Equilibrium solutions for the electric field and charge density of a continuously distributed charge

The present paper focuses on electromagnetic fluid dynamics following Physics of Fluids 37(8), 086123, 086124, 2025, that showed how an electric charge can be treated as an inviscid compressible fluid. Consequently, an electrostatic problem for stationary equilibrium of a continuously distributed charge is formulated in the form of finding both the electric field as well as the charge distribution. This is distinct from typical corresponding problems dealing with finding either the electric field or the charge distribution given one of them as input data. The adopted approach converts the typical linear problem into a nonlinear one. Maxwell and mass continuity equations representing the conservation of charge and mass, as well as the evolution of the electromagnetic fields are kinematic equations to be complemented by a momentum equation (inviscid and compressible Navier–Stokes equations) governing the dynamics of motion of the distributed charge. It is shown that while two types of possible stationary equilibria, one trivial (i.e. zero value of charge density and electric field) and the other one non-trivial (i.e. non-zero values of charge density and electric field) are possible, only the non-trivial one can materialize in reality. Oscillations may and do occur around the non-trivial equilibrium, but yield non-realistic results if they occur around the trivial equilibrium. The latter is the major reason for rejecting the trivial equilibrium and adopting the non-trivial one.