths, while the Rayleigh–Jeans law matches well
at long wavelengths but completely fails at short wavelengths.
Figure 5.3: The blackbody spectrum at 6,000 K as calculated with the Wien approximation, the Rayleigh–Jeans law and
the Planck law.
Both the Wien approximation (Eq. (5.16)) and the Rayleigh–Jeans law (Eq. (5.17))
can be directly derived from the Planck law:
For short wavelengths,
such that the −1 can be ignored and we arrive at the Wien approximation with C 1 = 2hc
2
and C 2 = hc/k B .
For long wavelengths we can use the approximation
which directly results in the Rayleigh–Jeans law.
The total radiant emittance of a blackbody is given by
where
is the Stefan–Boltzmann constant. Equation (5.19) is known as the Stefan-Boltzmann law.