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Wien's Law

  • Thread starter O_o
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O_o

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Hi,

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

For Wien's Displacement Law:

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

So if I take the derivative of [tex] P(\lambda,T) = \frac{8\pi kT}{\lambda^4} [/tex] with respect to [tex]\lambda[/tex] I have
[tex]8\pi kT} * (-4) * \lambda^{-5} = 0 [/tex] Where I'm guessing that everything except [tex] \lambda [/tex] 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

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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.
 

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