PU-204 · Electromagnetism II
Starting from the static laws inherited from Electromagnetism I, this unit shows how Faraday induction and the displacement current force the four Maxwell equations into their final unified form, then extracts their physical content: conservation of energy and momentum, electromagnetic waves, radiation from accelerating charges, and propagation through matter. The arc closes by demonstrating that the whole structure is the unique relativistic field theory of a conserved current, recasting E and B as one antisymmetric tensor and revealing electromagnetism as a consequence of gauge symmetry and special relativity.
Lectures
| L01 | From Statics to Dynamics: The Road to Maxwell — |
| L02 | Electromagnetic Induction and Motional EMF |
| L03 | Faraday's Law in Differential Form |
| L04 | Inductance, Magnetic Energy and Transients |
| L05 | The Flaw in Ampere's Law and the Displacement Current |
| L06 | Maxwell's Equations Complete |
| L07 | Scalar and Vector Potentials, Gauge Freedom |
| L08 | Energy in the Electromagnetic Field: Poynting's Theorem |
| L09 | Momentum and the Maxwell Stress Tensor |
| L10 | The Wave Equation in Vacuum |
| L11 | Plane Waves, Transversality and Polarization |
| L12 | Energy and Momentum Carried by Waves |
| L13 | Boundary Conditions at Interfaces |
| L14 | Reflection, Refraction and the Fresnel Equations |
| L15 | Waves in Conductors and Skin Depth |
| L16 | Dispersion and the Lorentz Oscillator Model |
| L17 | Guided Waves and Cavities |
| L18 | Retarded Potentials and Causality |
| L19 | Jefimenko's Equations |
| L20 | Fields of a Moving Point Charge: Lienard-Wiechert |
| L21 | The Larmor Formula |
| L22 | Electric Dipole Radiation and Antennas |
| L23 | Radiation Reaction and Its Puzzles |
| L24 | Special Relativity Recap for Electromagnetism — |
| L25 | The Field Tensor and Covariant Maxwell Equations |
| L26 | How E and B Transform Between Frames |
| L27 | Magnetism as Relativity: The Moving Wire |
| L28 | The Field Lagrangian and Gauge Invariance |
| L29 | Synthesis: Electromagnetism as a Gauge Theory |
Derivations homed in this unit
Faraday's Law and the Flux Rule
Derive the integral and differential forms of Faraday's law, separating motional EMF from the induced curl of E and unifying them via the flux rule.
The Displacement Current from Charge Conservation
Show that Ampere's law is inconsistent with the continuity equation for time-varying charge and that adding Maxwell's displacement current uniquely repairs it.
Assembly of Maxwell's Equations
Collect Gauss, Gauss-for-magnetism, Faraday, and Ampere-Maxwell into the complete coupled set and state their integral/differential equivalence.
Potentials and Gauge Freedom
Introduce the scalar and vector potentials that automatically satisfy the homogeneous equations and derive the gauge transformations that leave E and B invariant.
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).
Structure of Monochromatic Plane Waves
Show plane-wave solutions are transverse with E, B, and k mutually orthogonal, in phase, with |E| = c|B|, and derive polarization states.
Poynting's Theorem and Field Energy
Derive the local conservation of electromagnetic energy, identifying the energy density and the Poynting flux from Maxwell's equations and the Lorentz force.
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.
Retarded Potentials and Jefimenko's Equations
Solve the inhomogeneous wave equations for the potentials in Lorenz gauge to obtain retarded solutions and the causal Jefimenko fields.
Fields of a Moving Point Charge
Derive the Lienard-Wiechert potentials and fields of an arbitrarily moving charge, separating the velocity (near) field from the acceleration (radiation) field.
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.
Electric Dipole Radiation
Compute the radiation fields and angular power distribution of an oscillating electric dipole and its total power.
Reflection, Refraction and Fresnel Equations
Apply the electromagnetic boundary conditions at a dielectric interface to derive Snell's law and the Fresnel reflection and transmission coefficients.
Waves in Conductors and Skin Depth
Solve Maxwell's equations in an ohmic medium to obtain the complex wavevector, attenuation, and skin depth of fields in conductors.
Dispersion from the Lorentz Oscillator Model
Model bound electrons as driven damped oscillators to derive the frequency-dependent permittivity, refractive index, and anomalous dispersion.
The Electromagnetic Field Tensor
Combine the potentials into a four-potential and the fields into the antisymmetric field tensor, writing Maxwell's equations in manifestly covariant form.
Relativistic Transformation of E and B
Derive how electric and magnetic fields mix under Lorentz boosts, showing magnetism as the relativistic consequence of moving charge.
Field Lagrangian and Gauge Invariance
Derive Maxwell's equations from the covariant field Lagrangian and show local gauge invariance forces the coupling to a conserved current.