Question

A rectangular solid of height h increases in optical density as its height increases, so the index of refraction of the solid increases with height according to:

n(y) = 1.10(4.00y + 1.00)

where y is the distance, in meters, from the origin (see diagram). A beam of light traveling in air (n = 1.00) in the x-y plane strikes the bottom of the tank at the origin, making an angle of incidence with the normal of ?1. Assume:

1. n varies only with y, not with x or z.
2. The light travels exclusively in the x-y plane.
3. The block is wide enough so the light leaves through the top and not through a side

a) Use Fermat

Answer 1

Euler's equations for extremum: f - y'(?f/?y') = constant
n=c/v
t = ? ds/v = ?(1.20/c)(2y+1)?(1+y'^2)dx

which would make my functional:

f=(2y+1)?(1+y'^2)

and using the Euler equations above I have from that:

(2y+1)(1+y'^2)^.5 - y'^2(2y+1)(1+y'^2)^-.5 = (2y+1)(1+y'^2)^-.5 = constant

and that's where I'm stuck...

(2y+1)(1+y'^2)^-.5 = constant seems like it should be easy enough to integrate with dy and dx somehow, but try as I might I'm totally stumped.

Answer 2