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First, let's look at what causes a rainbow. The basic concept is the same as a prism, where white light is separated into its separate components by refraction.
In refraction, light is "bent" because of it slows down or speeds up in different media. In our example, light will slow down in water, which (through some advanced math) you can show will change its angle. Well, the angle and the amount that the light slows down is dependent on its wavelength (color), so some wavelengths will be bent more than others.
Because different colors are bent different amounts, when you put white light into a prism at an angle, you will see each color separate out on its own.
Now, how does this apply to rainbows being round?
Remember, when you put light into a prism, the prism is a rectangular piece of glass or plastic. It doesn't exhibit any lens-like action, for the most part. However, when we're looking at the drops of water that refract light to make a rainbow, you can figure that they are pretty sphere-like, so light that goes through them will not only be refracted in one direction perpendicular to the incidence (think if it's coming in on the x-direction, it bends in the y direction), but in two (if it's coming in on the x-direction, it bends in both the y and z directions)!
See the link below for more info on the effect a spherical lens would have on parallel rays to see how they would be refracted (just to see what a lens would do. The equation given only holds for a thin lens), and keep in mind, this is in two directions perpendicular to the light's incidence, not one.
Because of these combined effects, the differential refraction of different wavelengths and the effect of the spherical lens on light, the rainbow will both be multicolored and circular!
I hope that helps!
To answer the question that why the shape of a rainbow curved and not straight, one should first understand that the rainbow is not something that is present physically in the sky. It is the image formed in the eye by refraction, total internal reflection & subsequent refraction of parallel rays of sun light coming from the back of the observer through the rain drops in our front.
If one closely observes the shape of the rainbow, it is like an arc of a circle with the center on the line determined by the observer's eye and the sun. Let us call this line the the rainbow-axis.
Now consider the plain of the rainbow as a plain perpendicular to the rainbow axis with four quadrants.
The rain-drops and sun-rays at an angle of zero will form the image at zero angle in the rainbow plain. Similarly the ones at 45 degrees will form the portion at 45 degrees and so on. Thus the rain drops in each quadrant will form the corresponding image to complete the rainbow.
Rainbow of sizes larger than a semi-circle have been observed in the valleys while standing on a hill. I have personally seen such rainbows with the secondary and tertiary rainbows as well.
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