Electrostatic Patch Effect in Cylindrical Geometry. III. Torques
aa r X i v : . [ g r- q c ] S e p Electrostatic Patch Effect in Cylindrical Geometry. III. Torques
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: May 28, 2018) Abstract
We continue to study the effect of uneven voltage distribution on two close cylindrical conductorswith parallel axes started in our papers [1] and [2], now to find the electrostatic torques. Wecalculate the electrostatic potential and energy to lowest order in the gap to cylinder radius ratio foran arbitrary relative rotation of the cylinders about their symmetry axis. By energy conservation,the axial torque, independent of the uniform voltage difference, is found as a derivative of theenergy in the rotation angle. We also derive both the axial and slanting torques by the surfaceintegration method: the torque vector is the integral over the cylinder surface of the cross productof the electrostatic force on a surface element and its position vector. The slanting torque consistsof two parts: one coming from the interaction between the patch and the uniform voltages, and theother due to the patch interaction. General properties of the torques are described. A convenientmodel of a localized patch suggested in [2] is used to calculate the torques explicitly in terms ofelementary functions. Based on this, we analyze in detail patch interaction for one pair of patches,namely, the torque dependence on the patch parameters (width and strength) and their mutualpositions. The effect of the axial torque is then studied for the experimental conditions of theSTEP mission.
PACS numbers: 41.20Cv; 02.30Em; 02.30Jr; 04.80CcKeywords: Electrostatics - Patch effect - Cylindrical capacitor - Torques - Precision measurements - STEP ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The actual distribution of charges on a metal does not guarantee its surface to be anequipotential because of the impurities and microcrystal structure of the material. Thisphenomenon, known as patch effect (PE; for its experimantal study see paper [3]), is re-sponsible for the mutual force and torque between two metallic surfaces at finite distances.The effect is the larger, the closer the surfaces, as first confirmed by the calculation of thepatch effect force for two parallel conducting planes [4].PE in a cylindrical geometry was studied in the first two parts of our paper, [1] and [2](henceforth referred to as CPEI and CPEII, for ”‘Cylindrical Patch Effect”’), where the PEenergy and force have been examined. Here we calculate the torque due to PE between twocoaxial cylinders. This calculation completes the study of the electrostatic interaction for acylindrical capacitor; our analysis largely benefits from the results found in CPEI and II.PE is important for any precision measurement if its set-up includes conducting surfacesin a closed proximity to each other. For the STEP experiment [5–8], where the differentialaxial motion of cylindrical test masses (TM) will be used to test the universality of free fallto an unprecedented accuracy of about 1 part in 10 , the axial torque is of a particularinterest (see section VI).We determine PE torques between the two boundary surfaces of an infinitely long cylin-drical capacitor. By energy conservation the rotation by an angle γ about the directionˆ γ of one of the conductors relative to the other causes an electrostatic torque in the samedirection which is given by the formula (see, for instance, [9]) T γ = − ∂W ( γ ) ∂γ , (1)where W ( γ ) is the electrostatic energy as a function of the rotation angle. However, dueto the specifics of cylindrical geometry, we can properly