arXiv:0903.0600v4 [physics.class-ph] 13 Aug 2011 oa nacmn fortertcludrtnig u als but l understanding, applications. theoretical interesting that ne our new prove capacitance of to of enhancement us matrix an allows to the it hand, of other mean properties the physical mathematical On the enlighten properties. capacita that such of view, of matrix of point the another of han from properties one the on well-known that permits some operators fact derive mat and fact. matrices the the a positive as such of of well from theory proof derived as consequences matrix, the physical positive and is a matical paper is capacitance this of of capacit matrix of is topic matrix i the main capacitance meaning of The problem physical of eigenvalue no the matrix to particular, given the In ally study [23]-[26]. theo poor to the rather operators of advanced employment and more the matrices bod other Nevertheless, rigid and of [16]-[20]. 22], mechanics topics [21, the mechanics as oth such quantum the Physics motion, exten of On is branches operators in and [1]-[15]. matrices used view positive hav of of capacitance theory points the of several hand, matrix from the studied and been capacitance of concept The Introduction 1 ievlepolm onayconditions. boundary problem, eigenvalue xmnd ial,w ics oeise ocrigtega the formulation. concerning the issues of some invariance discuss embeddin we successive Finally, in so conductors examined. Moreover, of obtained. systems is of un- conductors properties capacitan conductors of equivalent number of an arbitrary system and o an considered, a problem is of The constraints energy der space. internal euclidean the the “po minimizing to a matrix isomorphic constructing a by space” such visualized tial for easily are problem study eigenvalue properties and Many the fact, of this meaning of is physical implications eigen physical electrostatics null the in non-degenerate explore We capacitance a with of matrix matrix positive-singular the that prove We Abstract ihstentto n rprisncsayfrorsubseq our for necessary properties and e notation and study, the to intend lishes we that conductors of system trostatic h oiiiyadohrpoete ftemti fcapacit of matrix the of properties other and positivity The h ae sdsrbtda olw:scin2dfie h ele the defines 2 section follows: as distributed is paper The PACS: Keywords: ∗ † [email protected] [email protected] 12.v 21.n 14.k 14.b 02.30.Tb 01.40.gb, 01.40.Fk, 02.10.Yn, 41.20.Cv, eatmnod ´sc.UiesddNcoa eClmi.Bogot´ Colombia. de Nacional F´ısica. Universidad de Departamento aaiac,eetottc,pstv matrices, positive electrostatics, Capacitance, n ahmtclimplications mathematical and oof .Diaz A. Rodolfo usu- s value. efor ce sively ance. stab- are g yof ry uent ten- The to o to d ead uge nce the ing he- me er c- ila .Herrera J. William , ∗ w y a e f . 1 fsurfaces of netra odco.Teptnilo ahitra condu internal cavity by each the on denoted potential be is The could conductor. of surface external equipotential framework an consider such the us them, Let are rounds They sections. ca- of remaining of [27]. system the matrix in the developments Ref. our of in properties obtained some pacitance summarizes section FrameworkThis Basic 2 Sectio suggest contains C. readers. appendix D for and appendix 7 problems and Sec. conclusions of our topic summarizes the simplification other is many this the and because posible, em On attention successively special are deserves conductors. that f ded of conductors defined of number is configurations arbitrary hand, conducto capacitance with of equivalent system system an a a problem and for the constraints, energy studies internal with 6 the represents Section of capacitance minimization of w operator. matrix potentials” positive the of cap hermitian “space which of a in matrix constructing product by the th inner done of explores is positivity B This the appendix tance. with of Sec. along implications 5 invaria capacitance. physical gauge Section of the matrix to formulation. goal respect the the main with of the subtleties positivity some A, discusses the Appendix prove with along to 3 is Sec. In developments. n oal aallt them, to parallel locally and surface oeta fteeuptnilsraei eoe by denoted is surface equipotential the of potential suiu in unique is edfieasurface a define we h ufc fteeuptnil h hre nteconduct the on charges by The denoted equipotential. are the of surface the conditions h nqeestermesrsta h ouinfreach for solution the that ensures theorem uniqueness The hreacmltdi uhacvt by cavity a such denot we in surface accumulated equipotential charge the in conductor external an ial,w en h oa surface total the define we Finally, n byLpaeseuto ntevolumen the in equation Laplace’s obey yteetra surface external the by volume N e sdfieasto iesols uiir functions auxiliary dimensionless of set a define us Let +1 onsiwr ihrsett h qioeta surface. equipotential the to respect with inward points S V ∇ i S 2 T N onigotadwt epc otecnutr The conductor. the to respect with outward pointing S f j i endb h surface the by defined V odcosada qioeta ufc htsur- that surface equipotential an and conductors lgtybge hntesrae fteconductors the of surfaces the than bigger slightly 0 = ϕ S T i Q , h onaycniin 1 niaethat indicate (1) conditions boundary The . f , i i with S 1 = N . † j +1 ( , S S 2 N i i N , . . . , lgtysalradlclyprle to parallel locally and smaller slightly = ) +1 1 = n n the and δ .,N ..., , i ij sa ntvco omlt the to normal vector unit an is , seFg ) edfieaset a define We 1). Fig. (see . ,Colombia. a, S ( S ,j i, T T n fteei aiyof cavity a is there if and ne physical ance: = N ..tevlm delimited volume the i.e. 1 = V S nenlsurfaces internal Q S 1 N , . . . , T N + +1 ihteboundary the with . . . h ntvector unit the , + S 1) + N +1 ϕ . N n the and +1 f c of nce S i are s the e bed- i that ctor and aci- . 8 n (1) ors ith an ed f or of of rs a 4 e j 2 R. A. Diaz, W. J. Herrera
