Derivation of Stefan-Boltzmann Law from Wien's Law

[PhysicsItHertz]

Homework Statement

Derive Stefan-Boltzmann Law from Wien's Law.
Hint: You can use (without proof) R(T)=∫(-∞ to ∞) R(λ,T)dλ, p(λ,T)= 4/c R(λ,T).

Homework Equations

Stefan-Boltzmann Law:P=AσT^4
Wien's Law: λmax=(2.898*10^-3 m*K)/T.

The Attempt at a Solution

Let λmax=(2.898*10^-3 m*K)/T.

Using cross multiplication gives: T=(2.898*10^-3m*K)/λmax.

Raising both sides to the fourth power gives: T^4=(2.898*10^-3m*K)^4/(λmax)^4.

Multiplying both sides by λmax^4 gives: T^4*(λmax)^4=(2.898*10^-3m*K)^4.

Im not really sure if this going to go anywhere.
My idea was to just algebraically manipulate Wien's Law to equate to Stefan-Boltzmann Law.

The issue I have with doing this is that I am not sure what the hint even means (physically).

 

[Matterwave]

I don't think you can actually derive the Stefan-Boltzmann law from Wien's displacement law... the SB law requires an integral over all wavelengths of the Planck distribution basically. Wien's displacement law just tells you where the maximum of the Planck distribution is, and that is not enough information to do the problem. Are you sure the problem is asking you to derive SB from Wien's displacement law? o.o

 

[TSny]

Derive Stefan-Boltzmann Law from Wien's Law.
Hint: You can use (without proof) R(T)=∫(-∞ to ∞) R(λ,T)dλ, p(λ,T)= 4/c R(λ,T).

Could it be that you are meant to base the derivation on Wien's distribution law rather than Wien's displacement law? Your hint seems to indicate that you are to integrate over some distribution law.

You should be able to use Wien's distribution law to show that the total power radiated is proportional to T⁴. However, I don't think you will get the correct numerical value for the proportionality constant.

 

[Tirain]

I am currently working on the same question except for wien's law we are supposed to use wien's law: p(λ,T) = f(λ,T)/^5 (maybe op and I are in the same class lol). I also could use some help with this problem. Honestly I have no idea where to even begin.

 

[TSny]

I am currently working on the same question except for wien's law we are supposed to use wien's law: p(λ,T) = f(λ,T)/^5 (maybe op and I are in the same class lol). I also could use some help with this problem. Honestly I have no idea where to even begin.

Did you mean to write p(λ,T) = f(λT)/λ⁵, where f is some undetermined function of the product of λ and T? That's the most general form of Wien's displacement law. I suspect that the OP was also supposed to use this form of Wien's law. The Wien's distribution law that I linked to in a previous post is a special case this general law. But the one I linked to is only an approximate formula that is accurate in the range of short wavelengths. I had forgotten about the general law that you are using.

Anyway, you should be able to derive the Stefan-Boltzmann law from p(λ,T) = f(λT)/λ⁵. Think about the meaning of p(λ,T) and how you would use p(λ,T) to find the total power radiated for all wavelengths.