Electric& Magnetic Field Theory

Transcription

1 Kurt Edmund Oughstun Electric& Magnetic Field Theory August 25, 2015 Springer

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3 Contents 1 Introduction... 3 References The Electric Field Coulomb s Law and the Electric Field Intensity Charge Density Gauss Law The Electrostatic Scalar Potential and Work Poisson s Equation for the Scalar Potential The Concept of an Ideal Conductor The Electric Dipole, Quadrupole, and Multipoles The Static Electric Dipole The Linear Electric Quadrupole Static Electric Multipoles The Electrostatic Field Produced by an Arbitrary Static Charge Distribution The Concept of a Perfect Dielectric Electrostatic Energy Capacitance and Energy of Multi-Conductor Systems Thomson s Theorem Forces on a Conductor Problems References Electric Current Conservation of Charge Electric Conductivity Resistance and Conductance Joule s Law Problems References v

4 vi Contents 4 The Magnetic Field The Biot and Savart Law Ampére s Law The Differential Relations of Magnetostatics The Vector Potential for the Magnetostatic Field Magnetic Field of a Spatially Localized Current Distribution A Scalar Potential for the Magnetic Field in Source-Free Regions The Concept of a Perfect Magnetic Material Faraday s Law of Electromagnetic Induction Self-Inductance Mutual Inductance Magnetic Energy Problems References Transmission Line Analysis Distributed Resistance and Internal Inductance Transmission Lines Transmission Line Equations Problems References A The Dirac Delta Function A.1 The One-Dimensional Dirac Delta Function A.2 The Dirac Delta Function in Higher Dimensions References B Helmholtz Theorem References C Green s Functions References D Boundary Value Problems D.1 Boundary Value Problems in Rectangular Coordinates D.2 Boundary Value Problems in Two-Dimensional Angular Regions D.3 Boundary Value Problems in Cylindrical Coordinates: Cylinder Functions D.4 Boundary Value Problems in Spherical Coordinates: Legendre Polynomials and Spherical Harmonics D.4.1 Legendre s Equation and the Legendre Polynomials D.4.2 Spherical Coordinate Boundary Value Problems with Azimuthal Symmetry

5 Contents vii D.4.3 Associated Legendre Functions and the Spherical Harmonics D.4.4 Boundary Value Problems in Spherical Coordinates D.4.5 The Addition Theorem for the Spherical Harmonics References Index

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9 Chapter 1 Introduction Classical electromagnetic field theory has its origins in the historically separate theories of electric and magnetic fields. In order to maintain close connection with experimental observation, the fundamental field equations derived from Coulomb s law, the Biot-Savart law, and Ampére s law [1] are typically expressed in terms of an appropriate spatial integration over regions containing electric charge, both static and moving with a uniform velocity (steady electric current), as well as from Faraday s law [2] for accelerating charge (time-varying electric current). The divergence and Stokes theorems from vector analysis are then used to express these relations in differential form, which is more amenable to mathematical analysis. With the inclusion of a displacement current in Ampére s law in order to satisfy conservation of charge, Maxwell [3, 4] united electric and magnetic field theory, forming electromagnetic field theory. Maxwell s equations are described by the set of four vector differential relations D(r,t) = (r,t), (1.1) B(r,t) = 0, (1.2) E(r,t) = B(r,t)/ t, (1.3) H(r,t) = J(r,t)+ D(r,t)/ t, (1.4) in MKSA units. The first two equations are mathematical statements of Gauss law, the third equation expresses Faraday s law, and the fourth equation expresses Ampére s law. These four first-order differential relations are completed by the appropriate constitutive relations (or material equations) relating the induced electric displacement field vector D(r, t), the magnetic intensity field vector H(r,t), and the conduction current density J c (r,t), referred to as the induction field, to both the electric field intensity vector E(r,t) and the magnetic induction field vector B(r,t), these latter two field vectors being referred to as the primitive fields. 3

10 4 1 Introduction The divergence of Ampére s law (1.4) followed by substitution from Gauss law (1.1) yields the equation of continuity J(r,t)+ (r,t)/ t, (1.5) which is the differential expression of the conservation of charge. This result depends upon the inclusion of the displacement current D(r, t)/ t in Ampére s law, as was originally done by Maxwell [3, 4]. Finally, Maxwell s equation (1.1) (1.4) are connected to physical measurement through the Lorentz force relation [5] F(r,t) = (r,t)e(r,t)+j(r,t) B(r,t), (1.6) from which it is seen that an electric field accelerates charge whereas a magnetic field changes its direction. In free space (vacuum), the appropriate constitutive relations are simply D(r,t) = ǫ 0 E(r,t) and H(r,t) = µ 1 0 B(r,t) where ǫ 0 = F/m is the dielectric permittivity and µ 0 = 4π 10 7 W/Am the magnetic permeability of free space. In source-free regions of space, the two curl relations of Maxwell s equations then become E(r,t) = µ 0 H(r,t)/ t, (1.7) H(r,t) = ǫ 0 E(r,t)/ t, (1.8) with E(r,t) = H(r,t) = 0. The curl of the first equation (1.7) with substitution from the second equation (1.8) and use of the vector identity F = ( F) 2 F then yields the vector wave equation for the electric field 2 E(r,t) 1 c 2 2 E(r,t) t 2 = 0, (1.9) and similarly, the curl of (1.8) with substitution from (1.7) yields the vector wave equation for the magnetic field 2 H(r,t) 1 c 2 2 H(r,t) t 2 = 0, (1.10) where c = 1/ ǫ 0 µ 0 is the speed of light in vacuum. Taken together, these results unified electric and magnetic fields into a single electromagnetic field and established the fact that light is an electromagnetic wave. The electromagnetic field equations uncouple into separate, independent equations for the electric and magnetic field vectors only when the field is static (i.e., when the time derivative of the field quantities identically vanish). In that idealized case, the electrostatic field is described by the pair of relations D(r) = (r), E(r) = 0 and the steady-state magnetic field is described by the separate pair of relations B(r) = 0 and H(r) = J(r).

11 References 5 As stated by Einstein regarding Maxwell s theory of electromagnetism in his 1936 article on Physics and Reality [6], the electric field theory of Faraday and Maxwell represents probably the most profound transformation which has been experienced by the foundations of physics since Newton s time... The existence of the field manifests itself, indeed, only when electrically charged bodies are introduced into it. The differential equations of Maxwell connect the spacial and temporal differential coefficients of the electric and magnetic fields. The electric masses are nothing more than places of non-disappearing divergency of the electric field. Light waves appear as undulatory electromagnetic field processes in space. In regard to the contribution by Lorentz to this theory, Einsten [6] went on to state that his theory was built on the following fundamental hypothesis : Everywhere (including the interior of ponderable bodies) the seat of the field is the empty space. The participation of matter in electromagnetic phenomena has its origin only in the fact that the elementary particles of matter carry unalterable electric charges, and, on this account are subject on the one hand to the actions of ponderomotive forces and on the other hand possess the property of generating a field. The elementary particles obey Newton s law of motion for the material point. The word ponderomotive is an adjective describing the tendency to produce movement of a ponderable body. By a ponderable body is meant one having nonzero mass. Finally, as a critique of the resultant Maxwell-Lorentz theory, Einstein [6] pointed out that This is the basis on which H. A. Lorentz obtained his synthesisof Newton s mechanics and Maxwell s field theory. The weakness of this theory lies in the fact that it tried to determine the phenomena by a combination of partial differential equations (Maxwell s field equations for empty space) and total differential equations (equations of motion of points), which procedure was obviously unnatural. The unsatisfactory part of the theory showed up externally by the necessity of assuming finite dimensions for the particles in order to prevent the electromagnetic field existing at their surfaces from becoming infinitely great. The theory failed moreover to give any explanation concerning the tremendous forces which hold the electric charges on the individual particles. H. A. Lorentz accepted these weaknesses of his theory, which were well known to him, in order to explain the phenomena correctly at least as regards their general lines. Einstein then concluded that [6] What appears certain to me...is that, in the foundations of any consistent field theory, there shall not be, in addition to the concept of field, any concept concerning particles. The whole theory must be based solely on partial differential equations and their singularity-free solutions. References 1. A. M. Ampère, Memoir on the mutual action of two electric currents, Annales de Chimie et Physique, vol. 15, pp , 1820.

12 6 1 Introduction 2. M. Faraday, Experimental Researches in Electricity. London: Bernard Quaritch, J. C. Maxwell, A dynamical theory of the electromagnetic field, Phil. Trans. Roy. Soc. (London), vol. 155, pp , J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford: Oxford University Press, H. A. Lorentz, Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte, Ann. Phys., vol. 9, pp , A. Einstein, Out of My Later Years. New York: Philosophical Library, pp

13 Chapter 2 The Electric Field The beginning is the most important part of the work. Plato The electric field produced by a specified static distribution of charges is developed here based solely upon Coulomb s law. An historical development of this theory is given by Elliott [1]. 2.1 Coulomb s Law and the Electric Field Intensity The mathematicaldescriptionofthe electricfield begins with Coulomb slaw 1 for the force exerted by one point charge on another This so-called electric charge is an inherent, inseparable physical property of certain fundamental (elementary) particles (e.g., electrons, protons, and positrons) that are the closestphysicalapproximationtoanidealpoint charge.letq 1 be astationary point charge with position vector r 1 relative to a fixed origin O, and let q 2 be a separate, distinct point charge with position vector r 2 r 1 relative to the same origin O. Coulomb s law then states that the force F 21 exerted by q 2 on q 1 is given by F 21 = K q 1q 2 r 2 ˆr 21 (2.1) where r r 1 r 2 is the distance between the two point charges and where ˆr 21 (r 1 r 2 )/ r 1 r 2 = (r 1 r 2 )/r is the unit vector directed from q 2 to q 1. The force is repulsive if q 1 and q 2 are of the same sign and attractive if they are of the opposite sign, accounted for by the direction of the unit vector ˆr 21 and the sign of the product q 1 q 2 in Eq. (2.1). Reciprocity requires that an equal but oppositely directed force F 12 is exerted by q 1 on q 2, so 1 Charles Augustin de Coulomb ( ) demonstrated the inverse square law of electric force in 1785 using a torsion balance. His results were preceded by the experimental observations of Benjamin Franklin in 1755, Joseph Priestley in 1767, John Robison in 1769, and Henry Cavendish in In an experiment performed at the Worcester Polytechnic Institute in 1936, Plimpton and Lawton showed that this dependency deviated from the inverse square law by less than 2 parts in 1 billion. 7

