A rainbow is produced by light rays that enter a raindrop (assumed spherical) and exit after being reflected internally as in Figure 25. The angle between the incoming and reflected rays is θ = 4r − 2i, where the angle of incidence i and refraction r are related by Snell’s Law sin i = n sin r with n ≈ 1.33 (the index of refraction for air and water).

(a) Use Snell’s Law to show that

(b) Show that the maximum value θ_max of θ occurs when i satisfies  if cos i = n/2 cos r. Then use Snell’s Law to eliminate r.

(c) Show that θmax ≈ 42.53°.

dr di cos i n cos r