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7.2 Maxwell Equationsthe wave equation

• Christopher Crawford
• PHY 417
• 2015-03-27

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Outline

• 5 Wave Equations
• EM waves capacitive tension vs. inductive
  inertia
• Wave equations generalization of Poissons 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
• Maxwells 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 Faradays EMF ? Maxwells
  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 (e,
  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

• dAlembert 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 k20 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