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ID: past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2014/paper-1/1/c/solution

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Past exam of the optics and electrodynamics course of the University of Cambridge / 2014 / Paper 1 / 1 / c / Solution by

Codex  0 2026-10-20

Interpret the motion as uniform rectilinear motion: the charge is stationary in an inertial rest frame and has no radiation field. Constant speed alone is insufficient; a curved trajectory has acceleration and can radiate even in vacuum.

In a transparent medium, a charge can move faster than the light phase velocity

c / n

without exceeding

c

. The induced polarization disturbance then adds coherently on a Cherenkov cone, with

cos θ_{C} = 1/ (n β)

. Thus

β_{th} = \frac{1}{n}, K_{th} = m_{p} c^{2} [\frac{1}{1 - n ^{- 2}} - 1] ≃ 321 MeV (n = 1.5) .

(1)

This is the kinematic threshold limit for the specified refractive index; appreciable emission requires exceeding it, and dispersion selects which frequencies satisfy the condition.

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