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

 P: 13 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.
Emeritus
HW Helper
Thanks
PF Gold
P: 11,868
 Quote by 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.
Without an explicit form for f(λ,T), you can't integrate this, as you probably realized.

 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.
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.
P: 13
 Quote by vela 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.
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?

 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 11,868 Using Wien's radiation law to derive the Stephan-Boltzmann law and Wien's distributio Do you recognize that integral? Think gamma function. In any case, it's a definite integral, so it's just some number.
P: 13
 Quote by vela Do you recognize that integral? Think gamma function. In any case, it's a definite integral, so it's just some number.
I'll compute the integral and then leave the final expression for my instructor. Thanks a million for your help.
HW Helper
Thanks
PF Gold
P: 4,887
 Quote by 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. ... 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.
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.
P: 2
 Quote by TSny Wien's law is actually ρ(λ,T)=f(λT)/λ5 where f is an undetermined function of the . Using this, see if you can get the integral to yield a constant times T4.
@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.
 Sci Advisor HW Helper P: 11,948 That calls for a substitution (change of variable) which would throw out of the integral exactly T to the power of 4.
P: 2
 Quote by dextercioby That calls for a substitution (change of variable) which would throw out of the integral exactly T to the power of 4.
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.

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