§7.2 Maxwell Equations the wave equation Christopher Crawford PHY 311 2014-05-02

Final Exam Based on 5 formulations of electromagnetism Derivative chain – gauge, potentials, fields, sources Structure of and relations between different formulations Field calculation methods organized around formulations Cumulative – uniform weighting through whole semester Will be 50% longer than midterm exams Similar problems as midterms Essay question – structure of EM fields / media Proof – relation between formulations Integration – Coulomb / Biot-Savart / Potential Integral – Gauss / Ampère [or modified versions] Boundary value problem – see examples Components – capacitor, resistor, inductor

Outline Review – electromagnetic potential & displacement current propagate electromagnetic waves Capacitive ‘tension’ vs. inductive ‘inertia’ Unification of E and B – filling in the cracks Derivative chain – different representations of fields Wave equation and solution – Green’s fn. and eigenfn’s

Electromagnetic Waves Sloshing back and forth between electric and magnetic energy Interplay: Faraday’s EMF  Maxwell’s displacement current Displacement current (like a spring) – converts E into B EMF induction (like a mass) – converts B into E Two material constants  two wave properties

Review: Two separate formulations ELECTROSTATICS Coulomb’s law MAGNETOSTATICS Ampère’s law E+B: Faraday’s law; b) rho + J: conservation of charge; c) space + time

Review: One unified formulation ELECTROMAGNETISM Faraday’s law stitches the two formulations together in space and time Previous hint: continuity equation

Unification of E and B Projections of electromagnetic field in space and time That is the reason for the twisted symmetry in field equations

Unification of D and H Summary

Wave equation: potentials

Wave equation: gauge

Wave equation: fields

Wave equation: summary d’Alembert operator (4-d version of Laplacian)

Homogenous solution Separate time variable to obtain Helmholtz equation General solution for wave Boundary Value Problems

Particular solution Green’s function of d’Alembertian Wikipedia: Green’s functions