Read an Excerpt
Theory of Electromagnetic Wave Propagation
By Charles Herach Papas Dover Publications, Inc.
Copyright © 1988 Charles Herach Papas
All rights reserved.
ISBN: 978-0-486-14514-3
CHAPTER 1
The electromagnetic field
In this introductory chapter some basic relations and concepts of the classic electromagnetic field are briefly reviewed for the sake of easy reference and to make clear the significance of the symbols.
1.1 Maxwell's Equations in Simple Media
In the mks, or Giorgi, system of units, which we shall use throughout this book, Maxwell's field equations are
[nabla] x E(r, t) = - [[partial derivative]/[partial derivative]t] B(r,t) (1)
[nabla] x H(r,t) = J(r,t) + [[partial derivative]/[partial derivative]t] D(r,t) (2)
[nabla] x B(r,t) = 0 (3)
[nabla] x D(r,t) = ρ(r,t) (4)
where E(r, t) = electric field intensity vector, volts per meter
H(r, t) = magnetic field intensity vector, amperes per meter
D(r,t) = electric displacement vector, coulombs per meter2
B(r,t) = magnetic induction vector, webers per meter2
J(r,t) = current-density vector, amperes per meter2
ρ(r,t) = volume density of charge, coulombs per meter3
r = position vector, meters
t = time, seconds
The equation of continuity
[nabla] x J(r,t) = - [[partial derivative]/[partial derivative]t] ρ(r,t) (5)
which expresses the conservation of charge is a corollary of Eq. (4) and the divergence of Eq. (2).
The quantities E(r,t) and B(r,t) are defined in a given frame of reference by the density of force f(r,t) in newtons per meter3 acting on the charge and current density in accord with the Lorentz force equation
f(r,t) = ρ(r,t)E(r,t) + J(r,t) x B(r,t) (6)
In turn D(r,t) and H(r,t) are related respectively to E(r,t) and B(r,t) by constitutive parameters which characterize the electromagnetic nature of the material medium involved. For a homogeneous isotropic linear medium, viz., a "simple" medium, the constitutive relations are
D(r,t) = εE(r,t) (7)
H(r,t) = [1/μ] B(r,t) (8)
where the constitutive parameters ε in farads per meter and μ in henrys per meter are respectively the dielectric constant and the permeability of the medium.
In simple media, Maxwell's equations reduce to
[nabla] x E(r,t) = - μ [[partial derivative]/[partial derivative]t] H(r,t) (9)
[nabla] x H(r,t) = J(r,t) + ε [[partial derivative]/[partial derivative]t] E(r,t) (10)
[nabla] x H(r,t) = 0 (11)
[nabla] x E(r,t) = [1/ε] ρ(r,t) (12)
The curl of Eq. (9) taken simultaneously with Eq. (10) leads to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Alternatively, the curl of Eq. (10) with the aid of Eq. (9) yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
The vector wave equations (13) and (14) serve to determine E(r,t) and H(r,t) respectively when the source quantity J(r,t) is specified and when the field quantities are required to satisfy certain prescribed boundary and radiation conditions. Thus it is seen that in the case of simple media, Maxweir's equations determine the electromagnetic field when the current density J(r,t) is a given quantity. Moreover, this is true for any linear medium, i.e., any medium for which the relations connecting B(r,t) to H(r,t) and D(r,t) to E(r,t) are linear, be it anisotropic, inhomogeneous, or both.
To form a complete field theory an additional relation connecting J(r,t) to the field quantities is necessary. If J(r,t) is purely an ohmic conduction current in a medium of conductivity s in mhos per meter, then Ohm's law
J(r,t) = σ E(r,t) (15)
applies and provides the necessary relation. On the other hand, if J(r,t) is purely a convection current density, given by
J(r,t) = ρ(r,t) v(r,t) (16)
where v(r,t) is the velocity of the charge density in meters per second, the necessary relation is one that connects the velocity with the field. To find such a connection in the case where the convection current is made up of charge carriers in motion (discrete case), we must calculate the total force F(r,t) acting on a charge carrier by first integrating the force density f(r,t) throughout the volume occupied by the carrier, i.e.,
F(r,t) = ∫f(r + r',t)dV' = q[E(r,t) + v(r,t) x B(r,t)] (17)
where q is the total charge, and then equating this force to the force of inertia in accord with Newton's law of motion
F(r,t) = [d/dt] [mv(r,t)] (18)
where m is the mass of the charge carrier in kilograms. In the case where the convection current is a charged fluid in motion (continuous case), the force density f(r,t) is entered directly into the equation of motion of the fluid.
