A rainbow is produced by the reflection of sunlight by spherical drops of water in the air. $\textbf{Figure P33.60}$ shows a ray that refracts into a drop at point $A$, is reflected from the back surface of the drop at point $B$, and refracts back into the air at point $C$. The angles of incidence and refraction, $\theta_a$ and $\theta_b$ , are shown at points $A$ and $C$, and the angles of incidence and reflection, ua and ur , are shown at point $B$. (a) Show that $\theta^B_a$ = $\theta^A_b$ , $\theta^C_a$ = $\theta^A_b $, and $\theta^C_b$ = $\theta^A_a$ . (b) Show that the angle in radians between the ray before it enters the drop at $A$ and after it exits at $C$ (the total angular deflection of the ray) is $\Delta= 2\theta^A_a - 4\theta^A_b + \pi$. ($Hint$: Find the angular deflections that occur at $A$, $B$, and $C$, and add them to get $\Delta$.) (c) Use Snell's law to write $\Delta$ in terms of $A$ a and n, the refractive index of the water in the drop. (d) A rainbow will form when the angular deflection $\Delta$ is $stationary$ in the incident angle $\theta^A_a$ -that is, when $d\Delta/d\theta^A_a$ = 0. If this condition is satisfied, all the rays with incident angles close to $\theta^A_a$ will be sent back in the same direction, producing a bright zone in the sky. Let $\theta_1$ be the value of $\theta^A_a)$ for which this occurs. Show that $cos^2\theta_1$ = $1 \over 3$ ($n^2$ - 10). ($Hint$: You may find the derivative formula $d($arcsin$ u(x)22/dx$ = ${(1 - u^2)^{-1/2}}(du/dx)$ helpful.) (e) The index of refraction in water is 1.342 for violet light and 1.330 for red light. Use the results of parts (c) and (d) to find $\theta_1$ and $\Delta$ really never } for violet and red light. Do your results agree with the angles shown in Fig. 33.19d? When you view the rainbow, which color, red or violet, is higher above the horizon?