Deriving Wien's Law from Planck's Formula

  1. As a refresher exercise in modern physics, I want to derive Wien's displacement law:

    [tex]\lambda_{max}T=2.898x10^{-3}mK[/tex]

    from Planck's formula:

    [tex]R(\lambda)=(\frac{c}{4})(\frac{8\pi}{\lambda^4})(\frac{hc}{\lambda})(\frac{1}{\exp^(\frac{hc}{\lambda\kT})-1})[/tex]

    by differentiating R([tex]\lambda[/tex]) and setting dR/d[tex]\lambda[/tex] = 0. I get to an expression like this:

    [tex]\exp^(\frac{hc}{\lambda\kT})(hc - 5kT\lambda)+5kT\lambda=0[/tex]

    If it wasn't for the "5kT[tex]\lambda[/tex]" term by itself on the left-hand side of the equation, the solution would simply be:

    ([tex]\lambda[/tex]) (T) = hc / 5k

    which is Wien's law. There must be something wrong though, or maybe there's a trick involved that I'm not seeing?

    Thanks
     
  2. jcsd
  3. dextercioby

    dextercioby 12,324
    Science Advisor
    Homework Helper

    Yes, you're dealing with a typical transcendental equation, to which exact solutions cannot be found in most cases, this one included.

    Daniel.
     
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