Because Maxwell's equations in simple media form a linear system, no generality is lost by considering the "monochromatic" or "steady" state, in which all quantities are simply periodic in time. Indeed, by Fourier's theorem, any linear field of arbitrary time dependence can be synthesized from a knowledge of the monochromatic field. To reduce the system to the monochromatic state we choose exp (-iωt) for the time dependence and adopt the convention
C(r,t) = Re {Cω(r)e-iωt]} (15)
where C(r,t) is any real function of space and time, Cω(r) is the concomitant complex function of position (sometimes called a "phasor"), which depends parametrically on the frequency f(= ω/2π) in cycles per second, and Re is shorthand for "real part of." Application of this convention to the quantities entering the field equations (1) through (4) yields the monochromatic form of Maxwell's equations:
[nabla] x Eω(r) = iωBω(r) (20)
[nabla] x Hω(r) = Jω(r) - iωDω(r) (21)
[nabla] x Hω(r) = 0 (22)
[nabla] x Dω(r) = ρω(r) (23)
In a similar manner the monochromatic form of the equation of continuity
[nabla] x Jω(r) = iωρω(r) (24)
is derived from Eq. (5).
The divergence of Eq. (20) yields Eq. (22), and the divergence of Eq. (21) in conjunction with Eq. (24) leads to Eq. (23). We infer from this that of the four monochromatic Maxwell equations only the two curl relations are independent. Since there are only two independent vectorial equations, viz., Eqs. (20) and (21), for the determination of the five vectorial quantities Eω(r), Hω(r), Dω(r), Bω(r), and Jω(r), the monochromatic Maxwell equations form an under determined system of first-order differential equations. If the system is to be made determinate, linear constitutive relations involving the constitutive parameters must be invoked. One way of doing this is first to assume that in a given medium the linear relations Bω(r) = αHω(r), Dω(r) = βEω(r), and Jω(r) = γEω(r) are valid, then to note that with this assumption the system is determinate and possesses solutions involving the unknown constants α, β, and γ, and finally to choose the values of these constants so that the mathematical solutions agree with the observations of experiment. These appropriately chosen values are said to be the monochromatic permeability μω, dielectric constant εω, and conductivity σω of the medium. Another way of defining the constitutive parameters is to resort to the microscopic point of view, according to which the entire system consists of free and bound charges interacting with the two vector fields Eω(r) and Bω(r) only. For simple media the constitutive relations are
Bω(r) = μω Hω(r) (25)
Dω(r) = εω Eω(r) (26)
Jω(r) = σω Eω(r) (27)
In media showing microscopic inertial or relaxation effects, one or more of these parameters may be complex frequency-dependent quantities.
For the sake of notational simplicity, in most of what follows we shall drop the subscript ω and omit the argument r in the monochromatic case, and we shall suppress the argument r in the time-dependent case. For example, E(t) will mean E(r,t) and E will mean Eω(r). Accordingly, the monochromatic form of Maxwell's equations in simple media is
[nabla] x E = iωμH (28)
[nabla] x H = J - iωεE (29)
[nabla] x H = 0 (30)
[nabla] x E = [1/ε] ρ (31)
1.2 Duality
In a region free of current (J = 0), Maxwell's equations possess a certain duality in E and H. By this we mean that if two new vectors E' and H' are defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)
then as a consequence of Maxwell's equations (source-free)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)
it follows that E' and H' likewise satisfy Maxwell's equations (source-free)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)
and thereby constitute an electromagnetic field E', H' which is the "dual" of the original field.
This duality can be extended to regions containing current by employing the mathematical artifice of magnetic charge and magnetic current. In such regions Maxwell's equations are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)
and under the transformation (32) they become
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)
Formally these relations are Maxwell's equations for an electromagnetic field E', H' produced by the "magnetic current" [- or +] [square root of (μ/ε)] J and the "magnetic charge" ± [square root of (μ/ε)]. These considerations suggest that complete duality is achieved by generalizing Maxwell's equations as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)
where Jm and ρm are the magnetic current and charge densities. Indeed, under the duality transformation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)
Thus to every electromagnetic field E, H produced by electric current J there is a dual field H', E' produced by a fictive magnetic current J'm.
1.3 Boundary Conditions
The electromagnetic field at a point on one side of a smooth interface between two simple media, 1 and 2, is related to the field at the neighboring point on the opposite side of the interface by boundary conditions which are direct consequences of Maxwell's equations.
