The Mathematics of Rainbows

After reviewing a simple explanation. you can trace the path the sunlight makes through a rain drop and derive a formula for the angle between the incoming sunlight and the refracted ray.

A few basics first:

  • if you draw a line from the center of the circle through the surface it intersects the surface at 90 deg
  • since Snell's law always references the angle between the light ray and the line which is perpendicular to the surface where the light ray hits, this extended radius line is the reference for measuring angles in Snell's law and also for the law of reflection, i.e., the angle of incidence equals the angle of reflection.



    First surface

    So in the diagram above, the angle A is the incident angle and the angle B is the transmitted angle. If we assume the index of refraction for air is 1.0 and the index of refraction for the water is n then we have from Snell's law:

    sin(A) = n * sin(B)
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