CALC A secondary rainbow is formed when the incident light undergoes two internal re...

CALC A secondary rainbow is formed when the incident light undergoes two internal reflections in a spherical drop of water as shown in Fig. 33.19e. (See Challenge Problem 33.60.) (a) In terms of the incident angle θ^A_a and the refractive index n of the drop, what is the angular deflection Δ of the ray? That is, what is the angle between the ray before it enters the drop and after it exits? (b) What is the incident angle θ₂ for which the derivative of Δ with respect to the incident angle θ^A_a is zero? (c) The indexes of refraction for red and violet light in water are given in part (e) of Challenge Problem 33.60. Use the results of parts (a) and (b) to find θ₂ and Δ for violet and red light. Do your results agree with the angles shown in Fig. 33.19e? When you view a secondary rainbow, is red or violet higher above the horizon? Explain.

33.60 ... CALC A rainbow is produced by the reflection of sunlight by spherical drops of water in the air. 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, θ_a and θ_b, are shown at points A and C, and the angles of incidence and reflection, θ_a and θ_r, are shown at point B. (a) Show that θ^B_a = θ^A_b, θ^C_a = θ^A_b, and θ^C_b = θ^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 (Hint: Find the angular deflections that occur at A, B, and C, and add them to get Δ.) (c) Use Snell’s law to write _ in terms of u A a and n, the refractive index of the water in the drop. (d) A rainbow will form when the angular deflection Δ is stationary in the incident angle θ^A_a —that is, when dΔ/dθ^A_a = 0. If this condition is satisfied, all the rays with incident angles close to θ^A_a will be sent back in the same direction, producing a bright zone in the sky. Let θ₁ be the value of θ^A_a for which this occurs. Show that (Hint: You may find the derivative formula d(arcsin u(x))/dx = (1 - u²)^-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 θ₁ and Δ 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?