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Energy Density of Blackbody Radiation

  1. Jan 14, 2014 #1
    1. The problem statement, all variables and given/known data

    [tex]\mbox{Let} \ p(< \nu_{0}) \mbox{be the total energy density of blackbody radiation in all frequencies less than} \ \nu_{0}, \mbox{where} \ h \nu_{0} << kT. \mbox{Derive an expression for} \ p (< \nu_{0})[/tex]


    2. Relevant equations

    [tex] p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{e^\frac{h\nu}{kT} -1} dv [/tex]


    3. The attempt at a solution

    We want to find the total energy density, so that means we'll have to integrate the Planck's Law. The limits of integration will be from 0 to v-knot.

    [tex] h \nu < h \nu_{0} << KT \Rightarrow h \nu << KT [/tex]

    Which if we simplify for this limit gives:

    [tex] p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{e^\frac{h\nu}{kT} -1} dv [/tex]

    [tex] p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{\dfrac{h\nu}{kT} +1 -1} dv [/tex]

    [tex] p(v) dv = \dfrac{8 \pi h} {c^3} \dfrac {\nu^3}{\frac{h\nu}{kT} } dv [/tex]

    [tex] p(v) dv = \dfrac{8 \pi KT}{c^3} \nu^2 dv \ \mbox{Rayleigh-Jeans Formula} [/tex]

    [itex]p(<v_{0}) = \dfrac{8 \pi KT}{c^3}\int _ {0} ^{\nu_{0}} \nu^2 dv[/itex]

    [itex]p(<v_{0}) = \dfrac{8 \pi KT}{3 c^3}\nu_{0}^3[/itex]

    Is there anything wrong? Thank you very much!
     
    Last edited: Jan 14, 2014
  2. jcsd
  3. Jan 15, 2014 #2

    TSny

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    Homework Helper
    Gold Member

    Looks good to me.
     
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