Classical Electromagnetic Radiation, Third Edition
592
Classical Electromagnetic Radiation, Third Edition
592eBook
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Overview
Starting with a brief review of static electricity and magnetism, the treatment advances to examinations of multipole fields, the equations of Laplace and Poisson, dynamic electromagnetism, electromagnetic waves, reflection and refraction, and waveguides. Subsequent chapters explore retarded potentials and fields and radiation by charged particles; antennas; classical electron theory; interference and coherence; scalar diffraction theory and the Fraunhofer limit; Fresnel diffraction and the transition to geometrical optics; and relativistic electrodynamics. A basic knowledge of vector calculus and Fourier analysis is assumed, and several helpful appendices supplement the text. An extensive Solutions Manual is also available.
Product Details
| ISBN-13: | 9780486283425 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/22/2013 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 592 |
| File size: | 27 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Jerry Marion (1929–52) published the original edition of this volume in 1965 under the imprint of Academic Press.
Read an Excerpt
Classical Electromagnetic Radiation
By Mark A. Heald, Jerry B. Marion
Dover Publications, Inc.
Copyright © 2012 Mark A. Heald and Adelia MarionAll rights reserved.
ISBN: 978-0-486-28342-5
CHAPTER 1
Fundamentals of Static Electromagnetism
In this book we shall be concerned mainly with radiation phenomena associated with electromagnetic fields. We shall study the generation of electromagnetic waves, the propagation of these waves in space, and their interaction with matter of various forms. The fundamental equations that govern all of these processes are Maxwell's equations. These are a set of partial differential equations that describe the space and time behavior of the electromagnetic field vectors. By way of review, we shall first examine briefly the static and steady-state properties of the electromagnetic field. In Chapters 2 and 3 we shall discuss two topics that are usually not considered at great length in introductory accounts of electromagnetism—multipole analysis and solutions of Laplace's equation—because these subjects are of importance in radiation phenomena. In Chapter 4 we shall treat time-varying electromagnetic fields and arrive at the four partial differential equations—Maxwell's equations—which give a full description of the classical behavior of electromagnetic fields. The remainder of the book is concerned primarily with radiation problems.
To describe electromagnetism, we use four vector fields:
E [equivalent to] electric field or electric intensity (statvolt/cm)
D [equivalent to] dielectric displacement field or simply displacement (statvolt/cm)
B [equivalent to] magnetic field or magnetic induction or magnetic flux density (gauss)
H [equivalent to] magnetic intensity or magnetic field (oersted)
The fields E and B are the fundamental fields because they represent the microscopic space-time averages within matter. The fields D and H are auxiliary fields, which involve the electromagnetic properties of matter and prove to be useful in treating macroscopic problems. Surprisingly, the verbal names of these four fields are less well standardized in the literature than the symbolsE, D, B, and H.
* * *
1.1 UNITS
To discuss electromagnetic phenomena, it is necessary to adopt one of many possible systems of units. Most likely the reader is already familiar with the Système International (SI), the system most popular for practical or engineering problems. The SI units (ampere, volt, tesla, etc.) are the calibration of choice for laboratory instruments. But in the study of the interaction of electromagnetic fields with the fundamental constituents of matter (atoms, molecules, electrons, etc.), the Gaussian system of units is commonly preferred and is used in this book. The Gaussian system is an amalgam of two 19th-century systems: electric quantities are measured in electrostatic units (esu) and magnetic quantities are measured in electromagnetic units (emu).
To read the physics literature, one must be bilingual in these two systems. Each provides distinctive insights into the physics. The Gaussian system retains popularity because factors of the speed of light c appear explicitly and appropriately. SI's constants [member of]0 and μ0, while hiding the factors of c, have the perverse virtue of forcing the user to face up to the distinction between the fundamental and auxiliary fields (e.g., B vs. H). SI is the system of legal metrology. It regards the ampere as a fourth fundamental unit along with the meter, kilogram, and second (MKSA). This convention makes dimensional analysis easier than with the Gaussian units, in which the electromagnetic dimensions are fractional powers of the centimeter, gram, and second (CGS—see Problem 1-1). The systems also differ in the choice of where to place factors of 4π, a controversial esthetic called rationalization.
A conversion table between Gaussian and SI units is given in Appendix D. Appendix E summarizes the fundamental electromagnetic equations in both systems. The remainder of this chapter not only provides a review of the fundamental principles of steady-state electromagnetism but also accustoms the reader to the use of Gaussian units.
