= Kinetic-energy threshold for Cherenkov radiation
{title2=$K_{\rm th}=mc^2[(1-n^{-2})^{-1/2}-1]$}
At a specified <frequency> with transparent-medium <phase> index $n>1$, <Cherenkov radiation> requires $\beta>1/n$. Inserting the threshold <speed> into <relativistic kinetic energy> gives $K_{\rm th}=mc^2[(1-n^{-2})^{-1/2}-1]$. Equality is the kinematic limit; the cone angle and emission vanish there. A dispersive material has different thresholds in different spectral intervals. The result uses <phase velocity>, not <group velocity>.
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