Maths on a Mug 3

Yesterday my (by far) most popular tweet was a picture of a rainbow. So I thought I would do a little diagram of what causes us to see a rainbow. The colours are reversed for a double rainbow because there is a second reflection.

Rainbows are always a captivating natural phenomena that people can’t help but point out to one another and they arise from the interplay of light and water droplets in the atmosphere. When sunlight encounters a spherical water droplet, part of it reflects off the droplet’s surface, while another portion enters the droplet, undergoing refraction—a bending of light due to a change in medium. This refracted light travels through the droplet, reflects off the inner surface, and refracts again as it exits.

The degree of bending, or refraction, depends on the light’s wavelength. Shorter wavelengths (blue and violet light) refract more than longer wavelengths (red and orange light). This variation causes the dispersion of white sunlight into its constituent colors, forming a spectrum. The refractive index of water, which measures how much light slows down in the medium, varies slightly with wavelength—approximately 1.34 for violet light and 1.33 for red light. This small difference is sufficient to separate sunlight into the vivid colors observed in a rainbow.

The formation of a rainbow involves specific angles. As light enters and exits the droplet, the angles of incidence and refraction are related by Snell’s law, which states that the ratio of the sines of these angles is equal to the ratio of the refractive indices of the two media. For a given wavelength, there exists a minimum deviation angle at which the emerging light is most intense. For red light, this angle is approximately 42 degrees relative to the incoming sunlight. This concentration of light at specific angles results in the circular arc of a rainbow.

The observer’s position is crucial in perceiving a rainbow. The Sun must be behind the observer, and the water droplets must be in front. The angle at which the light exits the droplets and reaches the observer’s eye determines the specific color seen at each point along the arc. This geometric arrangement explains why rainbows are often seen when the sun is low in the sky, such as during early morning or late afternoon, and why they appear as circular arcs centered opposite the Sun.

A detailed description of the mathematics involved was written up in an excellent article in Plus Magazine.