Radiation in a Plasma. I. Čerenkov Effect

Abstract

We start with the linearized plasma equations containing an isotropic pressure term, plus extra source terms ${\mathrm{J}}^{s}$ and ${\ensuremath{\rho}}^{s}$ in the Maxwell equations. The fields of (${\mathrm{J}}^{s}, {\ensuremath{\rho}}^{s}$) can be decomposed into two modes. The electromagnetic (EM) mode has all the magnetic field and no charge accumulation; it is the ordinary EM field of (${\mathrm{J}}^{s}, {\ensuremath{\rho}}^{s}$) in a dispersive medium of relative dielectric constant ${\ensuremath{\epsilon}}_{r}=1\ensuremath{-}{(\frac{{\ensuremath{\omega}}_{p}}{\ensuremath{\omega}})}^{2}$. The plasma (P) mode has all the charge accumulation and no magnetic field; at great distances from the source, it becomes a longitudinal (radial) plasma wave with the usual dispersion relation for plane plasma waves. Various potentials for the EM and P modes are given by the inhomogeneous Klein-Gordon equation. The fields of a uniformly moving charged particle are found by a Lorentz transformation. When $(\frac{u}{{v}_{0}})l1$ ($u=\mathrm{particle}\mathrm{velocity}$, ${v}_{0}=\mathrm{rms}\mathrm{thermal}\mathrm{velocity}$), the EM and P fields are exponentially screened outside oblate spheroids foreshortened in the direction of motion. When $(\frac{u}{{v}_{0}})g1$, the $P$ field exists only within the Mach (\ifmmode \check{C}\else \v{C}\fi{}erenkov) cone trailing the particle. The frequency and angular spectra of the \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation are found, and the total radiated energy is found by assuming an arbitrary high-frequency cutoff due to Landau damping. The expression for total radiated energy agrees with that given by Pines and Bohm, except for the logarithmic terms.