Derive Stefan-Boltzmann law from Wien’s law.

[Tirain]

Homework Statement

Derive Stefan-Boltzmann law from Wien’s law

Hint: you can use (without proof) R(T) = -∞∫∞ R(λ,T)dλ , ρ(λ,T)=(4/c)*R(λ,T)

Homework Equations

wien's law: p(λ,T) = f(λ,T)/^5
Stefan's law : R(T) = σT^4

The Attempt at a Solution

Honestly I am having trouble just starting this problem. I know I would be able to work it out if I knew where to begin.

 

[DrClaude]

There is another currectly active thread on that same question https://www.physicsforums.com/showthread.php?t=769373

 

[Tirain]

Oh thanks!

 

[nrqed]

Homework Statement

Derive Stefan-Boltzmann law from Wien’s law

Hint: you can use (without proof) R(T) = -∞∫∞ R(λ,T)dλ , ρ(λ,T)=(4/c)*R(λ,T)

Homework Equations

wien's law: p(λ,T) = f(λ,T)/^5
Stefan's law : R(T) = σT^4

The Attempt at a Solution

Honestly I am having trouble just starting this problem. I know I would be able to work it out if I knew where to begin.

Well, the first step is: what is the function $f(\lambda,T)$ ?

 

[paranormal]

To derive the Stefan-Boltzmann law from Wien's law, we can use the relationship between the spectral radiance (R) and the spectral energy density (ρ) given by ρ(λ,T)=(4/c)*R(λ,T). Since we know that Wien's law states that p(λ,T) = f(λ,T)/^5, we can substitute this into the equation for ρ(λ,T) to get ρ(λ,T) = (4/c)*R(λ,T) = (4/c)*f(λ,T)/^5.

Next, we can use the definition of the Stefan-Boltzmann law, R(T) = σT^4, and substitute this into the equation for R(λ,T) to get R(λ,T) = (4/c)*σT^4.

Now, we can integrate both sides of the equation with respect to wavelength (λ) from -∞ to ∞ to get ∫∞-∞ R(λ,T)dλ = (4/c)*∫∞-∞ σT^4dλ. By the definition of spectral radiance, the left side of the equation becomes R(T), and by the definition of the Stefan-Boltzmann law, the right side becomes R(T) = σT^4. Therefore, we have derived the Stefan-Boltzmann law from Wien's law.