Relationship between Wien's law and Stefan-Boltzmann's lawI'm studying Quantum mechanics by Bransden and Joachain and in the introduction chapter it says:Wien showed that the spectral distribution function had to be on the form ρ ( λ , T ) = λ − 5 f ( λ T )where f ( λ T ) is a function of the single variable λ T. It is a simple matter to show that Wien's law includes Stefan-Boltzmann law R ( T ) = σ T 4 One of the exercises is to show this and I cannot understand how to.This is what I've tried:The relationship between spectral emittance and spectral distribution is ρ ( λ , T ) = 4 c R ( λ , T ) ,where c is the speed of light, which inserted in the above equation gives R ( λ , T ) = c 4 λ − 5 f ( λ T ) .Now, the total spectral emittance is the integral of R over all wavelengths so R ( T ) = c 4 ∫ 0 ∞ λ − 5 f ( λ T ) d λThis is where I'm stuck. Can anyone help me figure this out?

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Relationship between Wien&#039;s law and Stefan-Boltzmann&#039;s lawI&#039;m studying Quantum mechanics by Bransden and Joachain and in the introduction chapter it says:Wien showed that the spectral distribution function had to be on the form  ρ  (  λ  ,  T  )  =      λ          −      5        f  (  λ  T  )where   f  (  λ  T  ) is a function of the single variable   λ  T. It is a simple matter to show that Wien&#039;s law includes Stefan-Boltzmann law   R  (  T  )  =  σ      T    4  One of the exercises is to show this and I cannot understand how to.This is what I&#039;ve tried:The relationship between spectral emittance and spectral distribution is  ρ  (  λ  ,  T  )  =      4    c    R  (  λ  ,  T  )  ,where c is the speed of light, which inserted in the above equation gives  R  (  λ  ,  T  )  =      c    4        λ          −      5        f  (  λ  T  )  .Now, the total spectral emittance is the integral of   R over all wavelengths so  R  (  T  )  =      c    4        ∫    0    ∞        λ          −      5        f  (  λ  T  )  d  λThis is where I&#039;m stuck. Can anyone help me figure this out?

pradassas66b2d · Accepted Answer

The solution to your problem(1) is this :                    R        (        T        )                            =                  c          4                          ∫                      λ            =            0                                λ            =            ∞                                    λ                      −            5                          f        (        λ        T        )                  d                λ        =                  c          4                          ∫                      λ            =            0                                λ            =            ∞                                    T                      4                                                              f              (              λ              T              )                                      (              λ              T                              )                                  5                                                                              d                (        λ        T        )                ⟹                            (01)                    R        (        T        )                            =                  c          4                                                                    (                                                                  ∫                                      μ                    =                    0                                                        μ                    =                    ∞                                                                                                              f                      (                      μ                      )                                                              μ                                              5                                                                                                              d                                μ                )                            ⏟                                            A            =            constant                                    T                      4                          =                                                            (                                  c                  4                                A                )                            ⏟                                            σ                                            T                      4                          =        σ                          T                      4                              But, sincerely, trying to find this directly you are missing important facts about the &quot;before Planck&quot; adventure of the blackbody radiation theory. For example, you must try to find from Wien&#039;s Law (your first equation) why if you know the function     ρ  (  λ  ,      T          1        )   for a given temperature         T          1         then you know it for any temperature     T   or that         λ          m      a      x        ⋅  T  =  b  =  c  o  n  s  t  a  n  t   (Wien&#039;s Displacement Law), see Emilio Pisanty answer therein : Showing Wien&#039;s Displacement Law from Wien&#039;s Law.

Relationship between Wien's law and Stefan-Boltzmann's law I'm studying Quantum mechanics by Brans

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