Proving Wien's Law Includes Stefan-Boltzmann & Wien's Displacement Laws

[O_o]

Hi,

I'm supposed to prove that Wien's Law: P(\lambda,T) = \frac{f(\lambda T)}{\lambda^5} includes Stefan-Botlzmann's Law R(T) = \sigma T^4 and Wien's Displacement Law: \lambda_{max} T = b

For Wien's Displacement Law:

I know that I would have to find when P(\lambda ,T) graphed against \lambda has a slope of 0. So I think I need to find the derivative with respect to \lambda. But the only two equations for P(\lambda,T) I have are P(\lambda,T) = \frac{f(\lambda T)}{\lambda^5} and P(\lambda,T) = \frac{8\pi kT}{\lambda^4}

So if I take the derivative of P(\lambda,T) = \frac{8\pi kT}{\lambda^4} with respect to \lambda I have
8\pi kT} * (-4) * \lambda^{-5} = 0 Where I'm guessing that everything except \lambda is being held constant and I don't know what to do from there.

Any hints or corrections of things I said would be appreciated. Thanks.

 

[James R]

I think you need to start with Planck's equation, which may be part of the f() function, above. I don't think your expression for P in terms of T and lambda is correct (although it might represent Wien's guess). That function has no peak - it gives an "ultraviolet catastrophe" as lambda goes to zero - there is no lambda_max.