This appendix derives the Lorentz force and torque on localized charge and current distributions in terms of their multipole moments. Starting from the general Lorentz force density, it introduces a comoving reference point X(t), rewrites the charge and current in a particle frame, and expands the external fields about that point. This yields a multipole expansion of the total force in terms of the net charge, electric dipole moment, charge quadrupole tensor, and magnetic dipole moment, and identifies the leading contributions, q0 E(X,t), (p⋅∇)E, and ∇(μ⋅B). The appendix defines the electric and magnetic dipole moments and discusses when higher-order multipoles can be neglected for slowly varying fields. It then interprets the force law as an effective equation of motion for a localized system carrying internal structure. A final section derives the torques on electric and magnetic dipoles, τE = p × E and τB = μ × B, summarizing how localized systems respond translationally and rotationally to external electromagnetic fields.

This chapter develops a description of magnetic matter and shows how magnetizable media are incorporated into classical electromagnetism. It begins from the near field of localized magnetic dipoles, introduces the magnetic dipole moment and magnetization density (𝑟, 𝑡), and derives the near-field Maxwell equations in the presence of magnetic matter. Linear magnetic materials are then defined through constitutive relations among 𝐵, 𝐻, and 𝑀, with permeability, paramagnetism, and diamagnetism characterized in terms of microscopic dipole moments and their alignment with external fields. Next, the chapter derives magnetic energy density in linear materials and explains how inserting magnetic cores enhances inductance. It then analyzes forces and torques between current distributions, the energy required to assemble prescribed current configurations, and the relationship between these forces and the underlying dipole–dipole interaction. Finally, returning to Faraday’s law, it interprets inductors and their field energy in terms of an effective “electromagnetic mass” associated with the motion of charge carriers.

The dipole moment is the elementary magnetic quantity, and magnetization is its mesoscopic volume average. The primary magnetic field B is related to the auxiliary magnetic field H. Sources of magnetic field are electric currents and magnetized material. The field produced by a given distribution of magnetization can be calculated by integrating the dipole field due to each volume element, or by using the equivalent distributions of electric currents or magnetic charge. Internal, external and demagnetizing fields are distinguished. The internal field may be defined on a mesoscopic or a macroscopic scale. Magnetic forces and energies are related to magnetization and externally applied field, and force densities to derivatives of the thermodynamic free energy.