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§7.2 Maxwell Equations the wave equation

§7.2 Maxwell Equations the wave equation. Christopher Crawford PHY 417 2015-03-27. Outline. 5 Wave Equations E&M waves: capacitive ‘tension’ vs. inductive ‘inertia’ Wave equations: generalization of Poisson’s eq. 2 Potentials, 1 Gauge, 2 Fields

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§7.2 Maxwell Equations the wave equation

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  1. §7.2 Maxwell Equationsthe wave equation Christopher Crawford PHY 417 2015-03-27

  2. Outline • 5 Wave Equations • E&M waves: capacitive ‘tension’ vs. inductive ‘inertia’ • Wave equations: generalization of Poisson’s eq.2 Potentials, 1 Gauge, 2 Fields • Solutions of Wave Equations – separation of variables • Helmholtz equation – separation of time • Spatial plane wave solutions – exponential, Bessel, Legendre • “Maxwell’s equations are local in frequency space!” • Constraints on fields • Dispersion & Impedance

  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: Poisson [Laplace] equation ELECTROMAGNETISM • Nontrivial 2nd derivative by switching paths (ε, μ)

  5. Wave Equation: potentials • Same steps as to get Poisson or Laplace equation • Beware of gauge-dependence of potential

  6. Wave equation: gauge

  7. Wave equation: fields

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

  9. Separation of time: Helmholtz Eq. • Dispersion relation

  10. Helmholtz equation: free wave • k2 = curvature of wave; k2=0[Laplacian]

  11. General Solutions • Cartesian • Cylindrical • Spherical

  12. Maxwell in frequency space • Separate time variable to obtain Helmholtz equation • Constraints on fields

  13. Energy and Power / Intensity • Energy density • Poynting vector • Product of complex amplitudes

  14. Boundary conditions • Same as always • Transmission/reflection: • Apply directly to field, not potentials

  15. Oblique angle of incidence

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