# Deriving Wien's Displacement Law

1. Mar 29, 2017

### binbagsss

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

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

3. The attempt at a solution

$k=\beta\bar{h}$:

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

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

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

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

Many thanks .

2. Mar 29, 2017

### Staff: Mentor

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.