Chapter 33, Problem 77P

Rainbow. Figure 33-67 shows a light ray entering and then leaving a falling, spherical raindrop after one internal reflection (see Fig, 33-21a). The final direction of travel is deviated (turned) from the initial direction of travel by angular deviation θ_dev(a) Show that θ_dev.(a) Show that θ_dev is

θ_dev = 180°+2θ_i-4θ_r,

where θ_i is the angle of incidence of the ray on the drop and θ_r is the angle of refraction of the ray within the drop, (b) Using Snell's law, substitute for θ_r in terms of θ_i, and the index of refraction n of the water. Then, on a graphing calculator or with a computer graphing package, graph θ_dev versus θ_i for the range of possible θ_i, values and for n = 1.331 for red light (at one end of the visible spectrum) and n = 1.333 for blue light (at the other end).

The red-light curve and the blue-light curve have different minima, which means that there is a different angle of minimum deviation for each color. The light of any given color that leaves the drop at that color’s angle of minimum deviation is especially bright because rays bunch up at that angle and the bright red light leaves the drop at one angle and the bright blue light leaves it at another angle.

Determine the angle of minimum deviation from the θ_dev curve for (c) red light and (d) blue light. (e) If these colors form the inner and outer edges of a rainbow (Fig. 33-21a), what is the angular width of the rainbow?

Figure 33-67 Problem 77.