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I don't understand the conversion between the frequency and wavelength forms of the Rayleigh-Jeans equation for spectral radiance. According to the Wikipedia article on the Rayleigh-Jeans law:

Rayleigh-Jeans law

However, if λ=c/f and we substitute c/f for lambda in the left side of the equation it does not come out to be the same as the right side of the equation. In other words:

Rayleigh-Jeans conversion

Which seems to be different than the right side of the first equation above.

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  • $\begingroup$ Have you read the section in the very Wikipedia article you link? $\endgroup$
    – ACuriousMind
    Commented Jan 2, 2015 at 15:38
  • $\begingroup$ This is a very common query, probably a duplicate exists. The solution is that $B_{\lambda}\ d\lambda = B_{\nu}\ d\nu$, so $B_{\nu} = B_{\lambda}\ d\lambda/d\nu$. $\endgroup$
    – ProfRob
    Commented Jan 2, 2015 at 15:54
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    $\begingroup$ @RobJeffries That is Planck's Law, not Rayleigh-Jeans Law. $\endgroup$ Commented Jan 2, 2015 at 16:06
  • $\begingroup$ The Rayleigh-Jeans law is just the low frequency approximation of the Planck distribution! The solution to your problem is as I wrote above or as given in the duplicate. $\endgroup$
    – ProfRob
    Commented Jan 2, 2015 at 16:08
  • $\begingroup$ Or here -physics.stackexchange.com/questions/37702/… $\endgroup$
    – ProfRob
    Commented Jan 2, 2015 at 16:10

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As ACuriousMind notes, the answer is indeed in the same Wikipedia article I link. The wavelength-based radiance is given in units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit wavelength, whereas the frequency-based radiance has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit frequency.

The units of measure for the radiance differs for the two equations in their standard forms and this must be accounted for in converting between the two equations.

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  • $\begingroup$ You could start every answer on this site with "if you had just read X ..." $\endgroup$
    – dllahr
    Commented Mar 7, 2023 at 9:56

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