imagine a rotation only about thesymmetry axis of the two infinite cylinders. Tilting, say, the inner one about any otherdirection leads to the intersection of the cylinders at some finite distance, i. e., to thebreaking of the problem geometry. For this reason, we employ also a different method ofthe torque calculation. The force, d ~F , due to the electrical field ~E acting on a small area dA of a conductor with the charge density σ is given by d ~F = σ ~EdA , (2)2see our comment [10]). The resulting element of the torque about the origin at distance ~r from dA is then d ~T = ~r × d ~F , (3)which expression, integrated all over the body surface, gives the general expression of thetorque acting on the conductor.The energy, the field and the surface charge density which are needed in the formulas (1)and (3) are expressed through the electrostatic potential in the gap. For typical experimentalconditions, such as in the STEP configuration [6, 7], the gap, d = b − a , is much smallerthan either of the cylinder radii, a < b . This, first, justifies the model of infinite cylinders,especially if the patches are predominantly far from the real cylinder edges, and, second, itallows for a significant simplification of results to lowest order in d/a .In the next section we solve the boundary value problem (BVP) for the potential in thegap with general voltage distributions on the cylinder surfaces. Based on this, we find theenergy in section III, and then the longitudinal PE torque by the formula (1). In section IVwe derive all the three components of the PE torque by the surface integration method usingformula (3) (the two expressions for the axial torque agree precisely). In section V a modelof the localized patch potential introduced in CPEII is described, ending with closed–formexpressions for the torques. The latter are then calculated and analyzed in the case whena single patch is present at each of the cylinders. In section VI we apply our results to theSTEP experiment set–up, coming up with a clear picture of the test mass rotation motion.The details of calculations are given in two appendices. II. ELECTROSTATIC POTENTIAL
We employ both Cartesian and cylindrical coordinates in two frames related to the innerand outer cylinders as shown in fig. 1. In the outer frame an arbitrary point is labelled bythe vector radius ~r ′ , Cartesian coordinates { x ′ , y ′ , z } , or cylindrical coordinates { ρ, ϕ ′ , z } ;in the inner frame the corresponding quantities are ~r , { x, y, z } , { ρ, ϕ, z } . The origins ofthe frames coincide so that the primed and unprimed coordinates are related simply by arotation, by some angle γ , about the z axis: x ′ = x cos γ + y sin γ, y ′ = − x sin γ + y cos γ, z ′ = z , (4)3r, in cylindrical coordinates, ρ ′ = ρ, ϕ ′ = ϕ − γ, z ′ = z . The surfaces of the inner andouter cylinders are thus described by the equations ρ = a and ρ = b , respectively, and areassumed to carry arbitrary distributions of electrostatic voltage. Hence the electrostaticpotential, Φ, satisfying the Laplace equation in the gap between the cylinders,∆Φ = 0 , ρ > a, ρ < b, ≤ ϕ < π, | z | < ∞ , (5)satisfies also the boundary conditions of the first kind at the cylinder surfaces: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 ) = V − + H ( ϕ − γ, z ) . (6)Here V − = const is the uniform potential difference, so all voltages are counted from theuniform voltage of the inner cylinder taken as zero. The non–uniform potential distributions,i.e., the patch voltages, are described by arbitrary smooth enough functions G ( ϕ, z ) and H ( ϕ, z ). Same as in CPEI, II we assume these functions squarely integrable; whereverproper, we