Relationship between temperature and wavelength? I am investigating the relationship between wavelength and temperature.
As seen the figure below of Planks law

What is the relationship  between the lambda(max) and Temperature? or in simpler terms, you see the red dots joined together by a dotted line, what is the equation of the dotted line?
Reference:
http://www.informationphilosopher.com/solutions/scientists/planck/
 A: Planck's law of black-body radiation can be stated in many different ways, depending on whether one is interested in the spectral energy density per volume or per area. It can also be expressed in terms of radiation wavelength or frequency.
The energy of a photon is 
$$ \epsilon = h\nu = \frac{hc}{\lambda}$$
I will not derive Planck's law here. It can be found in any standard textbook on statistical physics or on numerous websites. Instead let us accept that the spectral density per volume (the unit is $\frac{\text{Energy}}{\text{Volume}\cdot\text{Wavelength}})$
$$ u(\lambda)=\frac{C}{\lambda^5} \frac{1}{\exp(D/T\lambda)-1}$$
where $C$ and $D$ are constants incorporating factors of $h,c,k_B$.
This function has a maximum depending on temperature. Taking the derivative wrt. $\lambda$ (excercise!) and equating to zero one finds an expression like
$$ \exp(D/T\lambda)(4-D/T\lambda)-4=0$$
which is rather hard to solve. But observe that the expression depends only on $x=T\lambda$ not on each quantity separately! We may thus look for the solution $x_{max}$ and substitute back to obtain $$\lambda_{max}=\frac{x_{max}}{T}$$
Whatever the numerical value of $x_{max}$ (it is $2.897\, \text{Km}$), we definitely know that $\lambda_{max}$ is inverse proportional to temperature. This is called Wien's Displacement Law.
You may now - like the diagram probably shows - be interested in the (spectral) photon density that is radiated. Easy. Simply divide the energy density by the energy per photon:
$$ n(\lambda) = \frac{u(\lambda)\lambda}{hc} $$
If you repeat the argument above to find the maximum, you'll find that the number of photons peaks at $\lambda^n_{max}$ which is also inverse to temperature.
$$\lambda^n_{max} \propto T^{-1}$$
