QM- Stefan's constant vs radiation constant

[cep]

QM-- Stefan's constant vs radiation constant

Homework Statement

(a) Using Planck's formula for the energy density ρ(λ,T), prove that the total energy density is given by ρ(T)=aT⁴ where a = 8π⁵k⁴/(15h³c³). (b) Does this agree with the Stefan-Boltzmann law for the total emissive power?

The Attempt at a Solution

I had no problem with the proof in part (a), starting with the equation
ρ(T)dλ = 8πhc/λ⁵ * dλ/(e^hc/λkT-1) and integrating over λ 0→∞.

However, I am confused by the question in part (b). The answers are obviously related. I know that a, the radiation constant, is equal to 4σ/c, and I know you can derive the precise Stefan-Boltzmann equation from Planck's formula. I also suspect the professor is looking for an answer other than "no" or "sort-of." Does anyone know where the difference between that derivation and the one I completed and the one that yields P = σT⁴ is? What is the utility difference between the radiation constant and Stefan's constant?

Thanks guys!

 

[diazona]

Hmm... have a look at the Wikipedia article on Planck's law and see if it suggests anything