modern electrodynamics (andrew zangwill)

Distilled notes on Zangwill’s amazing Modern Electrodynamics — key results, proof sketches, solution strategies, and physical insight, written for review and reference at the graduate level.

How to Use These Notes

Not as a textbook replacement. These notes assume you’ve worked through the material once (which I highly recommend) and want a fast way back in — or that you’re solving problems and need the key tricks without re-reading 30 pages.

Collapsible sections (▶) contain proof sketches and worked examples. Each problem includes a brief description so the notes are self-contained without the textbook open.

Chapters

Chapter 1 — Mathematical Preliminaries

Vector calculus in Cartesian, spherical, and cylindrical coordinates. Einstein summation, Kronecker delta, Levi-Civita identities.

Integral theorems — divergence, Stokes, Green’s identities and their variants.

Delta functions — representations, 3D generalizations, ${\nabla }^2(1/r)=-4\pi \delta (\mathbf{r})$. Fourier series and transforms. Time-averaging theorem.

Helmholtz theorem — existence and uniqueness of the transverse/longitudinal decomposition, finite-volume variants. Tensors, pseudovectors, Lagrange multipliers.

7 end-of-chapter problems with solution sketches.

Chapter 2 — The Maxwell Equations

Electric charge and current — conservation, continuity equation, the convective derivative subtlety with point charges.

Building up Maxwell’s equations — from Coulomb and Biot–Savart through Faraday’s law to the displacement current. Differential and integral forms. Magnetism as a relativistic effect of electrostatics.

Matter — spatial averaging, macroscopic fields, auxiliary fields $\mathbf{D}$ and $\mathbf{H}$, constitutive relations, matching conditions.

Limits of validity — QED corrections, modified Coulomb/Gauss laws, magnetic monopoles, photon mass.

5 end-of-chapter problems with solution sketches.

Chapter 3 — Electrostatics

Coulomb’s law and superposition — scalar potential, Poisson’s equation, conservative nature of the electrostatic field.

Gauss’s law — symmetry-based calculations, solid angle, field line topology. Earnshaw’s theorem and its consequences for stability.

Energy — potential energy, total energy, interaction energy, Green’s reciprocity, ionization potential of metal clusters. The $\lambda$-scaling trick.

Electric stress tensor — force as a surface integral, force on charged surfaces.

14 end-of-chapter problems with solution sketches.

Chapter 4 — Electric Multipoles

Primitive Cartesian expansion — Taylor expansion of $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$, monopole, dipole, and quadrupole moments. Convergence in $R/r$.

Electric dipole — point dipole limit, singular charge density ${\rho }_D=-\mathbf{p}\cdot \nabla \delta$, the contact term in the field. Force, torque, and energy in external fields. Dipole layers, potential discontinuity, and the solid-angle formula.

Electric quadrupole — primitive vs. traceless tensors, principal axes, Maxwell’s construction. Nuclear fission as a quadrupole instability (Bohr–Wheeler).

Spherical multipole expansion — Legendre polynomials, spherical harmonics, addition theorem. Interior and exterior moments, azimuthal simplifications. Traceless Cartesian vs. spherical: $2\ell +1$ components and irreducibility.

14 end-of-chapter problems with solution sketches.

Chapter 5 — Conducting Matter

Thomson’s theorem — energy minimization as the origin of electrostatic induction. Polarizability of a conducting sphere.

Surface charge and edge singularities — matching conditions at conductor surfaces, the conducting disk via ellipsoidal limits, square-root divergence.

Screening and shielding — cavity field vanishing, exterior field independence from interior charge position.

Capacitance — self-capacitance, the capacitance matrix, Maxwell inequalities, coefficients of potential, Green’s reciprocity applications. Coulomb blockade in quantum dots.

Forces on conductors — electrostatic pressure, energy methods at constant charge vs. constant potential, the Legendre transform, battery argument.

Real conductors — screening length, Debye–Hückel and Thomas–Fermi models.

14 end-of-chapter problems with solution sketches.

Chapter 6 — Dielectric Matter

Polarization — free vs. bound charge, $\mathbf{P}(\mathbf{r})$ and its non-uniqueness, why Lorentz averaging fails, the modern (Resta–Vanderbilt) theory via polarization currents.

Fields from polarization — point-dipole representation, Poisson’s formula for uniform $\mathbf{P}$, the uniformly polarized sphere.

Displacement field $\mathbf{D}$ — Gauss’s law for free charge, matching conditions, constitutive relations, simple dielectrics ($\kappa$, $\chi$, $ϵ$). Screening of embedded charges. Parallel-plate capacitor examples.

Clausius–Mossotti — local field, polarizability, the conductor limit $\kappa \to \infty$.

Energy — total energy $U_E$, polarization energy $\Delta U_E$, fixed-charge vs. fixed-potential. Classical quark confinement model.

Forces — Coulomb force on isolated samples, why it fails for sub-volumes, Helmholtz force, Maxwell stress tensor for dielectrics.

17 end-of-chapter problems with solution sketches.

Chapter 7 — Laplace's Equation

Uniqueness theorems — why guessing works; Dirichlet, Neumann, and conductor charge specification.

