§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 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

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

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

Wave equation: gauge

Wave equation: fields

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

Separation of time: Helmholtz Eq. Dispersion relation

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

General Solutions Cartesian Cylindrical Spherical

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

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

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

Oblique angle of incidence