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.