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Hi I am very new to this aspect of physics and I am having trouble with the derivation of the Rayleigh-Jeans from the steps shown at the hyperphysics web site. I have emailed Dr. Rod Nave who is listed as the person-in-charge at the site but it appears there is no one at that email perhaps.

It would really be much appreciated if anyone can have a look at the uploaded image file here and follow the queries I make on them.

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First a couple of pointers to show that the final result on the HyperPhysics website is not unreasonable.


In the derivation given in the HyperPhysics website the number of modes is found to be $N = \dfrac{8 \pi L^3}{3 \lambda^3}$ and because $\nu \lambda = c$ where $\nu$ is the frequency and $c$ is the speed of light the equation for the number of modes becomes $N = \dfrac{8 \pi L^3 \nu^3}{3 c^3}$.

Following through the derivation one finds

$$\dfrac{\text{Number of modes per unit frequency}}{\text{Cavity volume}} = \dfrac {8 \pi \nu^2}{c^3}$$

In this method of derivation no negative signs crop up and the result is positive.


Deriving the expression via wavelength as on the HyperPhysics website gives a positive value for $$\dfrac{\text{Number of modes per unit wavelength}}{\text{Cavity volume}} $$ which is what one would expect?


The text in the HyperPhysics website gives a reason for "ignoring" the negative sign - "The negative sign sign reveals that the number of modes decreases with increasing wavelength".
In the derivation you want to know by how much the number of nodes changes with a change of wavelength.
The fact that the Mathematics tells you that it is a decrease in the number is not what you required, it is the magnitude of the change which is needed.
You could have done that just as well by decreasing the wavelength by an amount $-d \lambda$ which would have given you a positive value after differentiating.

Rather than just ignore the negative sign the HyperPhysics website includes another minus sign to make the required quantity positive.


Using the derivation via frequency you can switch to the wavelength relationship by using

$$\dfrac{dN}{d\lambda}=\dfrac {dN}{d\nu}\dfrac{d\nu}{d\lambda} = \dfrac {dN}{d\nu} \left ( -\dfrac{c}{\lambda^2} \right)$$ and then the minus sign turns up.

That minus sign arrives because $d \lambda = - \dfrac{1}{\nu^2} d\nu$ so an increase in the frequency produces a decrease in the wavelength and both result in a change in the number of modes which is what you are trying to find not whether the number of modes increases or decreases.

If the graph of $N$ against $\lambda$ had been drawn with the $\lambda$ axis pointing to the left instead of to the right would the same confusion have occurred?

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