Black body radiation and maximum spectral density

[fede.na]

Hi, I've got a simple question regarding the maximum of the spectral energy density in Planck's black body radiation. It turns out that if you calculate which frequency has the most power associated with it (i.e. maximize $R(\nu)$), then you do it with wavelength as well, and compare, they're not the same wave, meaning that $\lambda \nu \neq c$.

How can this happen? I suspect that it has to do with the energy density being associated with differential intervals of $\nu$, rather than values of $\nu$ itself... but nevertheless, I can't figure it out. Common sense tells me $\lambda \nu = c$ should work, because it should be the same wave that carries the maximum energy. Maximizing spectral energy density either in frequency or in wavelength should yield the same result since there's no physical meaning in the variable change, it's just math. Isn't it?

My math goes as follows. The power per unit area (and solid angle) emitted by a black body is:


$$ R_T (\nu) \operatorname{d}\!\nu = \frac{2\pi}{c^2} \frac{h\nu ^3}{e^{\frac{h\nu}{KT}} -1 }\operatorname{d}\!\nu $$

In wavelengths:

$$ R_T(\lambda)\operatorname{d}\!\lambda = 2\pi h c^2 \frac{\lambda ^{-5}}{e^{\frac{hc}{\lambda KT}}-1}\operatorname{d}\!\lambda$$


After maximization, I get:

$$ \nu _{max} = 2.821 · \frac{KT}{h} ; \lambda _{max} = \frac{hc}{4.965·KT} \rightarrow
\nu _{max} \lambda _{max} = 0.568c$$

Another suspicion I have (if it isn't the same one) is that I'm not interpreting correctly the differentials present in the expressions.

BTW if someone could tell me what you call this quantity $R_T (\nu)$ in English it'd be great, in Spanish it's something like "Radiance" or "Emittance", just translating by how it sounds. Meanwhile I'll just say spectral energy density.

Thanks!

 

[phyzguy]

Well, you've done the math correctly, so you've answered your own question of "How can this happen?" I guess it just tells you that when your common sense says one thing and the math says another, you should believe the math.

Perhaps one way to see it is to divide up the function into a finite number of wavelength and frequency bins Δλ and Δnu instead of using infinitesimals. If you do this, you'll see that because of the non-linear relation between wavelength and frequency, bins of fixed width Δλ don't translate into bins of fixed width Δnu, but instead translate into bins of width c/nu^2 Δnu. So as nu gets smaller, the bins get wider and wider, and contain more spectral energy. This causes the maximum to be shifted realtive to its position when you use bins of fixed Δnu.