will also assume them, as done in conditions (10), (A11) (A12) and (C5) ofCPEI.For any squarely integrable function u ( ϕ, z ) we have its Fourier expansion with the Fouriercoefficient u n ( k ): u ( ϕ, z ) = 12 π Z ∞−∞ dk ∞ X n = −∞ u n ( k ) e i ( kz + nϕ ) , u n ( k ) = 12 π π Z ∞ Z −∞ dϕ dz u ( ϕ, z ) e − i ( kz + nϕ ) . (7)For any two such functions u ( ϕ, z ) and v ( ϕ, z ) the useful Parceval identity holds,( u, v ) ≡ π Z ∞ Z −∞ dϕ dz u ( ϕ, z ) v ∗ ( ϕ, z ) = Z ∞−∞ dk ∞ X n = −∞ u n ( k ) v ∗ n ( k ) ; (8)here and elsewhere the star denotes complex conjugation.According to the boundary condition (6), we split the potential in two parts due to,respectively, the uniform boundary voltages and patches:Φ( ~r ) = Φ u ( ~r ) + Φ p ( ~r ) , (9)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 − ; (10)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 ) , (11)Φ p (cid:12)(cid:12)(cid:12)(cid:12) ρ = b = H ( ϕ − γ, z ) = 12 π Z ∞−∞ dk ∞ X n = −∞ H n ( k ) e − inγ e i ( kz + nϕ ) ;4unction Φ u is the classical solution for a cylindrical capacitor [9]; to l. o. in d/a it isΦ u ( ~r ) = ( a/d ) V − ln ( ρ/a ) . (12)Function Φ p is gotten by the standard separation of variables in cylindrical coordinates [seec.f. [11], Chs. 5, 6]. Its representation satisfying formally the Laplace equation is:Φ p ( ~r ) = 12 π Z ∞−∞ dk ∞ X n = −∞ [ A n ( k ) I n ( kρ ) + B n ( k ) K n ( kρ )] e i ( kz + nϕ ) , (13)where I n ( ξ ) , K n ( ξ ) are the modified Bessel functions of the 1st and 2nd kind, respectively[the Macdonald’s function K n ( ξ ) definition for the negative values of its argument is takenby the parity of I n ( ξ ); so, K n ( kρ ) stands for (sign k ) n K n ( | k | ρ )]. The unknown A n ( k ) , B n ( k )are found form the linear system implied by the boundary conditions (11): A n ( k ) I n ( ka ) + B n ( k ) K n ( ka ) = G n ( k ) ; (14) A n ( k ) I n ( kb ) + B n ( k ) K n ( kb ) = H n ( k ) e − inγ , n = 0 , ± , ± , . . . . It is the same system that has been effectively solved, to lowest order in d/a , in CPEI[Appendix A, coefficients A n ( k ) and B n ( k )], with the exception of e − ı nγ in the r.h.s. insteadof e ıkz . So, by replacing H n ( k ) e − ikz with H n ( k ) e − ı nγ in the answer (A13), CPEI, we get: A n ( k ) = − ad n K n ( kb ) h G n ( k ) − H n ( k ) e − ı nγ io , (15) B n ( k ) = ad n I n ( ka ) h G n ( k ) − H n ( k ) e − ı nγ io . Thus the electrostatic potential (13), to l. o. in d/a , is:Φ p ( ~r ) = − ad Z ∞−∞ dk ∞ X n = −∞ h G n ( k ) − H n ( k ) e − ı nγ i Ω n ( kρ ) e ı ( kz + nϕ ) ; (16)Ω n ( kρ ) = K n ( kb ) I n ( kρ ) − I n ( ka ) K n ( kρ ) . (17)Formulas (12), (16), and (17) allow us to calculate both the electric energy and field. III. AXIAL TORQUE BY THE ENERGY METHOD
The uniform potential (12) does not depend on γ , so the variation of the electrostaticenergy due to the rotation comes only from the patch potential (16), same as it happens withthe axial PE force (CPEII, section III). The axial torque, T z , is thus given by the formula(1) where W ( γ ) is replaced with W p ( γ ) that we calculate below.5 . Electrostatic Energy Denote D ∞ the infinite domain between the two cylinders of our capacitor. The patchenergy stored there, finite due to the locality of patch distributions, is: W p = ǫ Z D ∞ ( ∇ Φ p ) dV = ǫ Z ∞−∞ dz Z π bdϕ H ( ϕ − γ, 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 . (18)Here we used boundary conditions (11) and the fact that the potential is harmonic in thedomain D ∞ , see section III in CPEI for details. The double integrals above are calculatedvia Fourier coefficients of the potential and its derivative in ρ by the Parceval identity (8).The