A model for freely propagating transient electrical discharges, such as lightning and punch-through arcs, is developed in this paper. We describe the electromagnetic fields by Maxwell’s equations and we represent the interaction of electric fields with the medium to produce current by ∂J/∂t=ω2(E−E*Ĵ)/4π, where ω and E* are parameters characteristic of the medium, J≡current density, and Ĵ≡J/‖J‖. We illustrate the properties of this model for small-diameter, guided, cylindrically symmetric discharges. Analytic, numerical, and approximate solutions are given for special cases. The model describes, in a new and comprehensive fashion, certain macroscopic discharge properties, such as threshold behavior, quenching and reignition, path tortuosity, discharge termination with nonzero charge density remaining along the discharge path, and other experimentally observed discharge phenomena. Fields, current densities, and charge densities are quantitatively determined from given boundary and initial conditions. We suggest that many macroscopic discharge properties are properly explained by the model as electromagnetic phenomena, and we discuss extensions of the model to include chemistry, principally ionization and recombination.
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