7 2 Maxwell Equations the wave equation Christopher

§ 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 2

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 3

Review: Poisson [Laplace] equation ELECTROMAGNETISM • Nontrivial 2 nd derivative by switching paths (ε, μ) 4

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

Wave equation: gauge 6

Wave equation: fields 7

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

Separation of time: Helmholtz Eq. • Dispersion relation 9

Helmholtz equation: free wave • k 2 = curvature of wave; k 2=0 [Laplacian] 10

General Solutions • Cartesian • Cylindrical • Spherical 11

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

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

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

Oblique angle of incidence 15