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Maxwell's Equations

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Maxwell's Equations 2 1. Maxwell’s Equations the receiving antennas. Away from the sources, that is, in source-free regions of space, 1 Maxwell’s equations take the simpler form: ∂B ∇∇×E =− Maxwell’s Equations ∂t ∂D ∇∇×H = ∂t (source-free Maxwell’s equations) (1.1.2) ∇∇·D = 0 ∇∇·B = 0 The qualitative mechanism by which Maxwell’s equations give rise to propagating electromagnetic fields is shown in the figure below. 1.1 Maxwell’s Equations Maxwell’s equations describe all (classical) electromagnetic phenomena: ∂B ∇∇×E =− ∂t 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 ∂D ∇∇×H = J + electric field E, which through Amp`ere’s law generates a magnetic field, and so on. The ∂t (Maxwell’s equations) (1.1.1) cross-linked electric and magnetic fields propagate away from the current source. A ∇∇·D = ρ more precise discussion of the fields radiated by a localized current distribution is given in Chap. 15. ∇∇·B = 0 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 1.2 Lorentz Force for the electric and magnetic fields. The force on a charge q moving with velocity v in the presence of an electric and mag- The displacement current term ∂D/∂t in Amp`ere’s law is essential in predicting the netic field E, B is called the Lorentz force and is given by: existence of propagating electromagnetic waves. Its role in establishing charge conser- vation is discussed in Sec. 1.7. F = q(E + v × B) (Lorentz force) (1.2.1) 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. Newton’s equation of motion is (for non-relativistic speeds): The quantities D and B are the electric and magnetic flux densities and are in units of 2 2 dv [coulomb/m ] and [weber/m ], or [tesla]. D is also called the electric displacement, and m = F = q(E + v × B) (1.2.2) B, the magnetic induction. dt The quantities ρ and J are the volume charge density and electric current density where m is the mass of the charge. The force F will increase the kinetic energy of the (charge flux) of any external charges (that is, not including any induced polarization charge at a rate that is equal to the rate of work done by the Lorentz force on the charge, charges and currents.) They are measured in units of [coulomb/m3] and [ampere/m2]. that is, v · F. Indeed, the time-derivative of the kinetic energy is: The right-hand side of the fourth equation is zero because there are no magnetic mono- 1 dWkin dv pole charges. Eqs. (1.3.17)–(1.3.19) display the induced polarization terms explicitly. Wkin = m v · v ⇒ = m v · = v · F = q v · E (1.2.3) The charge and current densities ρ, J may be thought of as the sources of the electro- 2 dt dt magnetic fields. For wave propagation problems, these densities are localized in space; We note that only the electric force contributes to the increase of the kinetic energy— for example, they are restricted to flow on an antenna. The generated electric and mag- the magnetic force remains perpendicular to v, that is, v · (v × B)= 0. netic fields are radiated away from these sources and can propagate to large distances to 1.3. Constitutive Relations 3 4 1. Maxwell’s Equations Volume charge and current distributions ρ, J are also subjected to forces in the The next simplest form of the constitutive relations is for simple homogeneous presence of fields. The Lorentz force per unit volume acting on ρ, J is given by: isotropic dielectric and for magnetic materials: f = ρE + J × B (Lorentz force per unit volume) (1.2.4) D = E (1.3.4) where f is measured in units of [newton/m3]. If J arises from the motion of charges B = μH within the distribution ρ, then J = ρv (as explained in Sec. 1.6.) In this case, These are typically valid at low frequencies. The permittivity and permeability μ f = ρ(E + v × B) (1.2.5) are related to the electric and magnetic susceptibilities of the material as follows: By analogy with Eq. (1.2.3), the quantity v · f = ρ v · E = J · E represents the power = 0(1 + χ) per unit volume of the forces acting on the moving charges, that is, the power expended (1.3.5) by (or lost from) the fields and converted into kinetic energy of the charges, or heat. It μ = μ0(1 + χm) has units of [watts/m3]. We will denote it by: The susceptibilities χ, χm are measures of the electric and magnetic