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When charged particles travel through matter faster than light in that medium (v > c/n), they emit Cherenkov radiation. The shock-like radiation forms a cone with angle θ_c = arccos(1/(βn)). Provides evidence of superluminal particle motion relative to medium. Used in particle detectors.
From your study of electromagnetic waves in media, you know that light slows down inside a material: the phase velocity becomes v_ph = c/n, where n > 1 is the refractive index. Water has n ≈ 1.33, so light travels through water at about 75% of its vacuum speed. Special relativity forbids any object from exceeding c, but it says nothing about exceeding c/n — a particle can travel through water faster than light travels through water while still moving slower than c. When this happens, Cherenkov radiation is emitted.
The mechanism is best understood by analogy with a sonic boom. A supersonic aircraft moves faster than sound can propagate outward from it. The pressure waves pile up into a coherent shock front — a Mach cone — that trails the aircraft at a fixed half-angle determined by the ratio of the aircraft's speed to the sound speed. Replace "sound" with "light in the medium" and "aircraft" with "charged particle," and the physics is identical. As a fast-charged particle passes through a medium, it polarizes the atoms along its path. When v < c/n, the electromagnetic disturbances radiated from each point along the path are spherical waves that spread outward faster than the particle moves, and they largely cancel by destructive interference in the forward direction. When v > c/n, the particle outpaces its own electromagnetic wake; the disturbances can no longer cancel, and they constructively interfere along a cone that builds up coherently behind the particle.
The geometry is exact: the Cherenkov angle θ_c satisfies cos θ_c = c/(nv) = 1/(βn), where β = v/c. At threshold (v = c/n, β = 1/n), cos θ_c = 1 and θ_c = 0 — the cone is infinitely narrow (no radiation). As the particle speeds up, θ_c opens toward 90°. For highly relativistic particles (β → 1), the maximum angle is cos⁻¹(1/n): in water (n = 1.33), this gives θ_c ≈ 41°. Measuring θ_c tells you v, and since you can often measure the momentum p independently, you can determine the mass m — which is why Cherenkov detectors are powerful particle identification tools in high-energy physics experiments. The characteristic blue glow visible in water-cooled nuclear reactors is Cherenkov radiation from fast electrons (beta particles) produced in the reactor core.