Determine path light will travel in a index of refraction described as $n(y)=1+be^{-ry}$. Let the light start off at the origin (0,0) and go to the point (L,0). Determine the equation that must be solved to determine the constant in your path equation. Thus far I have used Euler-Lagrange relation applied to time but can't seem to deduce the full equation for the constant.


closed as off-topic by garyp, Yashas, Kyle Kanos, John Rennie, ZeroTheHero May 11 '17 at 16:07

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Keeping up with the site policy, I won't do your homework, just tell you how to go about this:

The general approach of dealing with these kind of situations where the refractive index is varying continuously, is to employ the Fermat's principle. The link contains most of the relevant information, but to summarize the approach, the principle will lead you to an equation of the type $$\delta \int n ds = 0$$ where $ds$ represents the length element along the light ray path. The same can be written in terms of your 2D co-ordinates, in order to segregate out the $x$ and $y$ dependent terms. More generally, substituting for the spatial dependence of $n$, one would arrive at the equation: $$\delta \int n(x,y) \sqrt{(1+(dy/dx)^2)} dx = 0$$

Thereafter, one can use the standard approach of calculus of variations - using Euler-Lagrange equation on this "action", to obtain the equations of motion. If the starting point is the origin, $n_{\rm origin} = 1 + b$ from the above equations, which can act as a boundary condition on $n$. (You need boundary conditions here since, EL equations would turn your problem into a differential equation.)

Hope that helps :)

  • 2
    $\begingroup$ perfect, I did everything you said and neglected boundary conditions! So silly, I can get it now $\endgroup$ – Kirin171 May 10 '17 at 20:00

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