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Mechanics - Calculus of variations

  • Thread starter lifeonfire
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  • #1
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Homework Statement


Suppose a ray of light travels from (x,y) = (-1,1) to (x,y) = (1,1) in a region where the index of refraction is n(y) = e^y.

(a) Find the path.


Homework Equations





The Attempt at a Solution


Is this okay?
The positions of the light ray are given by
initial (xi,yi) = (-1,1)
Final (xf,yf) = (+1,1)
Clearly the Y coordinate is invaiant
So
the refractive index of the region
n(y) = e^y
n(1) = e = 2.718
As the light ray travells, the index of refraction is invariant, so the ray will travell in straight line path from (xi,yi) to (xf,yf).
 

Answers and Replies

  • #2
phyzguy
Science Advisor
4,559
1,489
Not true. Since the light moves faster in regions of lower refractive index, it can get from x=-1 to x=1 faster by traveling a path that deviates from a straight line and spends more time in the region with y<1 where the propagation speed is faster. Although this path will be physically longer, the time of propagation will be less. Since light always takes the path of least time, it will not follow a straight line. Finding the exact path is a calculus of variations problem.
 

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