# Deriving Wien's Law from Planck's Formula

by Dorje
Tags: deriving, formula, planck, wien
 P: 26 As a refresher exercise in modern physics, I want to derive Wien's displacement law: $$\lambda_{max}T=2.898x10^{-3}mK$$ from Planck's formula: $$R(\lambda)=(\frac{c}{4})(\frac{8\pi}{\lambda^4})(\frac{hc}{\lambda})(\f rac{1}{\exp^(\frac{hc}{\lambda\kT})-1})$$ by differentiating R($$\lambda$$) and setting dR/d$$\lambda$$ = 0. I get to an expression like this: $$\exp^(\frac{hc}{\lambda\kT})(hc - 5kT\lambda)+5kT\lambda=0$$ If it wasn't for the "5kT$$\lambda$$" term by itself on the left-hand side of the equation, the solution would simply be: ($$\lambda$$) (T) = hc / 5k which is Wien's law. There must be something wrong though, or maybe there's a trick involved that I'm not seeing? Thanks
 Sci Advisor HW Helper P: 11,833 Yes, you're dealing with a typical transcendental equation, to which exact solutions cannot be found in most cases, this one included. Daniel.
 P: 1 http://en.wikipedia.org/wiki/Wien_approximation check this link you will see why the 5kt(lamda) shouldnt be there

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