14 8 2 The Electric Field that F 12 = F 21, a direct consequence of Eq. (2.1) since ˆr 12 = ˆr 21. In the rationalized MKSA (meter, kilogram, second, ampere) system, the unit of force is the newton (N), the unit of charge is the coulomb (C), and the constant appearing in Coulomb s law is given by K = 1 4πǫ N m 2 /C 2. Hereǫ F/misthepermittivity of free space,wherethefarad (F C/V) is the unit of capacitance and the Volt (V J/C) is a measure of work per unit charge. Coulomb s law in MKSA units then becomes F 21 = 1 4πǫ 0 q 1 q 2 r 2 ˆr 21 (N) (2.2) As written, Coulomb s law directly applies to any pair of point charges that are situated in vacuum and are stationary with respect to each other. It also applies in material media if F 21 is taken as the direct microscopic force between the two charges q 1 and q 2, irrespective of the other forces arising from all of the other charges in the surrounding material. The Coulombic force satisfies the principle of superposition. The electrostatic force exerted on a stationary point charge q 1 at r 1 by a system of stationary point charges q k at r k, k 1, is consequently given by the vector sum or linear superposition of all the Coulombic forces exerted on q 1 as F(r 1 ) = F k1 = q 1 q k 4πǫ 0 r 2 ˆr k1 (2.3) k 1 k 1 k1 with r k1 r 1 r k, where ˆr k1 r 1k /r 1k and r k1 = r 1k. The electric field intensity E(r) = E(x,y,z) at a point r = ˆ1 x x+ˆ1 y y+ˆ1 z z inspaceisdefinedasthelimiting force per unit charge exertedonatestcharge q at that point as the magnitude of the test charge goes to zero, viz. E(r) lim q 0 F(r) q (V/m) (2.4) The limit q 0 is introduced in order that the test charge does not influence the charge sources producing the electric field. The electric field is thus defined in such a way that it is independent of the presence of the test charge. Notice that this abstraction to a field concept (introduced by Michael Faraday) eliminates the mechanist requirement of action-at-a-distance that is embodied in Coulomb s law. From Eqs. (2.2) and (2.4), the electric field intensity at a fixed point r due to a single point charge q k situated at r k is E(r) = 1 4πǫ 0 q k R 2 ˆR, (2.5)

15 2.1 Coulomb s Law and the Electric Field Intensity 9 where R = r r k denotesthe vector from the source point P k at r k to the field point P at r with magnitude R = R, and where ˆR R/R is the unit vector along that direction. As a consequence of the principle of superposition, the electric field intensity at a fixed point r due to a system of stationary,discrete point charges q j located at the points r j, j = 1,2,...,n, is then given by the vector sum E(r) = 1 n q j ˆR j (2.6) 4πǫ 0 R 2 j=1 j where R j = r r j denotes the vector from the source point P j at r j to the field point P at r with magnitude R j and direction ˆR j R j /R j Charge Density The microscopic charge density ρ(r) is a scalar field whose value at any point r in space is defined by the limiting ratio q ρ(r) lim V 0 V (C/m 3 ) (2.7) where q is the net chargein the volume element V. From a microscopicperspective, the charge density ρ(r) is zero everywhere except in those regions occupied by fundamental charged particles. From a macroscopic perspective, the abrupt spatial variations in the microscopic charge density ρ(r), which are on the scale of interparticle distances, are removed through an appropriate spatial averaging procedure over spatial regions that are small on a macroscopic scale but whose linear dimensions are large in comparison with the particle spacing; a detailed description of this spatial-averaging procedure is presented in 4.1. The result is the macroscopic charge density (r) = ρ(r) (2.8) The electric field that is determined from such a macroscopic charge density is correspondingly a spatially-averaged field and, as such, is just what would be obtained through an appropriate laboratory measurement. Notice that the microscopic charge density can be obtained from the macroscopic density through the use of the Dirac delta function (see Appendix A). With the introduction of the charge density, the vector summation appearing in Eq. (2.6) may be replaced (in the appropriate limit) by a volume integration over the entire region of space containing the source charge distribution. Because q(r) = (r) V is the elemental charge contained in the volume element V at the point r, Eq. (2.6) becomes

16 10 2 The Electric Field 1 (r ) E(r) = lim V 0 4πǫ 0 R ˆR V 2 = 1 (r ) 4πǫ 0 R ˆRd 3 2 r, (2.9) where R = r r is the vector directed from the source point r to the field point r with magnitude R and direction ˆR, and where d 3 r = dx dy dz denotes the volume element at the source point r = (x,y,z ). Notice that the integral appearing in Eq. (2.9) is convergent for r = r. 2.2 Gauss Law Consider a point charge q at a fixed point in space together with a simplyconnected closed surface S. Let r denote the variable distance from q to a Fig. 2.1 A point charge q inside a closed surface S producing an electric field E(P) at a point P on S a distance r away. The differential element of surface area da at P has unit outward normal vector ˆn at an angle θ to E, subtending the solid angle dω at q. q dω S P da θ n v E(P) point P on the surface S with associated unit vector ˆr directed along the line from q to P, and let da denote the differential element of surface area at that point with outwardly directed unit normal vector ˆn, as illustrated in Fig The flux of E passing through the directed element of area da = ˆnda of S is then given by E ˆnda = 1 4πǫ 0 ˆn qˆr da (2.10) r2 The quantity (ˆr ˆn/r 2 )da is the differential element of solid angle dω subtended by da at the position of the point charge. With this identification, the flux of E passing through the directed element of area da = ˆnda of S becomes E ˆnda = (q/4πǫ 0 )dω. The total flux of E passing through S in the outward direction is then given by integrating this expression over the entire surface S, where S dω = 4πifq S and dω = 0ifq / S, resulting in S Gauss law for a single point charge { E ˆnda = 1 q, ifq S (2.11) ǫ 0 0, ifq / S S

17 2.2 Gauss Law 11 Notice that Gauss law does not provide a result if q is situated on S because the direction of the unit vector ˆr is then not uniquely defined. For a system of discrete point charges q j, the principle of superposition applies and Gauss law becomes E ˆnda = 1 q j, (2.12) ǫ 0 S where the summation extends over only those charges that are inside the region enclosed by the surface S. If the charge system is described by the charge density (r), one finally obtains the integral form of Gauss law S E ˆnda = 1 ǫ 0 j V (r)d 3 r (2.13) where V is the volume enclosed by the surface S. This derivation of Gauss law, as given in Eq. (2.11), depends only upon the following two mathematical properties of the electrostatic field embodied in Coulomb s law: (1) the inverse square law for the force between point chargesand(2) thecentralnatureofthe force.removeeither oneofthese two properties and Gauss law no longer follows. The generalized form of Gauss law in Eq. (2.13) relies upon the additional property of linear superposition. For a system of discrete point charges q j located at r = r j, the charge density is given by (r) = q j δ(r r j ), j where δ(r r j ) δ(x x j )δ(y y j )δ(z z j ) is the three-dimensional Dirac delta function (see Appendix A). With this substitution in Eq. (2.13), the integral form of Gauss law becomes E ˆnda = 1 q j δ(r r j )d 3 r = 1 ǫ 0 ǫ 0 S j which recaptures the microscopic form (2.12) of Gauss law for a system of discrete point charges with locations r j V. With application of the divergence theorem S E ˆnda = V ( E)d3 r, the integral form of Gauss law given in Eq. (2.13) becomes ( E /ǫ 0 )d 3 r = 0. V Because this expression holds for any region V, the integrand itself must then vanish, so that E(r) = (r) ǫ 0 (2.14) V j q j

18 12 2 The Electric Field which is the differential form of Gauss law. This single vector differential relation is not sufficient to completely determine the electric field vector E(r) for a given charge density (r). Helmholtz theorem (see Appendix B) states that a vector field can be specified almost completely (up to the gradient of an arbitrary scalar field) if both its divergence and curl are specified everywhere. The required curl relation for the electrostatic field follows from the integral form (2.9) of Coulomb s law, expressed here as E(r) = 1 4πǫ 0 (r ) r r r r 3d3 r, (2.15) the integration extending over all space. Because ( ) 1 r r = r r r r where 3 the gradient operator operates only on the unprimed coordinates, then E(r) = 1 4πǫ 0 (r ) r r d3 r. (2.16) Because the curl of the gradient of any well-behaved scalar function identically vanishes, Eq. (2.16) shows that which is Faraday s law for the electrostatic field. E(r) = 0 (2.17) 2.3 The Electrostatic Scalar Potential and Work The mathematical form of Coulomb s law in Eq. (2.16) suggests that a scalar potential for the electric field vector be defined as E(r) = V(r) (2.18) where the minus sign is introduced by convention, and V(r) = 1 4πǫ 0 (r ) r r d3 r (V) (2.19) the integration extending over the entire region of space containing the charge distribution under consideration. Notice that V(r) is not uniquely determined by Eq. (2.19) as one may add to it any quantity that has a zero gradient without changing E(r). In addition, note that the principle of superposition applies to the electrostatic potential V(r) just as it does to the electrostatic field vector E(r).

