#Perfect layers stick math full
The reason why you don't see the full circle is that the Earth gets in the way. But your eye can't distinguish the distances, all it sees is red light mingling together to form a circular arc that appears to be located somewhere in the distance. The rainbow comes from droplets that lie on the cone at different distances from your eye, some can be near and others can be far. You can try this by rolling a sheet of paper up into a cone shape and peering through the little hole at the tip. But when you look along the surface of a cone from its vertex, as your eye is doing, all you see is a circle. The droplets that light up red for you all lie on this cone: if they didn't, you wouldn't be catching their red rainbow ray. If you take all lines that emanate from your eye and make a angle with, what you get is a cone (see figure 9). This angle will of course be different for each frequency, or colour of light.īy staring hard at figure 4 you can convince yourself that the deviation is given by the formulaįigure 8: The deviated rainbow ray from your droplet makes a 42.52 degree angle with the line L. In other words, by what angle it is being turned as it is refracted, reflected and then refracted again (see figure 4). Using Snell’s law and the law of reflection (that the angle of incidence equals the angle of reflection), we can work out by how much a ray is deviated in terms of the angle at which it first hits the droplet. Catching rainbow raysīut why do we see each of the colours forming a perfect circular arc? To understand the shape of the rainbow, think of light from the Sun as coming down in parallel rays and striking a particular water droplet in the air. It's these different refraction angles for the different frequencies of light that gives a rainbow its colours. (The result is rounded to two decimal places.) Violet light with a refractive index of 1.34 has Thus, if a light ray hits the droplet so that, then red light with a refractive index 1.33 has
As air is very similar to a vacuum, the refractive index is very nearly equal to 1 for all frequencies. Here and are the angles shown in figure 3 and and are the refractive indices of air and water, respectively, for light with frequency. Snell’s law also tells us that the angle by which a ray is refracted is given by this equation: Since we’re assuming the droplet to be spherical, the normal in this case is just the extended radius of the droplet, connecting its centre to the point of incidence.
The law says that the refracted ray of light lies in the plane formed by the incident ray and the normal at the point of incidence – the normal is the line that passes through the point where the ray hits the droplet and is perpendicular to the surface of the droplet. Just how much the light of different frequencies is bent when entering the droplet is described by Snell’s law. The angles α and β are related by Snell's law. (The refractive index also varies slightly with temperature, but we can ignore this here.)įigure 3: The diagram shows the cross-section of the water droplet containing the incident ray, the refracted ray and the normal. But this small variation is enough to split sunlight into the beautiful spectrum of colours we see in a rainbow. The refractive index barely changes as the frequency varies: is around 1.34 for the violet end of the spectrum and around 1.33 for the red end. A measure of the slowing-down of light with frequency is given by the refractive index Its value depends not only on the frequency but also on the medium the light is entering (in this case water, as indicated by the subscript ). This is because the atomic structure of water interacts differently with waves of different frequencies. However, its speed will change by an amount that depends on the frequency. As the ray of light passes into water, the frequency, and therefore colour, remains the same. When light from the Sun travels through a vacuum (and to a very good approximation through air) all frequencies travel at the same speed, roughly 300,000 km per second. Figure 2: Light of different frequencies is refracted by different amounts.
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