Cherenkov radiation - phase velocity not group velocity

[JoePhysicsNut]

Why must the charged particle that leads to Cherenkov radiation travel faster than the phase velocity of light not the group velocity of light?

One of the sides of the triangle that is used to define cosθ is v=c/n i.e. the phase velocity. I don't see why it's one rather than the other.

Thanks!

 

[mfb]

Cherenkov radiation happens when the phase of the light cannot "keep up" with the particle, so you cannot get a static field around the particle (as seen by the particle).

One of the sides of the triangle that is used to define cosθ is v=c/n i.e. the phase velocity. I don't see why it's one rather than the other.

It is very similar to refraction of light.

 

[JoePhysicsNut]

Cherenkov radiation happens when the phase of the light cannot "keep up" with the particle, so you cannot get a static field around the particle (as seen by the particle).

Thanks for the reply! But again, is there a qualitative way of understanding why it's phase velocity not group velocity?

It is very similar to refraction of light.

True, refraction of light is also controlled by c/n - the phase velocity. Refraction can be understood from Fermat's principle, but again I don't know why it's one type of speed rather than the other.

 

[vanhees71]

It's not so much similar to the refraction of light but rather to the Mach cone of supersonic sound, here of course for the relativistic case of light propagation in a medium. You find the full calculation of the Cherenkov-radiation field in

A. Sommerfeld, Lectures on Theoretical Physics IV (Optics)

These are pretty old but very marvelous textbooks on classical theoretical physics. I really love those :-).

 

[sardar]

I can explain why the charged particle that leads to Cherenkov radiation must travel faster than the phase velocity of light, not the group velocity of light.

First, let's understand the difference between phase velocity and group velocity. Phase velocity is the speed at which a single wavefront travels, while group velocity is the speed at which a group of wavefronts (or a wave packet) travels. In other words, phase velocity is the speed of an individual particle in a wave, while group velocity is the speed of the overall wave.

In the case of Cherenkov radiation, the charged particle is traveling faster than the phase velocity of light, not the group velocity. This is because the phase velocity of light is always equal to the speed of light in vacuum, which is the maximum speed at which any particle can travel. Therefore, the charged particle must be traveling even faster than the speed of light in order to generate Cherenkov radiation.

To understand why this is the case, we need to look at the geometry of the situation. Cherenkov radiation is produced when a charged particle travels through a medium at a speed faster than the speed of light in that medium. This creates a shock wave of electromagnetic radiation, known as Cherenkov radiation.

In order for this shock wave to form, the charged particle must be traveling faster than the speed of light in the medium. This means that the phase velocity of the wavefronts created by the particle must also be faster than the speed of light in that medium. However, the group velocity of the overall wave can still be slower than the speed of light, as it is determined by the medium's refractive index.

In conclusion, the charged particle that leads to Cherenkov radiation must travel faster than the phase velocity of light, not the group velocity, in order to create the necessary shock wave and produce the characteristic blue light. This is due to the specific geometry and conditions required for Cherenkov radiation to occur.