Mechanics - Calculus of variations

  • Thread starter lifeonfire
  • Start date
  • #1
14
0

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
4,923
1,862
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.
 

Related Threads on Mechanics - Calculus of variations

  • Last Post
Replies
3
Views
1K
Replies
7
Views
2K
Replies
2
Views
799
Replies
1
Views
6K
Replies
14
Views
13K
Replies
8
Views
5K
Replies
2
Views
764
Replies
1
Views
4K
Replies
2
Views
2K
  • Last Post
Replies
0
Views
2K
Top