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Homework Help: Linearizing Stefan-Boltzmann equation

  1. Dec 12, 2007 #1
    1. The problem statement, all variables and given/known data
    I want to linearize both the Stefan-Boltzmann equation and the adiabatic relation to draw a relationship between δL/L(0) and δR/R(0).

    2. Relevant equations
    Stefan-Boltzmann equation in the form of L=4πσ(R^2)(T^4)
    Adiabatic relation TV^(γ-1) = constant

    3. The attempt at a solution
    I have L(0) + δL = 4πσ((R(0) + δR)^2)*((T(0) + δT)^4) for the S-B part, but don't know how to get it into this desired form:

    δL/L(0) = (2δR/R(0)) + (4δT/T(0))
    Last edited: Dec 12, 2007
  2. jcsd
  3. Dec 13, 2007 #2
    So you have:

    [tex]L_0+\delta L=4\pi\sigma(R_0+\delta R)^2(T_0+\delta T)^4[/tex]

    You will have to use a Taylor Expansion to expand the terms that can be considered very small. For example:

    [tex](a+x)^4=a^4(1+\frac{x}{a})^4 \approx a^4(1+\frac{4x}{a})[/tex]

    This only works when [tex]\frac{x}{a}[/tex] is very small compared to 1, as is the case in your example. If you're still having trouble, show your work and we can go from there.
    Last edited: Dec 13, 2007
  4. Dec 13, 2007 #3
    That helped a bunch, thanks.
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