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§7.2 Maxwell Equations the wave equation

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1 §7.2 Maxwell Equations the wave equation
Christopher Crawford PHY 417

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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