Electrostatic Patch Effect in Cylindrical Geometry. I. Potential and Energy between Slightly Non-Coaxial Cylinders
aa r X i v : . [ g r- q c ] S e p Electrostatic Patch Effect in Cylindrical Geometry. I.Potential and Energy between Slightly Non–Coaxial Cylinders
Valerio Ferroni and Alexander S. Silbergleit ICRANet, Dept. of Phys., Univ. ‘La Sapienza’, Rome, Italy current address : W.W.Hansen Experimental Physics Laboratory,Stanford University, Stanford, CA 94305-4085, USA ∗ Gravity Probe B, W.W.Hansen Experimental Physics Laboratory,Stanford University, Stanford, CA 94305-4085, USA † (Dated: October 25, 2018) Abstract
We study the effect of any uneven voltage distribution on two close cylindrical conductors withparallel axes that are slightly shifted in the radial and by any length in the axial direction. Theinvestigation is especially motivated by certain precision measurements, such as the Satellite Testof the Equivalence Principle (STEP).By energy conservation, the force can be found as the energy gradient in the vector of the shift,which requires determining potential distribution and energy in the gap. The boundary valueproblem for the potential is solved, and energy is thus found to the second order in the smalltransverse shift, and to lowest order in the gap to cylinder radius ratio. The energy consists ofthree parts: the usual capacitor part due to the uniform potential difference, the one coming fromthe interaction between the voltage patches and the uniform voltage difference, and the energy ofpatch interaction, entirely independent of the uniform voltage. Patch effect forces and torques inthe cylindrical configuration are derived and analyzed in the next two parts of this work.
PACS numbers: 41.20Cv; 02.30Em; 02.30Jr; 04.80CcKeywords: Electrostatics - Patch effect - Cylindrical capacitor - Potential and Energy - Precision measure-ments - STEP ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTIONA. Background and Motivation The electrostatic patch effect (PE) [1] is a nonuniform potential distribution on the surfaceof a metal. Charges in a metal move, due to a finite conductivity, until the electrostaticpotential becomes the same at all the surface points. However, a nonuniform dipole layermay form on the metal surface due to impurities or microcrystal structure. If it exists, thenthe potential on the surface is no longer uniform, and the electric field is not necessarilyperpendicular to the surface of the metal. In the (idealized) case of an isolated conductorthe net force and torque on the body is still equal to zero, but with two metallic surfaces ata finite distance, the net force and/or torque on each of them does not vanish, in general.This effect is larger, the closer the surfaces, as first confirmed by the calculation of the patcheffect force for two parallel conducting planes [2].This calculation of the patch effect force was particularly motivated by the LISA spaceexperiment to detect gravitational waves (see c.f. [3]). PE can similarly affect the accuracyof any other precision measurement if its set-up includes conducting surfaces in a closedproximity to each other. For instance, PE torques turned out one of the two major difficul-ties [4–6] in the analysis of data from Gravity Probe B (GP-B) Relativity Science Mission;the GP-B satellite was in flight in 2004–2005, to measure the relativistic drift of a gyroscopepredicted by Einstein’s general relativity (see c.f. [7]). This required theoretical calculationof PE torques [8] for the case of two concentric spherical conductors (the lab evidence of thepatch formation on the surfaces of the rotor and housing was found in [9]).The aim of this paper, consisting of three parts, is to study the force and torque due tovoltage patches on two close cylindrical conductors. In particular, in the STEP experimentalconfiguration [10–13] each test mass and its superconducting magnetic bearing form sucha pair of cylinders. The goal of STEP is the precise (1 part in 10 ) measurement of therelative axial acceleration of a pair of coaxial test masses, so the importance of properlyexamining cylindrical patch effect is evident.2 . Structure of the Paper We determine the PE forces (and axial torque, in the final part of this paper) betweentwo cylinders with parallel axes by the energy conservation argument. It implies, in general,that a small shift, ~r , of one of the conductors relative to the other causes an electrostaticforce between them given by (see, for instance, [14]) ~F ( ~r ) = − ∂W ( ~r ) ∂~r , (1)where W ( ~r ) is the electrostatic energy as a function of the shift.According to the formula (1), the force to linear order in a small shift, which is ourultimate goal, requires calculating W ( ~r ) to quadratic terms in ~r . Due to the specifics ofcylindrical geometry, a more general result for an arbitrary axial shift, z , but only smalltransverse shifts, x = x , y = x is available. So, we first find the energy in the form W ( ~r ) = W ( z ) + W µ ( z ) x µ + W µν ( z ) x µ x ν + O (cid:16) ρ (cid:17) , ρ ≡ q ( x ) + ( y ) . (2)Here and elsewhere in the paper we use the summation rule over repeated Greek indices: thesummation always runs from 1 to 2 (meaning the directions transverse to the cylinder axes).To compute the coefficients W ( z ) , W µ ( z ) , W µν ( z ), and then the force components bythe formulas (1), we need the distribution of the electrostatic potential in the gap betweenthe shifted cylinders to the quadratic order in the shift,Φ( ~r, ~r ) = Φ ( ~r, z ) + Φ µ ( ~r, z ) x µ + Φ µν ( ~r, z ) x µ x ν + O (cid:16) ρ (cid:17) , (3)since energy is a quadratic functional of the potential.A more rigorous view of expansions in the above formulas consists of assuming the trans-verse shift, ρ , small as compared to the nominal gap, d = b − a , between the coaxial cylinders( a < b are the inner and outer cylinder radii). The relevant small parameter is ρ /d , andthe expansions go in dimensionless quantities x µ /d, µ = 1 ,
