Deriving Wien's Displacement Law

[binbagsss]

Homework Statement

I have ##E(w)=w^3(e^{\beta\bar{h}w}-1)^{-1}##,

where ##\beta=K_{B}T##, ##K_{B}## boltzman constant

Homework Equations

Need to solve ##\frac{dE(w)}{dw}=0##

The Attempt at a Solution

[/B]
##k=\beta\bar{h}##:

##\frac{dE(w)}{dw}=3w^2(e^{kw}-1)^{-1}+w^3(e^{kw}-1)^{-2}ke^{kw}(-1)##
##=\frac{3w^{2}(e^{kw}-1)-kw^3e^{kw}}{(e^{kw}-1)^2}##

##\implies w^2(e^{kw}(3-kw)-3)=0##

##w\neq 0 \implies e^{kw}(3-kw)-3=0##

Is this right so far? I don't know how I'd solve this now...?

Many thanks .

 

[DrClaude]

Is this right so far? I don't know how I'd solve this now...?

That's correct, but there is no analytical solution to that equation. You have to use a graphical or numerical approach to find the value of ω.