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As I know the speed of the light doesn't change while it travels through a vacuum. But while it travels through a prism, it shows different deviation angles to different frequencies.

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So according to the $$ n = \frac {sin (A+D)/2}{sin (A/2)} $$ formula $n$ is different with the frequency of the light, when $n$ is different the velocity should be different through the medium.

Why does the velocity change with the frequency of the light while it travels through a medium but not in vacuum?

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Light’s velocity in a medium changes compared with its velocity in vacuum because the electrons in the atoms of the medium experience forces due to the light passing through, and, as they are accelerated, radiate light of their own which superposes with the incoming light. (Showing that this additional radiated light changes the phase velocity of the incoming light requires some math.)

The velocity shift is frequency-dependent because the electrons respond differently to different incoming light frequencies. They are basically driven harmonic oscillators. They have a natural frequency determined by the atomic or molecular physics of the medium, and when the driving frequency of the incoming light is close to their natural frequency they accelerate more easily. This is just like how a child on a swing moves more easily when you push the swing at the frequency it “wants” to swing at.

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