Chapter 4.5, Problem 65ES

Fermat’s principle (Exercise 64) also explains why light rays traveling between air and water undergo bending (refraction). Imagine that we have two uniform media (such as air and water) and a light ray traveling from a source $A$ in one medium to an observer $B$ in the other medium (Figure Ex-65). It is known that light travels at a constant speed in a uniform medium, but more slowly in a dense medium (such as water) than in a thin medium (such as air). Consequently, the path of shortest time from $A$ to $B$ is not necessarily a straight line, but rather some broken line path $A$ to $P$ to $B$ allowing the light to take greatest advantage of its higher speed through the thin medium. Snell’s law of refraction (biography on $\text{p}.226$ ) states that the path of the light ray will be such that $\frac{\sin {\theta }_1}{{\upsilon }_1}=\frac{\sin {\theta }_2}{{\upsilon }_1}$ where ${\upsilon }_1$ is the speed of light in the first medium, ${\upsilon }_2$ is the speed of light in the second medium, and ${\theta }_1$ and ${\theta }_2$ are the angles shown in Figure Ex-65. Show that this follows from the assumption that the path of minimum time occurs when $dt/dx=0$ .