Using Wien's radiation law to derive the StephanBoltzmann law and Wien's distributioby RedMech Tags: derive, distributio, radiation, stephanboltzmann, wien 

#1
Aug1512, 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 StefanBoltzmann 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. 



#2
Aug1512, 03:55 PM

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PF Gold
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#3
Aug1512, 04:26 PM

P: 13

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? 



#4
Aug1512, 04:32 PM

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Using Wien's radiation law to derive the StephanBoltzmann 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.




#5
Aug1512, 04:46 PM

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#6
Aug1512, 05:41 PM

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PF Gold
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#7
Sep1812, 04:22 PM

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#8
Sep1812, 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.




#9
Oct412, 11:24 PM

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