- #1

fluidistic

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## Homework Statement

Fermat's principle establishes that the path taken by a light ray between 2 given points is such that the time that the light takes is the smallest possible.

1)Demonstrate that a light ray which propagates through a medium with a constant refractive index follows a straight line.

2)Demonstrate that Snell's law holds for a ray of light passing by 2 points separated by a plane inter-phase which separates 2 different mediums (i.e. with different refractive indexes).

3)Determine the equation of the trajectory of a ray of light in the y-z plane assuming that the refractive index changes with z, i.e. n=n(z).

## Homework Equations

[tex]S=\int _{t_1} ^{t_2} L dt[/tex].

Optical path: [tex]L=\int _a ^b n(\vec r) ds[/tex]. I'd bet it should be [tex]L=\int _a ^b n(\vec r) d\vec s[/tex], right?

Euler-Lagrange equations: [tex]\frac{d}{dt} \left ( \frac{\partial L}{\partial \dot q} \right ) - \frac{\partial L}{\partial q}=0[/tex].

## The Attempt at a Solution

1)[tex]S=n \int _a ^b dr = n (b-a) \square[/tex].

2)[tex]S= \int _a ^b n(\vec r) d\vec r[/tex]. Here I'm not sure, but I think I should reach something like [tex]n_1 \int a ^c d\vec r + n_2 \int c ^b d\vec r[/tex] but I don't really trust it. I don't think I can even use the fact that [tex]n_1=\frac{c}{v _1}[/tex]. It's a problem arising in Classical Mechanics, not optics. And we're dealing with variational calculus and Euler-Lagrange's equations.

Do you have any idea how to tackle part 2? I'll try to do part 3 right after I've done part 2.

Thanks for any help.