Solving Maxwell’s Equations

We solve Maxwell’s macroscopic equations under the assumption that the sources of the electromagnetic fields are fully specified throughout space and time. Charge, current, polarization, and magnetization are thus assumed to have predetermined distributions as functions of the space-time coordinates (r, t). In this type of analysis, any action by the fields on the sources will be irrelevant, in the same way that the action on the sources by any other force-be it mechanical, chemical, nuclear, or gravitational-need not be taken into consideration. It is true, of course, that one or more of the above forces could be responsible for the presumed behavior of the sources. However, insofar as the fields are concerned, since the spatio-temporal profiles of the sources are already specified, knowledge of the forces would not provide any additional information. In this chapter, we use Fourier transformation to express each source as a superposition of plane-waves. Maxwell’s equations then associate each planewave with other plane-waves representing the electromagnetic fields. Inverse Fourier transformation then enables us to express the electric and magnetic fields as functions of the space-time coordinates.