On a charge conserving alternative to Maxwell’s displacement current

Abstract

Though sufficient for local conservation of charge, we show that Maxwell’s displacement current is not necessary. An alternative to the Ampere–Maxwell equation is exhibited and the alternative’s electric and magnetic fields and scalar and vector potentials are expressed in terms of the charge and current densities. The alternative describes a theory in which action is instantaneous and so may provide a good approximation to Maxwell’s equations where and when the finite speed of light can be neglected. The result is reminiscent of the Darwin approximation which arose from the study classical charged point particles to order (v/c)2 in the Lagrangian. Unlike Darwin’s, this approach does not depend on the constitution of the electric current. Instead, this approach grows from a straightforward revision of the Ampere equation that enforces the local conservation of charge.