2, so the remainder estimate O ( ρ ) means, in fact, O ( ρ /d ), etc. Also, for typical experimental conditions, such as theSTEP configuration [11, 12], the gap is much smaller than either of the radii. This justifiesthe model of infinite cylinders, and allows for a significant simplification of the results to acertain order in d/a . Thus two small parameters are actually involved in the problem, ρ /d ≪ , d/a ∼ d/b ≪ . (4)3e work all the way to quadratic order in the first of them, and give the final answer forthe PE forces to lowest order in the second one. However, some meaningful intermediateresults are valid without the last or both of these assumptions.All said pretty much defines the structure of the paper. In the next section the boundaryvalue problem (BVP) for the potential with general voltage distributions on the cylindersis solved, and the potential in the form (3) is found. Based on this, the energy represen-tation (2) is obtained explicitly in section III. The details of calculations, in places rathercomplicated and cumbersome, are found in the three appendices. The derivation of PEforces and torques and the analysis of their properties are given in the next two parts of thiswork. The results of this part can also be used whenever the solution to the BVP for theLaplace equation for the domain between two cylinders is needed, such as for instance, inmagnetostatics, thermostatics, stationary diffusion, etc. II. ELECTROSTATIC POTENTIAL BETWEEN TWOINFINITE CYLINDERS WITH PARALLEL AXESA. Boundary Value Problem
We employ both Cartesian and cylindrical coordinates in two frames of the inner and outercylinders as shown in fig.1. In the inner, or ‘primed’, frame the position of a point is givenby the vector radius ~r ′ , and Cartesian coordinates { x ′ , y ′ , z ′ } or cylindrical coordinates { ρ ′ , ϕ ′ , z ′ } . In the outer, or ‘unprimed’, frame the corresponding quantities are ~r , { x, y, z } , { ρ, ϕ, z } . The frame origins are separated by ~r , hence the primed and unprimed Cartesiancoordinates are related by ~r ′ = ~r + ~r ; x ′ = x + x , y ′ = y + y , z ′ = z + z ; (5)equivalenlty, in cylindrical components of the transverse shift [ ρ is defined in (2)], x ′ = x + ρ cos ϕ , y ′ = y + ρ sin ϕ ; tan ϕ = y /x , (6)As alternative writing we use x ≡ x , y ≡ x , x ≡ x , y ′ ≡ x ′ , etc.The surfaces of the inner and outer cylinders are ρ ′ = a and ρ = b , respectively. Theycarry arbitrary voltage distributions, so the electrostatic potential, Φ, satisfies the Laplace4quation in the gap,∆Φ = 0 , ρ ′ > a, ρ < b, ≤ ϕ < π, | z | < ∞ , (7)and the boundary conditions of the first kind at the boundaries:Φ (cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a = G ( ϕ ′ , z ′ ) , Φ (cid:12)(cid:12)(cid:12)(cid:12) ρ = b = V − + H ( ϕ, z ) ; (8) V − = const (9)the latter are formulated using two sets of coordinates, primed and unprimed. [The potentialdistribution can also be described by using either of them, such as Φ( ~r ′ ) ≡ Φ( ~r ′ , ~r ), orΦ( ~r ) ≡ Φ( ~r, ~r ); the second argument emphasizes the dependence on the shift between thecylinders.] Boundary voltages are split in two parts: the uniform potential difference, V − (all the voltages in the problem are counted from the uniform voltage of the inner cylindertaken as zero), and the non–uniform potential distributions (patch patterns) described byarbitrary smooth enough functions G ( ϕ ′ , z ′ ) and H ( ϕ, z ).The local nature of the patch distributions is stressed by requiring || G || = π Z ∞ Z −∞ dϕ ′ dz ′ | G ( ϕ ′ , z ′ ) | < ∞ , || H || = π Z ∞ Z −∞ dϕdz | H ( ϕ, z ) | < ∞ ; (10)these conditions are assumed valid throughout the paper. (Of course, primes can be droppedat the variables under the integrals, as done everywhere below). Later we will assumethe boundary functions more smooth, expressing the additional conditions as the squareintegrability of various derivatives of G ( ϕ ′ , z ′ ) and H ( ϕ, z ).The main tool in the solution of the boundary value problem (BVP) (7), (8) is the Fouriertransform in the axial and azimuthal variables. For any function u ( ϕ, z ) satisfying the squareintegrability condition the following representations hold: u ( ϕ, z ) = 12 π Z ∞−∞ dk ∞ X n = −∞ u n ( k ) e i ( kz + nϕ ) , u n ( k ) = 12 π π Z ∞ Z −∞ dϕdz u ( ϕ, z ) e − i ( kz + nϕ ) . (11)For any two such functions u ( ϕ, z ) and v ( ϕ, z ) the useful Parceval identity holds:( u, v ) ≡ π Z ∞ Z −∞ dϕdz u ( ϕ, z ) v ∗ ( ϕ, z ) = Z ∞−∞ dk ∞ X n = −∞ u n ( k ) v ∗ n ( k ) ; (12)here and elsewhere the star denotes complex conjugation. In a particular case u = v theidentity (12) shows that the square of the norm, || u || , of a function u , defined by the formula(10), is equal to the square of the norm of its Fourier coefficient u n ( k ).5inally, we note that the split in the boundary conditions implies the correspondingrepresentation of the potential, Φ( ~r ) = Φ u ( ~r ) + Φ p ( ~r ) , (13)where Φ u is originated by the uniform boundary voltage, and Φ p by the patch one. B. Solution for the Patch Potential