tial surface f = j (N + 1), in which the number of degrees of freedom is N(N + uipoten N+1 eq 1)/2, note that it is the same number of degrees of freedom of a n +1 SN+1 N N N real symmetric matrix. f= j × 1 For future purposes, we shall call the matrix with elements f= jN Cij and with i, j = 1,...,N the r-matrix (restricted matrix de- noted by C), while the Cij matrix with i, j = 1,...,N + 1 will f= ji be called the e-matrix (extended matrix denoted by Ce). Si ni 3 Discussion of the mathematical Figure 1: N conductors surrounded by an equipotential sur- face. The volume VST is the region in white. properties of the matrix
In this section we establish some additional mathematical prop- the fj functions depend only on the geometry. Since the func- erties of the matrix of capacitance. The central fact is that the tions fj acquire constant values on the surfaces Si with i = matrix of capacitance is a positive matrix. Basically, sections 2 1,...,N + 1, it is clear that fj is orthogonal to these surfaces. ∇ and 3 provide the mathematical framework whose physical im- The functions fj have some properties [27] plications will be explored in the remaining sections. N+1 Equations (1) and (3) tell us that the Cij elements are purely fj = 1; fj (Si) ni = (1 2δij ) fj (Si) ; 0 fj 1 geometrical. In addition, Eqs. (3) and (5) say that the e-matrix ∇ · − k∇ k ≤ ≤ j=1 is a real symmetric matrix in which the sum of elements of each X (2) row and column is null. From Eq. (6) the non-diagonal elements From these auxiliary functions we can construct a matrix that of the e-matrix are non-positive. The volume integral in Eq. provides a linear relation between the set of charges Qi and the (3) shows that the diagonal elements Ckk are strictly positive set of potentials ϕi in the following way for any well-behaved geometry. In particular, since CN+1,N+1 is positive, Eq. (7c) shows that at least one element of the form
Cij ε0 fj ni dS = ε0 fi fj dV (3) Ci,N+1 is different from zero (negative) for i = 1,...,N; thus ≡ − ∇ · ∇ ·∇ ISi ZVST rewriting Eq. (5) in the form N+1 N Qi = Cij ϕj (4) Cij = Ci,N+1 (9) j=1 − X j=1 X ij and some properties of the C matrix can be derived we see that if Ci,N+1 < 0 the sum of the elements of the i row − N+1 N+1 of the r-matrix is positive, if Ci,N+1 = 0 such a sum is null. Since at least one of the Ci,N elements is strictly negative, Cij = Cji, Cij = Cij = 0, (5) +1 j=1 i=1 we conclude that in the r-matrix the sum of elements on each X X row is non-negative and for at least one row the sum is positive. Cii 0 , Cij 0, (i = j). (6) ≥ ≤ 6 Because of the symmetry, all statements about rows are valid for The equations above are valid for i, j = 1,...,N + 1. The ex- columns. pressions below are valid for i, j = 1,...,N On the other hand, when VST is a connected region as in Fig. 1, the function fj should change progressively from its value 1 N N on conductor j up to the value zero in the conductor i with- Ci,N+1 0 , Cij 0 (7a) ≤ ≥ out taking local minima or maxima according to the proper- i=1 i=1 X X ties of Laplace’s equation. According with Eq. (2) the factor N 2 fj (Si) ni is positive for i = j and from Eq. (3) the non- Cjj Cij , CiiCjj Cij (7b) ∇ · 6 | | ≥ | | ≥ diagonal Cij factors must be strictly negative for a well-behaved i6=j X geometry. This discussion is not valid when the volume VST is N non-connected as in Fig. 2, we shall discuss this case in section CN+1,N+1 = Ci,N+1 , (7c) | | | | 7. When Cij < 0 for i = j, the discussion below Eq. (9), leads i=1 6 X to the fact that the sum of elements in each row of the r-matrix CN+1,N+1 Ci,N+1 (7d) | | ≥ | | is positive. and expressions for the internal electrostatic energy U of the In conclusion, for the e-matrix the sum of elements of each row system and of the reciprocity theorem can be obtained is null. Further, if VST is a connected region, all matrix elements of the e-matrix are non-null (for a well-behaved geometry), and N+1 N+1 N+1 N+1 for the r-matrix the sum of elements of each row is positive. The- 1 1 ′ ′ U = Cij ϕj ϕi = Qiϕi; Qiϕi = Qj ϕj (8) A B 2 2 orems and in Appendix A, show that under these conditions i,j i i=1 j=1 ❶ X X X X we find: The e-matrix is a real singular positive matrix, its ′ ′ null eigenvalue is non-degenerate and the other eigenvalues are where Qi, ϕi and Q , ϕ are two sets of charges and poten- { } { i i} positive. ❷ The r-matrix is a real positive-definite matrix∗. Its tials over the same configuration of conductors. The Cij ele- ments constitute a real symmetric matrix of dimension (N + 1) ∗The non-degeneration of the null eigenvalue of the e-matrix follows × Matrix of capacitance as a positive matrix 3 eigenvalues are all positive. ❸ The null eigenvalue of the e-matrix which is also equivalent to the non-invertibility. On the other is associated with (N + 1) dimensional eigenvectors of the form hand, the singularity of a matrix is also related with the linear − T dependence of the column (or row) vectors that constitute the φ (ϕ0, ϕ0, ..., ϕ0) (10) 0 ≡ matrix, this lack of independence in the case of Ce is manifested in the fact that no all charges can be varied