Electric and Magnetic Dipoles in the Lorentz and Einstein-Laub Formulations of Classical Electrodynamics

TL;DR

This paper compares the Lorentz and Einstein-Laub formulations of classical electrodynamics, focusing on how they describe electric and magnetic dipoles and their interactions with electromagnetic fields.

Contribution

It clarifies the differences between the Lorentz and Einstein-Laub models in representing dipoles and their force and torque calculations without relying on specific dipole models.

Findings

Maxwell's equations do not require explicit dipole models.

Lorentz formulation models dipoles as charge pairs or current loops.

Einstein-Laub provides direct force and torque calculations on dipoles.

Abstract

The classical theory of electrodynamics cannot explain the existence and structure of electric and magnetic dipoles, yet it incorporates such dipoles into its fundamental equations, simply by postulating their existence and properties, just as it postulates the existence and properties of electric charges and currents. Maxwell's macroscopic equations are mathematically exact and self-consistent differential equations that relate the electromagnetic (EM) field to its sources, namely, electric charge-density $\rho_{free}$, electric current-density $J_{free}$, polarization P, and magnetization M. At the level of Maxwell's macroscopic equations, there is no need for models of electric and magnetic dipoles. For example, whether a magnetic dipole is an Amperian current-loop or a Gilbertian pair of north and south magnetic monopoles has no effect on the solution of Maxwell's equations. Electromagnetic fields carry energy as well as linear and angular momenta, which they can exchange with material media - the seat of the sources of the EM field - thereby exerting force and torque on these media. In the Lorentz formulation of classical electrodynamics, the electric and magnetic fields, E and B, exert forces and torques on electric charge and current distributions. An electric dipole is then modeled as a pair of electric charges on a stick (or spring), and a magnetic dipole is modeled as an Amperian current loop, so that the Lorentz force law can be applied to the corresponding (bound) charges and (bound) currents of these dipoles. In contrast, the Einstein-Laub formulation circumvents the need for specific models of the dipoles by simply providing a recipe for calculating the force-density and torque-density exerted by the E and H fields on charge, current, polarization and magnetization.