Volume III — Classical Electrodynamics

Electrodynamics is the first great unification in physics, and it is worth pausing on how unlikely that was. Electricity and magnetism arrived as separate curiosities: amber that attracts dust, lodestones that point north. Light was a third thing entirely, the business of optics. That all three are the same phenomenon, governed by four equations that fit on a coffee mug, is not obvious from the outside. This volume is the story of how they fold into one.

We build it in the order the physics discloses itself. We begin with a single charge and the inverse-square law, which looks reassuringly like the gravity of Volume I (it is no accident that they share a form). From there the field acquires a potential, the potential obeys Laplace’s and Poisson’s equations, and we meet the computational heart of the volume: solving those equations on a grid, by relaxation, when no closed form exists. Then currents, magnetic fields, induction, and the moment where Maxwell adds one term for consistency and the equations suddenly predict light. By the end we reach the capstone, where the whole edifice is rewritten in a single relativistic object and electricity and magnetism are revealed as two views of one field, seen from different frames.

There is a name for what we are really learning here. In Landau and Lifshitz’s framing this whole subject is the classical theory of fields, and electrodynamics is its forerunning and richest classical example: a physics not of particles pulling on one another across a distance, but of a field that fills space and obeys local equations at every point. The volume assembles those equations one at a time, beginning with the very first, \(\nabla\cdot\mathbf E=\rho/\varepsilon_0\) (§3.3), which ties the field to its source right where the source sits. They accumulate until they close into Maxwell’s four, and the relativistic capstone then shows them to be a single law in spacetime. Keeping that arc in view, each new field equation a local law, the set of them the real content, is the best way to read the volume.

Computationally this volume asks something new of us. Mechanics was the world of the initial-value problem: give the state now, integrate forward. Electrostatics is the world of the boundary-value problem: fix the conditions on the walls, and solve for the field that fills the room. These need different tools, and we build them as they arise: numerical vector calculus introduced alongside the physics that needs it, relaxation solvers for the field equations, special functions for the symmetries that admit them, and the linear algebra and Fourier methods of Volume 0 fulfilled at last. The driven RLC circuit even turns out to be the damped, driven oscillator of §1.2 wearing different clothes (the same resonance, relabelled!).

We assume the vector calculus and the first encounter with E&M that a German second- or third-year has already had. The mathematics, as ever, is familiar; the novelty is the computational vantage point and the insistence that every field we draw be one we actually solved for. Work the notebooks in order: the volume is long by design, because electrodynamics rewards being seen whole rather than in pieces.

One note on the capstone. §3.12 belongs here as the summit of electrodynamics, but it leans on special relativity. If you are meeting relativity for the first time, you may prefer to read Volume IV’s special-relativity notebooks (§4.1–§4.5) first and return to §3.12 afterwards; we develop just enough relativity inline for it to stand on its own, but the fuller story is in Volume IV.

The volume does not stop at the capstone. What follows is longer than a coda and is best read as a second half, in which the fields we have built are put inside matter and the methods we have been quoting are finally derived.

The first pair lives where most fields actually do: inside matter. §3.13 works both halves of the story — bound charges summed into the polarized sphere’s exact interior field, a permittivity-jump relaxation solver benchmarked against the dielectric sphere, Clausius–Mossotti pricing argon’s permittivity from one atom, the magnetized cylinder unmasked as a solenoid, and the mean-field hysteresis that turns response into memory. The second takes the waves of §3.8 to the optics bench: §3.14 computes diffraction as the Fourier transform it is — Young’s fringes and the Airy resolution limit matched to their closed forms, the Fresnel number’s arc from shadow to far field, the Arago spot that decided the wave theory in 1818, and a lens revealed as an analog Fourier transformer.

Then the medium itself becomes the subject, in the order the physics discloses it. §3.15 asks what a wave does once it is inside something and what happens where two somethings meet: the driven oscillator of §1.2 relocated inside an atom and summed into a permittivity, absorption and the skin depth, the plasma frequency that decides why metals shine and why the ionosphere returns AM but not FM, and then Snell, Fresnel, Brewster and total internal reflection derived rather than quoted. §3.16 then removes the assumption almost every earlier notebook made without saying so — that a medium responds the same way in every direction. The permittivity becomes a tensor in its own right, a thing defined by how it transforms; the electric displacement stops being parallel to the field, exactly as angular momentum stopped being parallel to angular velocity in §2.6; and Neumann’s principle turns crystal symmetry into a component count, which makes the optical isotropy of cubic crystals a theorem rather than a coincidence. §3.17 then sends light through the crystal and watches what that costs: two waves for every direction, an ordinary index that ignores direction and an extraordinary one that does not, a ray that walks away from its own wave normal — which is the non-parallel D and E of the previous notebook, seen on an optical bench — and the wave plates that all of this makes possible.

The volume ends by going back for something it left behind. Three notebooks — §3.4, §3.5 and §3.9 — graded their solvers against series they quoted rather than derived, and each said so at the time. §3.18 derives them, and the separation constant turns out to be an eigenvalue of a Sturm–Liouville operator: one piece of mathematics that, run three times without modification, produces the box series in Cartesian coordinates, the multipole moments in spherical ones, and the cavity frequencies from three per-axis problems whose constants simply add. It closes where the method fails, on the Gibbs overshoot that will not go away.