Deriving Wien's Law from Planck's Formula

[Dorje]

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:

$$R(\lambda)=(\frac{c}{4})(\frac{8\pi}{\lambda^4})(\frac{hc}{\lambda})(\frac{1}{\exp^(\frac{hc}{\lambda\kT})-1})$$

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

 

[dextercioby]

Yes, you're dealing with a typical transcendental equation, to which exact solutions cannot be found in most cases, this one included.

Daniel.

 

[peter2020210]

http://en.wikipedia.org/wiki/Wien_approximation

check this link you will see why the 5kt(lamda) shouldn't be there