19 2.3 The Electrostatic Scalar Potential and Work 13 The convergence properties of the integral appearing in Eq. (2.19) are essential for the proper formulation of a given electrostatic problem. To that end, consider the behavior of the scalar potential inside a differential element of space containing charge. Let the volume element d 3 r be a spherical shell of thickness dr and radius r centered at the point P. Then d 3 r = 4πr 2 dr and the element of charge in this spherical shell produces a potential dv = 4πr 2 dr = rdr, 4πǫ 0 r ǫ 0 which remains finite as r 0. The scalar potential for the electrostatic field then converges for sufficiently well-behaved charge density functions (r). Consider now determining the work done in transporting a test charge q from a point A to another point B through an externally produced electrostatic field E(r). The electric force acting on the test charge q at any point in the field is given by Coulomb s law as F(r) = qe(r), so that the work done in moving the test charge q slowly 2 from A to B is given by the path integral B B W = F dl = q E dl, (2.20) A A where the minus sign indicates that this is the work done on the test charge against the action of the field. With Eq. (2.18) this expression becomes B B W = q V dl = q dv = q(v B V A ), (2.21) A A whichshowsthatthe quantityv(r) canbe interpretedasthepotentialenergy of the charge q in the electrostatic field. The negative sign in Eq. (2.19) is then seen to indicate that E(r) points in the direction of decreasing potential, and hence, decreasing potential energy. The relations appearing in Eqs. (2.20) (2.21) show that the path integral of the electrostatic field vector E(r) between any two points is independent of the path and is the negative of the potential difference between them, viz. B V B V A = E dl (2.22) A Notice that the electrostatic field only determines the difference between the electrostatic potential at the two points. If the path is closed, then E dl = 0 (2.23) 2 Sufficiently slow such that there are negligible accelerations resulting in negligible energy loss due to electromagnetic radiation so that the process is reversible.

20 14 2 The Electric Field With the application of Stokes theorem C E dl = ( E) ˆnda to this S result one immediately obtains E = 0, which is just Eq. (2.17). The electrostatic field E(r) is then seen to be an irrotational vector field and is therefore conservative. For an open system, it is convenient to choose the potential at infinity to be zero. The electrostatic potential at a point P is then given by P V(P) = E dl (2.24) which is referred to as the absolute potential. The set of points in space that are at a constant potential defines an equipotential surface. Because V(r) = constant on an equipotential, then dv = V V V dx+ dy + dz = 0. (2.25) x y z Because E = V, the electric field is everywhere perpendicular to the equipotential surfaces. Lines of force are then defined such that they are everywhere perpendicular to the equipotentials with E(r) tangent to them Poisson s Equation for the Scalar Potential Substitution of Eq. (2.18) into the differential form (2.14) of Gauss Law yields Poisson s equation 3 2 V(r) = (r) ǫ 0 (2.26) Because 2( r r 1) = 4πδ(r r ), the integral solution of Poisson s equation is given by Coulomb s law as [cf. Eq. (2.19)] V(r) = 1 4πǫ 0 (r ) r r d3 r, (2.27) the integration extending over the entire region containing charge. In chargefree regions of space the electrostatic potential satisfies Laplace s equation 2 V(r) = 0, (2.28) which states that the electrostatic potential can never have an extremum in a charge-free region. 3 Siméon Denis Poisson ( ) extended [2] Laplace s equation [3] in 1813 to include regions occupied by charge.

21 2.3 The Electrostatic Scalar Potential and Work 15 The continuity of the charge density (r) in Poisson s equation is not sufficient to ensure the existence of the second partial derivatives of the potential V(r). This is provided by the following definition due to Hölder [4]. Definition 2.1 (Hölder Condition). A function f(q) of the coordinates of a point Q is said to satisfy a Hölder condition at the point P iff there exists three positive constants A, α, and r 0 such that f(q) f(p) Ar α Q r r 0, (2.29) where r is the distance between the points P and Q. If a region R exists in which f(q) satisfies a Hölder condition at every point P for a fixed set of values for A, α, and r 0, then the function f(q) is said to satisfy a Hölder condition uniformly, or a uniform Hölder condition, in R. The following theorem due to Hölder [4] then holds. 4 Theorem 2.1. Let V(r) be the potential of a source distribution with piecewise continuous density (r) in a regular region R. Then at any interior point P 0 R at which satisfies a Hölder condition, the derivatives of V(r) exist and satisfy Poisson s equation Boundary Value Problems & Uniqueness The general boundary value problem for the electrostatic potential V(r) corresponding to a given charge distribution (r) in a particular region of space R amounts to determining a solution to Poisson s equation (2.26) [or Laplace s equation (2.28) if = 0] that satisfies the given boundary conditions specified on the system of surfaces S enclosing the region R. The boundary conditions that lead to a unique solution of Poisson s equation are: Dirichlet Boundary Conditions: The specification of V(r) on a closed surface S forms a Dirichlet problem for the region R enclosed by S. Neumann Boundary Conditions: The specification of the normal derivative V/ n on a closed surface S forms a Neumann problem for the region R enclosed by S. Mixed Boundary Conditions: The specification of Dirichlet boundary conditions on a portion S 1 of S and Neumann boundary conditions on the remainders 2 ofs, where S 1 S 2 =, formsamixed problem forthe region R enclosed by S = S 1 S 2. The proof of uniqueness with these boundary conditions proceeds in the usual manner. Suppose that there are two solutions V 1 (r) and V 2 (r), each satisfying Poisson s equation in the region R and both satisfying the same boundary condtions on S. Then the difference function V(r) V 1 (r) V 2 (r), 4 The proof may be found in Kellogg [5].

22 16 2 The Electric Field r R, satisfies Laplace s equation in R; that is 2 V(r) = 0 r R and either V(r) = 0 on S (Dirichlet boundary conditions), or V(r)/ n = 0 on S (Neumann boundary conditions), or V(r) = 0 on S 1 and V(r)/ n = 0 on S 2 with S = S 1 S 2 and S 1 S 2 = (mixed boundary conditions). Application of Green s first integral identity R V(r) then gives R ( V 2 V + V V ) d 3 r = ( φ 2 ψ + φ ψ ) d 3 r = S φ ψ n d2 r to S V V n d2 r, where V/ n = V ˆn. As 2 V = 0 in R and either V = 0 or V/ n = 0 on S, then V 2 d 3 r = 0, R which then implies that V = 0 everywhere in the region R. Consequently, V is a constant in R. It then follows that: for Dirichlet boundary conditions, V = 0 on S so that the constant must be zero and V 1 (r) = V 2 (r) in R and the solution is unique; for Neumann boundary conditions, V/ n = 0 on S and, apart from an unimportant arbitrary additive constant, the solution is unique; for mixed boundary conditions, V = 0 on part of S so that the constant must be zero and V 1 (r) = V 2 (r) in R and the solution is unique. A solution when both V and V/ n are specified arbitrarily on a closed boundary surface S, known as Cauchy boundary conditions, does not always exist because there are separate unique solutions for the Dirichlet and Neumann boundary value problems and these will not, in general, be consistent. The formal solution of Poisson s equation using Green s functions is presented in Appendix C. This representation naturally leads to the powerful method of images approach for solving field problems in the presence of conducting surfaces. For completeness, the solution of Laplace s equation in specific separable coordinate systems by the separation of variables method is also presented in Appendix C Average Electrostatic Potential Over a Sphere Consider a spherical surface S of radius a > 0 carrying a uniform surface charge density s with total charge Q = 4πa 2 s, as illustrated in Fig Because s is spherically symmetric, the electrostatic field vector is radially directed from the center O of the sphere and is a function of the radial distance R alone, so that E(r) = ˆ1 R E(R). By Gauss law, the electric field intensity at an observation point P a distance R > a from the center O of the 5 A thorough resource on the solution of boundary value problems for both electric and magnetic fields may be found in the classic text by Smythe [6].

23 2.3 The Electrostatic Scalar Potential and Work 17 Fig. 2.2 Spherical surface S with origin O and radius a. The exterior point P is at a distance R > a from O, and the variable distance from the surface S to P is denoted by r. S O a r R P sphereisgivenbye(r) = Q 4πǫ 0R andtheabsolutepotentialisv(r) = Q 2 4πǫ. 0R The potential is also given by Coulomb s law as V(R) = s S 4πǫ 0rda, where s = Q/4πa 2. Upon equating these two expressions, one obtains Q 4πǫ 0 R = Q da 4πa 2 S 4πǫ 0 r, (2.30) which then results in the geometrical identity 1 R = 1 da 4πa 2 S r, R > a (2.31) Theorem 2.2. The average value of 1 r over a spherical surface S, where r is the distance from a point on the surface S to an exterior point P, is equal to, where R is the distance from the center O of the sphere S to P. 1 R If the surface charge density s = Q/4πa 2 is removed and a point charge Q is placed at the exterior point P, the potential at the center O of the sphere is then given by the left-hand side of Eq. (2.30). The right-hand side of this equation is then just the average potential on the spherical surface. Hence: Theorem 2.3 (Mean Value Theorem). The average potential over any spherical surface is equal to the potential at the center of the sphere if there are no charges inside the sphere. Corollary 2.1. It is impossible to have a potential maximum or minimum in a charge-free region. Consider again a spherical surface S of radius a carrying a uniform surface charge density s = Q/4πa 2 with total charge Q, as illustrated in Fig Application of Gauss law to any concentric spherical surface S of radius R < a shows that, at any point P inside S, the electrostatic field intensity is zero because there isn t any enclosed charge. The electrostatic potential V(R) at any point P interior to S must then be equal to the potential at the surface, so that

