The electromagnetic wave emitted by, say, a radio or radar antenna is a coherent superposition of the individual radiation fields of a large number of electrons surging back and forth along the antenna. In principle, we could calculate the radiation field of such an antenna by performing a summation of the individual radiation fields of all the accelerated point charges on the antenna. For this purpose, we would need to keep track of the detailed motions of all the point charges. This is an impossible task, and it is much more convenient to describe the system of point charges in terms of the current density j _?. In this chapter, we will develop a formalism for calculating the electric and magnetic radiation fields directly from the current density. Our procedure will involve two steps: first use Eq. (13.3) to write the four-vector potential A generated by the given current density j _?, and then calculate the E and B fields from this potential. As we will see, any time dependence of the current will lead to the emission of radiation.

The formalism of this chapter is useful not only for finding the radiation emitted by an antenna but also for finding the radiation emitted by an atom or by a nucleus. We cannot calculate the radiation field of an atom or a nucleus from the detailed motion of the electrons, protons, or neutrons because the positions and velocities of these subatomic particles are not well...