# 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 ...**

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