24 18 2 The Electric Field Fig. 2.3 Spherical surface S with origin O and radius a. The interior point P is at a distance R < a from O, and the variable distance from the surface S to P is denoted by r. S P R r O a V(R) = Q 4πǫ 0 a = S which results in the geometrical identity s 4πǫ 0 r da = Q 4πa 2 S da 4πǫ 0 r, (2.32) 1 a = 1 da 4πa 2 S r, R < a (2.33) If the surface charge density s = Q/4πa 2 is removed and a point charge Q placed at the interior point P, then the final expression in Eq. (2.33) is seen to be the average potential taken over the spherical surface S. This then establishes the following theorem: Theorem 2.4. The average electrostatic potential taken over any spherical Q surface is equal to 4πǫ 0a, where a > 0 is the radius of the sphere and Q is the total enclosed charge, provided that there is no charge outside the sphere. 2.4 The Concept of an Ideal Conductor In order to complete the formal mathematical structure of the electrostatic field, it is necessary to introduce the concept of an ideal conductor. Definition 2.2 (Ideal Conductor). An ideal conductor is defined as a medium inside and on the surface of which electric charge freely flows under the influence of an externally applied electric field. Such an idealized medium is necessary not only from a mathematical point of view as it provides a convenient approach by which idealized boundary conditions may be introduced into the theory, but also from a physical point of view since an ideal conductor may be viewed as a material system whose electrical properties can be approached in some physically realizable manner. Because the electrostatic field is itself an idealization, it is entirely compatible with the concept of an ideal (or perfect) conductor. As an immediate consequence of this definition, the electrostatic field inside an ideal conductor

25 2.4 The Concept of an Ideal Conductor 19 must be zero as any nonzero internal field would induce a current flow. The propagation of such a current either in or on an ideal conductor would necessarily involve a dissipation of energy in the field and hence cannot occur in any static state of the field. This result them leads to the following equivalent definition of an ideal conductor. Definition 2.3 (Ideal Conductor). An ideal conductor may be defined as a medium that is incapable of sustaining an electrostatic field in its interior. Application of Gauss law [see Eq. (2.15)] to the interior region V of an ideal conductor results in the immediate conclusion that (r) = ǫ 0 E(r) = 0; r V. (2.34) Hence, any net static charge on an ideal conductor must reside on its surface. When a conductor is charged, the charges arrange themselves in such a manner that the fields they produce in the interior region of the conductor body aremutuallybalancedandthe netinteriorelectricfieldiszero.ifanidealconductor is placed in an external electrostatic field, the charges flow temporarily within it (this has now temporarily become a non-static arrangement) so as to set up a surface charge distribution which produces an additional electric field that, when added to the initial external electrostatic field, results in a zero field inside the conductor once static conditions are reestablished. The electrostatic field external to the conductor body is altered in this process as one electrostatic arrangement is replaced by another. Fig. 2.4 A perfectly conducting body with a cavity. The dotted surface S, to which Gauss law is applied to determine the enclosed charge, lies entirely within the body and encloses the cavity. S As a practical application of the properties of an ideal conductor, consider a conducting body containing a cavity that is placed in an externally generated electrostatic field. Application of Gauss law to any surface S that encloses the cavity and is completely contained within the conducting body, as depicted in Fig. 2.4, yields Q enc = ǫ 0 S E ˆnda = 0 for the charge enclosed, since the electrostatic field vanishes on S. If charges are completely contained within the cavity, then the induced surface charge residing on the inner cavity surface exactly cancels their net charge. Hence, if no charge is enclosed within the cavity then there is no surface charge induced on the inner cavity surface. The following two special cases are then realized:

26 20 2 The Electric Field 1. No externally applied electrostatic field: A charge q inside the cavity then induces an equal but opposite surface charge on the inside surface of the conductor. In order to preserve the net charge neutrality of the conductor body itself, a net surface charge q must then appear on the outside surface of the conductor body which, in turn, produces an external E-field. 2. An externally applied electrostatic field: Let there now be no charge enclosed within the cavity. Configure the Gaussian surface S such that it is entirely within the conductor body and is situated an infinitesimal distance from the outer surface of the conductor body. Because Q enc = 0, the entire net charge induced on the conductor must be external to S and hence, resides on the outside surface of the conductor body. Thus, when a hollow conductor with zero enclosed charge is placed in an externally generated electrostatic field, a surface charge is induced only on the outside surface of the conductor body. Consequently, a charge situated in the region exterior to a hollow conductor does not produce an electrostatic field in the interior cavity of the conductor body, whereas a charge situated in the interior cavity of the conductor body produces an electrostatic field in the region external to the conductor. The first of these two results forms the basis of electrostatic shielding by which the enclosed region may be shielded from external fields. The boundary conditions imposed on the external electrostatic field vector E(r) at the surface of an ideal conductor follow directly from the fact that E identically vanishes inside the conductor body. Application of Gauss and Faraday s laws [see Eqs. (2.14) and (2.17)] to any point on the conductor surface yields ˆn E(r) = s(r)/ǫ 0, r S, (2.35) ˆn E(r) = 0, r S, (2.36) respectively, where ˆn is the unit outward normal vector to the conductor surfaces andwhere s(r)denotesthesurfacechargedensityons.thesecond relation states that the tangential component of the external electrostatic field must vanish on the conductor surface if there is to be no induced current flow along the surface. Hence, the electrostatic field vector must be everywhere normal to the conductor surface S so that, from Eq. (2.35), E(r) = ˆn s(r) ǫ 0, r S. (2.37) Furthermore, because E(r) = V(r), this result shows that the electrostatic scalar potential of the field must be constant on the surface of an ideal conductor; that is, the surface of an ideal conductor is an equipotential surface of the electrostatic field. The distribution of charge over the conductor surface is then given by s = ǫ 0ˆn E = ǫ 0ˆn V, so that

27 2.5 The Electric Dipole, Quadrupole, and Multipoles 21 V(r) s(r) = ǫ 0, r S, (2.38) n the derivative of the potential being taken along the outward normal to the surface. The total charge on the conductor is then given by V Q s = ǫ 0 da. (2.39) n Consider now the scalar potential V(r) that is established in some region D of space by a system of chargedconductorsand (or) other external sources. The potential then satisfies Laplace s equation in D. Assume that this potential hasamaximum value at somepoint P D that is not onthe boundaryof some subregion where there is no field. The point P may then be surrounded byaclosedsurfacethatiscontainedentirelywithintheregiond andonwhich thenormalderivativesatisfiestheinequality V/ n < 0;ifP isonthe boundary of a subregion where there is no field, then 2 V = 0 in the region about that point and the field domain may accordingly be extended (in a mathematical sense) across the boundary without altering the field. Integration over the surface then gives V nda < 0. However, application of the divergence theorem followed by Laplace s equation yields V n da = 2 Vd 3 r = 0, and one has obtained a contradiction. Hence, in any charge-free region of space the electrostatic potential V(r) can assume maximum and minimum values only at the boundaries of regions where there is a field. This then shows that a test charge q that is introduced into an electrostatic field cannot be in static equilibrium because there is no point where its potential energy qv would have a minimum. This result may then be generalized as: Theorem 2.5 (Earnshaw s Theorem). A charged body placed in an electrostatic field cannot be maintained in stable equilibrium under the influence of the electrostatic forces alone. S 2.5 The Electric Dipole, Quadrupole, and Multipoles The properties of the electrostatic field produced by simple charge configurations leads to important results concerning the dielectric properties of material media and so are now considered in some detail The Static Electric Dipole The electric dipole consists of a positive and negative charge of equal magnitude Q separated by a distance s, as depicted in Fig With the z-axis

28 22 2 The Electric Field z P(r,θ,φ) Fig. 2.5 Static electric dipole formed by two point charges of equal but opposite charge ±Q separated by the fixed distance s. The origin of coordinates O is taken at the midpoint of the line joining the two point charges and the field observation point P is located at the spherical polar coordinates (r, θ, φ). x +Q s/2 O s/2 -Q φ θ v 1 r r r b r a y along the dipole axis through the two point charges and the origin O at the midpoint between them, the electrostatic potential at any point P(r,θ,φ) is given by the superposition of the potential due to each charge alone as V(r,θ,φ) = Q 4πǫ 0 ( 1 r b 1 r a ). (2.40) Application of the law of cosines to the pair of triangles ±QOP in Fig. 2.5 gives r± 2 = r2 + (s/2) 2 ± rscosθ with r a = r + and r b = r, so that ( ( ) 2 ( ) 3 r r ± = 1 1 s 2 2 4r ± s 2 r )+ cosθ 3 s 2 8 4r ± s 2 r cosθ 15 s r ± s 2 r cosθ +. Upon arranging terms in ascending powers of s/r, there results r = 1 s r ± 2r P 1(cosθ)+ s2 4r 2P 2(cosθ) 15s3 8r 3 P 3(cosθ)+, (2.41) wherep 0 (cosθ) = 1,P 1 (cosθ) = cosθ, P 2 (cosθ) = 1 2 (3cos2 θ 1),P 3 (cosθ) = 1 2 (5cos3 θ 3cosθ), etc.,arethe Legendrepolynomials(seeAppendixC). The electrostatic dipole potential is then given by V(r,θ,φ) = Qs 4πǫ 0 r 2P 1(cosθ) [1+ 15s2 4r 2 P 3 (cosθ) P 1 (cosθ) + ], (2.42) which falls off as 1/r 2, the potential due to a point charge falling off as 1/r. The dipole moment of the charge pair is defined as p Qs = Qsˆ1 s (2.43) withmagnitudep = Qs,wheretheunitvector ˆ1 s isdirectedfromthenegative to the positive charge along the dipole axis. With this definition, the first-

