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Deriving the rayleigh-jeans limit of planck law of radiation

  1. Sep 21, 2014 #1
    Hello there,

    Plack law of radiation
    B(\nu) = \frac{2\,h\,\nu^3}{c^2(e^{h\nu/kT}-1)}

    I want to show that for small frequencies, Reyleigh-Jeans law:
    B(\nu) = \frac{2\nu^2kT}{c^2}
    is correct.

    I take the limit of Planck law as ##\nu \to 0## using l'hopital rule:
    \lim_{\nu \to 0} \frac{2\,h}{c^2} \frac{\nu^3}{e^{h\nu/kT}-1} \stackrel{\text{l'H}}{=} \lim_{\nu \to 0} \frac{2\,h}{c^2} \frac{3\nu^2kT}{e^{h\nu/kT}h} = 0

    I am off by a factor of 3. What is wrong with my maths?

    Thank you for your time.

    Kind regards,
  2. jcsd
  3. Sep 21, 2014 #2
    Oh, I got it worked out. I just write, ##e^{h\nu/kT} \approx 1 + h\nu/kT##, and substitute this into Planck law of radiation.
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