independently as 4 Gauge invariance of the formula- can be seen from the expression Qint = QN+1 (12) tion − where Qint is the total charge of the internal conductors while We shall see that the properties of the matrix of capacitance QN+1 is the charge accumulated on the surface of the cavity of leads automatically to the gauge invariance of the linear relation the external conductor‡. Further, the linear dependence of the between charges and potentials. An outstanding result is that e-matrix can be visualized by observing that it has the same the gauge invariance involving the e-matrix, is closely related degrees of freedom as the r-matrix. This fact induces us to find with the existence of a null eigenvalue. expressions involving the r-matrix only. For this, we can rewrite We have two possible scenarios here, in the first the equipo- Eq. (4) by following the procedure that leads to Eq. (45) tential surface is the surface of the cavity of a conductor that en- N N closes the others. In the second, the equipotential surface is just Qk = Ckm (ϕm ϕN+1) CkmVm (13) − ≡ a geometrical place in the vacuum. The uniqueness theorem guar- m=1 m=1 antees the same solution in both cases but only in the interior of X X these relations are valid for k = 1,...,N + 1. However, since Eq. the equipotential surface. In the equipotential surface itself we (12) shows that QN+1 is not independent, we can restrict them can see that in the first case there is a charge QN accumulated +1 to k = 1,...,N. Rewriting Eq. (13) in matrix form with this in the cavity, while in the second case there is no charge in such a restriction we get surface at all. The problem lies in the fact that the electric field is not well-behaved in the surface of the cavity because of the Q = CV; V (V1,V2, ..., VN ) , Vi ϕi ϕN+1 (14) accumulation of surface charge [23]-[26], it is precisely because ≡ ≡ − this relation is written in terms of voltages instead of potentials, of this fact that we defined surfaces slightly different from the so it is clearly gauge invariant. Further, the relation is invertible real surfaces on each conductor (in which fj are well-defined). ∇ because the r-matrix C is positive-definite. It worths emphasiz- So all the observables (charges, potentials, electric fields) are the ing that all expressions obtained from now on in terms of voltages same in the interior of the equipotential surface for both scenar- and the r-matrix, are valid only if the voltages are taken with ios, but the surface charge and the electric field differ in both respect to the ϕN potential. cases when they are evaluated on the equipotential surface it- +1 self†. Anyway, the internal charges and any other observables not defined on the equipotential surface, are calculated in both 5 Physical implications of the posi- scenarios with the same set of Cij coefficients. From the discussion above, we see that when we have a set tivity of the matrix of free conductors, the simplest equipotential surface that we can define is the one lying at infinity with zero potential, which By constructing an appropriate inner product in a “space of po- is equivalent for most of the purposes to consider a cavity of a tentials”, we shall derive from another point of view some well- grounded external conductor in which all the dimensions of the known results, such as the reciprocity theorem and the positive cavity tend to infinity. nature of the internal energy. As a new result, we give a physical Further, we shall see that the linear relation between charges meaning to the eigenvalues and eigenvectors of the capacitance and potentials in Eq. (4) is gauge invariant by shifting the po- matrix, as well as their relation with the internal energy. Fi- ′ tential throughout the space as ϕ ϕ + ϕ0 with ϕ0 being a nally, we suggest some ways to determine experimentally the set → non-zero constant. This gauge transformation must keep all ob- of eigenvalues and eigenvectors of C, and how these eigenvectors servables unaltered, in particular the charge Qk on each surface and eigenvalues provide information about C. of the conductors. Writing Eq. (4) in matrix form and using Eq. To facilitate the derivation and interpretation of the results (10) we have let us define the following quantities
′ 1 1 Q = Ce (φ + φ0)= Ceφ = Q (11) cij Cij ; Φi Qi ≡ k0 ≡ k0 where we used the fact that φ0 is an eigenvector of Ce with null where k0 is a constant defined such that cij are dimensionless. eigenvalue. This gauge invariance says that there is an infinite From these definitions Eq. (4) could be rewritten in the form number of solutions (sets of potentials) for the linear equations N+1 (4) with given values of the charges, this fact is related in turn Φi = cij ϕj ; Φ = ceφ (15) with the non-invertibility of Ce. In other words, gauge invariance i=1 is related with the existence of an eigenvector with null eigenvalue X ‡This can be shown from Gauss’s law or directly from the formal- from theorem A or alternatively from theorem B, in appendix A, after ism presented here (see Ref. [27]). If the equipotential surface is a establishing the positive-definite nature of the r-matrix. geometrical place in the vacuum, Eq. (12) must be interpreted as a †Of course the potential on the equipotential surface is the same in numerical equality between the total internal charge and the quantity both cases by definition. on the right-hand side of Eq. (4) with i = N + 1. 4 R. A. Diaz, W. J. Herrera