29 2.5 The Electric Dipole, Quadrupole, and Multipoles 23 term approximation of the expression (2.42) for the absolute electrostatic dipole potential becomes V(r) p ˆ1 r 4πǫ 0 r 2, r3 s 3, (2.44) the correction factor being given by the bracketed quantity in Eq. (2.42). An ideal point dipole is obtained in the limit as s 0 with fixed dipole moment p and so is given by Eq. (2.44) without approximation. Equipotential surfaces V(r,θ) = constant for an ideal point dipole are indicated by the dotted curves in Fig. 2.6, with darker shading indicating increasing potential magnitude. Notice that the absolute potential is positive (negative) in the half-space that the dipole moment p is directed into (out of), the potential vanishing in the equatorial plane θ = π/2. In addition, notice that the Axis of Symmetry (Dipole Axis) Fig. 2.6 Electrostatic field lines of force (dashed curves) and equipotentials (dotted curves) for an ideal point dipole. The full three-dimensional field structure is obtained by rotating the figure about the dipole axis. The darkness of shading between neighboring equipotentials indicates increasing potential magnitude. V > 0 V = 0 V < 0 r dr E r ds rdθ E θ E(r,θ) dipole potential identically vanishes in the equatorial plane (θ = π/2) since r a = r b [ for all s 0. Finally, + ] for a non-ideal dipole (s > 0), the correction factor 1+ 15s2 P 3(cosθ) 4r 2 P 1(cosθ) appearing in Eq. (2.42) begins to deviate significantly from unity when r decreases below 2s. For smaller values of r, a series expansion of 1/r ± in powers of r/s is appropriate. The electrostatic field vector for a point dipole is obtained from the negative gradient of either Eq. (2.44) or Eq. (2.42) with the result E(r) = 3(p r)r r2 p 4πǫ 0 r 5 = p 4πǫ 0 r 3 [ˆ1 r 2cosθ+ ˆ1 θ sinθ ], (2.45) the second form of the result applying when the point dipole p = pˆ1 z is located at the origin as in Fig As indicated in Fig. 2.6, the lines of force

30 24 2 The Electric Field for this field are specified by dr rdθ = Er E θ = 2cosθ sinθ so that dr r = 2cosθdθ sinθ = 2d(sin θ) sinθ, with solution r = Asin 2 θ. Electrostatic field lines are indicated by the dashed curves in Fig. 2.6, the electrostatic dipole field vector being in the direction from higher to lower potential, as indicated by the arrows in the figure The Linear Electric Quadrupole The linear electric quadrupole consists of an inverted pair of electric dipoles of equal dipole moment magnitude p = Qs with centers displaced by the dipole separation s along their common axis, as illustrated in Fig With the Fig. 2.7 Linear electric quadrupole formed by two colinear dipoles of equal but opposite moment ±p with centers displaced by the dipole separation s along their common axis. The origin of coordinates O is taken at the midpoint of the line joining the two dipoles and the field observation point P is located at the spherical polar coordinates (r, θ, φ). z +Q s -2Q s +Q v 1 r θ r b r r a P(r,θ,φ) z-axis taken along the linear quadrupole axis with origin O at the midpoint of the arrangement, the potential at any point P(r,θ,φ) is given by the superposition of the potential due to each charge alone as V(r,θ,φ) = Q 4πǫ 0 ( 1 r a + 1 r b 2 r ) = Q 4πǫ 0 r ( r r a + r r b 2 ). (2.46) From the expansion given in Eq. (2.41) [with s 2 replaced by s] one has that r r a 1 s r P 1(cosθ) + s2 r P 2 2 (cosθ) and r r b 1 + s r P 1(cosθ) + s2 r P 2 2 (cosθ), provided that (s/r) 3 1. With these substitutions, the expression (2.46) for the absolute electrostatic potential of a linear quadrupole becomes V(r,θ,φ) 2Qs2 4πǫ 0 r 3P 2(cosθ), r 3 s 3, (2.47)

31 2.5 The Electric Dipole, Quadrupole, and Multipoles 25 which varies inversely as the cube of the radial distance r > 0 from the center of the quadrupole charge structure. Equipotential surfaces and electrostatic field lines for the ideal linear quadrupole are illustrated in Fig Axis of Symmetry (Quadrupole Axis) V > 0 Fig. 2.8 Electrostatic field lines of force (dashed curves) and equipotentials (dotted curves) for an ideal axial point quadrupole. The full three-dimensional field structure is obtained by rotating the figure about the quadrupole axis. The darkness of shading between neighboring equipotentials indicates increasing potential magnitude. V > 0 V = 0 V < 0 V = Static Electric Multipoles The field properties of static electric multipoles can be understood through the following geometric construction sequence (with s j s): The monopole field is a single point charge Q with potential V 0 Q/r. The dipole field is obtained by displacing a monopole through a distance s 1 and replacing it by one of equal but opposite sign, resulting in the potential V 1 Qs/r 2. The quadrupole field is obtained by displacing a dipole through a distance s 2 and replacing it by one of equal but opposite sign, resulting in the potential V 2 Qs 2 /r 3. The octupole field is obtained by displacing a quadrupole through a distance s 3 and replacingit byoneofequalbut oppositesign,resultinginthe potential

32 26 2 The Electric Field V 3 Qs 3 /r 4. For a 2 l -multipole, with l denoting the number of independent displacements s 1,s 2,,s l required to specify the charge arrangement, the potential is and the associated electric field then varies as V l Qs l /r l+1, (2.48) E l Qs l /r l+2. (2.49) For a linear point multipole, the angular dependence is described by the correspondinglegendre polynomial P l (cosθ) given in Eq. (C.70) ofappendix C. The electrostatic potential due to an ideal monopole is then given by V 0 (r,θ,φ) = Q 4πǫ 0 r P 0(cosθ) (2.50) with P 0 (ζ) = 1, that due to an ideal (point) dipole [cf. Eq. (2.44)] V 1 (r,θ,φ) = Qs 4πǫ 0 r 2P 1(cosθ) (2.51) with P 1 (ζ) = ζ, that due to an ideal linear (point) quadrupole [cf. Eq. (2.47)] V 2 (r,θ,φ) = Qs2 4πǫ 0 r 3P 2(cosθ) (2.52) with P 2 (ζ) = 1 2 (3ζ2 1), that due to an ideal (point) octupole with P 3 (ζ) = 1 2 (5ζ3 3ζ), and so-on. V 3 (r,θ,φ) = Qs3 4πǫ 0 r 4P 3(cosθ), (2.53) 2.6 The Electrostatic Field Produced by an Arbitrary Static Charge Distribution Consider now obtaining an expansion of the electrostatic field produced by an arbitrary (but fixed) charge distribution with density (r ) occupying a region τ in free-space and extending to a maximum distance r max from the origin O of a fixed coordinate system in terms of its multipole moments about that point, as illustrated in Fig It is assumed that O is positioned either within or is in close proximity to the charged region τ. The absolute electrostatic potential V(r) = V(x,y,z) is then given by V(x,y,z) = 1 (x,y,z ) 4πǫ 0 τ r d 3 r (2.54)

33 2.6 The Electrostatic Field Produced by an Arbitrary Static Charge Distribution 27 z Fig. 2.9 An arbitrary (but fixed) charge distribution (r ) occupying a finite region τ in freespace. The unit vector ˆ1 r is directed from the origin O (assumed to be located either within τ or in close proximity to it) to the field point P(x,y,z) a distance r away, where r > r max. x O τ V 1 r P (x,y,z ) r r r y P(x,y,z) where r = [ (x x ) 2 +(y y ) 2 +(z z ) 2] 1/2 is the distance from the source point at P (x,y,z ) to the field point at P(x,y,z) a distance r from O. Because the source point P is near to the origin O and provided that the field point P is far removed from O such that r > r max, the quantity 1/r may be expanded in a Taylor series about the origin as 1 r = 1 O r +(x x +y y +z z ) 1 O r + 1 2! (x x +y y +z z ) 2 1 O r +, (2.55) where (x x +y y +z z ) 2 = x x +y 2 y +z z 2 +2x y 2 x y + 2x z 2 x z +2y z 2 y z. The first term appearing in the Taylor series expansion (2.55) is given by 1/r O = 1/r (x,y,z )=(0,0,0) = 1/r. For the second term, one has that ( / x )(1/r ) O = (x x )/r 3 O = l/r 2, and similarly that ( / y )(1/r ) O = m/r 2 and ( / z )(1/r ) O = n/r 2, where l x/r, m y/r, and n z/r are the cosines of the angles between the position vector r = ˆ1 r r and the x, y, and z-axes, respectively. For the third term, one has that ( 2 / x 2 )(1/r ) O = (3l 2 )/r 3, ( 2 / y 2 )(1/r ) O = (3m 2 )/r 3, ( 2 / z 2 )(1/r ) O = (3n 2 )/r 3, ( 2 / x y )(1/r ) O = ( 2 / y x )(1/r ) O = 3lm/r 3, ( 2 / x z )(1/r ) O = ( 2 / z x )(1/r ) O = 3ln/r 3, and ( 2 / y z )(1/r ) O = ( 2 / z y )(1/r ) O = 3mn/r 3. With these substitutions in Eq. (2.55), the expression (2.54) for the electrostatic potential at the field point P(r) = P(x,y,z) may be expressed as V(x,y,z) = V 0 (x,y,z)+v 1 (x,y,z)+v 2 (x,y,z)+, (2.56) where each term V j (x,y,z), j = 0,1,2,3,..., corresponds to the related term in the Taylor series expansion of 1/r. The zeroth-order term in this expansion, V 0 (x,y,z) = 1 4πǫ 0 r τ (x,y,z )d 3 r = Q 4πǫ 0 r, (2.57)

