Question

A plane electromagnetic wave propagates in a medium moving with constant velocity `Vlt lt c` relative to an interial frame `K`. Find the velocity of that wave in the frame `K` if the refractive index of the medium is equal to `n` and the propagation direction of the wave coincides with that of the medium.

Answer 1

In vaccum inertial frames are all equivalent, the velocity of light is `c` in any frame. This equivalence of inertial frames does not hold in material media and here the frame I which the medium is at rest is singled out. It is in his frame that the velocity of light is `(c )/(n)` where `n` is the refractive in desc of light for that medium.
The velocity of light in the frame in which the medium is moving is then by the law of addition of velocities
`((c)/(n)+v)/(1+(c)/(n).v//c^(2)) = ((c)/(n)+v)/(1+(v)/(cm)) = ((c)/(n)+v) (1-(v)/(cn)+.......)`
`=(c)/(n)+v-(v)/(n^(2)) +.... = (c)/(n)+v (1-(1)/(n^(2)))`
This is the velocity of light in the medium in a frame in which the medium is moving with velocity `v lt lt c`.