Chapter 9.1, Problem 1P

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. Thus solve the following problems:

Derive the optical law of reflection. Hint: Let light go from the point $A=\left(x_1,y_1\right)$ to $B=\left(x_2,y_2\right)$ via an arbitrary point $P=$ $(x,0)$ on a mirror along the $x$ axis. Set $dt/dx=(n/c)dD/dx=$0, where $D=$ distance A P B, and show that then $\theta =\varphi$.