According to the formulas (7), (8) and (13), the BVP for the patch potential is:∆Φ p = 0 , ρ ′ > a, ρ < b, ≤ ϕ < π, | z | < ∞ ; (14)Φ p (cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a = G ( ϕ ′ , z ′ ) = 12 π Z ∞−∞ dk ∞ X n = −∞ G n ( k ) e i ( kz ′ + nϕ ′ ) ; (15)Φ p (cid:12)(cid:12)(cid:12)(cid:12) ρ = b = H ( ϕ, z ) = 12 π Z ∞−∞ dk ∞ X n = −∞ H n ( k ) e i ( kz + nϕ ) . (16)Since the boundary functions are real, their Fourier coefficients satisfy important relations, G n ( k ) = G ∗− n ( − k ) , H n ( k ) = H ∗− n ( − k ) ; (17)which are necessary and sufficient conditions for the corresponding imaginary parts to vanish.The dimension of functions G n ( k ) and H n ( k ) is volt · meter , since k is in inverse meters .The standard separation of variables in the Laplace equation in cylindrical coordinates(see c.f. [15], Chs. 5, 6) provides the following representation of the potential:Φ p ( ~r ′ ) = 12 π Z ∞−∞ dk ∞ X n = −∞ h A n ( k ) I n ( kρ ′ ) + B n ( k ) K n ( kρ ′ ) i e i ( kz ′ + nϕ ′ ) , (18)where I n ( ξ ) , K n ( ξ ) are the modified Bessel functions of the 1st and 2nd kind, respectively.In fact, the Macdonald’s function K n ( ξ ) requires some definition for the negative values ofits argument, which is taken here according to the parity of I n ( ξ ): the symbol K n ( kρ ′ )stands actually for (sign k ) n K n ( | k | ρ ′ ).The unknowns A n ( k ) , B n ( k ) are to be determined from the boundary conditions; thefirst of them, condition (15) at the inner cylinder, can apparently be fulfilled immediately: A n ( k ) I n ( ka ) + B n ( k ) K n ( ka ) = G n ( k ) , n = 0 , ± , ± , . . . . (19)6o satisfy the remaining boundary condition (16) we express the ba-sis solutions of the Laplace equation in the primed cylindrical coordinates, I n ( kρ ′ ) e i ( kz ′ + nϕ ′ ) , K n ( kρ ′ ) e i ( kz ′ + nϕ ′ ) , in terms of the set of same solutions in the unprimedcoordinates of the outer cylinder. This re–expansion approach has been successfully usedfor solving various BVPs with the basis solutions in spherical coordinates [16], [17], andin spherical and cylindrical coordinates [16], [18]. The needed re-expansions of cylindricalsolutions are readily available from [19]; the corresponding transformation formulas, alongwith all the detailed calculations, are found in Appendix A. In particular, the potential is:Φ p ( ~r ) = 12 π Z ∞−∞ dk ∞ X n = −∞ h ˜ A n ( k ) I n ( kρ ) + ˜ B n ( k ) K n ( kρ ) i e i ( kz + nϕ ) ; (20)˜ A n ( k ) = e ikz ∞ X m = −∞ A m ( k ) I m − n ( kρ ) e i ( m − n ) ϕ ; (21)˜ B n ( k ) = e ikz ∞ X m = −∞ B m ( k )( − m − n I m − n ( kρ ) e i ( m − n ) ϕ ; (22) n, m = 0 , ± , ± , . . . . The boundary condition (16) at the outer cylinder can now be satisfied by requiring˜ A n ( k ) I n ( kb ) + ˜ B n ( k ) K n ( kb ) = H n ( k ) , n = 0 , ± , ± , . . . . (23)Written explicitly using the expressions (21), (22), this, along with the equation (19), givesan infinite system of linear algebraic equations for the coefficients A n ( k ) and B n ( k ): I n ( ka ) A n ( k ) + K n ( ka ) B n ( k ) = G n ( k ) , n = 0 , ± , ± , . . . ; I n ( kb ) ∞ X m = −∞ A m ( k ) I m − n ( kρ ) e i ( m − n ) ϕ + (24) K n ( kb ) ∞ X m = −∞ B m ( k )( − m − n I m − n ( kρ ) e i ( m − n ) ϕ = H n ( k ) e − ikz . The solution to this system provides the patch potential for any values of the parametersinvolved. However, all we need is a perturbative 2nd–order solution in a small transverseshift [recall the condition (4) and our summation convention]: A n ( k ) = A n ( k ) + A µn ( k ) x µ + A µνn ( k ) x µ x ν + O (cid:16) ρ (cid:17) ; (25) B n ( k ) = B n ( k ) + B µn ( k ) x µ + B µνn ( k ) x µ x ν + O (cid:16) ρ (cid:17) , n = 0 , ± , ± , . . . . The system (24) is perfect for these perturbations: all its non-diagonal matrix elementsare small due to the factor I m − n ( kρ ) ∼ O (cid:16) ρ | m − n | (cid:17) . The calculations are done, and the7xpressions for all the coefficients are determined in Appendix A. They are subsequentlysimplified there to l. o. in the second small parameter from (4), d/a ≪
1, allowing for theexpansion of the patch potential in both primed and unprimed coordinates in the form:Φ p = Φ p + Φ pµ ( x µ /d ) + Φ pµν ( x µ /d ) ( x ν /d ) + O (cid:16) ( ρ /d ) (cid:17) . (26)To save space, we introduce a special notation F ≡ π Z ∞−∞ dk ∞ X n = −∞ ; (27)using it, in the inner cylinder coordinates, to l. o. in d/a ≪
1, we have:Φ p ( ~r ′ ) = − ad F (h G n ( k ) − H n ( k ) e − ı kz i Ω n ( kρ ′ ) e ı (cid:16) kz ′ + nϕ ′ (cid:17)) ; (28)Φ pµ ( ~r ′ ) = ad F ( c ± µ h G n ± ( k ) − H n ± ( k ) e − ı kz i Ω n ( kρ ′ ) e ı (cid:16) kz ′ + nϕ ′ (cid:17)) ; (29)Φ pµν ( ~r ′ ) = − ad F nh c ± µ c ± ν (cid:16) G n ± ( k ) − H n ± ( k ) e − ı kz (cid:17) + δ µν / (cid:16) G n ( k ) − H n ( k ) e − ıkz (cid:17)i Ω n ( kρ ′ ) e ı (cid:16) kz ′ + nϕ ′ (cid:17)) ; µ, ν = 1 , n ( kρ ′ ) = K n ( kb ) I n ( kρ ′ ) − I n ( ka ) K n ( kρ ′ ) , (31)and terms like c ± G n ± and c ± c ± G n ± should be read as the sums c ± G n ± = c + G n +1 + c − G n − , c ± c ± G n ± = c + c + G n +1 + c − c − G n − ;(the definitions of c ± µ are given in Appendix A). For | k | → ∞ , the large–argument asymp-totics of the Bessel functions ([15], Ch. 5) leads toΩ n ( kρ ′ ) ∼ | k | − / (cid:20) e −| k | ( b − ρ ′ ) − e −| k | ( ρ ′ − a ) (cid:21) , (32)so the integrals converge and the above representations hold for a < ρ ′ < b .For brevity, we omitted the factor [1 + O ( d/a )] that expresses the correction in termsof our small parameter d/a in all the formulas (28)—(30). The estimates of the remain-der are only valid under some additional, as compared to (10), smoothness conditions on8 ( ϕ ′ , z ′ ) , H ( ϕ, z ), namely, the square integrability of their mixed second derivatives [seeformulas (A11), (A12), Appendix A].By a similar fashion, we obtain the potential in the coordinates of the outer cylinder, asexplained at the end of Appendix A:Φ p ( ~r ) = − ad F nh G n ( k ) e ı kz − H n ( k ) i Ω n ( kρ ) e ı ( kz + nϕ ) o ; (33)Φ pµ ( ~r ) = ad F n c ± µ h G n ± ( k ) e ı kz − H n ± ( k ) i Ω n ( kρ ) e ı ( kz + nϕ ) o ; (34)Φ pµν ( ~r ) = − ad F nh c ± µ c ± ν (cid:16) G n ± ( k ) e ı kz − H n ± ( k ) (cid:17) + δ µ ν / (cid:16) G n ( k ) e ı kz − H n ( k ) (cid:17)i × Ω n ( kρ ) e ı ( kz + nϕ ) o , µ, ν = 1 , , (35)with Ω n ( kρ ) defined in (31), and the same meaning of the terms c ± G n ± , etc. Once again wedropped the factor [1+ O ( d/a )], which is true under the same conditions (A11), (A12). Usingthe large argument asymptotics of the Bessel functions again, one shows the formulas (33)–(35) to hold in the domain a < ρ < b , so that the combination of the two representations,in the unprimed and primed coordinates, covers the whole domain of the gap between thecylinders, that is, a < ρ ′ , ρ < b . C. Alternative Method of Finding the Patch Potential
