It is said that the speed of light increases when moving from optically denser medium to rarer medium but as light can be considered as a particle how does it get the energy to increase it velocity
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Good question, +1. But it is better to say a photon is something like a particle and something like a wave. See this. How can a red light photon be different from a blue light photon? – mmesser314 Aug 02 '20 at 17:13
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The energy of the photon doesn’t change. It’s a great question when considering individual photons It’s possible they don’t slow down at all and instead weave back-and-forth between the atoms as they propagate through the medium. This causes them to take a longer path. – Bill Alsept Aug 03 '20 at 14:41
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Light can be considered a particle, known as a photon, but the energy of each photon is not related to its velocity. Instead, it is given by the Planck formula, $$ E=h\nu, $$ where $h$ is Planck's constant and $\nu$ is the frequency of the light. Since the frequency of the light remains constant as it travels from one medium to another, the energy of those photons is also unchanged.
Emilio Pisanty
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I would just add to Emilio Pisanty's comment by saying that the frequency remaining a constant as light travels across a boundary separating two media is required because it is required that the tangential component of E be continuous across the boundary.
In turn, that the tangential component of E must be continuous across the boundary separating two media can be found by computing the line integral along any path through the two media and using Stoke's theorem.
CGS
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1I believe when the OP asks about the energy they are specifically referring to the speed of the photon not the frequency. In other words how does it accelerate as it leaves the medium? – Bill Alsept Aug 03 '20 at 16:00
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