Separation of variables in Cartesian, spherical, and cylindrical coordinates. Legendre polynomials, Bessel functions, Fourier–Bessel series. The generating function trick for azimuthal symmetry.

2D methods — complex potential, conformal mapping (Joukowski, conducting strip, fringing fields), wedge singularity and edge divergences.

Variational methods — Thomson’s theorem, energy minimization, trial functions.

22 end-of-chapter problems with solution sketches.

Chapter 8 — Poisson's Equation

Method of images — planar, spherical, and cylindrical boundaries. Point charge near grounded plane (energy subtlety), dielectric interfaces (fill-all-space trick), multiple images between parallel planes (exponential screening). Spherical images, method of inversion, force on neutral conductors (attraction then repulsion). Line charge and conducting/dielectric cylinders, vector potential trick.

Green function method — Dirichlet and Neumann Green functions, reciprocity, the magic rule. Physical interpretation as grounded-conductor potential. Force on a charge inside an oddly shaped shell.

Computing $G_D$ — eigenfunction expansion (Sturm–Liouville completeness), direct integration (cylindrical and spherical representations, jump conditions, Bessel/Wronskian machinery), method of splitting (exterior cylinder, force on charge near grounded tube).

Complex potentials — line charges and conformal mapping, wire chamber, capacitance of wire array.

Poisson–Boltzmann equation — mobile charges, algebraic vs. exponential screening.

18 end-of-chapter problems with solution sketches.

Chapter 9 — Steady Current

Drude model and Ohm’s law — drift velocity, statistical friction, conductivity $\sigma =n{q}^2\tau /m$. Child–Langmuir law for space-charge-limited transport.

Laplace’s equation in ohmic media — uniform conductors, matching conditions, resistance, the $RC={ϵ}_0/\sigma$ relation and its applications (contact resistance, four-point probe).

Joule heating — model-independent $\mathrm{d}W/\mathrm{d}t=\int \mathbf{j}\cdot \mathbf{E}$, variational minimum-dissipation principle (Ohm’s law emerges without being assumed).

EMF and Kirchhoff’s laws — fictitious field ${\mathbf{E}}^{\prime }$, circuit relations, power budget. Surface charges on bends.

Current source distributions — Poisson equation for $f(\mathbf{r})$, point sources, Neumann Green function, surface potential encoding of internal sources.

Drift-diffusion — Fick’s law, Einstein relation connecting transport and screening, cell resting potential.

20 end-of-chapter problems with solution sketches.

Chapter 10 — Magnetostatics

$\mathbf{B}$-field foundations — $\nabla \cdot \mathbf{B}=0$ (no monopoles), Thomson’s theorem (no local maxima of $|\mathbf{B}|$), Biot–Savart from Helmholtz, irrotational currents produce no field.

Ampère’s law and symmetry — formal reflection/rotation arguments for axial vectors (line, sheet, toroidal solenoid), matching conditions.

Magnetic scalar potential — Laplace’s equation machinery in current-free regions, multi-valuedness and cut surfaces, solid-angle representation.

Vector potential — gauge freedom, Coulomb gauge, matching conditions, worked examples (cylindrical wire, ring, MRI shielding).

Field-line topology — lines that never close, magnetic reconnection, chaos via Hamiltonian analogy, helicity as linking invariant.

18 end-of-chapter problems with solution sketches.

TODO: Fix all broken cross-references (ch. N → chN, missing anchors, links to unwritten chapters ch11+ that will need content added once those chapters exist).

TODO: Add errata stuff somewhere.

TODO: Implement condenser for reference mode.

TODO: Can I change the reading time upon condenser acting?

TODO: Maybe no links to equations as I want each chapter (and to some extent a section) to be self-sufficient.

TODO: Think about condensing tricks to make it useful for even fastest recap or something similar.

TODO: Add a one-line technique tag under each problem heading (e.g. “Method of images, Bessel identity, multiple reflections.”) so problems are searchable by method via ctrl+F.

TODO: Add Key Results + Trick Bank. Plan:

1. Per-chapter [!info] Key Results block (recap-visible) near the top of each chapter — the 5–15 most important formulas with one-line labels, no derivations. Chapter-local, co-located with the narrative.
2. Chapter-level “when to use what” decision guide ([!info], recap-visible) for chapters with multiple competing methods (ch7, ch8, ch10).
3. A separate tricks.md page collecting the cross-cutting moves that recur across chapters — λ-scaling, Green’s reciprocity, fill-all-space, multiply-and-integrate, method of inversion, vector-potential trick for $\mathbf{D}$, shifted sines on $[-a,a]$, $y=\ln \tan (\theta /2)$ for sphere Laplace, etc. Organized by category (integration, ansatz, symmetry, energy) with one-line descriptions and canonical-example links back to chapters.

TODO: Add “Common Mistakes / Pitfalls” sections — short [!warning] callouts in the trickiest spots (e.g. confusing total vs. interaction vs. self-energy, forgetting B is axial under reflection, factor-of-2 in image energies). Some of these already exist scattered as [!warning]; do a pass and unify.

TODO: Add completely honest disclaimer about the extent to which AI was used.