The solution for the patch potential can also be obtained by the method of perturbationof the boundary [20] exploiting the first of the parameter conditions (4). Indeed, by thecoordinate transformation (5), using ( ρ /ρ ) ∼ ( ρ /a ) ≪
1, one finds ρ ′ = q ρ + ρ + 2 ρρ cos( ϕ − ϕ ) = ρ + ( ρ /ρ ) cos( ϕ − ϕ ) + . . . . Hence the boundary equation ρ ′ = a is equivalent to ρ = a − ǫ ( ϕ ) , (36) ǫ ( ϕ ) = a µ ( ϕ ) x µ + b µν ( ϕ ) x µ x ν + O (cid:16) ρ (cid:17) , (37)with the coefficients a µ , b µν found explicitly. Expanding the l.h.s of the boundary condi-tion (15) at the inner cylinder in a Taylor series according to the equality (36) we find:Φ p (cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a = Φ p (cid:12)(cid:12)(cid:12)(cid:12) ρ = a − ǫ = Φ p (cid:12)(cid:12)(cid:12)(cid:12) ρ = a − ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = a ǫ + 12 ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = a ǫ + O (cid:16) ρ (cid:17) = G n ( k ) . (38)9he potential and the boundary value function can be written in the formΦ p ( ~r ) = Φ p ( ~r ) + Φ pµ ( ~r ) x µ + Φ pµν ( ~r ) x µ x ν + O (cid:16) ρ (cid:17) , (39) G ( ϕ ′ , z ′ ) = G ( ϕ, z ) + G µ ( ϕ, z ) x µ + G µν ( ϕ, z ) x µ x ν + O (cid:16) ρ (cid:17) . (40)Introducing the expansions (37), (39) and (40) to the boundary condition (38) and equatingthe coefficients at the same order on either side, we arrive at the set of the inner boundaryconditions for the potentials of all orders:Φ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = a = G ( ϕ, z ); Φ pµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = a = G µ ( ϕ, z ) + ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = a a µ ( ϕ ) , µ, ν = 1 , pµν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = a = G µν ( ϕ, z ) + " ∂ Φ pµ ∂ρ a ν ( ϕ ) + ∂ Φ p ∂ρ b µν ( ϕ ) − ∂ Φ p ∂ρ a µ ( ϕ ) a ν ( ϕ ) ρ = a . These potentials satisfy also the following boundary conditions at the outer cylinder, asimplied by (16) and (39):Φ p (cid:12)(cid:12)(cid:12)(cid:12) ρ = b = H ( ϕ, z ); Φ pµ (cid:12)(cid:12)(cid:12)(cid:12) ρ = b = Φ pµν (cid:12)(cid:12)(cid:12)(cid:12) ρ = b = 0 , µ, ν = 1 , , (41)and of course they are solutions to the Laplace equation in the gap a < ρ < b .Thus we have a sequence of boundary value problems that can be solved by the standardseparation of variables in the unprimed cylindrical coordinates, obtaining the potential to therequired order. This approach turns out much more cumbersome than that of the previoussection. Nevertheless, it provides an important cross–check; for this reason we have carriedit out and eventually obtained exactly the same result for the patch potential. D. Potential Due to Uniform Voltages and the Final Result
It remains to determine the potential, Φ u , generated by the uniform boundary voltages.As seen from the equations (7), (8) and representation (13), the BVP for it is:∆Φ u = 0 , ρ ′ > a, ρ < b, ≤ ϕ < π, | z | < ∞ ; (42)Φ u (cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a = 0 , Φ u (cid:12)(cid:12)(cid:12)(cid:12) ρ = b = V − , where V − is the given constant voltage from the formula (9). This prob-lem is a two–dimensional one, since the uniform potential does not depend on10he axial coordinate. Its solution can be obtained by the method of sec-tion II B without even using the re-expansions of the three–dimensional cylindri-cal solutions. However, formally the BVP (42) is a particular case of the BVP(7), (8) in which the boundary data are specified through their Fourier coefficients as G n ( k ) = 0 , H n ( k ) = 2 πV − δ ( k ) δ n , (43)where δ ( k ) and δ n are the Dirac delta-function and the Kronecker symbol, respectively.With certain caution, the solution can be thus obtained from the results of section II B,namely, formulas (28)—(30) and (33)—(35). The appropriate calculations, with some phys-ical explanations, are found in Appendix B; here we give just the resulting expressions.The solution of the above BVP in the gap a < ρ ′ , ρ < b is of the form:Φ u = Φ u + Φ uµ (cid:16) x µ /d (cid:17) + Φ uµν (cid:16) x µ /d (cid:17) (cid:16) x ν /d (cid:17) + O (cid:16) ( ρ /d ) (cid:17) . (44)In the inner cylinder coordinates ( a < ρ ′ < b ):Φ u ( ~r ′ ) = V − ρ ′ − ad ; (45)Φ uµ ( ~r ′ ) = − ad V − ℜ (cid:18) c + µ e − ıϕ ′ (cid:19) (cid:16) ρ ′ /a − a/ρ ′ (cid:17) ; (46)Φ uµν ( ~r ′ ) = a d V − ( ℜ (cid:18) c + µ c + ν e − ıϕ ′ (cid:19) (cid:20)(cid:16) ρ ′ /a (cid:17) − (cid:16) a/ρ ′ (cid:17) (cid:21) + δ µν ρ ′ − aa ) . (47)In the outer cylinder coordinates ( a < ρ < b ):Φ u ( ~r ) = V − ρ − ad ; (48)Φ uµ ( ~r ) = − ad V − ℜ (cid:16) c + µ e − ıϕ (cid:17) ( ρ/a − a/ρ ) ; (49)Φ uµν ( ~r ) = a d V − (cid:26) ℜ (cid:16) c + µ c + ν e − ıϕ (cid:17) h ( ρ/a ) − ( a/ρ ) i + δ µν ρ − aa (cid:27) . (50)Note that Φ u ( ~r ′ ) and Φ u ( ~r ) are identical functions of just different arguments, which,of course, coincide to the lowest order ( ~ρ ′ ≡ ~ρ for ρ = 0). More importantly, the two–dimensional BVP (42) allows for a very simple exact solution in the bipolar coordinates { α, β } (see e.g. [21], 7.3): it is a linear function of just one coordinate α , a transversecoordinate in the asymmetric gap. The symmetric limit (coaxial cylinders) is, however, asingular one for these coordinates, since the focal distance of the coordinate system tendsto infinity. Nevertheless, it is still possible to expand the solution properly and reproduceexactly the above results, — another important check of our calculations.11 II. ELECTROSTATIC ENERGY
