Formation of the Rainbow Logan Bonecutter April 24, 2015

Abstract The theory of the rainbow arose in the man’s sense of wonder thousands of years ago. About 2,500 years ago, Aristotle suggested the first theory that explains the formation of the rainbow. Many scientists studied and suggested different theories to try to improve the understanding of the appearance of the rainbow. The theories of Aristotle, Descartes, Newton, Fermat, and Snell are specifically described in this presentation. Calculations of the angles of observation for different colors are given by using calculus. Even now in the beginning of the 21 st century, the formation of the rainbow is not understood in all details.

Rainbow

Greek Goddess of Rainbow Iris

Aristotle’s Theory Rainbow had a specific shape Explained the appearance of the rainbow as a reflection from the clouds Mathematicians and scientists have gained interest Water and sunlight are needed to have a rainbow Everyone sees rainbows differently Time of day can change the rainbow

Descartes Theory Almost 400 years ago Understood why the rainbow was located where it was

Isaac Newton Interested in light and colors Band of light is a color spectrum In glass, the speed of light is different for different colors, which explains that the colors coming out from the prism have different angles.

Fermat’s Least Time Principle The principle which states that the path taken between two points by a ray of light is the path that can be travelled in the least time. O In this picture, light is traveling from point A to point O and from point O to point B through fast and slow media. Since the speed of light in the fast medium is V 1 and the speed of light in the slow medium is V 2, the traveling times from A to O and from O to B are given by

Fermat’s Least Time Principle The total traveling time from point A to point B is given by To find the least time we calculate the critical points of this function by taking the derivative and setting it equal to zero. From the picture By combining the previous formulas, we deduce from Fermat’s Principle, Snell’s Law

Snell’s Law where V 1 and V 2 are speeds of light in the first and second media correspondingly where - Indexes of refraction n 1 and n 2 are defined by formulas - We can write Snell’s Law in the following form where c is the speed of light in vacuum (1) (2)

Angle of Deviation First rotation α - β Third rotation α - β Second rotation π - 2β Total rotation π + 2α - 4β

Angle of Deviation - Angle of deviation for the first rainbow - Using Snell’s law we have (3)

First Rainbow Minimum of Deviation - By differentiation - Solving this equation we will get To find the minimum of angle of deviation, we should find the critical numbers of the deviation angle, or we should set the derivative to zero and

Graph of Deviation Angle Graphing the dependence of the deviation angle α, we get this picture From the picture, we can see the minimum of deviation angle at α = = 59.4° (3)

Observation Angle The first rainbow observation angle is given by For the color red, plug in index of refraction, k= We get For the color violet, plug in index of refraction, k= We get - From these calculations, the first rainbow angles are different for different colors. This explains the colors of the rainbow. (4) (5) (6)

Nth Order Rainbow - Angle of deviation for the nth order rainbow - Using Snell’s Law - By differentiation We get this formula (7)

Nth Order Rainbow cont. - By solving this equation for cos(α) and sin(α) we get and (8)

Second Rainbow

Formation of Second Rainbow First rotation α - β Second rotation π - 2β Third rotation π - 2β Fourth rotation α - β Total rotation: 2π + 2α - 6β

Second Rainbow Minimum of Deviation - From formula (8), by taking n = 2, we get the minimum of deviation for the second rainbow Observation angle for the second rainbow is given by For the color red, plug in index of refraction, k= We get For the color violet, plug in index of refraction, k= We get (9) (10) (11)

Third Rainbow The minimum of deviation of the third rainbow is given by (8) when n = 3 Observation angle for the third rainbow is given by For the color red, plug in index of refraction, k= We get For the color violet, plug in index of refraction, k= We get (12) (13) (14)

Quadruple Rainbow By similar calculations, one can find angles of observation for the quadruple rainbow and higher order rainbows.

References 1. James Stewart. Calculus C. Boyer, The rainbow: from myth to mathematics, Princeton University Press, Amateur Scientist, Scientific American July 1977, page : How to create and observe a dozen rainbows in a single drop of water by Jearl Walker

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