Supplement 2.3: The Stefan-Boltzmann Law    (1/2)

Derivation from Planck's radiation law

The energy density U

The spectral distribution of the energy density of the radiation field for black bodies as a function of frequency is:

u f = 8π f 2 c 3 hf exp{ hf / kT }1

It is integrated over all frequencies to obtain the spectrally integrated energy density U:

U= f=0 u f df ,

what means:

U= 8πh c 3 f=0 f 3 exp{ hf / kT }1 df

The integration becomes more clear by introducing a new variable:

hf / kT =x

With the replacements

f= kT h x     and     df= kT h dx

one has:

U= 8π k 4 c 3 h 3 T 4 x=0 x 3 e x 1 dx

The integral cannot be solved elementarily. It becomes:

x=0 x 3 e x 1 dx= π 4 15
...for mathematicians

Hereby it follows:

U= 8 π 5 k 4 15 c 3 h 3 T 4
Equations

The specific emission M

The relation of energy and specific emission of an isotropic radiation field (what means, no direction of propagation is preferred) reads:

U= 4 c M

It follows for the specific emission of a black body:

M= 2 π 5 k 4 15 c 2 h 3 T 4 =σ T 4

With h=6.6 10 34 Js , k=1.38 10 23 J/K and c=3 10 10 m/s one obtaines the Stefan-Boltzmann constant:

σ=5.670 10 8 W m 2 K 4

The radiance L

The relation of energy density and radiance of an isotropic radiation field reads:

U= 4π c L

Hereby becomes:

L= σ π T 4