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

Adelyn Rodriguez

Adelyn Rodriguez

Answered question

2022-05-10

Wien's fifth power Law and Stephan Boltzmann's fourth power laws of emissive power
Wien's fifth power law says that emissive power is proportional to the temperature raised to the fifth power. On the other hand, the Stefan–Boltzmann law says emissive power is proportional to the temperature raised to the fourth power. How can both of these be true?

Answer & Explanation

Braxton Gallagher

Braxton Gallagher

Beginner2022-05-11Added 21 answers

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 , λ max T = b is constant. So, from Planck's law (e.g. see here):
I ( λ max , T ) λ max 5 e h c / ( λ max k T ) 1 = T 5 / b 5 e h c / ( k b ) 1 T 5
and integrating Plank's law we get, by changing variables to x = h c / ( λ k T ) so that λ 5 d λ T 4 x 3 d x (the negative is absorbed by changing the integration interval limits):
0 I ( λ , T ) d λ 0 T 4 x 3 d x e x 1 T 4

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