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Understanding Wave Equations in Electromagnetism: Insights from Maxwell's Equations

This outline presents an in-depth exploration of wave equations within the context of electromagnetism, specifically focusing on Maxwell's equations. It covers essential topics such as the generalization of Poisson's equation, solutions through variable separation, and the Helmholtz equation. Key concepts include the interplay of electric and magnetic fields, dispersion relations, and energy transfer via the Poynting vector. Additionally, it discusses boundary conditions and the implications of gauge dependence, providing a comprehensive understanding of wave propagation and electromagnetic behavior.

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Understanding Wave Equations in Electromagnetism: Insights from Maxwell's Equations

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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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