I am stuck a bit on a question to do with Cherenkov radiation.
The requirement stated for Cherenkov radiation to occur is that the charged particle travels at a velocity greater than light in the medium.
We are asked to derive the the condition that radiation only occurs when
$$ p > \frac{mc}{\sqrt{2\delta n}} $$,
where we are told that the refractive index of the medium is $n = 1 + \delta n$, where $\delta n \ll 1 $ and that $ 1- \beta \ll 1 $, where $\beta = \frac{v}{c}$.
I have worked out that as the speed of the particle, $v > c_{medium}$, then then $v > \frac{c}{n}$. But this just gives me that $p > \frac{mc}{n}$ which is not what is required. I am clearly missing something here (not sure if I need to consider relatavistic effects), but if someone could give me a hint to where I am going wrong that would be great.
Best Answer
You must take into account the Lorentz factor in momentum equation at relativistic limits-here I've assumed 'm' to be the rest mass of particle.