Classical Electricity and Magnetism: Second Edition

Classical Electricity and Magnetism: Second Edition

by Wolfgang K. H. Panofsky, Melba Phillips

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Overview

Compact, clear, and precise in its presentation, this distinguished, widely used textbook offers graduate students and advanced undergraduates a diverse and well-balanced selection of topics.
Subjects include the electrostatic field in vacuum; boundary conditions and relation of microscopic to macroscopic fields; general methods for the solution of potential problems, including those of two and three dimensions; energy relations and forces in the electrostatic field; steady currents and their interaction; magnet materials and boundary value problems; and Maxwell’s equations. Additional topics include energy, force, and momentum relations in the electromagnetic field; the wave equation and plane waves; conducting fluids in a magnetic field; waves in the presence of metallic boundaries; the inhomogeneous wave equation; the experimental basis for the theory of special relativity; relativistic kinematics and the Lorentz transformation; covariance and relativistic mechanics; covariant formulation of electrodynamics; and the Liénard-Wiechert potentials and the field of a uniformly moving electron.
The text concludes with examinations of radiation from an accelerated charge; radiation reaction and covariant formulation of the conservation laws of electrodynamics; radiation, scattering, and dispersion; the motion of charged particles in electromagnetic fields; and Hamiltonian formulation of Maxwell’s equations.

Product Details

ISBN-13: 9780486439242
Publisher: Dover Publications
Publication date: 01/26/2005
Series: Dover Books on Physics Series
Edition description: Second Edition
Pages: 526
Sales rank: 823,470
Product dimensions: 5.50(w) x 8.50(h) x (d)

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CLASSICAL ELECTRICITY AND MAGNETISM


By Wolfgang K. H. Panofsky, Melba Phillips, SECOND EDITION

Dover Publications, Inc.

Copyright © 1990 Wolfgang K. H. Panofsky
All rights reserved.
ISBN: 978-0-486-13225-9



CHAPTER 1

THE ELECTROSTATIC FIELD IN VACUUM


The interaction between material bodies can be described either by formulating the action at a distance between the interacting bodies or by separating the interaction process into the production of a field by one system and the action of the field on another system. These two alternative descriptions are physically indistinguishable in the static case. If the bodies are in motion, however, and the velocity of propagation of the interaction is finite, it is both physically and mathematically advantageous to ascribe physical reality to the field itself, even though it is possible to replace the field concept by that of "delayed" and "advanced" direct interaction in the description of electromagnetic phenomena. We shall formulate even the electrostatic interactions as a field theory, which can then be extended to the consideration of nonstatic cases.

1-1 Vector fields. Field theories applicable to various types of interaction differ by the number of parameters necessary to define the field and by the symmetry character of the field. In a general sense, a field is a physical entity such that each point in space is a degree of freedom. A field is therefore specified by giving the behavior in time at each coordinate point of a quantity suitable to describe the physical content.

The types of fields possible are restricted by various considerations. Fields are classified according to the number of parameters necessary to define the field and by the "transformation character" of the field quantities under various coordinate transformations. A "scalar" field is described by the time dependence of one quantity at each point in space, a "threedimensional vector field" by three such quantities. In general, an "nth-rank tensor field" requires the specification of dn components, where d is the dimensionality of the space in which the field is defined. A scalar field is a zero-rank tensor field, and a vector field is a first-rank tensor field.

The field description of a physical entity is independent of the particular choice of coordinate system used. This fact restricts the transformation properties of the field components under coordinate transformations. We consider two types of transformations of coordinates, "proper" and "improper" transformations. Proper transformations are those which leave the cyclic order of the coordinates invariant (i.e., do not transform a right- handed into a left-handed coordinate system in three dimensions); translation and rotation are proper transformations. Improper transformations, such as inversion of the coordinate axes and reflection of the coordinate system in a plane, change the cyclic order of coordinates.

A basic vector is the distance r connecting two points; the components of r may be designated by rα. The components Vα of a vector field V transform like the components rα under both proper and improper transformations. A scalar is invariant under proper and improper transformations. The components Pα of a pseudovector field P transform like the components rα under proper transformations, but change sign relative to rα under improper transformations. A pseudoscalar is invariant under proper transformations but changes sign under improper transformations.

The electric field is a three-dimensional vector field, i.e., a field definable by the specification of three components. The theory of vector fields was developed in connection with the study of fluid motion, a fact which is betrayed repeatedly by the vocabulary of the theory. We shall consider some general mathematical properties of such fields before specifying the physical content of the vectors.

