Wein's fifth power Law and Stephan Boltzmann's fourth power laws of emissive power

Question

Wien's fifth power law says that emissive power is proportional to temperature to the fifth power. But the Stefan–Boltzmann law says emissive power is proportional to temperature to the fourth power. How can both of these be true?

Answer 1

Wien's fifth power law applies to the maximum height of the emissive power density. The Stefan–Boltzmann law applies to the total emissive power (the integration of the emissive power density).

From Wien's displacement law (derived here), $\lambda_\max T = b$ is constant. So, from Planck's law (e.g. see here):

$I'(\lambda_\max, T) \propto \frac{\lambda_\max^{-5}}{e^{hc/(\lambda_\max k T)} -1} = \frac{T^5/b^5}{e^{hc/(k b)} -1} \propto T^5$

and integrating Plank's law (e.g. here except I keep in terms of wavelength for slightly longer) we get, by changing variables to $x=hc/(\lambda kT)$ so that $\lambda^{-5}d\lambda \propto -T^4x^3dx$ (the negative is absorbed by changing the integration interval limits):

$\int_0^\infty I'(\lambda,T)d\lambda \propto \int_0^\infty \frac{T^4 x^3dx}{e^{x}-1} \propto T^4$.