Theentropy(density)sis
s=−∂f
∂T=−
4 kB^4 T^3
3 ~^4 c^3 π^2∫∞0x^3
ex− 1
dxand theinternal energy(density)uis
u=f+Ts=− 3 f=−(kBT)^4
~^3 c^3 π^2∫∞0x^3
ex− 1
dx.The integral at the end is just a constant factorπ
4
15. This gives an internal energy ofu=π^2 (kBT)^4
15 ~^3 c^3,
from which theStefan-Boltzmann law, the intensityI(power/area) emitted by a black body can
be found to be
I=
2 π^5 (kBT)^4
15 h^3 c^2=σT^4 ,withσas the Stefan-Boltzmann constant.