the dimensionless cij factors contain the same information as the system. There is no configuration in which the system makes Cij . Similarly, Φi are quantities with dimension of potential a net work on the external agent. Note that all the analysis but with the physical information of the charges Qi (it is like above is consistent with the features coming from the equivalent a “natural unit” for the charge). The aim of settle the charges equation 1 and potentials with the same dimension is to interpret Eq. (15) u = E2 dV as a linear transformation in the configuration space ΦN+1 in 2 ZVST which each axis has dimensions of potential. This space would where E is the electric field generated by the configuration be isomorphic to RN+1 if we define an inner product of the form throughout the volume VST . N+1 Let us interpret the eigenvalue equation of c. It reads † (Φ,φ)= Φ φ = Φiϕi (k) (k) (k) (k) i=1 cV = λkV Φ = λkV (17) X ⇒ where we have taken into account that this is a real vector space. we use superscripts to label a given eigenvector and subscripts to The capacitance matrix is hermitian (real and symmetric) with label a given component of a fixed eigenvector. If there is a set respect to this inner product. Now let us take two sets of charges i of n indices such that all the λi’s are identical, this eigenvalue { } and potentials Φ,φ and Φ′,φ′ over the same configuration is n fold degenerate. According with Eq. (17), each eigenvector { } { } (k−) of conductors. Doing the inner product (Φ′,φ), using Eq. (15) V means a configuration of voltages for which each internal (k) (k) and taking into account the hermiticity of ce, we have charge Φi is related with its corresponding voltage Vi by the same constant of proportionality λk. Now, since the eigenval- ′ ′ ′ ′ ′ k Φ ,φ = ceφ ,φ = φ , ceφ = φ , Φ = Φ,φ ( ) ues are positive, each internal charge Φi and its corresponding voltage V (k) have the same signk. so that i ′ ′ Let us construct a complete orthonormal set of real dimen- Φ ,φ = Φ,φ sionless eigenvectors u(k) of c associated with the eigenvalues which is the reciprocity theorem shown in Eq. (8). From λk. We show in appendix B, Eq. (49) that the internal energy this point of view, this theorem is a manifestation of the her- associated with a set of voltages described by the vector V can miticity of the e-matrix. Of course, we can define a potential be written in terms of those eigenvectors and eigenvalues space ΦN , in which the N internal charges and N voltages form N 2 N dimensional vector arrangements and the r-matrix acts as 1 (n) − u = λn u , V (18) an hermitian operator. In this space the reciprocity theorem 2 acquires the form n=1 X Φ′, V = Φ, V′ The set u(n) defines principal axes in the potential space where in this case Φ′ and Φ refer to configurations of the inter- N n (n) o nal charges only. Now we shall rewrite the electrostatic internal Φ , and u , V is the projection of the vector V along with energy U of the system given by Eq. (8) in our new language the principal axis u(n). If the configuration of voltages in the sys- (k) (k) tem is of the form V = V0u (i.e. if the vector V is parallel 1 ∗∗ u = (φ, ceφ) 0 ; u U/k0 (16) to a principal axis) we find 2 ≥ ≡ 2 1 2 1 (k) the inequality comes from the positivity of the e-matrix. This u = λkV0 = λk V (19) expression is gauge invariant and can be written in terms of the 2 2 r-matrix and voltages (see appendix B)§ as follows so the eigenvalue is proportional to the internal energy associated with a set of voltages that forms the corresponding normalized 1 u = (V, cV) 0 eigenvector of the r-matrix. 2 ≥ Let us suggest now a possible application that illustrates the Because c is positive-definite, a zero energy is obtained only with importance of the eigenvectors and eigenvalues of c. Assume V = 0. The only configurations with zero energy are the ones that for a given configuration with N internal conductors, we with all potentials equal¶. Hence, for any geometry of the set of have calculated the matrix c, as well as N linearly independent (k) conductors and for any configuration of charges and potentials on eigenvectors V and their associated eigenvalues λk. We can them, the external agent that ensembles it, makes a net work on double-check the correctness of our procedure with the following experiment: Let us settle the experimental arrangement of con- §There is a subtlety with the concept of internal energy. The value ductors at the voltages defined by a given eigenvector V(p), we of an energy is not gauge invariant, but the internal energy is indeed (p) a difference of energies between an initial and a final configuration (or then measure the N charges Φi that each internal conductor (p) (p) a work to ensemble a given system) this value should then be gauge acquires. Now we calculate the N quotients Φi /Vi where p invariant. ¶This is in turn related with the fact that the null eigenvalue of the kWe insist at this point that it is true only if the voltages of the in- e-matrix is non-degenerate. If a degeneration of the null eigenvalue ternal conductors are defined with respect to the equipotential surface were present, we would have at least one eigenvector associated with that surrounds them. ∗∗ (k) the zero eigenvalue and linearly independent of the vector φ0 defined in Since u are dimensionless, V0 has units of potential. Note that Eq. (10). The existence of this eigenvector would imply the existence when V is parallel to a principal axis (i.e. becomes an eigenvector of of a configuration of different potentials with a null value of the internal c), all