Converting between λ and ν for Blackbody Radiation?

In summary: Scientists measure the spectral radiance of blackbodies by using a spectroradiometer. This equipment simultaneously measures the radiance at multiple wavelengths. The peak intensity occurs at a different wavelength or frequency depending on the blackbody type.
  • #1
702
24
Forgive me for this stupid question, but how do I convert between

b57bebb5337f0de6333ec9bc85688c08.png


and

46a1efc581519117de92da6afb5a8e78.png


I tried c = νλ but that doesn't work. This is the Rayleigh Jeans Law by the way.
 
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  • #2
tade said:
Forgive me for this stupid question, but how do I convert between

p?image=https%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fb%2F5%2F7%2Fb57bebb5337f0de6333ec9bc85688c08.png


and

p?image=https%3A%2F%2Fupload.wikimedia.org%2Fmath%2F4%2F6%2Fa%2F46a1efc581519117de92da6afb5a8e78.png


I tried c = νλ but that doesn't work.
pl. give the full expression of the quoted equation and where these two are being used.
 
  • #4
The Rayleigh–Jeans law agrees with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe,

see the full expressions
associated Rayleigh–Jeans limits are given by

c0313ce746394b13ace2e1e668e6f176.png

or

22ed0ae7e400ebb5306a059fb4b69c1f.png

now you can see the approximations- actually they are not exact expressions
 
  • #6
You only have to remember that these are probability distributions ##\mathrm{d} N/\mathrm{d} \nu## or ##\mathrm{d} N/\mathrm{d} \lambda##. Now you have ##\nu=c/\lambda##. This implies
$$B_{\nu}=\frac{\mathrm{d} N}{\mathrm{d} \nu}=\frac{\mathrm{d} N}{\mathrm{d} \lambda} \left|\frac{\mathrm{d} \lambda}{\mathrm{d} \nu}\right| = B_{\lambda} \frac{c}{\nu^2}.$$
Now with
$$B_{\lambda}=\frac{2 c k_B T}{\lambda^4}=\frac{2 k_B T \nu^4}{c^3} \; \Rightarrow\; B_{\nu}=\frac{2 k_B T \nu^2}{c^2},$$
and this was to be shown.
 
  • #7
Thank you guys.

For
b57bebb5337f0de6333ec9bc85688c08.png


and

46a1efc581519117de92da6afb5a8e78.png


the peak intensities occur at different wavelengths or frequencies.How do scientists measure the spectral radiance of blackbodies? Are there TWO types of equipment, one for [tex]B_\lambda[/tex] and the other for [tex]B_\nu[/tex], such that each device yields a peak at a different frequency?
 

1. How do you convert between wavelength (λ) and frequency (ν) for blackbody radiation?

The relationship between wavelength and frequency is given by the equation λν = c, where c is the speed of light. This means that as the wavelength decreases, the frequency increases, and vice versa. To convert between the two, you can use the formula λ = c/ν or ν = c/λ.

2. Why is it important to convert between wavelength and frequency for blackbody radiation?

Converting between wavelength and frequency allows us to better understand the behavior of blackbody radiation and its interactions with matter. It also allows us to make accurate measurements and predictions in various fields such as astronomy, atmospheric science, and material science.

3. Can you convert between wavelength and frequency for blackbody radiation using any units?

Yes, as long as the units are consistent. The speed of light, c, is typically measured in meters per second (m/s), so the units for wavelength and frequency should also be in meters (m) and inverse seconds (1/s) respectively. However, other units such as nanometers (nm) and terahertz (THz) can also be used.

4. How does the temperature of a blackbody affect the conversion between wavelength and frequency?

The temperature of a blackbody affects the wavelength and frequency of its radiation according to Wien's displacement law. This law states that the wavelength of maximum emission is inversely proportional to the temperature, while the frequency of maximum emission is directly proportional to the temperature. This means that as the temperature increases, the peak of the blackbody radiation shifts to shorter wavelengths and higher frequencies.

5. Is there a simple way to visualize the conversion between wavelength and frequency for blackbody radiation?

Yes, you can use a graph known as the Planck curve or blackbody radiation curve. This graph shows the intensity of blackbody radiation at different wavelengths or frequencies, and the peak of the curve corresponds to the wavelength or frequency of maximum emission. By changing the temperature on the graph, you can see how it affects the conversion between wavelength and frequency for blackbody radiation.

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