A rainbow is produced by the reflection of sunlight by spherical drops of water in the air. Figure $P 33.66$ 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, $\theta_{a}$ and $\theta_{r}$ are shown at point $B .$ (a) Show that $\theta_{a}^{B}=\theta_{b}^{A}, \theta_{a}^{C}=\theta_{b}^{A},$ and $\theta_{b}^{C}=\theta_{a}^{A} .$ (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 $\Delta=2 \theta_{a}^{A}-4 \theta_{b}^{A}+\pi .$ (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 $\theta_{a}^{A}$ 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 $\cos ^{2} \theta_{1}=\frac{1}{3}\left(n^{2}-1\right)$ . (Hint: You may find the derivative formula $d(\arcsin u(x)) / d x=$ $\left(1-u^{2}\right)^{-1 / 2}(d u / d x)$ 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 $(\mathrm{c})$ and $(\mathrm{d})$ to find $\theta_{1}$ and $\Delta$ for violet and red light. Do your results agree with the angles shown in Fig. 33.20 $\mathrm{d}$ ? When you view the rainbow, which color, red or violet, is higher above the horizon?