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.