All vector fields in three dimensions are uniquely defined if their circulation densities (curl) and source densities (divergence) are given functions of the coordinates at all points in space, and if the totality of sources, as well as the source density, is zero at infinity. Let us prove this theorem formally. Consider a three-dimensional vector field V(.x, y, z) such that

[nabla] x V = s, (1-1)

[nabla] x V = c. (1-2)

Equation (1-2) is self-consistent only if the circulation density c is irrotational, i.e., if

[nabla] x c = 0. (1-2')

We shall first show that if

V = -[nabla]φ + [nabla] x A, (1-3)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-4)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-5)

then V satisfies Eqs. (1-1) and (1-2).

It is necessary to examine the notation of Eqs. (1-4) and (1-5) before proceeding with the proof. The symbol xα stands for x, y, z at the field point; the symbol x'α stands for x', y', z' at the source point; the function r(xα, x'α) is the symmetric function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

representing the positive distance between field and source point. The reader should note carefully the functional relationships explicit in Eqs. (1-4) and (1-5). In integrals of this type these functional dependences will often not be fully stated; for example, we may write the volume integrals

φ = [1/4π]∫[s/r]dv', (1-4')

A = [1/4π]∫[c/r]dv', (1-5')

as a short notation. We shall sometimes use R for the radius vector from an origin of coordinates to the field point xα, and [xi] for that of a source point x'α, then r = |R - [xi]|.

Let us demonstrate that V as expressed by Eq. (1-3) is a solution of Eqs. (1-1) and (1-2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Laplacian operator [nabla]2 operates on the field coordinates; hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1-6)

Now we can show that

where δ(r), the Dirac δ-function, is defined by the functional properties

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-9)

if the point r = 0 is included in the volume of integration, and by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-10)

for any arbitrary function f so long as the volume of integration includes the point r = 0. The δ-function is not an analytic function but essentially a notation for the functional properties of the three defining equations. It will always be used in terms of these properties.

Since it is evident by direct differentiation that [nabla]2(1/r) = 0 for r ≠ 0, we have only to prove that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-11)

in order to verify Eq.(1-7). [In Eq.(1-11) the point r = 0, that is, xα = x'α, is included in the volume of integration.] By the application of Gauss's divergence theorem, applicable to any vector V,*

∫[nabla] x Vdv = ∫V x dS,

it is seen that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where Ω is the solid angle subtended at xα by the surface of integration S' over the variables x'α Since S' includes xα, we have simply ∫ dΩ = 4π, and Eq. (1-11) is verified. Hence from Eqs. (1-6) and (1-10),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-12)

which was to be proved.

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-13)

We shall be able to show that the first integral vanishes if c is bounded in space. If we anticipate this result, we see immediately, from Eq. (1-7), that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1-14)

so that Eq. (1-2) is also satisfied.

To prove that the first term of Eq. (1-13) vanishes, let us examine the coordinate variables involved in the integrand. The operator [nabla] has the components [partial derivative]/[partial derivative]xα. If we introduce the operator operating on the source coordinates, then for any arbitrary function g[r(xα, x'α)], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-15)

Therefore the first integral of Eq. (1-13) may, be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The differential operators now operate on the variables of integration and we may integrate by parts. Each component of I becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-16)

The second integral vanishes because the divergence of c is zero [Eq. (1-2')]. The first term can be transformed to a surface integral by means of Gauss's theorem; if c is bounded in space the surface may be taken sufficiently large so that c is zero over the entire integration. Hence Eq. (1-16) is zero, and the proof is complete.

We have thus proved that if the source density s and the circulation density c of a vector field V are given everywhere, then a solution for V can be derived from a scalar potential φ and a vector potential A. The potentials φ and A are expressed as integrals over the source and circulation densities.

It can be proved that this system of solutions is unique if the sources are bounded in space, i.e., there are no sources at infinity, and thus the fields themselves vanish at sufficiently large distance from the sources. Suppose that there are two functions, V1 and V2, which satisfy Eqs. (1-1) and (1-2). Their difference, the function W = V1 - V2, obeys the conditions

[nabla] x W = 0, (1-17)

[nabla] x W = 0, (1-18)

at every point in space and is zero at infinity. If we now show that W vanishes everywhere, we shall have proved that for finite sources there is only one solution for Eqs. (1-1) and (1-2). To prove this we note that if Eq. (1-18) is satisfied we can always put

W = -[nabla]ψ(1-19)

and, from Eq. (1-17),

[nabla]2ψ(1-20)

everywhere. If we apply Gauss's divergence theorem to the vector Ψ[nabla]Ψ we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-21)

The left side vanishes if the boundary is taken at sufficiently large distance from the sources, since Ψ tends to zero at least as 1/r, and the first term on the right is identically zero because of Eq. (1-20). Therefore Eq. (1-21) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-22)

and hence W = V1 - V2 = 0 everywhere. Thus V as given by Eq. (1-3) is unique.