Fourier coefficients of the derivatives are found from the formula (16); the calculationgoes the same way as in CPEI, Appendix C, and results in ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = a,b = − d Z ∞−∞ dk ∞ X n = −∞ h G n ( k ) − H n ( k ) e − ı nγ i e ı ( kz + nϕ ) . (19)To l. o. in d/a , this expression holds at both the inner and the outer boundary. Using theFourier coefficients, G n ( k ) and H n ( k ), of the boundary functions, we write formula (18) as: W p = ǫ a d Z ∞−∞ dk ∞ X n = −∞ (cid:12)(cid:12)(cid:12) G n ( k ) − H n ( k ) e − ınγ (cid:12)(cid:12)(cid:12) . (20)The only part of this that depends on γ , and thus contributes to the axial torque, is: W p ( γ ) = − ǫ ad Z ∞−∞ dk ∞ X n = −∞ ℜ h G n ( k ) H ∗ n ( k ) e ınγ i . (21) B. Axial Torque
Using (21), we calculate the axial torque by the formula (1): T z = − ∂W p ( γ ) ∂γ = − ǫ ad Z ∞−∞ dk ∞ X n = −∞ n ℑ h G n ( k ) H ∗ n ( k ) e ınγ i . (22)This representation is valid, to lowest order in d/a , for an arbitrary rotation γ . The torquedoes not vanish only if patches are at both boundaries [ G n , H n γ = 0, unless G n ( k ) = λH n ( k ), λ real, i. e., the patch distributionsat both cylinders are the same up to scaling. The expression of the axial torque perfectlymatches that of the axial force in the symmetric configuration [CPEII, formula (21)]; theonly difference is γ instead of z , and the factor n in place of k .6 V. ALL TORQUES BY THE SURFACE INTEGRATION METHODA. General Formulas for Electrostatic Torques
According to the formula (3), the patch effect torque on the outer cylinder is: ~T = b Z π dϕ Z ∞−∞ dz σ (cid:16) ~r × ~E (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = (23) ǫ a Z π dϕ Z ∞−∞ dz " ~r × ∂ Φ ∂ρ ˆ e ρ + 1 ρ ∂ Φ ∂ϕ ˆ e ϕ + ∂ Φ ∂z ˆ e z ! ∂ Φ ∂ρ ρ = b " O da ! . Here we expressed the electrical field through the potential, and the charge density as theproduct of ǫ and the normal component of the field, by the Gauss law. We also set b = a + d ≈ a , so the above holds to l. o. in d/a . The Cartesian components of the torqueare found using the well–known cylindrical unit vectors: ~T = ǫ a Z π dϕ Z ∞−∞ dz (" − zρ ∂ Φ ∂ϕ cos ϕ + ρ ∂ Φ ∂z − z ∂ Φ ∂ρ ! sin ϕ ˆ x − (24) " zρ ∂ Φ ∂ϕ sin ϕ + ρ ∂ Φ ∂z − z ∂ Φ ∂ρ ! cos ϕ ˆ y + ∂ Φ ∂ϕ ! ˆ z ) ∂ Φ ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b . Because of the bilinear structure of this expression, the splitting (9) of the potential in thesum of Φ u and Φ p implies formally three contributions to the torque: one from the uniformvoltages only, the other due to the interaction of patches and uniform voltages, and thethird one from the patches only, just like we had it for the force in CPEII. However, uniformvoltages do not give any torque, so the first contribution vanishes. Likewise, the interactionbetween the patches and uniform potential difference gives zero axial torque: the interactiontorque is perpendicular to the symmetry axis (slanting torque): T Intx = ǫ a ∂ Φ U ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b Z π dϕ Z ∞−∞ dz − z ∂ Φ p ∂ρ sin ϕ − zρ ∂ Φ p ∂ϕ cos ϕ + ρ ∂ Φ p ∂z sin ϕ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b ; (25) T Inty = ǫ a ∂ Φ U ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b Z π dϕ Z ∞−∞ dz z ∂ Φ p ∂ρ cos ϕ − zρ ∂ Φ p ∂ϕ sin ϕ − ρ ∂ Φ p ∂z cos ϕ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b . (26)In contrast with that, the torque due to the patch interaction generally has all the com-ponents: T px = ǫ a Z π dϕ Z ∞−∞ dz ( ∂ Φ p ∂ρ " − zρ ∂ Φ p ∂ϕ cos ϕ + ρ ∂ Φ p ∂z − z ∂ Φ p ∂ρ ! sin ϕ ρ = b ; (27)7 py = − ǫ a Z π dϕ Z ∞−∞ dz ( ∂ Φ p ∂ρ " zρ ∂ Φ p ∂ϕ sin ϕ + ρ ∂ Φ p ∂z − z ∂ Φ p ∂ρ ! cos ϕ ρ = b ; (28) T pz = ǫ a Z π dϕ Z ∞−∞ dz ∂ Φ p ∂ρ ∂ Φ p ∂ϕ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b . (29)Formulas (25)—(29) provide the general representation for the torque based on the surfaceintegration method. Below we use them, along with the expressions (12) and (16) of thepotential in the gap, to find the PE torque on the cylinder. B. Axial Torque by the Surface Integration Method
