How do you derive Stefan-Boltzmann and Wien's laws from Wien's distribution function?

[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.