An Electrostatic Wave

All electrostatic fields E (i.e., ones with no time dependence) can be derived from a scalar potential V (E = −∇V ) and hence obey ∇ × E = 0. The latter condition is sometimes considered to be a requirement for electrostatic fields. Show, however, that there can exist time-dependent electric fields for which ∇ × E = 0, which have been given the name “electrostatic waves”. In particular, show that a plane wave with electric field E parallel to the wave vector k (a longitudinal wave) can exist in a medium with no time-dependent magnetic field if the electric displacement D is zero. This cannot occur in an ordinary dielectric medium, but can happen in a plasma. (Time-independent electric and magnetic fields could, of course, be superimposed on the wave field.) Compare the potentials for the “electrostatic wave” in the Coulomb and Lorentz gauges. Discuss energy density and flow for such a wave. Deduce the frequency ω of the longitudinal wave in a hot, collisionless plasma that propagates transversely to a uniform external magnetic field B0 in terms of the (electron) cyclotron frequency,