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This is a question relating to Wien's displacement law for the Planck function. As we all know frequency and wavelength are related to the speed of light by:

$$\nu\lambda=c$$

However, why is it that:

$$\nu_{\mathrm{peak}}\lambda_{\mathrm{peak}}\neq{c}$$

Any explanations would be very much appreciated.

To all of the people wanting to know where this statement came from. It hasn't come from anywhere specific, is it a well known fact of the Planck function. $\lambda_{\mathrm{peak}}=0.290T^{-1}$ cm K and $\nu_{\mathrm{peak}}=5.88\times{10^{10}}T$ Hz K$^{-1}$.

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  • $\begingroup$ Welcome to Physics SE. Look around, and please take the tour. For this particular question, a little more background would be useful, in particular the source for the statement you disagree with. $\endgroup$
    – Jon Custer
    Commented Oct 6, 2015 at 18:22
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    $\begingroup$ It's in any physics textbook, but I can't find a sufficient explanation as to why this is. $\endgroup$
    – DarthPlagueis
    Commented Oct 6, 2015 at 18:28
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    $\begingroup$ @user3125347 - So, to repeat myself, what is your source for this statement? $\endgroup$
    – Jon Custer
    Commented Oct 6, 2015 at 18:28
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    $\begingroup$ Possible duplicate physics.stackexchange.com/q/91192 $\endgroup$
    – Timaeus
    Commented Oct 6, 2015 at 20:27
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    $\begingroup$ possible duplicate of The strange thing about the maximum in Planck's law $\endgroup$
    – Timaeus
    Commented Oct 6, 2015 at 20:28

2 Answers 2

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The maximum of the spectral flux per unit wavelength $$I(\lambda,T)$$ does not correspond by $\lambda\nu = c$ to the maximum of the spectral flux per unit frequency $$I(\nu,T)$$ since these two functions are related by $$ I(\lambda,T)\mathrm{d}\lambda = I(\nu,T)\mathrm{d}\nu$$ but are not the same function, so their maxima are not the same.

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  • $\begingroup$ our electrons went flying by each other... $\endgroup$
    – Jon Custer
    Commented Oct 6, 2015 at 18:35
  • $\begingroup$ Can I assume that I(nu,T) is interchangeable with B_nu(T)? $\endgroup$
    – DarthPlagueis
    Commented Oct 6, 2015 at 18:40
  • $\begingroup$ The only thing I'd add is that indeed the student is right to intuit that if $f(y)$ is maximized for some special $y^*$ then $f(c/x)$ is maximized for $c/x^* = y^*,$ so that indeed the problem is that these are not defined to be equal densities (as in the same function) but equivalent densities (as in the same physics). $\endgroup$
    – CR Drost
    Commented Oct 6, 2015 at 18:40
  • $\begingroup$ @user3125347: It's the spectral flux/radiance per unit frequency. If that's what your $B_\nu(T)$ is then yes, otherwise no. $\endgroup$
    – ACuriousMind ♦
    Commented Oct 6, 2015 at 18:42
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I believe that the issue is with the difference between evaluating the peak in Planck's law with respect to frequency vs with respect to wavelength. Since this is pointed out on the Wikipedia page, it seems a bit much to replicate the differentiation of the Planck distribution vs wavelength and frequency here. However, the point is that since wavelength and frequency are inversely related, the derivative of the energy density with respect to one or the other can, and will, result in a different answer.

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