We have gone into this formal proof in great detail not only because the theorems are of fundamental importance but also because the methods are of general usefulness throughout the study of electromagnetic fields. For convenience, let us summarize the results obtained:

(a) If the source density s and the circulation density c of a vector field V are given for a finite region of space and there are no sources at infinity, then V is uniquely defined.

(b) If V has sources s but no circulation density c, V is derivable from a scalar potential ?.

(c) If V has circulation density c but no sources s, V is derivable from a vector potential A.

(d) V is always derivable from a scalar and a vector potential.

(e) At points in space where s and c vanish, V is derivable from a scalar potential φ for which [nabla]2φ = 0, or from a vector potential A for which [nabla] × [nabla] × A = 0. We may add that at such points the field is said to be harmonic.

(f) If s and c are identically zero everywhere, V vanishes everywhere.

(g) The unique solution for V in terms of s and c is given by means of the potentials as expressed by the integrals (1-4) and (1-5).

(h) We have established a systematic notation for source and field coordinates. If we add the convention that the vector r points from source to field point we may extend our list of useful mathematical relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These properties of general vector fields will be indispensable in the physical considerations which follow. We shall have a consistent field theory representing the empirical laws of electricity and magnetism when we have written these laws as a set of equations giving the source and circulation densities, i.e., the divergence and curl, of the field vectors representing the electromagnetic fields. This is the fundamental program of classical electromagnetic theory.

1-2 The electric field. We shall first consider the electrostatic field in vacuum. The electric field is defined in terms of the force produced on a test charge q by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-23)

where F is the force (newtons) on the test charge q (coulombs). The definition is entirely independent of the system of units, but in the mks system* the electric field E defined by this equation is in volts per meter. The limit q [right arrow] 0 is introduced in order that the test charge will not influence the behavior of the sources of the field, which will then be independent of the presence of the test body. The requirement that the test charge be vanishingly small compared with all sources of the field raises a fundamental difficulty, since the finite magnitude of the electronic charge does not permit the limit q [right arrow] 0 to be carried out experimentally. This restriction therefore limits the practical validity of the definition to cases where the sources producing the field are equivalent to a large number of electronic charges. Definition (1-23) is thus entirely suitable only for macroscopic phenomena, and we shall have to exercise great care in applying it to the treatment of the elementary charges of which matter is actually composed. For microscopic processes the field cannot be defined "operationally" in terms of its effect; it must be described in terms of its sources, assuming that the macroscopic laws of the field sources are still valid.

The field E is a three-dimensional vector (not pseudovector) field if we adopt the convention that the charge q is a scalar (not a pseudoscalar). In terms of this convention the transformation character of both the electric and (later) the magnetic fields is defined.

1-3 Coulomb's law. The experimentally established law for the force between two point charges in vacuo was originally formulated as an action at a distance:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-24)

Here F2 is the force on charge q2 due to the presence of charge q1, and r is the radius vector position of charge q2 measured from an origin located at charge q1; [??] is a constant, 107/4πc2 farads/meter in this system of units; c is the experimental value of the velocity of propagation of plane electromagnetic waves in free space (see Appendix I), and all distances are measured in meters. In the mathematical identity on the right the gradient operator acts on the coordinates of the charge q2. The law applies equally to positive and negative charges, and indicates that like charges repel, unlike charges attract. If the test charge in Eq. (1-23) is assumed positive, comparison with Eq. (1-24) yields immediately

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-25)

giving the electric field E at the position r due to a charge q at the origin of the radius vector. Here q corresponds to q1 of Eq. (1-24).

We can prove Gauss's flux theorem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-26)

as a direct consequence of Coulomb's law: Consider an element of surface dS, expressed as a vector directed along the outward normal of the element as shown in Fig. 1-1, at a distance r from a charge q at a point x'α. By taking the scalar product of dS and both sides of Eq. (1-25) we secure

FIG. 1-1. Elements of surface and solid angle contributing to the total electric flux of Gauss's theorem.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-26')

Since the integral of dΩ over a closed surface which includes the point x'α is just 4π, Gauss's theorem follows immediately. The principle of superposition enables us to sum the separate fields of any number of point charges, so that q of Eq. (1-26) is the total charge inside the boundary surface S.

