1 Maxwell’s Equations 1.1 Maxwell’s Equations Maxwell’s equations describe all (classical) electromagnetic phenomena: ∇ × E = − ∂ B ∂t ∇ × H = J + ∂ D ∂t ∇ · D = ρ ∇ · B = 0 (Maxwell’s equations) (1.1.1) The first is Faraday’s law of induction , the second is Amp` ere’s law as amended by Maxwell to include the displacement current ∂ D /∂t , the third and fourth are Gauss’ laws for the electric and magnetic fields. The displacement current term ∂ D /∂t in Amp` ere’s law is essential in predicting the existence of propagating electromagnetic waves. Its role in establishing charge conser- vation is discussed in Sec. 1.7. Eqs. (1.1.1) are in SI units. The quantities E and H are the electric and magnetic field intensities and are measured in units of [volt/m] and [ampere/m], respectively. The quantities D and B are the electric and magnetic ﬂux densities and are in units of [coulomb/m 2 ] and [weber/m 2 ], or [tesla]. D is also called the electric displacement , and B , the magnetic induction . The quantities ρ and J are the volume charge density and electric current density (charge ﬂux) of any external charges (that is, not including any induced polarization charges and currents.) They are measured in units of [coulomb/m 3 ] and [ampere/m 2 ]. The right-hand side of the fourth equation is zero because there are no magnetic mono- pole charges. Eqs. (1.3.17)–(1.3.19) display the induced polarization terms explicitly. The charge and current densities ρ, J may be thought of as the sources of the electro- magnetic fields. For wave propagation problems, these densities are localized in space; for example, they are restricted to ﬂow on an antenna. The generated electric and mag- netic fields are radiated away from these sources and can propagate to large distances to 2 1. Maxwell’s Equations the receiving antennas. Away from the sources, that is, in source-free regions of space, Maxwell’s equations take the simpler form: ∇ × E = − ∂ B ∂t ∇ × H = ∂ D ∂t ∇ · D = 0 ∇ · B = 0 (source-free Maxwell’s equations) (1.1.2) The qualitative mechanism by which Maxwell’s equations give rise to propagating electromagnetic fields is shown in the figure below. For example, a time-varying current J on a linear antenna generates a circulating and time-varying magnetic field H , which through Faraday’s law generates a circulating electric field E , which through Amp` ere’s law generates a magnetic field, and so on. The cross-linked electric and magnetic fields propagate away from the current source. A more precise discussion of the fields radiated by a localized current distribution is given in Chap. 15. 1.2 Lorentz Force The force on a charge q moving with velocity v in the presence of an electric and mag- netic field E , B is called the Lorentz force and is given by: F = q( E + v × B ) (Lorentz force) (1.2.1) Newton’s equation of motion is (for non-relativistic speeds): m d v dt = F = q( E + v × B ) (1.2.2) where m is the mass of the charge. The force F will increase the kinetic energy of the
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- Energy, Electric charge, Permittivity, Dielectric, unit volume