34 28 2 The Electric Field where Q is the net charge in τ, is called the monopole term because it is the potential one would have at P if the entire charge distribution were concentrated at O. This term is zero in the multipole expansion (2.56) only if thenetchargeqiszero.ifqisnon-vanishing,thenv 0 (x,y,z)isthedominant term in the multipole expansion as r because it decreases only as r 1. Notice that the value of Q is independent of the origin O. The first-order term in the multipole expansion (2.56) is the dipole term V 1 (x,y,z) = p ˆ1 r 4πǫ 0 r 2 (2.58) with ˆ1 r ˆ1 x l + ˆ1 y m + ˆ1 z n denoting the unit vector along the radial line from O to the field point P(r) = P(x,y,z), as indicated in Fig. 2.9, where p r (r )d 3 r (2.59) τ is the dipole moment of the charge distribution in τ taken with respect to O. Hence, the first-order term V 1 (r) describes the potential at the field point P(r) due to an effective dipole at the origin O with dipole moment p. The dipole moment of an extended charge distribution may also be defined as p = Q r, (2.60) where Q is the total net charge in the distribution, and where r τ r (r )d 3 r = τ (r = 1 r (r )d 3 r (2.61) )d 3 r Q τ is a vector extending from the origin O to the chargecentroid of the extended charge distribution. Notice that if Q = 0, then r and p, as given by Eq.(2.60), is indeterminate; however, Eq.(2.59) always determines the dipole moment unambiguously and is to be used in that singular case. Notice that in the Q = 0 case, the dipole moment is independent of the choice of origin. Finally, if Q 0, then the dipole moment of the charge distribution can always be made to vanish by choosing the origin O at the centroid of the charge distribution which is determined by setting r = 0. The second-order term in the expansion (2.56) is the quadrupole term [ 1 V 3 (r) = 4πǫ 0 r 3 3lmQ xy +3lnQ xz +3mnQ yz (3l2 1)Q xx (3m2 1)Q yy (3n2 1)Q zz ]. (2.62) The scalar quantities Q αβ define the quadrupole moment tensor Q = (Q αβ ), α,β = x,y,z, of the charge distribution, with

35 2.7 The Concept of a Perfect Dielectric 29 Q αβ = Q βα α β (r )d 3 r = Qα β. τ (2.63) This set of expressions for the quadrupole term is simplified considerably if the charge distribution possesses certain symmetries. For example, if the charge distribution in τ possesses cylindrical symmetry about the z-axis, then Q xy = Q yz = Q xz = 0 and Q xx = Q yy. It is then convenient to define a single scalar quadrupole moment Q of the charge distribution as Q 2(Q zz Q xx ) = (3z 2 r 2 ) (r )d 3 r, (2.64) τ where r 2 = x 2 +y 2 +z 2. Under these conditions, Eq. (2.63) becomes V 3 (r) = Q 3n 2 1 4πǫ 0 4r 3 = Q 3cos 2 θ 1 4πǫ 0 4r 3, (2.65) where n z/r = cosθ with θ denoting the angle between the positive z-axis and the line segment oflength r extending from the origino to the field point P = P(r,θ). The multipole expansion of the electrostatic potential due to a cylindrically-symmetric charge distribution (r ) is then given by [cf. Eqs. (2.50) (2.53)] V(r,θ) = Q 4πǫ 0 r + p 4πǫ 0 r 2P 1(cosθ)+ Q 8πǫ 0 r 3P 2(cosθ)+ (2.66) in terms of the Legendre polynomials P l (cosθ). The first nonvanishing term in this multipole expansion then dominates the behavior of V(r,θ) as r. 2.7 The Concept of a Perfect Dielectric In a perfect dielectric medium, all of the charged particles are bound either in atomic or molecular configurations. When an external electric field is applied, the positive and negative charges bound in each molecule are displaced in opposite directions and the molecular charge density of each molecule is accordingly distorted. Positive charge is displaced in the direction of E and negative charge in the opposite direction so that the induced molecular dipole moment is in the same direction as E; each molecule then produces an average electric field that is in a direction opposite to E. After static conditions are re-established, the multipole moments of each molecule will differ from their zero field (unperturbed) values. For a simple dielectric, the dominant multipole that is induced is the dipole, all higher-order multipoles being negligible by comparison. The dielectric is then said to be polarized by the external electric field with its molecules possessing induced dipole moments. In a nonpolar dielectric, the molecules have zero permanent dipole moments

36 30 2 The Electric Field so that, in the absence of an applied electric field, the dipole (and higherorder multipole) moments are all zero. In a polar dielectric, the molecules possess a nonzero permanent dipole moment such that, in the absence of an applied field, the molecular dipole moments are randomly oriented so that the spatially-averaged dipole(and higher-order multipole) moments are again all zero. Application of an external electric field in such a simple dielectric then produces a macroscopic electric polarization density P(r) given by the spatial average of the microscopic dipole moment as P(r) = j N j p j (r), (2.67) where p j (r) is the microscopic dipole moment of the jth-type of molecule comprising the dielectric, p j (r) is the spatial average of this microscopic dipole moment taken over a macroscopically small but microscopically large region centered at the point r, and where N j is the averagenumber density of j-type molecules in that region. Because the net charge in a perfect dielectric is zero, this macroscopic dipole moment density P(r) is independent of the choice of origin [see the discussion following Eq. (2.61)]. The connection between microscopic and macroscopic field quantities is obtained through the same spatial-averaging process. The macroscopic electric field vector E(r) for the electrostatic field is defined as the spatial average of the microscopic electric field vector e(r) as E(r) e(r). (2.68) There is then no distinction between microscopic and macroscopic fields in vacuum. When the spatial-averaging procedure is applied to Faraday s law (2.17) for the microscopic electrostatic field, the same equation results, viz. E(r) = 0. (2.69) This result then implies that the macroscopic field may likewise be expressed in terms of a macroscopic scalar potential V(r) as E(r) = V(r). (2.70) The spatial average of Gauss law (2.14) for the microscopic electrostatic field yields E(r) = 1 ǫ 0 ρ(r), (2.71) so that the divergence of the macroscopic electrostatic field vector is determined by the spatial average of the microscopic charge density in the dielectric. The proper description of this quantity deserves careful attention (see Ch. 3 of Lorrain and Corson [7] and Ch. 13 of Kittel [8] as well as Vol. I of Böttcher s two volume treatise on electric polarization [9]). Consider first determining the electrostatic field produced by a dielectric body with macroscopic dipole moment density P(r) at a field point exterior

37 2.7 The Concept of a Perfect Dielectric 31 to the body in vacuum. From Eq. (2.44), the potential dv(r) produced at the exterior point r due to the dipole P(r )d 3 r at the interior point r is given by (4πǫ 0 )dv(r) = (P(r ) ˆ1 r /r 2 )d 3 r = [P(r ) (1/r)]d 3 r, where ˆ1 r is the unit vector directed from the source point r to the field point r, separated by the (nonvanishing) distance r = [(x x ) 2 +(y y ) 2 +(z z ) 2 ] 1/2, and where operates on the primed (source) coordinates. The total potential at the exterior point r in vacuum is then given by V(r) = 1 4πǫ 0 = 1 4πǫ 0 = 1 4πǫ 0 P ( ) 1 d 3 r V r [ ( P V r [ P ˆn d 2 r r S ) d 3 r V V ] P d 3 r, r ] P d 3 r r where S is the boundary surface to the dielectric region V with outward unit normal vector ˆn. Comparison of this result with the form (2.9) of Coulomb s law shows that this external field is produced by both a surface polarization charge density sb (r) P(r) ˆn,r S, and a volume polarization charge density b(r) P(r),r V, so that V(r) = 1 4πǫ 0 [ S sb (r ) d 2 r + r V b(r ] ) d 3 r. r Consider next an arbitrary but fixed point ζ inside the dielectric body V such that a sphere of radius R centered at ζ can be constructed with surface S ζ lying entirely within V, thereby dividing the dielectric into two regions, the region V inside S ζ and the region V outside S ζ but inside S. The radius R is chosen sufficiently small such that the enclosed volume V is macroscopically small, in which case the macroscopic electric field vector E(r), dipole moment density P(r), and polarization charge density b = P do not vary appreciably in V. The potential at ζ due to the dipole distribution in the exterior region V is given by the preceding result with the surface integral taken over both S and S ζ and the volume integral taken over V, with electric field intensity E (ζ) = 1 4πǫ 0 [ S ˆr sb(r ) r 2 d 2 r + ˆr sb(r ] ) S ζ r 2 d 2 r + ˆr b(r ) V r 2 d 3 r, where ˆr is the unit vector from the dipole source to the field point ζ and r is the distance between these two points. The average electric field intensity E (ζ) due to the near dipoles enclosed by S ζ is given by the summation overj of the product of the molecular dipole field p j /(4πǫ 0 R 3 ), where p j denotes the average dipole moment of the j-type molecule in S ζ, times the number density N j of j-type molecules times

38 32 2 The Electric Field the volume (4/3)πR 3 enclosed by S ζ, so that E (ζ) = 4 3 πr3 j N j p j (r) 4πǫ 0 R 3 = P, 3ǫ 0 where P = j N j p j (r), as defined in Eq. (2.67). The integral taken over the spherical surface S ζ centered at ζ may be evaluated in the following manner. Construct an axis in the direction ˆ1 P parallel to the dipole moment density P and let θ be the angle between ˆ1 P and the line extending from a point on the surface S ζ to the point ζ, so that sb = P ˆn = P cosθ, where ˆn is the unit normal to the surface S ζ directed inwards toward the center ζ. Then ˆr sb(r ) S ζ r 2 d 2 r = 2πP π 0 cos 2 θsinθdθ = 4 3 πp, since ˆr = ˆ1 ζ cosθ for this geometry. This term then cancels the near dipole contribution E (ζ) = P/3ǫ 0 given above. The total electrostatic field intensity produced at an interior point of a polarized dielectric is then given by E(r) = 1 4πǫ 0 [ S ˆr sb(r ) r 2 d 2 r + V ˆr b(r ] ) r 2 d 3 r, where the integration over the region V has been extended to the entire volume V of the dielectric body by adding the contribution from the spherical region V which vanishes as it describes the electrostatic field at the center of a uniform spherical charge distribution. Taken together, these results show that the electrostatic potential and field both inside and outside a simple dielectric with free charge are given by V(r) = 1 [ 4πǫ 0 E(r) = 1 [ 4πǫ 0 S S s(r )+ sb (r ) d 2 r + r ˆr s(r )+ sb (r ) r 2 d 2 r + V V (r )+ b(r ] ) d 3 r r ˆr (r )+ b(r ) r 2 d 3 r where s is the free surface charge density on S and is the free volume charge density in V. For a simple dielectric, the spatial average of the microscopic charge density [see Eq. (2.71)] is thus seen to be given by, ], ρ(r) = (r) P(r), (2.72) where denotes the macroscopic charge density in the dielectric and P is the macroscopic polarization density defined in Eq. (2.67). The presence of the divergence of P(r) in this spatial average of the microscopic charge density

