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§7.2 Maxwell Equations the wave equation
Christopher Crawford PHY 417
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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
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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
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Review: Poisson [Laplace] equation
ELECTROMAGNETISM Nontrivial 2nd derivative by switching paths (ε, μ)
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Wave Equation: potentials
Same steps as to get Poisson or Laplace equation Beware of gauge-dependence of potential
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Wave equation: gauge
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Wave equation: fields
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Wave equation: summary
d’Alembert operator (4-d version of Laplacian)
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Separation of time: Helmholtz Eq.
Dispersion relation
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Helmholtz equation: free wave
k2 = curvature of wave; k2=0 [Laplacian]
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General Solutions Cartesian Cylindrical Spherical
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Maxwell in frequency space
Separate time variable to obtain Helmholtz equation Constraints on fields
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Energy and Power / Intensity
Energy density Poynting vector Product of complex amplitudes
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Boundary conditions Same as always Transmission/reflection:
Apply directly to field, not potentials
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Oblique angle of incidence
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