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

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RedMech
#1
Aug15-12, 03:23 PM
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.
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vela
#2
Aug15-12, 03:55 PM
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Quote Quote by RedMech View Post
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.
RedMech
#3
Aug15-12, 04:26 PM
P: 13
Quote Quote by vela View Post
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
#4
Aug15-12, 04:32 PM
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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.
RedMech
#5
Aug15-12, 04:46 PM
P: 13
Quote Quote by vela View Post
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
#6
Aug15-12, 05:41 PM
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Quote Quote by RedMech View Post
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
#7
Sep18-12, 04:22 PM
P: 2
Quote Quote by TSny View Post
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
#8
Sep18-12, 04:45 PM
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That calls for a substitution (change of variable) which would throw out of the integral exactly T to the power of 4.
Humdinger
#9
Oct4-12, 11:24 PM
P: 2
Quote Quote by dextercioby View Post
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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