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# Calculus of Variations - Fermat's principle watch

1. Given that in a horizontally stratified medium the refractive index is given by n(z)=SQRT(a-bz) where 'z' is the height and 'a' and 'b' are positive constants, prove that light rays travelling in a vertical plane follow inverted parabolas.

Now Fermat's theorem is the integral of n(z)dl, and it states that this optical path is minimised. I've tried doing this question in the following fashion:

I expressed Fermat's theorem as

the integral:n(z)((dy)^2+(dz)^2)^1/2=integral: [n(z)(1+y'^2)^1/2]dz then I used Euler's equation, and since df/fy=0 then d/dz(df/dy')=0 so df/dy'=constant (where 'f' is the functional in the integral). If df/dy' is a constant then you can get an expression for y' in terms of some constants and z but this doesnt give me the equation of a parabola.

Thanks.
2. Just worked this out.

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Updated: January 30, 2010
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