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For a rainbow to form, water droplets in the air act as millions of tiny prisms refracting the sunlight into its component colors. The reason it appears as curved is also due to the physics of light. As light passes from air to water it bends slightly due to a difference in the refractive indexes of air and water. Some of the refracted light bounces back from the water droplet at just the right angle to be seen by you, the observer. The orientation of these droplets with just the right angles in relation to the sun and the observer is a circle. Technically, the shape is best described as a cone with the rainbow at the circular end and the observer at the tip. However, you can never see the entire circle since the horizon gets in the way, thus the appearance of a semicircle. Incidentally, this is also why you can never reach the end of the rainbow. As soon as you move, the droplets that reflect the light back to you all change and you are technically seeing a different rainbow than before.
This is because light passes through a prism is because the light passing through the prismatic raindrops is not of a liner type. The light of the sun it coming through in a circular shape. As it passes through the millions of rain drops it spreads the visible spectrum and is reflected back, making the various wavelengths of light visible. You are actually looking at what should be a circle, but because of the horizon you are not able to see the bottom portion. Also if you look carefully you will see there are two different rainbows, which become visible when the light passing through the rain is at the correct angle.
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