Homework Help: QM- Stefan's constant vs radiation constant

1. Oct 4, 2011

cep

QM-- Stefan's constant vs radiation constant

1. The problem statement, all variables and given/known data

(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?

3. 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/(ehc/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!

2. Oct 4, 2011

diazona

Re: QM-- Stefan's constant vs radiation constant

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