This chapter develops a macroscopic description of electric dipole matter and shows how dielectrics are incorporated into classical electromagnetism. Starting from the near field of localized electric dipoles, it coarse-grains to a polarization density P(r,t), derives Maxwell’s equations in dielectric form, and connects the macroscopic polarization field to underlying dipole moments and local fields. Boundary value problems with piecewise constant polarization and permittivity are formulated using Green’s functions and solved for canonical planar and spherical geometries. Specializing to linear, isotropic dielectrics leads to constitutive relations, susceptibility and permittivity, and to the Clausius–Mossotti relation linking microscopic polarizability to macroscopic response. The chapter then analyzes energy storage in dielectrics, capacitance in simple geometries with and without dielectric fillings, and the forces exerted on polarizable media and on individual dipoles. A brief note on spherical harmonics introduces tools needed for multipole expansions and for evaluating fields of polarized spheres.

This appendix analyzes the forces on and energies of dielectric materials, building on the multipole Lorentz force law. Starting from the local force on moving charges, electric dipoles, and magnetic dipoles, it coarse-grains to macroscopic polarization and magnetization fields and derives formulas for the net force on a dielectric body in external fields. Using the constitutive relation, P = (ϵ – ϵ0)E, it discusses averaged and nonlocal response and identifies simple limits in which forces can be written directly in terms of P, M, and the applied fields. The second part tackles the subtle question of electrostatic energy in dielectrics. This is done by separating the work needed to bring in free charge from the work of rearranging bound polarization charge, and shows that the electromagnetic contribution can be written as δW = ∫E⋅δD dV. For linear media, this reduces to the energy density 1/2 E⋅D.