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Deriving Stefan's fourth-power law

  1. Feb 21, 2013 #1
    1. The problem statement, all variables and given/known data

    The total intensity I(T) radiated from a blackbody (at all wavelengths λ) is equal to the integral over all wavelengths, 0 < λ < ∞, of the Planck distribution (4.28). (a) By changing variables to x = hc/λkBT, show that ¡(T) has the form I(T) = σT4 where a is a constant independent of temperature. This result is called Ste fan’s fourth-power law, after the Austrian physicist Josef Stefan.

    2. Relevant equations

    I(λ,T) = (2πhc^2)/(λ^5)*(1/(e^(hc/λkBT))-1)

    3. The attempt at a solution

    I understand that I need to substitute x into the equation and the easy part that I get:


    out of the second part. However the first part seems to be just inflating the equation by substitution to which it will merely increase continually.
    I did end up substituting:


    into the numerator to get:


    just not sure where to go from here.
  2. jcsd
  3. Feb 21, 2013 #2


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    You have I(λ,T) = A/[λ5(eb/λ- 1)] where A and b are constants which you can identify. b carries the temperature dependence.

    As you noted, the total intensity is an integral over λ of this expression. Then follow your idea of a change of variable of integration to x = b/λ. When the smoke clears, you should get a bunch of constants times an integral over x. Don't worry about doing the integral, it will just be some dimensionless number independent of T. All of the T dependence will come from the factors of b in the mess of constants in front of the integral.
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