Derivation of Stefan-Boltzmann Law from Wien's Law

In summary: You'll need to use the hint provided by the problem statement as well.In summary, to derive the Stefan-Boltzmann Law from Wien's Law, we can use the fact that the total power radiated is proportional to T^4 and integrate over the distribution law p(λ,T) = f(λT)/λ^5. This will result in the Stefan-Boltzmann Law, P=AσT^4, where A is a constant and σ is the Stefan-Boltzmann constant.
  • #1
PhysicsItHertz
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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).
 
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  • #2
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
 
  • #3
PhysicsItHertz said:
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.
 
  • #4
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.
 
  • #5
Tirain said:
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.
 

Related to Derivation of Stefan-Boltzmann Law from Wien's Law

1. What is Wien's Law?

Wien's Law is a basic law of physics that describes the relationship between the wavelength of light emitted by a blackbody and its temperature. It states that as the temperature of a blackbody increases, the peak wavelength of its emitted radiation shifts towards shorter wavelengths.

2. What is the Stefan-Boltzmann Law?

The Stefan-Boltzmann Law is a physical law that relates the total energy emitted by a blackbody to its temperature. It states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature.

3. How are Wien's Law and the Stefan-Boltzmann Law related?

Wien's Law and the Stefan-Boltzmann Law are related because Wien's Law provides the peak wavelength of radiation emitted by a blackbody, while the Stefan-Boltzmann Law describes the total energy emitted. This means that by combining these two laws, we can calculate the total energy emitted by a blackbody at a given temperature.

4. Can the Stefan-Boltzmann Law be derived from Wien's Law?

Yes, the Stefan-Boltzmann Law can be derived from Wien's Law. This can be done by using the peak wavelength of radiation given by Wien's Law and integrating it over all wavelengths to obtain the total energy emitted, which is then equated to the Stefan-Boltzmann Law.

5. Why is the Stefan-Boltzmann Law important?

The Stefan-Boltzmann Law is important because it is a fundamental law of physics that helps us understand the behavior of blackbodies and the energy they radiate. It also has many practical applications, such as in calculating the energy output of stars and in engineering designs that involve heat transfer and radiation.

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