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Wien's Displacement Law stated that for a blackbody emitting radiation,

$$\lambda_{max}=\dfrac{1}{T}$$

where $T$ is the temperature of the body and $\lambda_{max}$ is the maximum wavelength of radiation emitted.

Due to the relationship between wavelength, frequency and the speed of light, a value of maximum wavelength would give a value of minimum frequency, and vice versa.

I then saw on the Wikipedia page for Wien's Displacement Law that

$$f_{max}=\dfrac{\alpha k_BT}{h},$$

where $\alpha=2.82...$, $k_B$ is Boltzmann's Constant, $T$ is the temperature of the body and $h$ is Planck's Constant.

How can this relationship for maximum frequency be shown?

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    $\begingroup$ You've misstated Wien's law. There is no maximum wavelength - the possible wavelengths in the thermal radiation span interval from $0$ to $\infty$. The meaning of $\lambda_m$ in the Wien law is that it gives wavelength at which the spectral distribution function $I(\lambda,T)$ is maximum. $\endgroup$
    – Ján Lalinský
    Commented Oct 13, 2014 at 18:42
  • $\begingroup$ Ahh, yes, I misread the question. In light of what @JánLalinský pointed out I retract my previous comment. (P.S. Jan you might consider posting that as an answer) $\endgroup$
    – David Z
    Commented Oct 13, 2014 at 19:13
  • $\begingroup$ @DavidZ the question is misconcepted, so I prefer to just comment and give some time to questioner to consider rewriting it. $\endgroup$
    – Ján Lalinský
    Commented Oct 13, 2014 at 19:56
  • $\begingroup$ OK that makes sense, I'm in my first month of university so perhaps some explanations are simplified at this level and then become more technical as the course progresses. $\endgroup$
    – ODP
    Commented Oct 20, 2014 at 12:37

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The Wien displacement law gives the maximum of a function, so the way to compute it is to start with the Planck function in frequency domain, $$ B(\nu,T)=\frac{8\pi\nu^2}{c^3}\frac{h\nu}{e^{h\nu/kT}-1} $$ Take the derivative with respect to $\nu$, set it equal to zero and solve for $\nu$. You'll likely have to use some numerical methods (e.g., iterative searching) to find the value.

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