How a charge conserving alternative to Maxwells displacement current entails a Darwin like approximation to the solutions of Maxwells equations

Abstract

Though sufficient for local conservation of charge, Maxwells displacement current is not necessary. An alternative to the Ampere-Maxwell equation is exhibited and the alternatives electric and magnetic fields and scalar and vector potentials are expressed in terms of the charge and current densities. The magnetic field is shown to satisfy the Biot_Savart Law. The electric field is shown to be the sum of the gradient of a scalar potential and the time derivative of a vector potential which is different from but just as tractable as the simplest vector potential that yields the Biot_Savart Law The alternative describes a theory in which action is instantaneous and so may provide a good approximation to Maxwells equations where and when the finite speed of light can be neglected. The result recalls the Darwin approximation which arose from the study classical charged point particles to order (v/c)2 in the Lagrangian. Unlike Darwin, 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.