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

摘要

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 ρ_(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- and torque-densities exerted by the E and H 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 S_L = μ_0~(-1) E × B, and the linear and angular momentum densities of the EM field are p_L = ε_0E × B and L_L = r × p_L, whereas in the Einstein-Laub formulation the corresponding entities are S_(EL) = E × H, p_(EL) = E × H/c~2, and L_(EL) = r × p_(EL). (Here μ_0 and ε_0 are the permeability and permittivity of free space, c is the speed of light in vacuum, B = μ_0H + M, and r 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 wave-function 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 m_1 and m_2 as being proportional to -m_1·m_2. In contrast, the Einstein-Laub formalism, which is ignorant of such particle-particle interactions, needs to account for them separately.