Question

I have been stuck with this for the whole week HELP PLEASE: Prove the Stefan-Boltzmann law by showing that the total power radiated per unit area of blackbody is given by: P=integral from 0 to infinity I(lambda;T) d lambda = OT^4.

Answer 1

\[I (\lambda , T) = \frac{2h \lambda^3}{c^2} \frac{1}{(e^{\frac{h \lambda }{k T}}-1)}\]
is the intensity of the ligth emitted from an object \[ I(\lambda, T)*A \delta \lambda \delta \Omega\] is the power emitted from an object of area A through the angle Omega. Power per unit area is:\[\frac{P}{A} = \int\limits_{0}^{\infty} I (\lambda, T) \delta \lambda \int\limits \delta \Omega\] integrate Omega over a hemisphere which looks like: \[ \frac{P}{A} = \int\limits_{0}^{\infty} I (\lambda, T) \delta \lambda \int\limits_{0}^{2 \pi} \delta \theta \int\limits_{0}^{\pi/2} cos\phi sin \phi \delta \phi \] The two integrals on the right nicely turn out to be just Pi, which leaves you with only: \[ \frac{P}{A}= \frac{2\pi h }{c^2 } \int\limits_{0}^{\infty} \frac{\lambda^3}{e^{\frac{h \lambda}{KT}}-1} \delta \lambda \]
from here, do a substitution letting:
\[ u = \frac{h \lambda}{k T} \]

Solve your integrals to get:
\[\frac{2 \pi^5k^4}{15c^2h^3} T^4\] and that giant fraction of constants outfront gets simplified into one constant , sigma