Denote D L a finite domain between the two cylinders cut at z = ± L ; D ∞ is then thewhole infinite volume of the gap between the cylinders. The electrostatic energy stored in D L is expressed through the potential as W L = ǫ Z D L ( ∇ Φ) dV = ǫ Z D L dV h ( ∇ Φ u ) + 2 ( ∇ Φ u · ∇ Φ p ) + ( ∇ Φ p ) i , according to representation (13). The energy consists, naturally, of three terms: the one dueto the uniform potential, the contribution coming from the interaction between the uniformand patch potentials, and the energy of patches. The uniform potential does not dependon the axial coordinate, so the uniform part of energy (and eventually the force from it, seesection I B) is proportional to the cylinder height, 2 L ; i.e., it is the energy per unit lengththat is finite. We thus write this part of the energy in the form: W u ( L ) = W u ( L, ~r ) = ǫ L Z C ( ∇ ⊥ Φ u ) dA ; (51) C is the gap annulus at z = const , ∇ ⊥ is the corresponding two–dimensional gradient.In the two remaining contributions one can take the limit L → ∞ , assuming, as usual, alocal character of the patches: W int = W int ( ~r ) = ǫ Z D ∞ ( ∇ Φ u · ∇ Φ p ) dV ; (52) W p = W p ( ~r ) = ǫ Z D ∞ ( ∇ Φ p ) dV . (53)Using the results of the previous section, we are going to calculate the quantities (51)—(53)one by one in the three sections below; some details of these calculations can be foundin Appendix C. In these calculations we systematically employ the following convenientformulas.Let D be some domain with the boundary B , and u ( ~r ) and v ( ~r ) be some functions squarelyintegrable over D with their derivatives. Let also u ( ~r ) satisfy the Laplace equation in D ;then by integrating by parts we find0 = Z D v ∆ u dV = Z B v ∂u∂n dA − Z D ( ∇ v · ∇ u ) dV , where n is the outward normal to the boundary B . Thus Z D ( ∇ v · ∇ u ) dV = Z B v ∂u∂n dA, ∆ u = 0 in D ; (54)12or v = u , in particular, Z D ( ∇ u ) dV = Z B u ∂u∂n dA, ∆ u = 0 in D . (55)Moreover, if both functions are harmonic in D , then we can represent the integral (54) intwo symmetric forms, Z D ( ∇ v · ∇ u ) dV = Z B v ∂u∂n dA = Z B u ∂v∂n dA, ∆ u = ∆ v = 0 in D . (56)Formulas (54)—(56) hold in the space of any dimension D , in particular, for D = 2 , A. Uniform Energy
The definition (51) and formula (55) with u = Φ u give: W u ( L, ~r ) = ǫ L Z C ( ∇ ⊥ Φ u ) dA = ǫ L Z π bdϕ Φ u ∂ Φ u ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = ǫ LaV − Z π dϕ ∂ Φ u ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b [1 + O ( d/a )] , where we used both boundary conditions (42). Substituting the expansion (44) for thepotential and then carrying out the straightforward integration using expressions (48)–(50),we obtain ( A L is the area of the piece of cylinder with the height 2 L ): W u ( L, ~r ) = W u ( L ) + W uµ ( L ) ( x µ /d ) + W uµν ( L ) ( x µ /d ) ( x ν /d ) + O h ( ρ /d ) i ; (57) W u ( L ) = ǫ LaV − Z π dϕ ∂ Φ u ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = 2 πLǫ ad (cid:16) V − (cid:17) = ǫ A L ( V − ) d ; W uµ ( L ) = ǫ LaV − Z π dϕ ∂ Φ uµ ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = 0 ; (58) W uµν ( L ) = ǫ LaV − Z π dϕ ∂ Φ uµν ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = πLǫ ad (cid:16) V − (cid:17) δ µν = ǫ A L ( V − ) d δ µν . The zero order term here is, of course, the classical energy of a plane capacitor, the same asfor the cylindrical one to l.o. in d/a ; the first order term vanishes due to symmetry.13 . Interaction Energy
We apply formula (54) to the expression (52) with u = Φ p and v = Φ u , taking intoaccount both boundary conditions in the problem (42), which gives: W int ( ~r ) = ǫ Z D ∞ ( ∇ Φ u · ∇ Φ p ) dV = ǫ aV − Z ∞−∞ dz Z π dϕ ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b [1 + O ( d/a )] . As in the uniform case above, we substitute here the expansion (26), to get: W int ( ~r ) = W int + W intµ ( x µ /d ) + W intµν ( x µ /d ) ( x ν /d ) + O h ( ρ /d ) i ; (59) W int = ǫ aV − Z ∞−∞ dz Z π dϕ ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b ; W intµ = ǫ aV − Z ∞−∞ dz Z π dϕ ∂ Φ pµ ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b ; W intµν = ǫ aV − Z ∞−∞ dz Z π dϕ ∂ Φ pµν ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b . (60)The derivatives of the potential at ρ = b are found in Appendix C, formulas (C2)—(C3).Each of them has the form of the double Fourier transform (11), u ( ϕ, z ) = 12 π Z ∞−∞ dk ∞ X n = −∞ u n ( k ) e i ( kz + nϕ ) , where u ( ϕ, z ) is any one of these derivatives. By inverting this formula and setting then k = 0 and n = 0 in the result, one obtains u n ( k ) = 12 π π Z ∞ Z −∞ dϕdz u ( ϕ, z ) e − i ( kz + nϕ ) , Z ∞−∞ dz Z π dϕ u ( ϕ, z ) = 2 πu (0) . The last one is exactly the integral that stands in each of the expressions (60). Thuscombining it with the mentioned formulas for the derivatives, after some not very tediousalgebra using the relation ( c − µ ) ∗ = c + µ for the coefficients involved, we find: W int = − πǫ ad V − [( G (0) − H (0))] ,W intµ = +4 πǫ ad V − ℜ h c + µ ( G (0) − H (0)) i , (61) W intµν = − πǫ ad V − n ℜ h c + µ c + ν ( G (0) − H (0)) i + ( δ µν /
