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My question is with respect to Newton's experiment of using two identical glass prisms [in which one is inverted with respect to the first one]. When he allowed all the colors of the spectrum to pass through the second prism, he found a beam of white emerging from the other side of the second prism.

enter image description here

And I know that refraction is due to different speeds of different wavelengths of light. So, How can those colors recombine to form a beam of white light (since different colors have different speeds)?

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The speed of light in a vacuum is constant for all wavelengths. In other media (like glass), it can vary. The speed of light in a particular medium doesn't depend on its history, only on what the medium is. Questions such as " Will it speed back up to the speed of light?" don't make sense - the speed of light depends only on what it's currently traveling in.

The different wavelengths do move at different speeds in air, but the difference is so small that white light remains "white" - all the colors move at effectively the same speed.

The colors in that figure are combining to white light because they have traveled the same path length. As you can see, the purple beam traveled the least distance in the first prism, but it also travels the furthest distance in the second prism. The red beam does the reverse - it traveled the furthest distance in the first prism and the least distance in the second. Added together the two colors traveled the same path length, so they recombine.

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The speed of light in a medium depends on the (real part of the) refractive index $n$ of the medium. The speed of light in a medium is given by $$c = c_0/n$$ where $c_0$ is the speed of light in vacuum. Since the refractive index $n$ of the prism depends on the wavelength, we obtain dispersion. However, as soon as the light leaves the prism the refractive index $n_{air}$ must be used. Since $n_{air}\approx 1$ for all wavelengths in the visible range, there exists no dispersion. All wavelength move with the same speed $c = c_0/n_{air}$.

Don't think of the refractive index as a friction. If the index of refraction exceeds one it slows down the speed of light, but no energy is lost -- again, we exclude absorption and concentrate on the real part of the refractive index.

Maybe one more idea to get your head around: Suppose white light enters a medium with "significant" dispersion, but the angle of incidence is $0°$. In this case the different wavelength have different velocities, but the refraction angle is zero. Hence, the rays with different wavelength reach our detector at different times, however, they are overlapping. Thus, the following image sketches helps us to understand the physics lightRay

As said before, the colours are not shifted in their position. Nevertheless, in the upper image I shifts the three colours in $y$-direction to explain the "superposition picture", while the lower image shoes the "experienced color":

  • At time $t_0=0$ the blue light hits the detector. Hence, we detect "blue".
  • At time $t_1 = s_1/c_{green} = n_{green} \cdot s_1/c_0$ the green light ray hits the detector. Hence, we detect the superposition of "blue" and "green". We experience magenta.
  • Finally, at time $t_2 = s_2/c_{red} = n_{red}\cdot s_2/c_0$ the red light ray hits the detector. Hence, we detect the superposition of "blue", "green", and "red". We experience white light again.
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