Sec.4.1 - The Calculus of ranbows
Applied Project in Sec.4.1, Calculus by Stewart
Chinese version: 彩虹的微積分學
Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain the shape, location, and colors of rainbows.
Question 1:
The figure shows a ray of sunlight entering spherical raindrop at A. Some of the light is reflected, but the line AB shows the path of the part that enters the drop. Notice that the light is refracted toward the normal line AO and in fact Snell"s Law says that

Show that the minimum value of deviation is
The significance of the minimum deviation is that when

Question 2:
Problem 1 explains the location of the primary rainbow, but how do we explain the colors? Sunlight comprises a range of wavelengths, from the red range through orange, yellow, green, blue, indigo, and violet. As Newton discovered in his prism experiments of 1666, the index of refraction is different for each color, (The effect is called dispersion.) For red light the refractive index
Question 3:
Perhaps you have seen a fainter secondary rainbow above the primary bow. That results from the part of a ray that enters a raindrop and is refracted at A, reflected twice (at B and C), and refracted as it leaves the drop at D (see the figure at the left). This time the deviation angle

Show that
Taking

Question 4:
Show that the colors in the secondary rainbow appear in the opposite order from those in the primary rainbow.