# Phase velocity > c

1. Sep 20, 2006

### quasar987

In a plasma for instance, the dispersion relation is $k^2=\omega ^2 \epsilon_0 \mu_0 -\epsilon_0\mu_0 \omega_p^2$, where I have made the approximation that the permitivity and permeability in a plasma are about those in empty space. Let's take the case where k is real (which happens as soon as $\omega >\omega_p$). From this, I can calculate the phase velocity:

$$v_{\phi}=\frac{\omega}{k}=\sqrt{c^2+\frac{\omega_p^2}{k^2}}>c$$

On the other hand, the group velocity turn out to be c, or a little less than c if we disregard the approximations.

But this is for a monochromatic plane wave. It's not like we have a wave packet where the modulation envelope moves with the group velocity. The wave does spread at the phase velocity, transporting with it an energy density. So energy is carried at a speed greater than c. :grumpy:

Last edited by a moderator: Sep 20, 2006
2. Sep 20, 2006

### Staff: Mentor

IIRC, for particles (matter), the group velocity (true particle velocity) is always < c and the phase velocity always > c.

In a plasma the EM fields propagate at c, but the particles are still constrained by group velocities < c, and in terrestrial plasmas, the energies are on the order of keV (maybe up to 100-200 keV).

3. Sep 20, 2006

### Claude Bile

My response would be that you can't get a purely monochromatic wave, due to the fact that you can't get a wave that is infinitely long in time.

It's one of those difficulties that appears only when infinites and delta functions etc. are included in the analysis, remove the infinites and things work again .

Claude.

4. Sep 21, 2006

### pseudovector

There are a lot of such paradoxes, where something goes faster than light. The solution is that you can never use them to send information or energy faster than the speed of light.