Rainbows happen due to something called refraction – when it’s raining and sunny at the same time, sunlight will hit a raindrop, and get split up into its constituent parts before it leaves the raindrop on the other side (you see the same thing when you shine a light through a glass prism). The light gets separated out based on different wavelength intensities; red light has longer wavelengths, which means it doesn’t get bent very much when passing through the water, but blue/purple light has a shorter wavelength, which makes it much more likely to get bent through the droplet.
Think of it as a group of people of different strengths trying to push straight through a crowd – the stronger people (the red light) can push through quite easily, so the direction in which they’re walking doesn’t change much. The weaker people (the blue/purple light) get pushed around a bit more though, which means they leave the crowd in a different direction on the other side.
Anyway, the reason we see 7 colours in a rainbow is because we’re seeing what’s called the ‘visible spectrum’ of light, which runs from red to violet. This spectrum is actually musch wider – on one end, there’s ultraviolet light, and on the other, infra-red – but human eyes don’t have the right sort of light detectors to be able to see those bits! Birds, however, can see ultraviolet light – so a rainbow would look very different to them!
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