Maxwell’s equations are essential mathematical tools in the analysis of
electromagnetics and antennas problems. In this chapter, Maxwell's equation for general
time-varying electromagnetic fields are derived from the basic laws of
electromagnetics, and presented in integral and point forms. The special cases for timeharmonic
and static fields are obtained from the general equations. The vector potential
concept that has been introduced in Chapter 5 for static fields, is generalized in this
chapter for time-varying fields. The electric and magnetic vector potentials are
important quantities in determining the electromagnetic fields radiated from electric and
magnetic current sources. By solving Helmholtz equations, general formulations for the
electric and magnetic vector potentials are presented in terms of electric and magnetic
current sources respectively. The chapter includes also the application of multi-pole
expansion technique to obtain the vector potential for some time-varying fields
problems, derivation of boundary conditions for time-varying fields, derivation of
Poynting vector, and discussion of electromagnetic power flow. The topics of the
chapter are supported by numerous illustrative examples and figures in addition to
solved problems and homework problems at the end of the chapter.
Keywords: Ampere’s law, boundary conditions, displacement current, electric
field intensity, electric flux density, electric vector potential, Faraday’s law,
Gauss’ laws, magnetic field intensity, magnetic flux density, magnetic vector
potential, Maxwell’s equations, Poynting vector, time-varying fields.
