= Solution
In a transparent uniform medium, constructive phase interference along the trajectory requires $\omega-\mathbf k\cdot\mathbf v=0$. Since $k=n\omega/c$, the <Cherenkov cone angle> obeys
$$
\boxed{\cos\theta_C=\frac1{n\beta}.}
$$
Here $n\beta=2\sqrt{1-10^{-4}}>1$, so emission is allowed and
$$
\boxed{\theta_C=\arccos\left[\frac1{2\sqrt{1-10^{-4}}}\right]
=59.99835^\circ.}
$$
The <electron> is not an isolated vacuum particle: it interacts with the dielectric polarization of the medium. Here the angle describes the <photon> direction relative to the trajectory; the conical <wavefront> surface has the complementary angle.
This angle is the classical or negligible-recoil result. At finite photon energy, energy-momentum bookkeeping in a nondispersive medium gives a recoil correction proportional to $\hbar\omega/E$; no photon frequency is supplied for such a correction. In a dispersive medium use $n(\omega)$ separately for each transparent frequency band.
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