Black body radiation and maximum spectral density

In summary, the conversation discusses the maximum of spectral energy density in Planck's black body radiation and the discrepancy between maximizing it in frequency and wavelength. It is explained that this difference occurs due to the non-linear relationship between wavelength and frequency, causing bins of fixed width in wavelength to translate into wider bins in frequency as frequency decreases. This results in a shift in the position of the maximum of spectral energy density.
  • #1
fede.na
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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, 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 [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!
 
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  • #2
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.
 

1. What is black body radiation?

Black body radiation refers to the electromagnetic radiation emitted by an idealized object that absorbs all of the radiation that falls on it. This means that a black body does not reflect or transmit any radiation, but instead absorbs all of it. This radiation is dependent on the temperature of the object and follows a specific distribution known as the Planck distribution.

2. What is the maximum spectral density in black body radiation?

The maximum spectral density, also known as the peak or maximum emission, occurs at a specific wavelength for each temperature of a black body. This wavelength is known as the peak wavelength and is inversely proportional to the temperature of the black body. As the temperature increases, the peak wavelength decreases, meaning that the radiation shifts to shorter wavelengths.

3. How is maximum spectral density calculated?

The maximum spectral density is calculated using Wien's displacement law, which states that the peak wavelength is equal to a constant divided by the temperature of the black body. This constant is known as Wien's displacement constant and has a value of 2.898 x 10^-3 meters * Kelvin.

4. What is the relationship between temperature and maximum spectral density?

As mentioned earlier, the relationship between temperature and maximum spectral density is inverse. This means that as the temperature of a black body increases, the peak wavelength decreases, resulting in a shift towards shorter wavelengths. This is due to the fact that higher temperatures result in higher energy levels and therefore, shorter wavelengths of radiation.

5. How is black body radiation used in scientific research?

Black body radiation plays an important role in many areas of scientific research, including astrophysics, cosmology, and materials science. It is used to study the properties of stars and galaxies, as well as the early universe. In materials science, black body radiation is used to understand the thermal properties of materials and to develop more efficient energy sources. Additionally, black body radiation is used in the design and calibration of various instruments, such as thermal cameras and spectrometers.

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