A point source of light P that is located in a medium 1 with refractive index n1, and its
image O in the medium 2 with refractive index n2 > n1. The
boundary between the two media crosses the plane of the drawing along the curve B. Taking the position of the image as an origin for the Cartesian system of coordinates (x, y), the position of the source can be defined as l on the axis x, and the position of the crosspoint C between the surface B and the axis x as f0. In the 2D – problem, the position of P becomes (l, 0), and the position of C becomes (f0, 0). Basing on the Fermat’s principle, find the equation of the boundary curve B as a function y(x), for particular l and f0.
Fermat's Principle: Light always travels the path of least time.
Optical length: n1 * l1 = n2 * l2 given two mediums with refractive indexes of n1 and n2, an the length of the light path l1 and l2, respectively.
The Attempt at a Solution
I have attempted to set up the equation as follows.
Now, we can use f0 and l to set up the optical length. By Fermat's Principle, we should have the optical length of every other path equal to this optical length, which is (l-f0) * n1 + f0 * n2
Let a point X(x, y) be on the boundary curve B. Thus, using the distance formula, we can set up the following equation:
n1 * sqrt ((l-x) ^2 + y^2) + n2 * sqrt (x^2 + y^2) = (l-f0) *n1 + f0 * n2.
The problem is, I cannot solve this equation. I have tried Wolfram Alpha, but it cannot solve for this equation in terms of x. PLEASE HELP ALL PHYSICS GENIUSES.