Classical Electrodynamics

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Classical Electrodynamics

• Jingbo Zhang
• Harbin Institute of Technology

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Chapter 2Electromagnetic Waves
Section 1 Wave Equations
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Review

• Maxwells Equations

These four coupled first order partial
differential equations can be rewritten as two
uncoupled second order partial equations.
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1 Wave Equation for E

• To derive the wave equation for E in the volume
  with no net electric charge and no electromotive
  force,

we have a second order partial equations for E
uncoupled with B,
which is the homogeneous wave equation.
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2 Wave Equation for B

• In the same way, we can get the wave equation for
  B, under the same conditions on last page,

Noticed, it has exactly the same form as the wave
equation for E,
Such the similarity could be seen as the
electromagnetic duality in another way.
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• In vacuum, s 0, the wave equations for electric
  and magnetic fields have the form,

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3 The time-independent wave equation

• If the electric field is a time-harmonic wave, we
  can write it in a Fourier component Ansatz,

thus, we can get the time-independent wave
equation for electric field,
introduce the relaxation time of the medium,
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• For the time-harmonic wave equation,

in the limit of long relaxation time, we
get time-independent propagating wave equation,
in the limit of short case, it tends
to time-independent diffusion equation,
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Homework 2.1

• Derive the wave equation for the fields E and B
  from the Maxwells equations in vacuum in which
  the electric charge and current vanished.

• Derive the wave equation for the fields E
  described by the electromagnetodynamic equations
  under the assumption of vanishing the net
  electric and magnetic charge densities and in
  absence of electromotive and magnetomotive
  forces.