# Frequency and Wavelength peak for Wien's displaement law of a blackbody [duplicate]

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$^{-1}$.

• 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. Commented Oct 6, 2015 at 18:22
• It's in any physics textbook, but I can't find a sufficient explanation as to why this is. Commented Oct 6, 2015 at 18:28
• @user3125347 - So, to repeat myself, what is your source for this statement? Commented Oct 6, 2015 at 18:28
• Possible duplicate physics.stackexchange.com/q/91192 Commented Oct 6, 2015 at 20:27
• possible duplicate of The strange thing about the maximum in Planck's law Commented Oct 6, 2015 at 20:28

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.
• 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). Commented Oct 6, 2015 at 18:40
• @user3125347: It's the spectral flux/radiance per unit frequency. If that's what your $B_\nu(T)$ is then yes, otherwise no. Commented Oct 6, 2015 at 18:42