observables become simpler as in the case of the axis of rotation energy. in the rigid body motion. Matrix of capacitance as a positive matrix 5 is fixed. If our calculation of c was correct, these quotients must where β is the multiplier. Writing the internal energy as be equal (within experimental uncertainties) and must coincide N N with the eigenvalue λp. We can proceed in the same way with 1 1 U = (V, CV)= VkCkj Vj (26) each eigenvector. Further, if we measure in each of these config- 2 2 k j=1 urations the internal energy of the arrangement, we can contrast X=1 X these experimental values with the ones yielded by Eq. (19). replacing Eqs. (24, 26) into Eq. (25) and using the symmetry of Though the inverse problem could be difficult in practice, it the matrix, we find deserves to say that we can in principle determine eigenvectors N and eigenvalues experimentally (adjusting voltages until we find Cij Vj = βCi,N+1 ; i = 1,...,N (27) constant quotients between voltages and charges). If we can j=1 determine a complete set of eigenvectors and eigenvalues exper- X imentally, the matrix of capacitance can be obtained through a and applying a sum over i on Eq. (27) similarity transformation. Defining X as the matrix of eigenvec- N N N tors and Λ as the matrix of eigenvalues (we use a 3 3 matrix × Vj Cij = β Ci,N+1 , (28) for illustration) j=1 i=1 i=1 X X X N V V V Vj CN+1,j = βCN+1,N+1 (29) 11 12 13 − − (1) (2) (3) j=1 X V V V V21 V22 V23 (20) ≡ ≡ X V31 V32 V33 where we have used (5). Subtracting Eqs. (29, 23) and solving λ1 0 0 for β we find Λ 0 λ2 0 (21) Q0 ≡ β = (30) 0 0 λ3 − CN+1,N+1 the c matrix can be obtained by the relation Eq. (27) can be rewritten as T −1 CV = βvc ; v (C1,N+1,C2,N+1, ..., CN,N+1) (31) c = XΛX (22) c ≡ For a given β, the solution of Eq. (31) is unique because the 6 Minimization of the internal en- r-matrix C is invertible. It is easy to check that V =(Va, ..., Va), is a solution of Eq. (31), inserting this solution in Eq. (31), we ergy get
Problems of minimization of energy under constraints are very N useful in Physics. We illustrate by an example a process of mini- Va Cij = βCi,N+1 , i = 1,...,N j=1 mization of the internal electrostatic energy of a set of conductors X VaCi,N+1 = βCi,N+1 , i = 1,...,N (32) under the contraint of constant internal charge. The procedure − followed in this section is based on the properties of the c matrix − where we have used (5). From (32) we have developed here, and on the Lagrange’s multipliers method, lead- ing naturally to an equivalent capacitance between the external Va = β (33) conductor and the set of internal conductors. Such a procedure − can be extended to more complex constraints. As an impor- Thus, the configuration of N voltages that minimizes the energy tant remark, our example shows that the e-matrix can be useful with a fixed value of Qint, is given by for practical calculations, despite it does not contain additional VT =( β, β, ..., β) (34) degrees of freedom with respect to the r-matrix. − − − For N internal conductors inside the cavity of an external con- This kind of solution for V is expected because the configura- ductor, let us find the configuration V of voltages that minimizes tion of minimal energy is obtained when we connect all the in- the internal energy with the constraint that the total internal ternal conductors among them by conducting wires, this proce- charge Qint is a constant Q0. Since Qint = QN+1 and taking − dure clearly keeps Qint constant and equates internal potentials. into account that Eq. (13) is also valid for k = N + 1, we have Since all potentials of the interior conductors are the same, we N can define a single voltage between the external conductor and
Qint = CN+1,j Vj = Q0 (23) the internal ones, this voltage is β . Since Qint = QN+1 = Q0 − | | − j=1 we can figure out the system as equivalent to a system consisting X of two conductors with charges Q0 and voltage β . Thus, we the function Z (V) that defines the constraint is ± | | are led naturally to an equivalent capacitance for this system of N potentials and charges Z (V) CN+1,j Vj Q0 = 0 (24) ≡− − j=1 Q0 X Q0 = Ceq β Ceq = = CN+1,N+1 | | | | ⇒ β from the Lagrange’s multipliers method we have
∂U ∂Z where we have used Eq. (30). It can be checked that the internal + β = 0 ; i = 1,...,N (25) energy U for the configuration described by (34) is ∂Vi ∂Vi 6 R. A. Diaz, W. J. Herrera
see that the sum of elements in each row of the r-matrix is null 1 2 1 2 U = CN ,N V = Ceqβ (35) except for the N row, for which the sum is positive. Therefore, +1 +1 a − 2 2 the r-matrix of a chain of embedded conductors satisfies the con- as expected. A brief comment with respect to the e-matrix is ditions of theorem C in appendix A. Consequently, for a chain in order. This matrix has no additional degrees of freedom with of embedded conductors, the r-matrix continues being positive- respect to the r-matrix so that we can formally write all results definite, and the e-matrix is still singular positive. Combining in terms of the elements Cij of the r-matrix. Notwithstanding, these facts with theorem B of appendix A, we obtain that the the extended elements could be useful for explicit calculations. null eigenvalue of the e-matrix is non-degenerate†. It is again Assume for instance that for the problem in the present section, consistent with the fact that the only configuration of null inter- we want to calculate the total internal charge for a given voltage nal energy is the one associated with all conductors at the same of the system, and the equivalent capacitance. These calculations potential. can be done with the following expressions
