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Prove that Planck function increases with temperature

  1. Apr 13, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove that the Planck function increases monotonically with temperature.

    2. Relevant equations

    Bv(T) = 2hv3c-2(ehv/kT - 1)-1

    3. The attempt at a solution

    I first went through this piece-by-piece, but I am not a mathematician so I don't know if this constitutes "proof"!

    1) First consider ehv/kT and note that this function will decrease as T increases.

    2) This means (ehv/kT - 1) will also decrease.

    3) Therefore (ehv/kT - 1)-1 will increase with T.

    4) The other elements are independant of T, so the Planck function will increase with T.

    I also thought of taking the derivative with respect to T of the Planck function, to see if the gradient ever reached zero to indicate a stationary point. After doing a couple of substitutions, I got

    dBv(T)/dT = 2h2v4c-2k-1(e-hv/kT + 1)-1

    I'm not sure that is correct, and I think it tells me that gradient gets smaller as T increases. This doesn't help my argument much!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 13, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    The above argument sounds good to me.
    That's not the correct result for the derivative. Ignoring the constants out front, you have

    B = (x-1)-1 where x = ehv/kT.

    Try using the chain rule: dB/dT = (dB/dx)##\cdot##(dx/dT)
     
  4. Apr 14, 2013 #3
    Thanks TSny, I managed to compute the derivative correctly and that can also be put in terms of positive, increasing functions of T so I am sure that is adequate proof.
     
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