= Solution
At $p=600\ \mathrm{MeV}/c$, the <velocities> are $\beta=p/\sqrt{p^2+m^2c^2}$, giving approximately $1.0000$, $0.98485$ and $0.53874$ for <electrons>, <muons> and <protons>. The glass threshold is $\beta>1/1.55=0.64516$. Thus the first two produce <Cherenkov radiation>, with internal angles
$$
\theta_e\simeq49.82^\circ,\qquad\theta_\mu\simeq49.07^\circ,
$$
whereas a primary <proton> at this <momentum> is below threshold.
If an optical arrangement transports those cone angles without changing them, a plane $L=2\ \mathrm m$ downstream would intersect rings of radius
$$
R=L\tan\theta_C=L\sqrt{n^2\beta^2-1},\qquad\boxed{R_e\simeq2.369\ \mathrm m,\quad R_\mu\simeq2.307\ \mathrm m}.
$$
These are \b[conditional geometric estimates], not the ray trace for a flat slab in air. The $0.1\ \mathrm m$ emission depth alone spreads the radius by order $0.12\ \mathrm m$, already larger than the approximately $0.062\ \mathrm m$ separation. The slab is one <radiation length>: <electrons> lose appreciable <energy> by <bremsstrahlung> and begin an <electromagnetic shower>, producing a broader and less clean pattern. <Muons> mainly ionize and retain a comparatively clean cone, with some <Multiple Coulomb scattering>. A below-threshold primary <proton> gives no direct cone, although energetic <delta electrons> and other secondary interactions can produce light.
There is an essential optical qualification. For a normal beam through a parallel-faced glass slab followed by air, the internal critical angle is $\theta_{\rm crit}=\arcsin(1/1.55)\simeq40.18^\circ$. Both emission angles exceed it. <Snell's law> would require $\sin\theta_{\rm air}=1.55\sin\theta_C>1$, so the ideal photons undergo <total internal reflection> at the downstream face and do not form those direct downstream rings. Optical coupling, a prism or different surface geometry must be specified to extract them. The stated material parameters alone do not define a unique detected pattern; this flat-slab geometry is a concrete counterexample to silently assuming straight transmitted rays.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2016-cherenkov-glass-interface.png]
{title=Total internal reflection prevents direct downstream rings from a flat glass-air slab}
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