Electromagnetism: Maxwell Equations, Wave Propagation and Emission

Chapter 15

Emission of Radiation

The purpose of this last chapter is to study the emission of waves by time-dependent sources: moving charges and simple harmonic currents in antennas. As in the case of a sustained oscillator, the source is taken into account by a term f(r, t) on the right-hand side of the equation of propagation. In mechanics, knowledge of the forces and the equations of motion is not sufficient to determine the motion; the initial conditions are needed. In wave theory, knowledge of the equation of propagation and the sources f(r, t) at each point r and at any time t is not sufficient to determine the wave. We need the initial conditions, i.e. the values of u and its time derivative at the initial time t = 0 and at each point in space. If the medium is bounded, we need also the boundary conditions. In this chapter, we assume that the medium is infinite, linear, and isotropic of electric susceptibility ε and magnetic permeability μ, and that the source is restricted to a small region, so that the solution and its gradient vanish rapidly at large distances.

15.1. Retarded potentials and fields

We have seen that the fundamental laws of electromagnetic phenomena are Maxwell’s equations [9.12] to [9.15]. We have also seen in section 9.3 that it is often practical to use the vector potential A and the scalar potential V such that

The homogeneous Maxwell equations ∇.B ...