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Deriving Stefan's fourth-power law

  • #1

Homework Statement



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.

Homework Equations



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

The Attempt at a Solution



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

1/(e^x)-1

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:

hc=xλkBT

into the numerator to get:

=2π(kBTx)c/λ^5

just not sure where to go from here.
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
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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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