4) ( G (0) − H (0)) o . In these calculations we have also used the property (17) as applied to the Fourier coefficient f n ( k ) = G n ( k ) e ıkz − H n ( k ) ; (62)recall that ℜ ( · ) denotes the real part of ( · ).14 . Patch Energy We now apply the formula (55) with u = Φ p to the definition (53) of the patch energy.Making use of the boundary conditions (15) and (16), we obtain: W p ( ~r ) = ǫ Z D ∞ ( ∇ Φ p · ∇ Φ p ) dV = ǫ Z ∞−∞ dz Z π bdϕ H ( ϕ, z ) ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b − Z ∞−∞ dz ′ Z π adϕ ′ G ( ϕ ′ , z ′ ) ∂ Φ p ∂ρ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a . Combining this with the expansion (26) for the patch potential we arrive at the energy ofpatches written as in the formula (2): W p ( ~r ) = W p + W pµ ( x µ /d ) + W pµν ( x µ /d ) ( x ν /d ) + O h ( ρ /d ) i , (63)where, to l. o. in d/a , W pξ = ǫ a Z ∞−∞ dz Z π dϕ H ( ϕ, z ) ∂ Φ pξ ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b − Z ∞−∞ dz ′ Z π dϕ ′ G ( ϕ ′ , z ′ ) ∂ Φ p ∂ρ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a , (64)for ξ = 0 , µ, µν , i. e., for all the functional coefficients in the expression (63). These doubleintegrals can be calculated immediately employing the Parceval identity (12), for which oneneeds the Fourier coefficients of each of the two factors in the integrand in (64). The firstfactors H ( ϕ, z ), G ( ϕ ′ , z ′ ) give no such problem, while the second ones are the derivatives ofthe potentials at the boundaries. Their expressions are found in Appendix C: formulas (C2),(C3) provide them at ρ = b , while (C6), (C7) work for ρ ′ = a . All these expressions havea structure of the double Fourier transform with the explicit Fourier coefficients. Thereforewe can write, first for the zeroth order term in the energy expansion (63): W p = ǫ a Z ∞−∞ dk ∞ X n = −∞ ( H ∗ n ( k ) " − f n ( k ) d − G ∗ n ( k ) " − f n ( k ) d e − ıkz ) = ǫ a d Z ∞−∞ dk ∞ X n = −∞ | f n ( k ) | , (65)where we used the notation (62) again, as well as the formulas (C2) and (C6).Similar calculations of W pµ , W pµν are based on the formulas (C2), (C6), and (C3), (C7)respectively; they exploit relations P ∞ n = −∞ c − µ f ∗ n ( k ) f n − ( k ) = P ∞ n = −∞ c − µ f ∗ n +1 ( k ) f n ( k ) and15 c − µ ) ∗ = c + µ resulting in W pµ = − ǫ a d Z ∞−∞ dk ∞ X n = −∞ ℜ h c + µ f ∗ n ( k ) f n +1 ( k ) i ; (66) W pµν = ǫ a d Z ∞−∞ dk ∞ X n = −∞ h δ µν / | f n ( k ) | + 2 ℜ (cid:16) c + µ c + ν f ∗ n ( k ) f n +2 ( k ) (cid:17)i . Replacing now f n ( k ) with its expression (62) in the formulas (65), (66) allows us to obtainthe final answer for the energy of the patches: W p = ǫ a d Z ∞−∞ dk ∞ X n = −∞ | G n ( k ) e ıkz − H n ( k ) | ; W pµ = − ǫ ad Z ∞−∞ dk ∞ X n = −∞ ℜ h c + µ (cid:16) G ∗ n ( k ) e − ıkz − H ∗ n ( k ) (cid:17) (cid:16) G n +1 ( k ) e ıkz − H n +1 ( k ) (cid:17)i ; (67) W pµν = ǫ a d Z ∞−∞ dk ∞ X n = −∞ n δ µν / | G n ( k ) e ıkz − H n ( k ) | +2 ℜ h c + µ c + ν (cid:16) G ∗ n ( k ) e − ıkz − H ∗ n ( k ) (cid:17) (cid:16) G n +2 ( k ) e ıkz − H n +2 ( k ) (cid:17)io . Remarkably, the expression for the zeroth order energy W p is perfectly similar to that inthe spherical case [see [8], the last expression in the formula (3) there]. The similarity getseven more pronounced when rewriting the formula (67) for W p using the Parceval identity: W p = ǫ a d Z ∞−∞ dz Z π dϕ [ V a ( ϕ, z ) − V b ( ϕ, z )] ;here V a ( ϕ, z ) ≡ G ( ϕ, z ) and V b ( ϕ, z ) ≡ H ( ϕ, z ) are the patch voltages taken in the coordinateof the outer cylinder.The calculation of the energy is now finished; we have all its parts in exactly the formneeded for the calculation of the electrostatic force. Acknowledgments
This work was supported by ICRANet (V.F.) and by KACST through the collaborativeagreement with GP-B (A.S.). The authors are grateful to Remo Ruffini and Francis Everittfor their permanent interest in and support of this work, and for some valuable remarks.John Mester has helped us a lot by providing information about STEP, and by encouragingto carry out this analysis. A special thanks to David Hipkins, a great motivator for thepatch effect analysis. We are extremely greatful to Paul Worden for his input regardingthe STEP requirements, and other remarks. We thank Dan DeBra and Sasha Buchman forfruitfull discussions and insightful suggestions.16 ppendix A: Calculation of the Patch Potential
The re–expansion formulas for cylindrical solutions to the Laplace equation in shiftedcoordinates are found in [19], and . They read: I n ( kρ ′ ) e inϕ ′ = ∞ X m = −∞ I n − m ( kρ ) e i ( n − m ) ϕ I m ( kρ ) e imϕ ; K n ( kρ ′ ) e inϕ ′ = ∞ X m = −∞ ( − n − m I n − m ( kρ ) e i ( n − m ) ϕ K m ( kρ ) e imϕ , where ρ and ϕ are the polar components of the shift as defined in (2) and (6). Therefore ∞ X n = −∞ h A n ( k ) I n ( kρ ′ ) + B n ( k ) K n ( kρ ′ ) i e inϕ ′ = ∞ X n = −∞ I n ( kρ ) e inϕ ∞ X m = −∞ I m − n ( kρ ) e i ( m − n ) ϕ A m ( k ) + ∞ X n = −∞ K n ( kρ ) e inϕ ∞ X m = −∞ ( − m − n I m − n ( kρ ) e i ( m − n ) ϕ B m ( k ) , where we have changed the order of summation and then switched the indeces m and n .When substituted in the representation (18) of the patch potential Φ p in primed coordinatestaking into account z ′ = z + z , this gives exactly the representation (20)—(22) in theunprimed coordinates of the outer cylinder.We are now to construct the 2nd order perturbative solution (25), " A n ( k ) B n ( k ) = " A n ( k ) B n ( k ) + " A µn ( k ) B µn ( k ) x µ + " A µνn ( k ) B µνn ( k ) x µ x ν + O (cid:16) ρ (cid:17) , (A1)to the infinite system of linear algebraic equations (24). As mentioned in section II B,its matrix coefficients contain the factor I m − n ( kρ ) ∼ O (cid:16) ρ | m − n | (cid:17) , so only the terms m = n, n ± , n ± I ( kρ ) = 1 + ( kρ ) / O (cid:16) ( kρ ) (cid:17) , I ± ( kρ ) = ( kρ ) / O (cid:16) ( kρ ) (cid:17) ,I ± ( kρ ) = ( kρ ) / O (cid:16) ( kρ ) (cid:17) , as well as the convenient representations ( c +1 ≡ . , c +2 ≡ . i ) ρ = δ µν x µ x ν ; ( ρ / e iϕ = c + µ x µ , ( ρ / e iϕ = c + µ c + ν x µ x ν , (A2)we