Converting between λ and ν for Blackbody Radiation?!

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  • Thread starter tade
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  • #1
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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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Answers and Replies

  • #2
963
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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
963
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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
vanhees71
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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
552
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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?
 

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