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What are Rainbows?
How exactly do rainbows form?

When light travels through one medium to another medium that is denser, such as from air to water, it slows down and exits the new medium at a different angle than the one it entered with. This is called refraction.
In water droplets, sunlight can enter and bounce off the inner surface as a kind of mirror and exit at a sharp angle. Since each color has a different wavelength, each one is slowed to a different degree and is refracted at a slightly different angle. This is called dispersion.
Red light will exit the droplet at 42 degrees from the angle at which the sunlight entered; however, blue light will exit at an angle of 40 degrees, with the other colors being in between.
Rainbows only happen when the sunlight is coming from behind you and is low in the sky. The sunlight shines through the raindrops, but actually only one color will refract at the exact angle necessary for it to reach your eye. So, in one part of the sky, all the raindrops will bounce red light into your eye, while the other colors will scatter too high or too low. But in another part of the sky, the blue light will be the one to reach you. This combination of raindrops refracting certain wavelengths of light that hit your eye is what causes a rainbow.
What creates the bow of the rainbow?
Actually, rainbows form as a circle in front of you at an angle of 40-42 degrees from your line of sight. Meaning that the Earth covers part of the circle, and hence why you see only the upper part.
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