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Question:
If the phenomenon of rainbows is the formation of small spectrums on tiny raindrops, why do we see the whole rainbow as a BIG spectrum
Answer:

Rainbow is certainly formed by dispersion of light by the small raindrops, but there are phenomena lke reflection and refraction are also involved in the process. The dispersed light gets reflected within the raindrop and is is seen outside the droplet in air. Again, in air, there is refraction of the light, and because of this we see a big spectrum in the sky.

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