The Three Quasistatic Limits of the Maxwell Equations

TL;DR

This paper explores the three distinct quasistatic limits of Maxwell's equations, providing a unified perturbation framework, deriving their integral solutions, and clarifying their physical applicability in macroscopic media.

Contribution

It generalizes the orderings of Maxwell's equations to include all three quasistatic limits using perturbation expansion and gauge conditions.

Findings

Derived integral solutions for the three limits.

Unified framework for magneto-, electro-, and electromagnetic-quasi-static limits.

Clarified the physical regions where each limit applies.

Abstract

It is shown that the Galilean limit (V << c, or L/T <<c)) of the Maxwell equations admits three different limits: the magneto-quasi-static, electro-quasi-static, and electromagnetic-quasi-static limits, in addition to the two obvious static limits. The first two quasi-static limits have been previously identified as Galilean Electromagnetics, while the latter is also known as the Darwin approximation. Using a perturbation expansion, a generalization of Rappetti and Rousseaux [Applied Numerical Mathematics, 79, 92] orders the vacuum Maxwell equations and obtains all three limits. To order the equations, the dimensionless version of the Maxwell equations are derived using a modification of Jackson's review of EM unit systems [Jackson, Classical Electrodynamics, Wiley, 1999, 3rd ed.] The perturbation expansion is repeated for the potential form of the Maxwell equations to emphasize the importance of gauge conditions. The integral solutions of the potentials are derived for the three limits, and the generalized Coulomb and Biot-Savart equations are derived from these solutions. It is shown that although the forms are the same as the static equations, the quasi-static forms of the Maxwell equations are recovered. The induction term is recovered when the time derivative of the vector potential is kept. The displacement current is recovered when the Lorenz gauge is used. The equivalence of this approach and Jackson's derivation [Amer. J. of Phys., 70, 917 (2002)] of the Darwin approximation is shown. The regions of applicability of the quasi-static forms of the Maxwell equations are discussed in terms of macroscopic media.