39 2.7 The Concept of a Perfect Dielectric 33 accounts for any spatial nonuniformity in this vector field and is referred to as the polarization or bound charge density b, where b(r) P(r) (2.73) for all points r in the interior of the dielectric body. In addition, associated with this macroscopic dipole moment density is the surface polarization charge density sb (r) P(r) ˆn (2.74) for all points on the surface of the dielectric body with outward unit normal vector ˆn. With this substitution, the spatial average of Gauss law becomes D(r) = (r), (2.75) where D(r) ǫ 0 E(r)+P(r) (2.76) is the electric displacement vector (in C/m 2 ) for a simple dielectric. Because E(r) = 1 ǫ 0 D(r) 1 ǫ 0 P(r), (2.77) the macroscopic electric field intensity E(r) inside a simple dielectric is then seen to be given by the sum of two vector fields: the field D(r)/ǫ 0 associated with the spatially-averaged molecular charge density (typically zero) plus any free charge f that is externally supplied, where (D(r)/ǫ 0 ) = (r)/ǫ 0 ; and the field P(r)/ǫ 0 associated with the bound or polarization charge of the dielectric, where ( P(r)/ǫ 0 ) = b(r)/ǫ 0. The field lines of the electric displacement vector D then begin and end only on externally supplied (free) charge as well as on the spatially-averaged molecular charge when this latter quantity is nonzero, whereas the lines of force of the macroscopic electric field vector E begin and end on either free or bound (polarization) charge. For a simple dielectric, the macroscopic electric polarization density P(r) is linearly related to and in the same direction as the macroscopic electric field intensity E(r) at that point, so that P(r) = ǫ 0 χ e E(r), (2.78) where χ e is the electric susceptibility of the simple dielectric. Notice that χ e is dimensionless and is real-valued in the static case. Taken together, Eqs. (2.76) and (2.78) give D(r) = ǫ 0 (1+χ e )E(r) = ǫ 0 ǫ r E(r) = ǫe(r), (2.79) where ǫ is the dielectric permittivity of the medium, given by

40 34 2 The Electric Field ǫ ǫ 0 (1+χ e ), (2.80) and where ǫ r = ǫ/ǫ 0 is the relative permittivity of the dielectric. Notice that ǫ is real-valued in the static case. If the dielectric material is spatially inhomogeneous so that χ e = χ e (r) and ǫ = ǫ(r) vary with position, then substitution of Eq. (2.71) into Gauss law (2.75) results in the expression ǫ(r) 2 V(r) + ǫ(r) V(r) = (r), which may be rewritten as 2 V(r)+ ( ln(ǫ(r)) ) V(r) = (r) ǫ(r). (2.81) If the dielectric is spatially homogeneous so that ǫ is independent of position within the material, then the above expression reduces to Poisson s equation 2 V(r) = (r) ǫ, (2.82) which further reduces to Laplace s equation in charge-free regions of space. Boundary conditions on the electrostatic field vectors across an interface S separating two simple dielectrics with permittivities ǫ 1 and ǫ 2 may be obtained by direct application of the integral form of Gauss law to a simple closed surface with identical faces on opposite sides of S and Faraday s law to a simple closed circuit with identical segments on opposite sides of S with the results ˆn (D 2 (r) D 1 (r) ) = s(r), r S, (2.83) ˆn ( E 2 (r) E 1 (r) ) = 0, r S, (2.84) where ˆnistheunitnormaltothesurfaceatthepointr,directedfrommedium 1 into medium Electrostatic Energy Whenever two charges q a and q b are brought within a distance R ab of each other, work is expended against the Coulombic force [Eq. (2.2)] in consummating the process. Once the charges are in place, the persistence of this force makes the energy stored in the electrostatic field potentially available whenever demanded. If it is assumed that the charges are moved slowly enough into place (i.e. reversibly), then their kinetic energies may be neglected and any loss due to electromagnetic radiation effects, significant if rapid charge accelerations occur, may then be neglected. Consider then the energy stored in a fixed configuration of n charges, given by the reversible work required to assemble the static charge configuration. Assume that all n chargesq 1,q 2,...,q n areinitially locatedat infinity in their zero potential state. Upon bringing just q 1 from infinity to its final position P 1, no workis expended because no other chargesare present. The workdone

41 2.8 Electrostatic Energy 35 in bringing q 2 from infinity to P 2 is then given by U 2 = q 2 V (1) 2 = q 1 V (2) 1, where V (1) 2 = q 1 /(4πǫ 0 R 12 ) denotes the potential at P 2 due to the charge q 1 at P 1, and where V (2) 1 = q 2 /(4πǫ 0 R 21 ) denotes the potential at P 1 due to the charge q 2 at P 2. The work done in bringing a third charge q 3 in from infinity to P 3 is then given by U 3 = q 3 V (1) 3 +q 3 V (2) 3 = q 1 V (3) 1 +q 2 V (3) 2, and so on for the remaining charges q 4,q 5,...,q n, taking note of the symmetry relation q k V (j) k = q j V (k) j (2.85) where V (j) k denotes the electrostatic potential at P k due to the charge q j at P j. The total energy U e = U 1 +U 2 + +U n stored in the assembled charge configuration can then be written in two different ways: First by adding the first forms of the above equations, giving U e = q 2 V (1) 2 + q 3 V (1) 3 + q 3 V (2) 3 + q 4 V (1) 4 +q 4 V (2) 4 +q 4 V (3) 4 + +q n V n (1) +q n V n (2) + +q n V (n 1) N,orbyaddingthe second forms of the above equations, giving U e = q 1 V (2) 1 +q 1 V (3) 1 +q 2 V (3) q n 1 V (n). The average q 1 V (4) 1 +q 2 V (4) 2 +q 3 V (4) 3 + +q 1 V (4) 1 +q 2 V (n) of these two expressions then yields the result U e = 1 2 n q k V k (J) (2.86) k=1 for the potential energy of the assembled charge configuration, where q k denotes the charge of the k th particle located at the fixed point P k, and where V k denotes the absolute potential at P k due to all of the chargesin the configuration except q k. This result has rather general applicability provided that the potential V k is properly determined. Notice further that this expression does not include the self-energy of the individual charges, this being defined as the energy that would be liberated if each charge was allowed to expand to an infinite volume. Because of this, Eq. (2.86) identically vanishes for a single point charge, as required. For a continuous volume charge density (r), the expression given in Eq. (2.86) for the electrostatic potential energy generalizes to U e = 1 2 V (r)v(r)d 3 r (2.87) with analogous expressions for surface S(r) and line l(r) charge densities. Notice that this generalized expression for the electrostatic potential energy includes the self-energies of the charges. For example, for a uniform spherical charge distribution = 3Q/(4πr 3 0 ) for r r 0 with total charge Q and radius r 0, the absolute electrostatic potential inside the sphere is found to be given by N

42 36 2 The Electric Field V(r) = Q ( r 2 8πǫ 0 r0 3 0 r 2) + Q ; r r 0. 4πǫ 0 r 0 From Eq. (2.87), the self-energy of this spherical charge distribution is then U se = 3Q r0 [ Q 2r πǫ 0 r0 3 ( r 2 0 r 2) + Q ]r 2 dr = 3Q2, 4πǫ 0 r 0 20πǫ 0 r 0 so that U se as r 0 0 with fixed Q 0, showing that it requires infinite energy to construct an ideal point charge. On the other hand, U se = ( /ǫ 0 )r0 2 0asr 0 0withfixedchargedensity ;thegeneralizedexpression (2.87) for the electrostatic potential energy should then be viewed from this latter point of view. Consider finally deriving an expression for the electrostatic energy in terms of the field quantities alone. From Poisson s equation for a spatially homogeneous simple dielectric [see Eq. (2.82)], the charge density may be expressed as (r) = ǫ 2 V(r) at every point in the field which, when substituted in Eq. (2.87), yields U e = ǫ V(r) 2 V(r)d 3 r, (2.88) 2 V where V is any volume containing all of the charges in the system. From Green s first integral identity ( V φ 2 ψ+ φ ψ ) d 3 r = S φ ψ ˆnd2 r with φ(r) = ψ(r) = V(r), oneobtains ( V V 2 V +( V) 2) d 3 r = S V V ˆnd2 r, so that U e = ǫ [ ] V V ˆnd 2 r ( V) 2 d 3 r (2.89) 2 S Because V can be any volume that contains all of the charges in the system, the surface S may be chosen at an arbitrarily large distance from the charge distribution. Furthermore, because V(r) falls off at least as fast as 1/r as r, then V(r) falls off at least as fast as 1/r 2 as r, and because the surface area of S increases as r 2 in that limit, then the surface integral appearing in Eq. (2.89) decreases at least as fast as 1/r as r and can be made arbitrarily small by choosing S sufficiently distant from the source charge distribution. Because V(r) = E(r), the electrostatic energy is then given by U e = 1 2 V D(r) E(r)d 3 r = ǫ 2 V V E(r) 2 d 3 r (2.90) where the volume V must only be large enough to include all regions where the electrostatic field E(r) produced by the charge distribution is nonzero; if this is not satisfied, then the electrostatic energy is given by Eq.(2.89). Notice that this expression includes the self-energies of all the charges in the system. The integrand in Eq. (2.90) is defined as the electrostatic energy density