N N N
Qint = Qk = Va Cij = Ceq Va 8 Conclusions i=1 i=1 j=1 X X X N N N We have studied an electrostatic system consisting of a set of
Ceq = CN+1,N+1 = Ci,N+1 = Cij N conductors with an equipotential surface that encloses them. − i=1 i=1 j=1 The associated matrix of capacitance has dimensions (N + 1) X X X × (N + 1) (extended or e-matrix) even if the equipotential surface Therefore, in terms of the elements of the r-matrix all the goes to infinity. It is usual in the literature to work with the N(N + 1)/2 coefficients must be evaluated with Eq. (3), to matrix of dimension N N (restricted or r-matrix), this practice × calculate Qint and Ceq . In contrast, by using the e-matrix, only is correct only if the voltage of the conductors is taken with the CN+1,N+1 coefficient should be calculated through Eq. (3) to respect to the potential of the equipotential surface. We prove find such observables. This difference becomes more important that the e-matrix is a real positive and singular matrix, this as N increases. Similar advantages of using the e-matrix appear is consistent with the fact that gauge invariance requires the in more general contexts (see appendix A of Ref. [27]). existence of a null non-degenerate eigenvalue of this matrix. The r-matrix is a real positive-definite matrix so all its eigenvalues are 7 The case of a chain of embedded positive. By constructing a “potential space” with inner product, we conductors can derive some results such as the reciprocity theorem and the non-negativity of the electrostatic internal energy of the system, Let us consider a system of conductors in successive embedding from another point of view. A given eigenvector V(p) of the (p) as described in Fig. 2 and in appendix C. We encounter such r-matrix corresponds to a set of voltages Vi , such that if we systems quite often in Physics. We observe that in systems as (p) settle the internal conductors at these voltages, the charges Qi the one in Fig. 1, the volume VST consists of a “single piece”, acquired by each internal conductor i, are such that the quotient but in successively embedded conductors such a volume con- (p) (p) λp = Qi /Vi is the same for all internal conductors and corre- sists of “several disjoint pieces” as shown in Fig. 2. The main (p) sponds to the eigenvalue associated with Vi . The positivity of consequence coming from this difference is the fact that some (p) the eigenvalues λp guarantees that each charge Q posseses the elements of the matrix of capacitance are null for embedded sys- i V(p) tems. This fact simplifies considerably the calculation of such a same sign as the associated voltage i . In addition, a given eigenvalue is proportional to the internal energy associated with matrix. However, we shall prove that the other properties of the the set of voltages generated by its corresponding eigenvector. matrix of capacitance remains the same. Section 3 shows that the e-matrix is singular positive and the Moreover, a complete set of orthonormal eigenvectors of the r- matrix defines principal axes in the “potential space”. It worths r-matrix is positive-definite. The first fact was independent of emphasizing that eigenvectors and eigenvalues can be measured the connectivity of VS . In contrast, the second fact was de- T experimentally, and provide information about the matrix of ca- rived from the statement that all non-diagonal Cij elements were pacitance. strictly negative. However, it is shown in appendix C that in the case of a chain of embedded conductors (see Fig. 2), some of The problem of the minimization of the internal energy is stud- ied under the constraint of constant value of the total internal the non-diagonal elements Cij are null because the volume VST is disconnected. Then we should check whether the r-matrix is charge. In this case we can define an equivalent capacitance for still positive-definite for the chain of embedded conductors. any number of internal conductors. From this problem, we real- ized that although the e-matrix has the same degrees of freedom Appendix C shows that elements of the form Ci,i±1 are non- zero in general. Appealing to an argument analogous to the one as the r-matrix, such extension could lead to great simplifications presented in Sec. 3 we can show that for a well-behaved geometry of some practical calculations. Further, systems of successive embedded conductors are ana- of our embedded conductors, Cii > 0 and Ci,i±1 < 0, while the remaining elements vanish∗. With these properties and the fact lyzed showing that some coefficients of capacitance are null for that the sum of elements of each row of the e-matrix is null, we these systems, allowing an important simplification for practical
∗ † Of course if i = 1 (or i = N + 1) the element Ci,i−1 (or Ci,i+1) Note that in this case, theorem A of appendix A cannot be used does not exist. For a given i, at least one of them exists. to establish the non-degeneration of the null eigenvalue. Matrix of capacitance as a positive matrix 7 calculations. This fact is related with connexity properties of the volume in which Laplace’s equation is considered. Moreover, we T 2 prove that for these configurations of embedded conductors the N CN = ckknk k e-matrix is still positive singular with a non-degenerate eigen- X value and the r-matrix is positive-definite. 1 2 2 2 + ckm[n + n (nk nm) ] 2 k m − − } Finally, the properties of the matrix of capacitance shown m X here, can be useful for either a formal understanding or prac- T 1 2 1 2 tical calculations in electromagnetism. It worths observing the N CN = ckk + ckm nk + ckmnm 2 2 k (" m # m ) similarity in structure between the matrix of capacitance and the X X X inertia tensor. 1 2 ckm (nk nm) (37) − 2 − k m X X
