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I understand that the speed of light for different wavelengths is different in a non-vacuum. Many of the explanations point to how different photons interact differently with the atoms in the material. However, I would like to ask how I can understand the reasons of variable speeds using the equation $v = f \lambda$.

Is the frequency for the different colours the same? If so, why?

Or else, how does the frequency differ from the frequency in a vacuum. For example, if the frequency and wavelength of red light in a vacuum is given by $f1$ and $\lambda1$ respectively, how will these variables change when in a medium with refractive index more than 1.

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The frequency in the medium is the same as in vacuum. The speed changes and this results in a change in wavelength, according to the formula.

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First, all colors have a different frequency. The lowest is red and the highest is indigo. You can't confuse frequency with speed although frequency can affect speed. Light travels at the same speed but those with less frequency have a longer wavelength while those with a larger frequency have a shorter wavelength.

Using a vacuum proves that all colors travel at the same speed because there is nothing else to prohibit or interfere with how it travels. Introducing a medium can change the way it travels by bringing in multiple factors, including the frequency, and density of the material it is traveling through.

There is an inverted relationship between the refractive index and the velocity of light. The velocity of light will be faster as the refractive index decreases. Light will move faster through a medium with a small refractive index. The velocity will be equal to the speed of light divided by the refractive index.

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