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Deriving Wien's Law from Planck's Formula

by Dorje
Tags: deriving, formula, planck, wien
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Dorje
#1
Apr4-06, 10:01 PM
P: 26
As a refresher exercise in modern physics, I want to derive Wien's displacement law:

[tex]\lambda_{max}T=2.898x10^{-3}mK[/tex]

from Planck's formula:

[tex]R(\lambda)=(\frac{c}{4})(\frac{8\pi}{\lambda^4})(\frac{hc}{\lambda})(\f rac{1}{\exp^(\frac{hc}{\lambda\kT})-1})[/tex]

by differentiating R([tex]\lambda[/tex]) and setting dR/d[tex]\lambda[/tex] = 0. I get to an expression like this:

[tex]\exp^(\frac{hc}{\lambda\kT})(hc - 5kT\lambda)+5kT\lambda=0[/tex]

If it wasn't for the "5kT[tex]\lambda[/tex]" term by itself on the left-hand side of the equation, the solution would simply be:

([tex]\lambda[/tex]) (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
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dextercioby
#2
Apr5-06, 04:07 AM
Sci Advisor
HW Helper
P: 11,927
Yes, you're dealing with a typical transcendental equation, to which exact solutions cannot be found in most cases, this one included.

Daniel.
peter2020210
#3
Dec5-11, 12:21 AM
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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