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§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 1. 2. 3. 4. 5. 6. – 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 2 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 3 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 4 Review: Two separate formulations ELECTROSTATICS •Coulomb’s law MAGNETOSTATICS •Ampère’s law 5 Review: One unified formulation ELECTROMAGNETISM • Faraday’s law stitches the two formulations together in space and time • Previous hint: continuity equation 6 Unification of E and B • Projections of electromagnetic field in space and time – That is the reason for the twisted symmetry in field equations 7 Unification of D and H Summary 8 Wave equation: potentials 9 Wave equation: gauge 10 Wave equation: fields 11 Wave equation: summary • d’Alembert operator (4-d version of Laplacian) 12 Homogenous solution • Separate time variable to obtain Helmholtz equation • General solution for wave Boundary Value Problems 13 Particular solution • Green’s function of d’Alembertian Wikipedia: Green’s functions