Wien's displacement law's proof

[jonathanpun]

When I was doing my assignment, I need to proof the Wien's Law.
The question given frequency dependent energy density function. So differentiate it respect to frequency v. Equate it with zero, and solve. i solved the value a=2.82144 = hv/KT=hc/(lambda)KT, i cannot get a correct value for the Wien's constant b = 2.898*10^-3, i only get 5.102*10^-3
But if i convert the frequency dependent Planck's energy density equation into wavelength dependent, and differentiate, solve. I get a*exp(a)-5*exp(a)+5=0 => a=4.96511=hc/(lambda)KT. Then i get a correct value for b.

So my question is why the Max. wavelength seems not corresponding to Max. frequency?

I have read the wikipedia about the law. but i don;t understand why it take ""the value 4 in this equation (midway between 3 and 5) yields a "compromise" wavelength-frequency-neutral peak, which is given for x = 3.92069039487...""

And it said "Because the spectrum from Planck's law of black body radiation takes a different shape in the frequency domain from that of the wavelength domain, the frequency location of the peak emission does not correspond to the peak wavelength using the simple relationship between frequency, wavelength, and the speed of light."

So what is the connection of the two value b obtained by two method?

 

[phyzguy]

Well, you just did the math. The frequency peak is different from the wavelength peak because the functional form of intensity vs wavelength is different from the functional form of intensity vs frequency, so when you find the maxima they are in slightly different places. I don't think there is any deeper explanation.

Since Wien's displacement law is stated in terms of wavelength, this is the correct function to use. I found this surprising too when I first encountered it, but the mathematics speaks for itself.