43 2.8 Electrostatic Energy 37 u e (r) 1 2 D(r) E(r) (J/m3 ) (2.91) which is associated with the field energy at each point in space Capacitance and Energy of Multi-Conductor Systems The total energy of the electrostatic field produced by a system of n charged ideal conductors embedded in a spatially homogeneous simple dielectric medium is given by Eq. (2.90) with the integration taken over the entire region V external to the conductors. Because E = V in V, this expression may be rewritten as U e = ǫ E Vd 3 r 2 = ǫ [ ] (VE)d 3 r V Ed 3 r 2 = ǫ (VE)d 3 r, 2 the second integral vanishing because E = 0 throughout V. Application of the divergence theorem transforms the remaining integral into a sum of surface integrals taken over each surface S j of the entire system of conductors which spatially bound the electrostatic field plus an additional integral over an infinitely remote surface. This latter integral vanishes if the system of charged conductors is situated within a finite region of space as the electrostatic field produced by them then diminishes with sufficient rapidity (E R 2 and V R 1 ) as R. With the subscript j denoting the j th conductor and V j the constant value of the electrostatic potential on that conductor, the above expression becomes U e = ǫ 2 n V j E ˆnda, S j j=1 where ˆn here denotes the unit outward normal to the conductor surface, directed from the conductor body into the field region (notice that this is opposite to the convention used in the divergence theorem, as reflected in the change of sign in the above equation). Because Q j = ǫ S j E ˆnda, one finally obtains U e = 1 n Q j V j (2.92) 2 j=1

44 38 2 The Electric Field which is analogous to the expression (2.86) for the electrostatic potential energy of a system of point charges. The charges Q j and potentials V j of the conductor bodies cannot both be arbitrarily prescribed and consequently must be related in some fashion. Because the field equations are linear and homogeneous, it is expected that these relations should also be linear. In order to prove this 6, consider a fixed geometrical arrangement of n conductor bodies. First, assume that all of the conductors are uncharged except for the k th conductor which carries the nonzero charge Q k. Let the corresponding solution of Laplace s equation in the region V exterior to the conductor bodies be denoted by V (k) (r) and let V (k) j, j = 1,2,...,n, denote the constant potential value on the j (th) conductor. Now let the charge on the k th conductor be changed to αq k, where α is a constant. The function αv (k) (r) then satisfies Laplace s equation in V. The electrostatic potential and all of its derivatives are thus everywhere changed by the factor α. In addition, because s = ǫ V/ n, it then follows that the charge density on each conductor surface S j is multiplied by the same factor α, in whichcasethe chargeonthek th conductorbody becomesαq k while the other conductors remain uncharged. Because a solution of Laplace s equation that satisfies a particular set of Dirichlet boundary conditions is unique (see ), the potential αv (k) (r) is then the correct solution of this modified system, establishing the linearity of the relationship between the charges and potentials of the conductor bodies. More importantly, however, is the conclusion that the potential of each conductor body is directly proportional to the charge Q k of the k th conductor. Hence V (k) j = p jk Q k, j = 1,2,...,n, where the coefficients p jk depend only upon the geometry of the conductor system. By superposition, the electrostatic potential on the j th conductor when all of the conductors are charged is given by V j = n k=1 V (k) j, so that V j = n p jk Q k (2.93) k=1 forj = 1,2,...,n,whichisthedesiredlinearrelationshipbetweenthecharges and potentials on the conductor bodies of the n conductor system. The coefficients p jk, which depend only upon the geometry of the multi-conductor system, are called coefficients of potential, where p jk is the potential of the j th conductor per unit charge on the k th conductor. Substitution of this result into Eq. (2.92) for the potential energy gives 6 The proof presented here follows that given in 3.12 and 6.5 of Reitz and Milford [10].

45 2.8 Electrostatic Energy 39 U e = 1 2 n j=1 k=1 n p jk Q j Q k (2.94) Hence, the electrostatic potential energy of a system of charged conductors is a quadratic function of the charges on the conductors comprising the system. Theorem 2.6. The coefficients of potential p jk for a multi-conductor system satisfy the three fundamental properties: p jk = p kj, p jk 0, (2.95) p jj p jk, k. Proof. (1). The first property follows from Eq. (2.94). The total differential of U e is given by ( ) ( ) ( ) Ue Ue Ue du e = dq 1 + dq dq n. Q 1 Q 2 Q n If the only charge in the n-conductor system that is changed is Q j, then ( ) Ue du e = dq j = 1 Q j 2 n (p jk +p kj )Q j dq j. The work expended (and hence the change in potential energy) transporting an element of charge dq j from a zero potential reservoir to the j th conductor is given by n du e = V j dq j = p jk Q k dq j. Comparison of this expression for du e with the preceding expression then gives 1 n 2 k=1 (p jk+p kj )Q k = n k=1 p jkq k. Because this result must be valid for all possible values of Q k, one must then have that 1 2 (p jk+p kj ) = p jk, and hence p jk = p kj, which proves the first property. (2).Thesecondpropertythatthecoefficientsofpotentialp jk arenon-negative followsfromthefactthatthepotentialduetoanetpositivechargeispositive. (3). In ordertoprovethe third property,let the j th conductorcarryapositive charge Q j while all the remaining conductors in the n conductor system remain uncharged. Because the k th conductor (k j) is uncharged, then the net number of electrostatic lines of force leaving this conductor body is zero. Two distinct cases must then be separately considered: Case I: There are no lines of force either leaving or impinging upon the k th conductor, in which case it is in an equipotential region, and hence, is shielded by another conductor. It is then either contained within the body of the charged j th conductor, in which case its potential is V j so that k=1 k=1

46 40 2 The Electric Field p jk = p jj, or it is located inside some other conductor l j in which case p jk = p jl and attention is then shifted to the l th conductor. Case II: The lines of electric flux leaving the k th conductor are balanced by the lines of flux impinging upon it. Because the origin of this electric flux is the charge Q j on the j th conductor, it is then possible to trace a given flux line that is impinging on the k th conductor back (possibly via other conductors) to the j th conductor. This means that the j th conductor is at a higher potential than the k th conductor, so that V j > V k when Q j > 0. From Eq. (2.93), this then implies that p jj > p jk, to which an equality sign must be added to cover Case I. This proves the third property, completing the proof of the theorem. The relation given in Eq. (2.93) is a set of n linear equations expressing the potentials V j on each of the n conductors in terms of the charges Q j residing on them. This system of equations may be inverted to yield a set of equations giving the charge values Q j on the conductors in terms of their potentials V j with the result Q j = n c jk V k (2.96) j=1 for j = 1,2,...,n. The coefficients c jj are the coefficients of capacitance (or capacity coefficients) whereas the coefficients c jk with j k are the electrostatic induction coefficients. The coefficients of capacitance and induction form a matrix C = (c jk ), with the coefficients of capacitance forming the diagonal and the coefficients of inductance the off-diagonal elements, that is the inverse of the matrix P = (p jk ) of the coefficients of potential p jk, so that C = P 1. (2.97) As a consequence of theorem 2.6, the following equivalent theorem holds.

47 2.8 Electrostatic Energy 41 Theorem 2.7. The coefficients of capacitance and induction c jk for a multiconductor system satisfy the three fundamental properties: c jk = c kj, c jj > 0, (2.98) c jk 0, j k. Substitution of Eq. (2.96) into Eq. (2.92) for the electrostatic potential energy yields U e = 1 n n c jk V j V k (2.99) 2 j=1 k=1 The electrostatic potential energy of a system of charged conductors is thus seen to be a quadratic function of either the charges or the potentials on the various conductors comprising the system. From Eq. (2.96), the capacitance of a conductor is seen to be given by the total charge on the conductor when it is maintained at unit potential, all other conductors in the system being held at zero potential. In that case Q j = c jj V j and c jj = Q j V j. (2.100) A pair of isolated conductors which can store equal and opposite charges ±Q, independently of whether other conductors in the system are charged, form what is known as a capacitor. This independence from the presence of other charges in the system implies that one of the conductor bodies shields the other, as illustrated in Fig. 2.10, so that the electrostatic potential contributed to each conductor of the pair by any external charge is the same; if this isn t the case, then one must consider the entire multi-capacitor system comprised of the n conductors. If two ideal conductors S 1 and S 2 form such a capacitor, application of Eq. (2.93) to that arrangement yields V 1 = p 11 ( Q)+p 12 Q+V E, V 2 = p 21 ( Q)+p 22 Q+V E, when +Q is on S 2 and Q is on S 1 and where V E denotes the common potential due to the presence of any external charges. The potential difference V = V 2 V 1 between the two conductors is then given by V = (p 11 +p 22 2p 12 )Q, (2.101) where the potential reference is taken on the negatively charged conductor in order to make V non-negative. The difference in electrostatic potential between the two conductor bodies of a capacitor is then seen to be proportional

48 42 2 The Electric Field to the stored charge Q. 7 This result may then be written as V = Q/C, where C Q V = (p 11 +p 22 2p 12 ) 1 (F) (2.102) is the capacitance of the capacitor. The capacitance is then seen to be the charge stored per unit of potential difference. The unit of capacitance in MKSA units is the farad (F) where F C/V. Fig A capacitor formed by a pair of conductors separated by a simple dielectric with permittivity ǫ with one conductor (S 1 ) being shielded by the other (S 2 ) from other charges in the n conductor system. + S ε S Consider now a simple capacitor consisting of two (perfectly) conducting bodies separated in space by a simple dielectric with permittivity ǫ and brought to a static charge state with net charge +Q on body 1 and Q on body 2. Such a capacitor then possesses the following properties: Free charges ±Q reside on the conductor surfaces, resulting in a surface charge density sj (r) on each body (j = 1,2 respectively) such that ±Q = sj (r)d 2 r. (2.103) S j Gauss law shows that the E-field lines originate normally from the surface S 1 of the positively charged body and terminate normally on the surface S 2 of the negatively charged body, with D(r) ˆnd 2 r = ±Q (2.104) S for any surface S enclosing either S 1 (+Q) or S 2 ( Q). As a consequence of the perpendicularity of E at each conductor surface, they are both equipotential surfaces, where V(r) = V 1 when r S 1 and V(r) = V 2 when r S 2. A single-valued potential difference V = V 1 V 2 then exists between the two conducting bodies, given by 7 By convention, the absolute value of the charge on one of the two conductor bodies is referred to as the charge on the capacitor.