1.2 THE LAWS OF COULOMB AND GAUSS
The first experimental fact we wish to invoke is that the force between two point charges at rest is directed along the line connecting the charges, and the magnitude of the force is directly proportional to the magnitude of each charge and inversely proportional to the square of the distance between the charges. This is Coulomb's force law and in Gaussian units assumes the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
for the force exerted on q1 by q2. (The Gaussian unit of charge is the statcoulomb or esu.) The quantity r is the distance between the charges, and er is the unit vector in the direction from q2 to q1. If the charges carry the same sign, the force is repulsive; if the signs are opposite, the force is attractive. The force on a test charge q defines the electric field vector according to
F = qE (1.2)
Thus coulomb's field law due to a source charge q' is
E = [q'/r2]er (1.3)
An important property of the electric field (indeed, of the electromagnetic field) is that it is linear. That is, the principle of superposition applies, and the field due to a number of charges is just the vector sum of the individual fields. Were it not for this property, the analysis of electromagnetic phenomena would be exceedingly difficult.
The notation required to express superposition is a little cumbersome. If there is more than one source charge producing the E field, then we must deal with two overlaid coordinate systems: one to express the location of the charges and one to express the location of the point where the field is being evaluated. As shown in Fig. 1-1, we will let the primed radius vector r' locate
the source point and the unprimed radius vector r locate the field point. Thus the vector distance from a particular source charge to a field point is (r - r'), and we will denote the unit vector in this direction by er-r'. This notation is such a nuisance however that we will shorthand the magnitude and unit vector by r and er when the meaning is clear from context, as in Eq. (1.3). But, to be explicit, the general form of Coulomb's (field) law for an arbitrary array of point charges is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
where the index i designates the ith charge at location r'i.
The integration of the normal component of the electric field E over a surface area [increment of S] is called the flux of E through this surface,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
where da = nda is the directed element of area. Now, if a charge q is enclosed by a Gaussian surface S (of arbitrary shape), the flux of E through this closed surface turns out to be equal to 4π times the enclosed charge. By superposition, if qencl is the total net charge enclosed within the surface, irrespective of its spatial distribution within the volume, we have the ITL∫ITL expression of Gauss' law for the electric field:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)
where the direction of da is that of the outward normal to the surface S (note that any charge outside the surface does not count!). The elementary formal proof is supplied in Problem 1-3. The factor of 4π comes from an integration over solid angle. Applications are given in Problems 1-4 through 1-8.
A vector field, such as E, is often pictorialized by drawing "lines of force" or field-lines, which are continuous curves everywhere parallel to the local direction of the field. Consider a set of field-lines that form the walls of a thin tubular region of space. Gauss' law shows that this construction is properly called a flux tube, and that the tube necessarily begins on positive charge and ends on negative charge. That is, a field-line is simply the limiting form of a flux tube of negligible cross section.
We may convert the integral relation of Eq. (1.6) into a differential relation as follows. The closed surface S defines the enclosed volume V, to which we can apply the divergence theorem, Eq. (A.53),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
And the enclosed charge q can be expressed as the volume integral of the charge-density function ρ(r) [charge-per-unit-volume],
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
With these substitutions, Eq. (1.6) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)
But this relation must be valid for any volume V, and so the integrands themselves must be equal at all points. The result is the differential expression of Gauss' law:
[div E = 4πρ] (total charge) (1.10)
This equation states that the divergence of E is zero except at locations occupied by charge, reaffirming that electrostatic field-lines always originate and terminate on electric charges. In Eq. (1.10)ρ is the total charge density; a different bookkeeping can be used in the presence of a dielectric medium [see Eq. (1.29)].
The Coulomb field, Eq. (1.3), is a "central force," and therefore the electrostatic field is conservative. That is, the line integral from one point to another is independent of path (see Problem 1-9). It follows that the line integral around any closed path must vanish:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)
Using Stokes' theorem, Eq. (A.54), this integral equation can be transformed into the equivalent differential statement of the conservative law,
[curl E = 0] (1.12)
The conservative property allows us to define a scalar potential Φ for the electric field, such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)
in which r0 is the location of an arbitrary reference point where Φ is defined to be zero, and the integral is carried out along any path from r0 to the "field point" r. The inverse operation gives the field in terms of the potential,
[E = - grad Φ] (1.14)
Gauss' law, Eq. (1.10), may now be expressed in terms of the potential as
div grad Φ = - 4 πρ
The scalar derivative operator div grad is more commonly written as [nabla]2, known as the Laplacian operator. Thus we have Poisson's equation,
[[nabla]2Φ = - 4 πρ] (1.15)
which expresses the physical content of Coulomb's law as a second-order differential equation for the scalar potential. In regions of space that contain no charge, Poisson's equation reduces to Laplace's equation,
[[nabla]2Φ = 0] (1.16)
Solutions of these important equations will be discussed in Chapter 3.
(Continues...)
Excerpted from Classical Electromagnetic Radiation by Mark A. Heald, Jerry B. Marion. Copyright © 2012 Mark A. Heald and Adelia Marion. Excerpted by permission of Dover Publications, Inc..
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