How Does Wien's Law Relate to Total Emissive Power and Maximum Wavelength?

[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. Homework 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]

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. Homework 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.

 

[RedMech]

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?

 

[vela]

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

 

[RedMech]

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.

 

[TSny]

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.

 

[Humdinger]

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.

 

[dextercioby]

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

 

[Humdinger]

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.