1. Classical Electrodynamics Jingbo Zhang Harbin Institute of Technology

2. Chapter 2 Electromagnetic Waves Section 1 Wave Equations

3. Jingbo Zhang Section 1 Wave EquationsChapter 2 Sept 1, 2004 Classical Electrodynamics Review  Maxwell’s Equations These four coupled first order partial differential equations can be rewritten as two uncoupled second order partial equations.

4. Jingbo Zhang Section 1 Wave EquationsChapter 2 Sept 1, 2004 Classical Electrodynamics 1 Wave Equation for E  To derive the wave equation for E in the volume with no net electric charge and no electromotive force, which is the homogeneous wave equation. we have a second order partial equations for E uncoupled with B,

5. Jingbo Zhang Section 1 Wave EquationsChapter 2 Sept 1, 2004 Classical Electrodynamics 2 Wave Equation for B  In the same way, we can get the wave equation for B, under the same conditions on last page, Such the similarity could be seen as the electromagnetic duality in another way. Noticed, it has exactly the same form as the wave equation for E,

6. Jingbo Zhang Section 1 Wave EquationsChapter 2 Sept 1, 2004 Classical Electrodynamics  In vacuum, σ = 0, the wave equations for electric and magnetic fields have the form,

7. Jingbo Zhang Section 1 Wave EquationsChapter 2 Sept 1, 2004 Classical Electrodynamics 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,

8. Jingbo Zhang Section 1 Wave EquationsChapter 2 Sept 1, 2004 Classical Electrodynamics  For the time-harmonic wave equation, in the limit of short case, it tends to time-independent diffusion equation, in the limit of long relaxation time, we get time-independent propagating wave equation,

9. Jingbo Zhang Section 1 Wave EquationsChapter 2 Sept 1, 2004 Classical Electrodynamics Homework 2.1  Derive the wave equation for the fields E and B from the Maxwell’s 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.