If we apply the divergence theorem to E,

∫SE x dS = ∫[nabla] x Edv,

and make use of the fact that the total volume integral of the charge density ρ is simply the total charge q, the application of the flux theorem enables us to put Eq. (1-25) into the form

[nabla] x E = ρ/[??] (1-27)

Here ρ is the charge per unit volume at the point where the electric field is E. Since the curl of the gradient of a scalar is zero, it further follows from Eq. (1-25) and the principle of superposition that

[nabla] x E = 0. (1-28)

The electrostatic field is thus irrotational. That the electrostatic field is completely defined by a charge distribution then follows from the theorem that a vector field is uniquely determined by the curl and the divergence of the field.

It is instructive to note that Eqs. (1-27) and (1-28) follow directly from Coulomb's law in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-30)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-27')

Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-28')

since the curl of a vector expressible as a gradient vanishes identically. For a point source q we have directly from Eq. (1-25):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-27")

1-4 The electrostatic potential. Since the static field is irrotational, it may be expressed as the gradient of a scalar function. We may define an electrostatic potential φ by the equation

E = -[nabla]φ. (1-31)

In Cartesian coordinates the components of the field parallel to the xα axes respectively are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-32)

The application of the general vector relation known as Stokes' theorem,

∫([nabla] x E) x dS = [??]E x dl, (1-33)

where dl is the infinitesimal vector length tangent to a closed path of integration, leads to

[??]E x dl = 0, (1-34)

since the curl of E is everywhere zero. This shows that the electrostatic field is a conservative field: no work is done on a test charge if it is moved around a closed path in the field. Since the work done in moving a test charge from one point to another is independent of the path, we can uniquely define the work necessary to carry a unit charge from an infinite distance to a given point as the potential of that point. If one considers fields of less than three dimensions, i.e., sources extending to infinity in one or more directions, this definition will lead to difficulties and a point other than infinity must be taken as a reference point. So long as only finite sources are considered, however, this definition of potential is both adequate and convenient.

The substitution of Eq. (1-31) in Eq. (1-27) leads at once to Poisson's equation

[nabla]2φ = -ρ/[??], (1-35)

and in a region of zero charge density to Laplace's equation,

[nabla]2 = 0, (1-36)

The fundamental problem of electrostatics is to determine solutions of Poisson's equation appropriate to the conditions under particular consideration.

1-5 The potential in terms of charge distribution. The electrostatic potential at a given point was defined by Eq. (1-31) in terms of the electric field at that point. Since the source density of the electrostatic field is just ρ/εo, we know from Eq. (1-4) that the potential is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in terms of the charge density at all points in space. Field theory, however, permits us to find a solution even if ρ(x'α) is known only within an arbitrary surface S; the effect of the other sources is then replaced by the knowledge of the boundary values of the potentials or their derivatives on the surface S.

To obtain an explicit expression for φ(xα) in terms of ρ within S and φ and [nabla]φ on S, we make use of Green's theorem, which states

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-37)


(Continues...)

Excerpted from CLASSICAL ELECTRICITY AND MAGNETISM by Wolfgang K. H. Panofsky, Melba Phillips, SECOND EDITION. Copyright © 1990 Wolfgang K. H. Panofsky. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Prefaces
1. The Electrostatic Field in Vacuum
2. Boundary Conditions and Relation of Microscopic to Macroscopic Fields
3. General Methods for the Solution of Potential Problems
4. Two-Dimensional Potential Problems
5. Three-Dimensional Potential Problems
6. Energy Relations and Forces in the Electrostatic Field
7. Steady Currents and Their Interaction
8. Magnetical Materials and Boundary Value Problems
9. Maxwell’s Equations
10. Energy, Force, and Momentum Relations in the Electromagnetic Field
11. The Wave Equation and Plane Waves
12. Conducting Fluids in a Magnetic Field (Magnetohydrodynamics)
13. Waves in the Presence of Metallic Boundaries
14. The Inhomogeneous Wave Equation
15. The Experimental Basis for the Theory of Special Relativity
16. Relativistic Kinematics and the Lorentz Transformation
17. Covariance and Relativistic Mechanics
18. Covariant Formulation of Electrodynamics
19. The Liénard-Wiechert Potentials and the Field of a Uniformly Moving Electron
20. Radiation from an Accelerated Charge
21. Radiation Reaction and Covariant Formulation of the Conservation Laws of Electrodynamics
22. Radiation, Scattering, and Dispersion
23. The Motion of Charged Particles in Electromagnetic Fields
24. Hamiltonian Formulation of Maxwell’s Equations
Appendixes
Bibliography
Index

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