PU-305 · Classical Electrodynamics
Beginning from the empirical force laws of Coulomb and Ampère, the unit assembles Maxwell's equations as a consistent, relativistically-covariant field theory and extracts their physical consequences: energy and momentum transport, electromagnetic waves, and radiation from accelerating charges. The intellectual arc moves from static fields in matter, through the dynamics of coupled fields and currents, to the recognition that electromagnetism is intrinsically a special-relativistic theory whose gauge and Lorentz structure prefigures all of modern field theory.
Lectures
| L01 | The Empirical Foundations: Charge, Coulomb, and Fields — |
| L02 | Vector Calculus Machinery and the Delta Function |
| L03 | Gauss's Law in Integral and Differential Form |
| L04 | The Scalar Potential and Poisson's Equation |
| L05 | Energy in the Electrostatic Field |
| L06 | Boundary-Value Problems and Uniqueness |
| L07 | Method of Images and Green's Functions |
| L08 | The Multipole Expansion |
| L09 | Dielectrics: Polarization and the D Field |
| L10 | Magnetostatics: Biot–Savart and Ampère's Law |
| L11 | The Vector Potential |
| L12 | Magnetic Matter: Bound Currents and H |
| L13 | Faraday's Law and Electromagnetic Induction |
| L14 | Charge Conservation and the Displacement Current |
| L15 | Maxwell's Equations: The Complete System |
| L16 | Poynting's Theorem and Energy Flow |
| L17 | Field Momentum and the Maxwell Stress Tensor |
| L18 | Electromagnetic Waves in Vacuum |
| L19 | Waves in Matter and Dispersion |
| L20 | Reflection, Refraction, and the Fresnel Equations |
| L21 | Waveguides and Cavities |
| L22 | Potentials, Gauge Freedom, and the Lorenz Gauge |
| L23 | Retarded Potentials and Causality |
| L24 | Fields of a Moving Point Charge |
| L25 | The Larmor Formula and Dipole Radiation |
| L26 | Radiation Reaction and Its Puzzles |
| L27 | Special Relativity Recap for Electrodynamics — |
| L28 | The Covariant Formulation of Maxwell's Equations |
| L29 | Transformation of Fields Between Frames |
| L30 | The Field Lagrangian and Gauge Symmetry: A Bridge Forward |
Derivations homed in this unit
Laplacian of 1/r as a Delta Function
The identity nabla-squared(1/r) = -4-pi delta^3(r) is established, giving the distributional foundation for electrostatic Green's functions.
Gauss's Law from Coulomb's Law
The differential form div-E = rho/epsilon0 is derived from the inverse-square Coulomb field via the divergence theorem and the delta-function identity for 1/r.
Electrostatic Potential and Irrotational E
curl-E = 0 for static fields is shown, permitting E = -grad(phi) and reducing electrostatics to the Poisson equation with a uniqueness theorem for boundary data.
Energy Stored in the Electric Field
The total work to assemble a charge distribution is recast as an integral of epsilon0 E-squared/2 over all space, localizing energy in the field.
Multipole Expansion of the Potential
Expands the potential of a bounded charge distribution in inverse powers of distance, identifying monopole, dipole, and quadrupole terms via Legendre polynomials.
Bound Charge and the Displacement Field
The macroscopic field of polarized matter is shown equivalent to bound charges rho_b = -div P, motivating D = epsilon0 E + P and its Gauss law.
Ampère's Circuital Law from Biot–Savart
Derives that the curl of the magnetostatic field equals mu-zero times current density, equivalently the line integral equals the enclosed current.
Vector Potential and Bound Currents
B = curl-A is introduced from div-B = 0, and magnetized matter is reduced to bound currents J_b = curl-M, defining the H field.
Faraday's Law and the Induced Field
The flux rule is generalized to curl-E = -dB/dt, showing time-varying magnetic fields source a non-conservative electric field.
Displacement Current and Maxwell's Equations
Charge conservation via the continuity equation forces the displacement-current term in Ampère's law, completing the self-consistent Maxwell system.
Poynting's Theorem
Maxwell's equations yield local energy conservation with flux S = E x H/mu0, identifying electromagnetic energy transport.
Maxwell Stress Tensor and Field Momentum
Derive electromagnetic momentum conservation, expressing the force on charges as the divergence of the stress tensor plus the rate of change of field momentum.
Electromagnetic Wave Equation in Vacuum
Decouple Maxwell's equations in vacuum to obtain wave equations for E and B propagating at c = 1/sqrt(mu0 eps0).
Fresnel Equations at an Interface
Matching E and B boundary conditions at a dielectric interface yields the reflection and transmission amplitudes and Brewster's angle.
Retarded Potentials in Lorenz Gauge
The potentials are shown to satisfy inhomogeneous wave equations whose causal solutions are the retarded potentials.
Larmor Formula for Radiated Power
Extract the radiation field from an accelerating charge and integrate the Poynting flux to obtain the total power radiated as proportional to acceleration squared.
Covariant Formulation of Electrodynamics
Maxwell's equations are written as d-mu F^{mu-nu} = mu0 J^nu, demonstrating manifest Lorentz covariance and unifying E and B into one tensor.
Transformation of E and B under Boosts
The tensor F^{mu-nu} yields the mixing of electric and magnetic fields between inertial frames, showing they are one relativistic object.