Now, since Σk = ckk + ckm 0, we have Acknowledgments m ≥ 1 P 1 c + c c (38) We thank Divisi´on Nacional de Investigaci´on de Bogot´a(DIB), kk 2 km 2 km m ≥− m of Universidad Nacional de Colombia (Bogot´a) for its financial X X support. it is convenient to separate the sets k′ and k′′ in Eq. (38) { } { } 1 1 c ′′ ′′ + c ′′ = c ′′ (39) k k 2 k m 2 k m m − m X X A Some special types of matrices 1 1 c ′ ′ + c ′ > c ′ (40) k k 2 k m 2 k m m − m This appendix concerns the study of a special type of matrices. X X Let define Σk as the sum of the elements on the k row of a given we examine first the case in which k′ is empty. In that case − { } matrix. We shall make the following there are no equations of the type (40), and all indices accomplish Definition: A sp-matrix, is a square real matrix of finite the equation (39). Using (39) in Eq. (37) and the fact that dimension in which cmk = ckm 0 for k = m, and in which ckm 0, we find ≤ 6 ′ ≤ Σk is non-negative for all k. We denote with a single prime k ′ { ′′} the set of indices for which Σk > 0 and with double prime k T 1 2 1 2 { } N CN = ckm nk + ckmnm the set of indices for which Σk′′ = 0. If no prime is used, either − 2 2 k (" m # m ) situation could happen. X X X 1 2 Theorem A: If C is a sp-matrix, then C is a positive matrix + ckm (nk nm) 2 | | − with respect to the usual complex inner product. ➊ If k′ is k m { T} X X empty, the matrix is singular and vectors of the form N = 0 using the symmetry of the matrix and taking into account that (n,...,n) are eigenvectors of C with null eigenvalue. Further, k,m are dumb indices, the first two terms on the right-hand side the null eigenvalue is non-degenerate if all elements of the matrix vanish and we find are non-null. ➋ If k′′ is empty, the matrix is positive-definite. { } Proof T 1 2 : We should prove that N CN = ckm (nk nm) 0 (41) 2 | | − ≥ k m X X V†CV 0 (36) ≥ Equation (41) shows that the bilinear form is always non- negative and that NT CN = 0 for non-zero vector arrangements for an arbitrary vector V, and we should look under what condi- of the form NT (n, n, . . . , n). Consequently, the matrix is sin- 0 ≡ tions exists at least one non-zero vector V for which this bilinear gular positive. We can check that N0 is an eigenvector of C, with expression is null. Rewriting V = N + iM with N, M being real null eigenvalue. If all elements ckm are non-null, Eq. (41) shows vector arrangements, and using the symmetry of C, the bilinear that this is the only linearly independent solution, so that the form in Eq. (36) becomes NT CN + MT CM. Therefore, it suf- zero eigenvalue is non-degenerate. fices to prove the positivity (or non-negativity) of the bilinear Now we examine the case in which k′′ is empty, so there { } form with real vector arrangements. Let N be a non-zero real are no equations of the type (39), and all indices accomplish the vector, the associated bilinear form is equation (40). Replacing (40) into Eq. (37), using the symmetry of C, the fact that ckm 0, and that N =0 we find ≤ 6 T N CN = nkckknk + ckm nknm T 1 2 N CN > ckm (nk nm) 0 (42) k k m6=k 2 | | − ≥ X X X k m X X for the remaining of this appendix, we assume that indices and the bilinear form becomes positive if and only if N = 0. 6 labeled with different symbols are strictly different. We Hence, the sp-matrix is positive-definite when k′′ is empty. { } rewrite the bilinear form as QED. 8 R. A. Diaz, W. J. Herrera
Theorem B: Let C be a matrix of dimension (N + 1) such a replacement also shows that the equality holds if nN = 0, × (N + 1), such that Σi = 0 for all rows. This matrix has eigen- while the strict inequality holds if nN = 0. Using the symmetry T 6 vectors of the form N =(n,...,n) associated with a null eigen- of the matrix and the fact that ckm 0 we have 0 ≤ value. Let Cr be the N N submatrix of C consisting of the × T 1 2 elements Cij of C with i, j = 1,...,N. If Cr has no null eigen- N CN ckm (nk nm) ≥ 2 | | − ‡ C k m values , the null eigenvalue of is non-degenerate. X X Proof : The condition Σi = 0 for i = 1,...,N + 1 gives Since C is singular, a non-trivial solution must exist for the bi- N+1 linear form to be null. For this, the equality must hold in this Cik = 0 ; i = 1,...,N + 1 (43) relation, therefore nN = 0. The term on the right written in k=1 terms of the non-zero elements of the matrix yields X Eigenvectors of C with null eigenvalues must give N−1 T c12 2 ck,k+1 2 N CN = | | (n1 n2) + | | (nk nk+1) N+1 2 − 2 − k=2 Ciknk = 0 ; i = 1, ..., N + 1 (44) X N−1 k=1 ck,k−1 2 cN,N−1 2 X + | | (nk nk−1) + | | n 2 − 2 N−1 assuming nk = n for all k and using condition (43), Eq. (44) k=2 T X is satisfied. Thus, N0 is an eigenvector associated with a null eigenvalue. From the condition (43) we also find for this expression to be zero each term in these sums must be zero. Since all matrix elements involved in this expression are N+1 N non-zero, the last sum says that nN−1 = 0, while the other sums Ciknk = Ciknk + Ci,N+1nN+1 say that n1 = n2 = ... = nN−1. Since nN was already zero, it k k X=1 X=1 shows that the only solution is the trivial one, contradicting the N N singularity of the matrix. QED. = Ciknk + Cik nN+1 − k k ! X=1 X=1 N+1 N B Some properties of the internal Ciknk = Cik (nk nN+1) ; i = 1, .., N + 1 (45) − k=1 k=1 energy X X replacing (45) in (44) the latter becomes The equation (16) for the internal energy u in “natural units” N can be written in terms of voltages instead of potentials with the