We denote by n a unit vector which is normal to the interface and directed from medium 1 into medium 2, and we distinguish quantities in medium 1 from those in medium 2 by labeling them with the subscripts 1 and 2 respectively. From an application of Gauss' divergence theorem to Maxwell's divergence equations, [nabla] · B = ρm and [nabla] · D = ρ, it follows that the normal components of B and D are respectively discontinuous by an amount equal to the magnetic surface-charge density ηm and the electric surface-charge density η in coulombs per meter:
n x (B2 - B1) = ηm n x (D2 - D1) = η (40)
From an application of Stokes' theorem to Maxwell's curl equations, [nabla] × E = -Jm + iωμH and [nabla] - H = J -iωεE, it follows that the tangential components of E and H are respectively discontinuous by an amount equal to the magnetic surface-current density Km and the electric surface-current density K in amperes per meter:
n x (E2 - E1) = -Km n x (H2 - H1) = K (41)
In these relations Km and K are magnetic and electric "current sheets" carrying charge densities η m and η respectively. Such current sheets are mathematical abstractions which can be simulated by limiting forms of electromagnetic objects. For example, if medium 1 is a perfect conductor and medium 2 a perfect dielectric, i.e., if σ1 = ∞ and σ1 = 0, then all the field vectors in medium 1 as well as ηm and Km vanish identically and the boundary conditions reduce to
n x B2 = 0 n x D2 = η n x E2 = n x H2 = K (42)
A surface having these boundary conditions is said to be an "electric wall." By duality a surface displaying the boundary conditions
n x D2 = 0 n x B2 = ηm n x E2 = - Km n x H2 = (43)
is said to be a "magnetic wall."
At sharp edges the field vectors may become infinite. However, the order of this singularity is restricted by the Bouwkamp-Meixner edge condition. According to this condition, the energy density must be integrable over any finite domain even if this domain happens to include field singularities, i.e., the energy in any finite region of space must be finite. For example, when applied to a perfectly conducting sharp edge, this condition states that the singular components of the electric and magnetic vectors are of the order σ-½, where d is the distance from the edge, whereas the parallel components are always finite.
1.4 The Field Potentials and Antipotentials
According to Helmholtz's partition theorem any well-behaved vector field can be split into an irrotational part and a solenoidal part, or, equivalently, a vector field is determined by a knowledge of its curl and divergence. To partition an electromagnetic field generated by a current J and a charge σ, we recall Maxwell's equations
[nabla] x H = J -iωD (44)
[nabla] x E = iωB (45)
[nabla] x D = ρ (46)
[nabla] x B = 0 (47)
and the constitutive relations for a simple medium
D = εE (48)
B = μH (49)
From the solenoidal nature of B, which is displayed by Eq. (47), it follows that B is derivable from a magnetic vector potential A:
B = [nabla] x A (50)
This relation involves only the curl of A and leaves free the divergence of A. That is, [nabla] · A is not restricted and may be chosen arbitrarily to suit the needs of calculation. Inserting Eq. (50) into Eq. (45) we see that E - iωA is irrotational and hence derivable from a scalar electric potential φ:
E = -[nabla]φ + iωA (51)
This expression does not necessarily constitute a complete partition of the electric field because A itself may possess both irrotational and solenoidal parts. Only when A is purely solenoidal is the electric field completely partitioned into an irrotational part [nabla]φ and a solenoidal part A. The magnetic field need not be partitioned intentionally because it is always purely solenoidal.
By virtue of their form, expressions (50) and (51) satisfy the two Maxwell equations (45) and (47). But in addition they must also satisfy the other two Maxwell equations, which, with the aid of the constitutive relations (48) and (49), become
[1/μ] [nabla] x B = J - iωεE and [nabla] x E = ρ/ε (52)
When relations (50) and (51) are substituted into these equations, the following simultaneous differential equations are obtained, relating φ and A to the source quantities J and ρ:
[nabla]2 φ = iω[nabla] x A = ρ/ε (53)
[nabla]2A + k2A = -μJ + [nabla]([nabla] x A - iωεμφ) (54)
where k2 = ω 2με. Here [nabla] x A is not yet specified and may be chosen to suit our convenience. Clearly a prudent choice is one that uncouples the equations, i.e., reduces the system to an equation involving φ alone and an equation involving A alone. Accordingly, we choose [nabla] x A = iωεμφ or [nabla] x A = 0.
(Continues...)
Excerpted from Theory of Electromagnetic Wave Propagation by Charles Herach Papas. Copyright © 1988 Charles Herach Papas. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
