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In my textbook it says that Cauchy's equation is $$\mu(\lambda)=A+\frac{B}{\lambda^{2}}+\frac{C}{\lambda^{4}}+ \cdots$$

But what comes after $\frac{C}{\lambda^{4}}$? There is literally nothing given in my book as to what comes after $\frac{C}{\lambda^{4}}$. I even searched the entire Internet and no where did I find what comes after $\frac{C}{\lambda^{4}}$. Please tell me. I am so confused.

Buzz
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RIPAN BARUAH
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1 Answers1

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The equation is an empirical relationship, there is no derivation. The equation could continue with inverse powers of $\lambda^6$, $\lambda^8$ etc., however often only the $A$ and $B$ terms are necessary to obtain a good approximation for wavelengths in the visible part of the spectrum. The Sellmeier equation, developed after Cauchy's equation, can provide a better approximation at longer wavelengths than visible.

Nick
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  • So, should the equation be $μ(λ) = Α+\frac{B}{λ^{2}}+\frac{C}{λ^{4}}+ \frac{D}{λ^{6}} +\frac{E}{λ^{8}} + \frac{F}{λ^{12}}+\frac{G}{λ^{14}}+......$? Are the powers of $λ$ multiples of $2$? – RIPAN BARUAH May 02 '21 at 03:12
  • You missed out 10 but yes, the equation has even powers of $\lambda$. – Nick May 02 '21 at 08:26
  • Oh sorry, I missed $10$. One more question. Are $A, B, C, D, E, .... $etc. all constants? If they are constants, on what the factors do their values depend upon? – RIPAN BARUAH May 02 '21 at 11:34
  • They're all empirical constants. You do an experiment to measure some data, then calculate values of A, B etc that give the best fit to the data. – Nick May 02 '21 at 12:21
  • Do the values of $A, B, C, .....$ depend on the medium characteristics? – RIPAN BARUAH May 02 '21 at 13:27
  • Yes they're different for different materials. Values of A and B for some glasses are given on the wikpedia page. https://en.wikipedia.org/wiki/Cauchy%27s_equation – Nick May 02 '21 at 13:31
  • Consider editing this answer in response to the edited question. – Emilio Pisanty May 04 '21 at 14:01