A beam of light is sent into a material that
becomes progressively more dense so that the speed of light changes
continuously. In plane cartesian coordinates, the speed of light is a
function of x alone: v = v(x). Assume that the light beam leaves the
origin (x=0, y=0) at a 45o angle (dy/dx = 1).
a. Use Fermat's Principle and the calculus of
variations to derive a differential equation for the path taken by the beam: y =
y(x). Note that an element of arc length can be written as: ds = [ 1 + (dy/dx)2 ]1/2.
Solve as far as you can without knowing more about v(x).
b. If the speed of light is specifically
v(x) = c / ( 1 + x/L )1/2, where L is a characteristic length,
compute the path analytically and plot it.