Chapter 4.5, Problem 64ES

Fermat’s principle (biography on $\text{p}.213$ ) in optics states that light traveling from one point to another follows that path for which the total travel time is minimum. In a uniform medium, the paths of “minimum time� and “shortest distance� turn out to be the same, so that light, if unobstructed, travels along a straight line. Assume that we have a light source, a flat mirror, and an observer in a uniform medium. If a light ray leaves the source, bounces off the mirror, and travels on to the observer, then its path will consist of two line segments, as shown in Figure Ex-64. According to Fermat’s principle, the path will be such that the total travel time $t$ is minimum or, since the medium is uniform, the path will be such that the total distance traveled from $A$ to $P$ to $B$ is as small as possible. Assuming the minimum occurs when $dt/dx=0,$ show that the light ray will strike the mirror at the point $P$ where the “angle of incidence� ${\theta }_1$ equals the “angle of reflection� ${\theta }_2$ .