Mathematics of Rainbows MATH 171 Freshman Seminar for Mathematics Majors J. Robert Buchanan Department of Mathematics 2010
What is a Rainbow? A rainbow is created by water, sunlight, and the principles of reflection and refraction. These latter two principles are the result of light traveling from one point to another along the path of minimal time. Image courtesy of The Pennsylvania State Climatologist.
Preferred Rainbow
Law of Reflection Theorem (Law of Reflection) The angle of incidence and the angle of reflection of a reflected ray are equal. P a Q Θ i Θ r b x d
Fermat s Principle The Law of Reflection is a consequence of Fermat s Principle which states that light travels between two points along the path of minimum time. Let c be the speed of light, a be the distance of point P from the reflector, b be the distance of point Q from the reflector, and d be the horizontal distance between P and Q. We will assume that light from point P strikes the reflector a distance x horizontally from P. Where should x be located so that the total travel time of the light is minimized?
Solution If we let T (x) be the total travel time, then T (x) = 1 x c 2 + a 2 + 1 (d x) c 2 + b 2. We must minimize T with 0 x d. We can use the Extreme Value Theorem which states the minimum of a continuous function on an interval occurs either at the endpoints of the interval or at a critical number inside the interval. T x (x) = c x 2 + a d x 2 c (d x) 2 + b 2 The critical number occurs where x c d x = x 2 + a 2 c (d x) 2 + b 2 sin θ i = sin θ r θ i = θ r.
Law of Refraction (1 of 2) The Law of Refraction is sometimes known as Snell s Law. Theorem (Snell s Law) When light travels between two different mediums with speeds c i and c r, respectively, the angle of incidence θ i and the angle of refraction θ r are related by the equation, sin θ i = c i c r sin θ r. The ratio c i /c r is sometimes called the index of refraction.
Law of Refraction (2 of 2) P a Θ i x Medium 1 Medium 2 Θr b d Using Fermat s Principle, derive Snell s Law.
Solution If we let T (x) be the total travel time, then T (x) = 1 c i x 2 + a 2 + 1 c r (d x) 2 + b 2. We must minimize T with 0 x d. We can use the Extreme Value Theorem which states the minimum of a continuous function on an interval occurs either at the endpoints of the interval or at a critical number inside the interval. T x (x) = c i x 2 + a d x 2 c r (d x) 2 + b 2 The critical number occurs where x d x = c i x 2 + a 2 c r (d x) 2 + b 2 sin θ i = c i c r sin θ r.
Primary Rainbow (1 of 3) A primary rainbow results from a ray of light originating at the sun passing through a raindrop. We will assume that a raindrop is a sphere. The light ray is refracted as it enters the raindrop, it is then reflected off the back of the raindrop, and then is refracted once more as it leaves the raindrop and travels to an observer s eye. The spread of colors in a rainbow is caused by different wavelengths of light (corresponding to different colors) having different indices of refraction.
Primary Rainbow (2 of 3) The index of refraction for red light is approximately 1.3318 while the index of refraction for violet light is approximately 1.3435.
Primary Rainbow (3 of 3) 1 For the primary rainbow let α be the angle of incidence as the light ray enters the raindrop from the sun. Let β be the angle of refraction. Derive an expression for the amount of clockwise rotation (call this rotation R) a ray experiences as it passes through a raindrop to produce the primary rainbow. 2 Use implicit differentiation and Snell s Law to find dr/dα. 3 Show that R is minimized when k cos α = 2 1 3 where k is the index of refraction for a wavelength of light. 4 Approximately what are the rotation angles for red light and violet light?
Secondary Rainbow (1 of 2) A secondary rainbow results from a ray of light taking a different path through the raindrop than the primary path. The light ray is refracted as it enters the raindrop, it is then reflected twice off the inside of the raindrop, and then is refracted once more as it leaves the raindrop and travels to an observer s eye.
Secondary Rainbow (2 of 2) 1 For the secondary rainbow let α be the angle of incidence as the light ray enters the raindrop from the sun. Let β be the angle of refraction. Derive an expression for the amount of counterclockwise rotation (call this rotation R) a ray experiences as it passes through a raindrop to produce the secondary rainbow. 2 Use implicit differentiation and Snell s Law to find dr/dα. 3 Show that R is minimized when k cos α = 2 1 8 where k is the index of refraction for a wavelength of light. 4 Approximately what are the rotation angles for red light and violet light?