Chapter 9.1, Problem 3P

The speed of light in a medium of index of refraction $n$ is $v=ds/dt=c/n.$ Then the time of transit from $A$ to $B$ is $t={\displaystyle {\int }_A^{B}d}t=c^{-1}{\displaystyle {\int }_A^{B}n}ds.$ By Fermat’s principle above, $t$ is stationary. If the path consists of two straight line segments with $n$ constant over each segment, then $\begin{array}{l}\hfill \\ {\displaystyle {\int }_A^{B}n}ds=n_1{d}_1+n_2{d}_2\hfill \end{array}$ and the problem can be done by ordinary calculus. solve the following problems:

Show that the actual path is not necessarily one of minimum time. Hint: In the diagram, $A$ is a source of light; C D is a cross section of a reflecting surface, $c$ and $B$ is a point to which a light ray is to be reflected. A P B is to be the actual path and $A{P}^{\prime }B,A{P}^{\prime \prime }B$ represent varied paths. Then show that the varied paths:

(a) Are the same length as the actual path if C D is an ellipse with $A$ and $B$ as foci.

(b) Are longer than the actual path if C D is a line tangent at $P$ to the ellipse in (a).

(c) Are shorter than the actual path if C D is an are of a curve tangent to the ellipse at $P$ and lying inside it. Note that in this case the time is a maximum!

(d) Are longer on one side and shorter on the other if C D crosses the ellipse at $P$ but is tangent to it (that is, C D has a point of inflection at $P$).