- #1

- 10

- 0

## Main Question or Discussion Point

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 [itex]R(\nu)[/itex]), then you do it with wavelength as well, and compare, they're not the same wave, meaning that [itex]\lambda \nu \neq c[/itex].

How can this happen? I suspect that it has to do with the energy density being associated with differential intervals of [itex]\nu[/itex], rather than values of [itex]\nu[/itex] itself... but nevertheless, I can't figure it out. Common sense tells me [itex]\lambda \nu = c[/itex] should work, beacause 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 [itex]R_T (\nu)[/itex] 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!

How can this happen? I suspect that it has to do with the energy density being associated with differential intervals of [itex]\nu[/itex], rather than values of [itex]\nu[/itex] itself... but nevertheless, I can't figure it out. Common sense tells me [itex]\lambda \nu = c[/itex] should work, beacause 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 [itex]R_T (\nu)[/itex] 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!