Using Wien's radiation law to derive the Stephan-Boltzmann law and Wien's distributio

RedMech

1. The problem statement:

Using Wien's law ρ(λ,T)=f(λ,T)/λ^5, show the following:

(a) The total emissive power is given by R = aT4 (the Stefan-Boltzmann law),
where a is a constant.
(b) The wavelength λmax at which ρ(λ,T) - or R(λ,T) - has its maximum is such that λ*T = b (Wien's displacement law), where b is a constant.

2. Relevant equations:

Wien's radiation law:
ρ(λ,T)=f(λ,T)/λ^5
ρ(λ,T)=c1/(λ^5*exp{c2/λT})

3. The attempt at a solution:

So I tried integrating Wien's equation from zero to infinity
ρ(total)dλ=c/4∫ρ(λ,T)dλ=c/4∫[f(λ,T)/λ^5]dλ. But I got nowhere.

Then I used the full expression of wien's law and tried the integration again
ρ(total)dλ=c/4∫[c1/(λ^5*exp{c2/λT})]dλ
I still didn't know what to do. So please help.

vela

Without an explicit form for f(λ,T), you can't integrate this, as you probably realized.

This approach should work. How did you try to integrate this? I'd try a substitution like u=1/λ and see where it goes.

RedMech

I substituted x=c2/λT for the sake of the exponential term.
dx=[-c2/λ^2T]dλ. The integral has become w=(c1*c*T^4)/4c2^4∫[x^3/e^x]dx (Please note that for c1 and c2, the 1 and 2 are subscripts of c. The independent c is the speed of light)

How is this equation looking?

vela

Do you recognize that integral? Think gamma function. In any case, it's a definite integral, so it's just some number.

RedMech

I'll compute the integral and then leave the final expression for my instructor. Thanks a million for your help.

TSny

Wien's law is actually ρ(λ,T)=f(λT)/λ⁵ where f is an undetermined function of the product of λ and T. Using this, see if you can get the integral to yield a constant times T⁴.

Humdinger

@TSny, I was wondering if you might be able to give me a small hint in regards to how to proceed with this problem only using the ρ(λ,T)=f(λT)/λ⁵ form of Wien's law. I tried integration by parts but that just led to a more convoluted expression. I see that you underlined the phrase "product of λ and T" but I'm still not sure how to handle the f(λT) term in the integral.

dextercioby

That calls for a substitution (change of variable) which would throw out of the integral exactly T to the power of 4.

Humdinger

Thank you dextercioby, my mistake was in assuming that I need to find the unknown function f(λT). I was able to figure out the answer based on your hint.