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Deriving Wien's Displacement Law

  1. Mar 29, 2017 #1
    1. The problem statement, all variables and given/known data

    I have ##E(w)=w^3(e^{\beta\bar{h}w}-1)^{-1}##,

    where ##\beta=K_{B}T##, ##K_{B}## boltzman constant


    2. Relevant equations

    Need to solve ##\frac{dE(w)}{dw}=0##

    3. The attempt at a solution

    ##k=\beta\bar{h}##:

    ##\frac{dE(w)}{dw}=3w^2(e^{kw}-1)^{-1}+w^3(e^{kw}-1)^{-2}ke^{kw}(-1)##
    ##=\frac{3w^{2}(e^{kw}-1)-kw^3e^{kw}}{(e^{kw}-1)^2}##

    ##\implies w^2(e^{kw}(3-kw)-3)=0##

    ##w\neq 0 \implies e^{kw}(3-kw)-3=0##

    Is this right so far? I don't know how I'd solve this now...?

    Many thanks .
     
  2. jcsd
  3. Mar 29, 2017 #2

    DrClaude

    User Avatar

    Staff: Mentor

    That's correct, but there is no analytical solution to that equation. You have to use a graphical or numerical approach to find the value of ω.
     
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