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Simple Thermodynamic Proof stumps me!

  1. Jul 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that the average number of photons per unit volume in the cavity for a temperature T is given by

    n=( I / Pi^2 . (kT/c(h-bar))^3 )

    where I = integral from infinity to 0. dx x^2 ((exp (x)-1))^-1 = 2.404


    2. Relevant equations

    You may assume the mean number of photons occupying a state at energy (strange looking e)= hbar omega when the radiation has a temperature T is

    <N>=1 over exp (strange looking e/kT)-1

    You may also assume that the number of available states in the angular frequency rage omega to omega plus delta omega in a cavity of volume V is given by

    g(omega)d-omega = V/c^3 . omega^2 / pi^2 d-omega

    3. The attempt at a solution

    I know I need to do N/V however I have no idea how available states are useful to me, perhaps I could rearrange for V, but then I dont' know what g(omega) can be used for.
     
  2. jcsd
  3. Jul 18, 2009 #2
    You are given the no. of photons per state, then you're given the number of states in the range [tex]\omega[/tex] to [tex]d\omega[/tex], how would you find the total number of photons?
     
  4. Jul 22, 2009 #3
    Um... I would times the number of states by number of photons. Is that what g(omega) d omega is refering to? So that means, If I rearrange, and have the above times by c^3 divided by omega squared and times by pi squared.. I should a formula for V...

    Theres a d omega on both sides, can I just cancel them out or are they important?
     
  5. Jul 22, 2009 #4
    Not quite...g(w) is not a constant. You'll have to integrate over the whole range of w. Besides, how is the energy of the photon related to w?
     
  6. Jul 23, 2009 #5
    Would I integrate V/c^3 . omega^2 / Pi^2 with respect to d omega? The limits are from omega to omega + d omega, Would I just integrate it without limits to get a formula :

    V. omega^3 = g(omega) omega
    -----------
    3 C^3 pi^2

    and then plug something into E=H(bar) omega.

    ps. is there anyway of making these formulas more neat on the forums? I 'm sure I've seen people make actually integral signs come up..
     
  7. Jul 25, 2009 #6
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