Cik (nk nN+1)=0; i = 1, ..., N + 1 (46) r-matrix. By using the fact that φ0 is an eigenvector of ce with − k=1 null eigenvalue, and the hemiticity of ce, Eq. (16) becomes X in particular Eq. (46) holds for i = 1, ..., N. With this restriction 2u = (φ, ce (φ φ0)) =(ceφ, (φ φ0)) Eq. (46) becomes − − 2u = (ce (φ φ ) , (φ φ ))=(φ φ , ce (φ φ )) − 0 − 0 − 0 − 0 CrV = 0 ; V (n1 nN+1, n2 nN+1, ..., nN nN+1) (47) ≡ − − − defining a vector arrangement of N + 1 voltages Ve we find since Cr has no null eigenvalues, the only solution for Eq. (47) T 2u =(Ve, ceVe) ; V (V1, ..., VN ,VN+1) , Vi ϕi ϕ0 is V = 0. Hence the only type of solutions for N are of the form e ≡ ≡ − N = (nN+1, nN+1, ..., nN+1) which are all linearly dependent. writing this bilinear form explicitly and expanding the sums we Hence, the null eigenvalue is non-degenerate. It is immediate find that these solutions satisfy Eq. (46) for i = N + 1 as well. Note N+1 N+1 that C is not necessarily symmetric or real. QED. 1 ′′ u = cij ViVj Theorem C: Let C be a N N sp-matrix such that k = 2 × { } i=1 j=1 1, 2,...,N 1 and the terms X X { − } N N 1 ci,i = ci ,i , ci,i = ci ,i ; i = 2, ..., N 1 = cij ViVj + VN+1KN+1 +1 +1 −1 −1 2 − i=1 j=1 X X are non-zero, while the remaining non-diagonal terms vanish. N Then C is positive-definite. 1 KN+1 ci,N+1Vi + cN+1,N+1VN+1 ≡ 2 Proof: Assume C as singular and arrive to a contradiction. i=1 Replacing Eqs. (39, 40) in Eq. (37) we obtain X choosing ϕ0 = ϕN+1 we find VN+1 = 0, hence
T 1 2 1 2 N N N CN ckm nk + ckmnm 1 1 ≥ ("− 2 # 2 ) u = cij ViVj = (V, cV) k m m 2 2 X X X i=1 j=1 1 2 X X ckm (nk nm) T − 2 − V N i i N k m (V1, ..., V ) , V ϕ ϕ +1 X X ≡ ≡ − ‡ If Cr is a normal matrix (or if it can be brought to the canonical thus the internal energy can be written in terms of the r-matrix form), it is equivalent to say that Cr is non-singular. and the voltages in a simple way as long as the latter are defined Matrix of capacitance as a positive matrix 9
with respect to the external potential ϕN+1. Since V is gauge (see Fig. 2). In addition, we define Vk with k = 1,...,N as the invariant u also is. volume formed by the points exterior to the conductor k and On the other hand, remembering that we can always construct interior to the cavity of the conductor k + 1 that contains the a complete orthonormal set of real dimensionless eigenvectors conductor k. Let us examine the non-diagonal elements Ckm (k) e ue of the e-matrix associated with the eigenvalues λk, we can assuming from now on that k
j4 of capacitance (see Ref. [27]). n S4 From the results above, we see that for N + 1 successively 4 embedded conductors with N 2, we have ≥
Cii = (Ci,i−1 + Ci,i+1) ; i = 2,...,N S − S1 2 V1 j1 C11 = C12 ; CN+1,N+1 = CN,N+1 j3 n n S − − V2 1 2 3 V n3 3 j2 n3 How many degrees of freedom do we have for the e-matrix?. n2 D Suggested Problems Figure 2: A chain of embedded conductors with N = 3. S S For checking the comprehension of the present formulation and The surfaces 2, 3, have an inner and an outer part. its advantages, we give some general suggestions for the reader.
Let us study a set of N + 1 conductors which are successively 1. Show all the properties stated here for the r-matrix and embedded. We label them k = 1,...,N + 1 from the inner to e-matrix with specific examples. the outer. Observe that the surface Sk for each conductor with k = 2,...,N has an inner and an outer part, but for SN+1 we §Remember that the surfaces are slightly different from the surfaces only define an inner part and for S1 we only define an outer part of the conductors for the gradient to be well-defined. 10 R. A. Diaz, W. J. Herrera
2. Look for differences and similarities between the matrix of [12] H. J. Wintle, The capacitance of the cube and square plate by capacitance in electrostatics and the inertia tensor in me- random walk methods, Journal Electrostat. 62 (2004) 51-62. chanics, from the physical and mathematical point of view. [13] D. Palaniappan, Classical image treatment of a geometry 3. From (φ, cφ) k we find (u, cu) = 1 with u φ/√k. composed of a circular conductor partially merged in a di- ≡ ≡ This defines the equation of an ellipsoid, describe how to electric cylinder and related problems in electrostatics, J. find the length of the axes of the ellipsoid in the ΦN and Phys. A: Math. Gen. 38 (2005) 6253-6270. ΦN+1 spaces for the r-matrix and the e-matrix respectively. [14] S. Ghosh and A. Chakrabarty, Estimation of capacitance Describe the principal axes in these “potential spaces”. of different conducting bodies by the method of rectangular 4. By setting ∂u/∂ϕi = 0, prove that in the absence of con- subareas, J. Electrostat. 66 (2008) 142-146. straints, the only local minimum of the internal energy is [15] Y. Xiang, Further study on electrostatic capacitance of an given by sets of the type φ . 0 inclined plate capacitor, Journal Electrostat. 66 (2008) 366- 5. Prove Eq. (35) for the minimal internal energy under the 368. constraint of constant internal charge. [16] F. Ninio, A simple proof of the Perron-Frobenius theorem 6. Let a,b be two positive numbers. Consider the 4 4 { } × for positive symmetric matrices, J. Phys. A: Math. Gen. 9 matrix given by (1976) 1281-1282.
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