![]() 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). Perhaps you have seen a fainter secondary rainbow above the primary bow. So the rainbow really consists of seven individual bows corresponding to the seven colors 3. By repeating the calculation of Problem I for these values of k, show that the rainbow angle is about 42.3" for the red bow and 40.6" for the violet bow. (The effect is called dispersion.) For red light the refractive index is k = 1.3318whereas for violet light it is k 1.3435. As Newton discovered in his prism experiments of 1666, the index of refraction is different for each color. Problem I 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. (This angle is called the rainbow angle.) as su 2. ![]() TUULUVOSE 3.4 The figure below shows that the angle of elevation from the observer up to the highest point on the rainbow is 180-138 42. It is the concentration of rays coming from near the direction of minimum deviation that creates the brightness of the primary rainbow. This means that many rays with a 59.4 become deviated by approximately the same amount. The significance of the minimum deviation is that when a = 59.4' we have D'(a) -0.50 0. Thus D(a) = (a - b) + (1 - 2B) + = 1 + 2a - 48 Formation of the primary rainbow Show that the minimum value of the deviation is D(a) - 138 and occurs when a 59.4'. (Notice that it is refracted away from the normal line.) The angle of deviation D(a)is the amount of clockwise rotation that the ray has undergone during this three-stage process. part of it is reflected, but for the time being we are more interested in the part that leaves the raindrop at C. The angle of incidence equals the angle of reflection.) When the ray reaches C. At B some of the light passes through the drop and is refracted into the air, but the line BC shows the part that is reflected. Bis the angle of refraction, and is the index of refraction for water. Notice that the light is refracted toward the normal line AO and in fact Snell's Law says that sing = ksing where is the angle of incidence. Some of the light is reflected, but the line AB shows the path of the part that enters the drop. The figure shows a ray of sunlight entering a spherical raindrop at A. ![]() In this project we use the ideas of Descartes and Newton to explain the shape, location, and colors of rainbows. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. Math 20 - Calculus I $4.2 - Extra Credit Applied Project: The Calculus of Rainbows Show your work! Good luck! (40pts) Rainbows are created when raindrops scatter sunlight.
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