Proceedings Article | 8 February 2015
KEYWORDS: Magnetism, Particles, Electrons, Scattering, Electrodynamics, Maxwell's equations, Polarization, Electromagnetism, Chemical species, Ferromagnetics
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 𝜌𝜌free, electric current-density 𝑱𝑱free,
polarization 𝑷𝑷, and magnetization 𝑴𝑴. 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, 𝑬𝑬 and 𝑩𝑩, 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- and torque-densities exerted by the 𝑬𝑬 and 𝑯𝑯 fields on charge, current, polarization and magnetization.
The two formulations, while similar in many respects, have significant differences. For example, in the Lorentz
approach, the Poynting vector is 𝑺𝑺𝐿𝐿 = 𝜇𝜇0
−1𝑬𝑬 × 𝑩𝑩, and the linear and angular momentum densities of the EM field
are 𝓹𝓹𝐿𝐿 = 𝜀𝜀0𝑬𝑬 × 𝑩𝑩 and 𝓛𝓛𝐿𝐿 = 𝒓𝒓 × 𝓹𝓹𝐿𝐿, whereas in the Einstein-Laub formulation the corresponding entities are
𝑺𝑺𝐸𝐸𝐸𝐸= 𝑬𝑬 × 𝑯𝑯, 𝓹𝓹𝐸𝐸𝐸𝐸= 𝑬𝑬 × 𝑯𝑯⁄𝑐𝑐2, and 𝓛𝓛𝐸𝐸𝐸𝐸= 𝒓𝒓 × 𝓹𝓹𝐸𝐸𝐸𝐸. (Here 𝜇𝜇0 and 𝜀𝜀0 are the permeability and permittivity of free
space, 𝑐𝑐 is the speed of light in vacuum, 𝑩𝑩 = 𝜇𝜇0𝑯𝑯 + 𝑴𝑴, and 𝒓𝒓 is the position vector.) Such differences can be
reconciled by recognizing the need for the so-called hidden energy and hidden momentum associated with Amperian
current loops of the Lorentz formalism. (Hidden entities of the sort do not arise in the Einstein-Laub treatment of
magnetic dipoles.) Other differences arise from over-simplistic assumptions concerning the equivalence between
free charges and currents on the one hand, and their bound counterparts on the other. A more nuanced treatment of
EM force and torque densities exerted on polarization and magnetization in the Lorentz approach would help bridge
the gap that superficially separates the two formulations.
Atoms and molecules may collide with each other and, in general, material constituents can exchange energy,
momentum, and angular momentum via direct mechanical interactions. In the case of continuous media, elastic and
hydrodynamic stresses, phenomenological forces such as those related to exchange coupling in ferromagnets, etc.,
subject small volumes of materials to external forces and torques. Such matter-matter interactions, although
fundamentally EM in nature, are distinct from field-matter interactions in classical physics. Beyond the classical
regime, however, the dichotomy that distinguishes the EM field from EM sources gets blurred. An electron’s wavefunction
may overlap that of an atomic nucleus, thereby initiating a contact interaction between the magnetic dipole
moments of the two particles. Or a neutron passing through a ferromagnetic material may give rise to scattering
events involving overlaps between the wave-functions of the neutron and magnetic electrons. Such matter-matter
interactions exert equal and opposite forces and/or torques on the colliding particles, and their observable effects
often shed light on the nature of the particles involved. It is through such observations that the Amperian model of a
magnetic dipole has come to gain prominence over the Gilbertian model. In situations involving overlapping particle
wave-functions, it is imperative to take account of the particle-particle interaction energy when computing the
scattering amplitudes. As far as total force and total torque on a given volume of material are concerned, such
particle-particle interactions do not affect the outcome of calculations, since the mutual actions of the two
(overlapping) particles cancel each other out. Both Lorentz and Einstein-Laub formalisms thus yield the same total
force and total torque on a given volume—provided that hidden entities are properly removed. The Lorentz
formalism, with its roots in the Amperian current-loop model, correctly predicts the interaction energy between two
overlapping magnetic dipoles 𝒎𝒎1 and 𝒎𝒎2 as being proportional to −𝒎𝒎1 ∙ 𝒎𝒎2. In contrast, the Einstein-Laub
formalism, which is ignorant of such particle-particle interactions, needs to account for them separately.