find: I ( kρ ) = 1 + k δ µν x µ x ν + O (cid:16) ( kρ ) (cid:17) ; (A3)17 ( kρ ) e iϕ = k c + µ x µ + O (cid:16) ( kρ ) (cid:17) , I − ( kρ ) e − iϕ = k c − µ x µ + O (cid:16) ( kρ ) (cid:17) , c − µ ≡ ( c + µ ) ∗ ; I ( kρ ) e iϕ = k c + µ c + ν x µ x ν + O (cid:16) ( kρ ) (cid:17) , I − ( kρ ) e − iϕ = k c − µ c − ν x µ x ν + O (cid:16) ( kρ ) (cid:17) . Here we introduced the notation c − µ for a uniform writing. Substituting expressions (A1)and (A3) in the system (24) allows us to obtain the following sequence of equations for theunknown coefficients in expansions (A1): I n ( ka ) A ξn ( k ) + K n ( ka ) B ξn ( k ) = G ξn ( k ) ,I n ( kb ) A ξn ( k ) + K n ( kb ) B ξn ( k ) = H ξn ( k ) , where n = 0 , ± , ± , . . . , and the index ξ assumes three values ξ = 0 , µ, µν , with µ, ν = 1 , G n ( k ) = G n ( k ) , G µn ( k ) = G µνn = 0 ; (A4) H n ( k ) = H n ( k ) e − ikz , H µn ( k ) = − k c ± µ h I n ( kb ) A n ± ( k ) − K n ( kb ) B n ± ( k ) i ; H µνn ( k ) = − k h I n ( kb ) c ± µ A νn ± ( k ) − K n ( kb ) c ± µ B νn ± ( k ) i − k " I n ( kb ) c ± µ c ± ν A n ± ( k ) + δ µ ν A n ( k ) ! + K n ( kb ) c ± µ c ± ν B n ± ( k ) + δ µ ν B n ( k ) ! , and terms like c ± A n ± and c ± c ± A n ± should be read as c ± A n ± ( k ) = c + A n +1 ( k ) + c − A n − ( k ) , c ± c ± A n ± ( k ) = c + c + A n +1 ( k ) + c − c − A n − ( k ) . Solving the above linear systems leads to the following set of answers ( ξ = µ, µν ): A n ( k ) = G n ( k ) K n ( kb ) − H n ( k ) K n ( ka ) e − ı kz D n ( k ) ,B n ( k ) = H n ( k ) I n ( ka ) e − ı kz − G n ( k ) I n ( kb ) D n ( k ) ; (A5) A ξn ( k ) = − H ξn ( k ) K n ( ka ) D n ( k ) , B ξn ( k ) = H ξn ( k ) I n ( ka ) D n ( k ) ; (A6)where D n ( k ) = K n ( kb ) I n ( ka ) − K n ( ka ) I n ( kb ) . (A7)Expressions (A5)—(A7) and (A1) provide the solution for the patch potential in the form(18) to the 2nd order in ρ /d and any value of d/a . We now simplify all the expressions tol. o. in this small second parameter. To do this, we employ the Taylor expansions: I n ( kb ) = I n ( ka ) + I ′ n ( ka ) kd + O h ( kd ) i , b = a + d, d → K n ( kb ) = K n ( ka ) + K ′ n ( ka ) kd + O h ( kd ) i ; (A8)18here the primes denote the derivatives with respect to the whole argument. With the helpof these expansions and the known formula for the Wronskian of the Bessel functions, W ( I n ( ξ ) , K n ( ξ )) = − /ξ , (A9)we simplify the denominator (A7) to lowest order in d/a as: D n ( k ) = kd h I n ( ka ) K ′ n ( ka ) − K n ( ka ) I ′ n ( ka ) i + da O da ! + O n ( ka ) da ! = − da O da ! + O n ( ka ) da ! . (A10)The second term in the remainder could be dropped if k and n were bounded. But inour formulas both run from minus to plus infinity, so we need to keep this remainder tomake the asymptotic expressions uniform for the whole range of k and n . However, whensubstituted in the sums over n and integrals over k , as in formula (18), these terms give riseto contributions O (( d/a ) ) provided that the proper sums/integrals converge, Z ∞−∞ dk ∞ X n = −∞ k n | G n ( k ) | < ∞ , Z ∞−∞ dk ∞ X n = −∞ k n | H n ( k ) | < ∞ . (A11)By the Parceval identity (12), the latter conditions are equivalent to π Z ∞ Z −∞ dϕ ′ dz ′ | ∂ G/∂ϕ ′ ∂z ′ ) | < ∞ , π Z ∞ Z −∞ dϕdz | ∂ H/∂ϕ∂z ) | < ∞ . (A12)Moreover, in the expressions of A n , A µn , A µνn we use the expansion K n ( ka ) = K n ( kb ) − K ′ n ( ka ) kd + O h ( kd ) i , while for B n we employ the first of the formulas (A8). This is done, in fact, to provide theproper convergence of integrals over k and series in n at infinity, see section II B.For the zero order coefficients, we thus have: A n ( k ) = − ad K n ( kb ) h G n ( k ) − H n ( k ) e − ı kz i + O da ! + O n ( ka ) da ! ,B n ( k ) = ad I n ( ka ) h G n ( k ) − H n ( k ) e − ı kz i + O da ! + O n ( ka ) da ! . (A13)From now on, all our asymptotic formulas have the same two–term remainder estimates,which we simply drop. 19y the formulas (A13) and definition (A4), we obtain thus H µn ( k ) = 1 d c ± µ h G n ± ( k ) − H n ± ( k ) e − ı kz i , and the expressions (A6) become A µn ( k ) = ad K n ( kb ) c ± µ h G n ± ( k ) − H n ± ( k ) e − ı kz i ,B µn ( k ) = − ad I n ( ka ) c ± µ h G n ± ( k ) − H n ± ( k ) e − ı kz i . (A14)By the same token, formula (A4) simplifies to H µνn ( k ) = 1 d ( c ± µ c ± ν h G n ± ( k ) − H n ± ( k ) e − ı kz i + δ µν h G n ( k ) − H n ( k ) e ı kz i) , and the coefficients (A6) are written as A µνn ( k ) = − ad ( K n ( kb ) " c ± µ c ± µ h G n ± ( k ) − H n ± ( k ) e − ı kz i + δ µ ν h G n ( k ) − H n ( k ) e − ı kz i ; (A15) B µνn ( k ) = ad ( I n ( ka ) " c ± µ c ± µ h G n ± ( k ) − H n ± ( k ) e − ı kz i + δ µ ν h G n ( k ) − H n ( k ) e − ı kz i . Expressions (A13)—(A15), when substituted in the representation (18), provide exactly theformulas (28)–(30) for the potential in the primed coordinates.To obtain the potential in the unprimed coordinates we need, as seen from the formula(20), to compute the coefficients ˜ A n , ˜ B n to the 2nd order in the transverse shift, in a completesimilarity with the formula (A1): " ˜ A n ( k )˜ B n ( k ) = " ˜ A n ( k )˜ B n ( k ) + " ˜ A µn ( k )˜ B µn ( k ) x µ + " ˜ A µνn ( k )˜ B µνn ( k ) x µ x ν + O (cid:16) ρ (cid:17) , (A16)To get them, we simply use the definitions (21) and (22) of the coefficients ˜ A n , ˜ B n , andintroduce there the expansions (A1) with the known coefficients (A13)—(A15), as well asthe expansions (A3). This is a tedious and cumbersome, but straightforward calculation,which ends with the representations (33)–(35).20 ppendix B: Calculation of the Potential due tothe Uniform Boundary Voltages The uniform potential is obtained substituting the Fourier coefficients of the boundaryfunctions (43), G n ( k ) = 0 , H n ( k ) = 2 πV − δ ( k ) δ n , in the formulas found for the patch potential, (28)–(30) and (33)–(35). The zeroth order(coaxial configuration) potential in the primed coordinates then is [recall the definitions (31)and (27)]: Φ u ( ~r ′ ) = ad V − lim k → h K ( kb ) I ( kρ ′ ) − I ( ka ) K ( kρ ′ ) i . Since I ( ξ ) → , K ( ξ ) ∼ ln(2 /ξ ) when ξ →