polarization dPloss properties of the material. For example, we have for the electric flux density: = J · E (ohmic power losses per unit volume) (1.2.6) dV D = E = 0(1 + χ)E = 0E + 0χE = 0E + P (1.3.6) In Sec. 1.8, we discuss its role in the conservation of energy. We will find that elec- tromagnetic energy flowing into a region will partially increase the stored energy in that where the quantity P = 0χE represents the dielectric polarization of the material, that region and partially dissipate into heat according to Eq. (1.2.6). is, the average electric dipole moment per unit volume. In a magnetic material, we have B = μ0(H + M)= μ0(H + χmH)= μ0(1 + χm)H = μH (1.3.7) 1.3 Constitutive Relations where M = χmH is the magnetization, that is, the average magnetic moment per unit The electric and magnetic flux densities D, B are related to the field intensities E, H via volume. The speed of light in the material and the characteristic impedance are: the so-called constitutive relations, whose precise form depends on the material in which 1 μ the fields exist. In vacuum, they take their simplest form: c = √ ,η= (1.3.8) μ D = 0E The relative permittivity, permeability and refractive index of a material are defined by: (1.3.1) B = μ0H μ √ rel = = 1 + χ, μrel = = 1 + χm ,n= relμrel (1.3.9) 0 μ0 where 0,μ0 are the permittivity and permeability of vacuum, with numerical values: 2 so that n = relμrel. Using the definition of Eq. (1.3.8), we may relate the speed of light −12 and impedance of the material to the corresponding vacuum values: 0 = 8.854 × 10 farad/m (1.3.2) −7 1 1 c0 c0 μ0 = 4π × 10 henry/m c = √ = √ = √ = μ μ00relμrel relμrel n (1.3.10) The units for 0 and μ0 are the units of the ratios D/E and B/H, that is, μ μ μ μ μ n η = = 0 rel = η rel = η rel = η 0 0 n 0 coulomb/m2 coulomb farad weber/m2 weber henry 0 rel rel rel = = , = = · · volt/m volt m m ampere/m ampere m m For a non-magnetic material, we have μ = μ0, or, μrel = 1, and the impedance becomes simply η = η0/n, a relationship that we will use extensively in this book. From the two quantities 0,μ0, we can define two other physical constants, namely, More generally, constitutive relations may be inhomogeneous, anisotropic, nonlin- the speed of light and the characteristic impedance of vacuum: ear, frequency dependent (dispersive), or all of the above. In inhomogeneous materials, the permittivity depends on the location within the material: 1 8 μ0 c0 = √ = 3 × 10 m/sec ,η0 = = 377 ohm (1.3.3) μ00 0 D(r,t)= (r)E(r,t) 1.3. Constitutive Relations 5 6 1. Maxwell’s Equations In anisotropic materials, depends on the x, y, z direction and the constitutive rela- In Sections 1.10–1.15, we discuss simple models of (ω) for dielectrics, conductors, tions may be written component-wise in matrix (or tensor) form: and plasmas, and clarify the nature of Ohm’s law: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ D E J = σE (Ohm’s law) (1.3.15) ⎢ x ⎥ ⎢ xx xy xz ⎥ ⎢ x ⎥ ⎣ Dy ⎦ = ⎣ yx yy yz ⎦ ⎣ Ey ⎦ (1.3.11) In Sec. 1.17, we discuss the Kramers-Kronig dispersion relations, which are a direct Dz zx zy zz Ez consequence of the causality of the time-domain dielectric response function (t). Anisotropy is an inherent property of the atomic/molecular structure of the dielec- One major consequence of material dispersion is pulse spreading, that is, the pro- tric. It may also be caused by the application of external fields. For example, conductors gressive widening of a pulse as it propagates through such a material. This effect limits and plasmas in the presence of a constant magnetic field—such as the ionosphere in the the data rate at which pulses can be transmitted. There are other types of dispersion, presence of the Earth’s magnetic field—become anisotropic (see for example, Problem such as intermodal dispersion in which several modes may propagate simultaneously, 1.10 on the Hall effect.) or waveguide dispersion introduced by the confining walls of a waveguide. In nonlinear materials, may depend on the magnitude E of the applied electric field There exist materials that are both nonlinear and dispersive that support certain in the form: types of non-linear waves called solitons, in which the spreading effect of dispersion is = = + + 2 +··· D (E)E , where (E) 2E 3E (1.3.12) exactly canceled by the nonlinearity.
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