Find the conditions under which a charged particle moving uniformly through a medium with refractive index `n` emits light (the Vavilov-Cherenkov effect). Find also the direction of that radiations.

Explanation

The field induced by a charged particle moving with velocity `V` excites the atoms of the medium turning them into sources of light waves. Let us consider two arbitrary points `A` and `B` along the apth of the particle. The light waves emitted from these points when the particle passes them reach the point `P` simultaneously and particle each other provided thet are in phase which is the case is general if the time taken by the light wave to propagate from the point `A` to the point `C` is equal to that taken by the particle to fly over the distance `AB`. Hence we obtain
`cos theta = (v)/(V)`
where `v = (c)/(n)` is the phase velocity of light. It is evident that the radiation is possible only if `Vgtv` i.e. when the velocity of the particle exceeds teh phase velocity of light in the medium.