0, we furthermore obtainΦ u ( ~r ′ ) = ad V − ln( ρ ′ /b ) = ad V − h ln( ρ ′ /a ) + O ( d/a ) i . (B1)The logarithm above is a remnant of the exact solution for the coaxial case,[ V − ln( ρ ′ /a ) / ln( b/a )], with the logarithm in the denominator taken to lowest order in d/a ≪ d = b − a ). Moreover, the upper log may be simplified in a simi-lar way:ln( ρ ′ /a ) = ln ρ ′ − aa ! = ρ ′ − aa [1 + O ( d/a )] ;the potential in a narrow capacitor does not feel the curvature of the electrodes to l. o.,therefore it is a linear function of the transverse coordinate. Introducing the last expressionto the formula (B1) we finally obtain:Φ u ( ~r ′ ) = V − ρ ′ − ad [1 + O ( d/a )] . (B2)By the same token, the linear and quadratic potentials are obtained from expressions(29) and (30) as:Φ uµ ( ~r ′ ) = − ad V − lim k → (cid:26) c ± µ e ∓ ıϕ ′ h K ∓ ( kb ) I ∓ ( kρ ′ ) − I ∓ ( ka ) K ∓ ( kρ ′ ) i(cid:27) , and Φ uµν ( ~r ′ ) = ad V − lim k → (cid:26) c ± µ c ± ν e ∓ ıϕ ′ h K ∓ ( kb ) I ∓ ( kρ ′ ) − I ∓ ( ka ) K ∓ ( kρ ′ ) i + δ µν / h K ( kb ) I ( kρ ′ ) − I ( ka ) K ( kρ ′ ) io . I − n ( ξ ) = I n ( ξ ) , K − n ( ξ ) = K n ( ξ ), these may be written as ( ℜ ( · ) is the real part of ( · )):Φ uµ ( ~r ′ ) = − ad V − lim k → (cid:26) ℜ ( c + µ e − ıϕ ′ ) h K ( kb ) I ( kρ ′ ) − I ( ka ) K ( kρ ′ ) i(cid:27) ; (B3)Φ uµν ( ~r ′ ) = ad V − lim k → (cid:26) ℜ ( c + µ c + ν e − ıϕ ′ ) h K ( kb ) I ( kρ ′ ) − I ( ka ) K ( kρ ′ ) i + δ µν / h K ( kb ) I ( kρ ′ ) − I ( ka ) K ( kρ ′ ) io . (B4)The near zero asymptotics of Bessel functions allows one to transform (B3), (B4) to:Φ uµ ( ~r ′ ) = − ad V − (cid:26) ℜ ( c + µ e − ıϕ ′ ) h ρ ′ /b − a/ρ ′ i(cid:27) = − ad V − (cid:26) ℜ ( c + µ e − ıϕ ′ ) h ρ ′ /a − a/ρ ′ i + O ( d/a ) (cid:27) ; (B5)Φ uµν ( ~r ′ ) = a d V − ℜ (cid:26) c + µ c + ν e − ıϕ ′ (cid:20)(cid:16) ρ ′ /b (cid:17) − (cid:16) a/ρ ′ (cid:17) (cid:21) + δ µν ln (cid:16) ρ ′ /b (cid:17)(cid:27) = a d V − ℜ ( c + µ c + ν e − ıϕ ′ (cid:20)(cid:16) ρ ′ /a (cid:17) − (cid:16) a/ρ ′ (cid:17) (cid:21) + δ µν ρ ′ − aa + O ( d/a ) ) , (B6)where we used the same transformation of a logarithm as when deriving formula (B2) fromexpression (B1).Formulas (B2), (B5), and (B6) are the final expressions for the uniform potential in theinner cylinder coordinates as given in formulas (45)–(47). The potential in the unprimedcoordinates is now gotten in exactly the same way without any new difficulties: one juststarts, respectively, from the expressions (33)–(35) of the patch potential in the unprimedcoordinates, and follows the steps described above; in this way, the final results (48)–(50)are found. Appendix C: Calculation of the Energy
Here we provide the intermediate results needed for the energy computation. The first ofthem is the derivatives ∂ Φ p /∂ρ, ∂ Φ pµ /∂ρ, ∂ Φ pµν /∂ρ at the outer cylinder boundary ρ = b re-quired to calculate W int by the formulas (60). First of all we differentiate in ρ the expressions(33)–(35), i.e., the function Ω n ( kρ ) in them. The result is: ∂ Ω n ( kξ ) ∂ξ = ∂∂ξ [ K n ( kb ) I n ( kξ ) − I n ( ka ) K n ( kξ )] = k h K n ( kb ) I ′ n ( kξ ) − I n ( ka ) K ′ n ( kξ ) i , (C1)22here the prime indicates the derivative with respect to the whole argument, as usual. Atthe outer boundary, exploiting the relation I n ( ka ) = I n ( kb ) − kdI ′ n ( ka ) + O (cid:16) ( kd ) (cid:17) , and the formula (A9) for the Wronskian, we can write the derivative (C1) ∂ Ω n ( kξ ) ∂ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = b = k h − W ( I n ( kξ ) , K n ( kξ )) | kξ = kb + O [( kd ) ] i = 1 b + O k a da ! . Combining this with the representations (33)—(35), we obtain: ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = − d F n f n ( k ) e ı ( kz + nϕ ) o ; ∂ Φ pµ ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = 1 d F n c ± µ f n ± ( k ) e ı ( kz + nϕ ) o ; (C2) ∂ Φ pµν ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = − d F nh c ± µ c ± ν f n ± ( k ) + ( δ µ ν / f n ( k ) i e ı ( kz + nϕ ) o . (C3)Here F is the integration—summation operator defined in (27), and, for brevity, f n ( k ) ≡ G n ( k ) e ıkz − H n ( k ) . (C4)The correction in the formulas (C2), (C3) is given by [1 + O ( d/a )] under the conditions ∞ Z −∞ dk ∞ X n = −∞ | k | | G n ( k ) | < ∞ , ∞ Z −∞ dk ∞ X n = −∞ | k | | H n ( k ) | < ∞ . (C5)The derivatives ∂ Φ p /∂ρ ′ , ∂ Φ pµ /∂ρ ′ , ∂ Φ pµν /∂ρ ′ at the inner cylinder ρ ′ = a are alsoneeded, by the expressions (64). They are derived from the formulas (28)—(30) exactly asabove, the only significant variation is in transforming the formula (C1). Using the secondof expansions (A8) and formula (A9), we find: ∂ Ω n ( kξ ) ∂ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ = a = k h − W ( I n ( kξ ) , K n ( kξ )) | kξ = ka + O [( kd ) ] i = 1 a + O k a da ! . With this small change, the desired formulas are found as: d ∂ Φ p ∂ρ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a = −F (cid:20) f n ( k ) e ı ( k ( z ′ − z )+ nϕ ′ ) (cid:21) ; d ∂ Φ pµ ∂ρ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a = F (cid:20) c ± µ f n ± ( k ) e ı ( k ( z ′ − z )+ nϕ ′ ) (cid:21) (C6) d ∂ Φ pµν ( ~r ′ ) ∂ρ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ ′ = a = −F h c ± µ c ± ν f n ± ( k ) + ( δ µ ν / f n ( k ) i e ı ( k ( z ′ − z )+ nϕ ′ ) , (C7)with the same notations and same order of the remainder as in the formulas (C2), (C3),assuming again condition (C5). 23
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