We compute the axial component (29) of the torque employing the Parceval identity (8): T pz = ǫ a Z ∞−∞ dk ∞ X n = −∞ (cid:20) − d (cid:16) G ∗ n ( k ) − H ∗ n ( k ) e inγ (cid:17) in H n ( k ) e − inγ + (30)12 d (cid:16) G n ( k ) − H n ( k ) e − inγ (cid:17) in H ∗ n ( k ) e inγ (cid:21) = − ǫ ad Z ∞−∞ dk ∞ X n = −∞ ℑ h nG n ( k ) H ∗ n ( k ) e inγ i . We used Fourier coefficients of the two derivatives of the potential, the radial one (19), andthe angular one obtained by differentiating the second of boundary condition (11): ∂ Φ p ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = 12 π Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) h in H n ( k ) e − inγ i . (31)The axial torque (31) and (22) by the surface integration and energy method, respectively,is exactly the same; this is an important cross–check of our calculations. C. Slanting Torques by the Surface Integration Method
1. Uniform and patch potential interaction
We start with calculating the torque due to the interaction of uniform and patch poten-tials. We first substitute the uniform field, V − /d , in the formula (25): T Intx = ǫ ad V − Z π dϕ Z ∞−∞ dz − z ∂ Φ p ∂ρ sin ϕ − zρ ∂ Φ p ∂ϕ cos ϕ + ρ ∂ Φ p ∂z sin ϕ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b , to get the x component. The last term vanishes after integrating in z , so: T Intx = ǫ ad V − Z π dϕ Z ∞−∞ dz − z ∂ Φ p ∂ρ sin ϕ − zρ ∂ Φ p ∂ϕ cos ϕ ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b .
8y the definition (7) of the Fourier transform, this double integral is equal to 2 π timesthe Fourier coefficient of the integrand at n = k = 0. The needed Fourier coefficients aredetermined in Appendix A, formulas (A7) and (A8), so: T Intx = − π ǫ ad V − ℜ " ∂∂k (cid:16) G ( k ) − H ( k ) e − iγ (cid:17) k =0 " O da ! . (32)This final compact formula for the torque has been obtained by employing the property G − n ( − k ) = G ∗ n ( k ) , H − n ( − k ) = H ∗ n ( k ) of Fourier coefficients of real functions. The estimateof the remainder in the expression (32) holds for patch distributions G ( ϕ, z ) and H ( ϕ, z )satisfying conditions (A11), (A12) and (C5), CPEI, and also such that the products zG ( ϕ, z ),and zH ( ϕ, z ) are squarely integrable, see formula (A10) in Appendix A. The validity of theseconditions is assumed everywhere below, including the final expressions of all the torques.A similar calculation for the y component starts from the formula (26) and uses theFourier coefficients (A6) and (A9). It results, to lowest order in d/a , in: T Inty = 4 π ǫ ad V − ℑ " ∂∂k (cid:16) G ( k ) − H ( k ) e − iγ (cid:17) k =0 , (33)with the same remainder as in formula (32).
2. Patch potentials interaction
Now we go for the expressions of the torque caused by the interaction between patches.The x component of this torque, T px , given by the formula (27), with the help of the Parcevalidentity becomes [the proper Fourier coefficients are found in (19), (A7), (A8), and (A12)]: T px = − ǫ a d Z ∞−∞ dk ∞ X n = −∞ " G ∗ n ( k ) − H ∗ n ( k ) e ı nγ × ∂∂k "(cid:16) G n − ( k ) − H n − ( k ) e − i ( n − γ (cid:17) − (cid:16) G n +1 ( k ) − H n +1 ( k ) e − i ( n +1) γ (cid:17) O da ! . This formula can be simplified further: integrating by part in the first of the two products andshifting there the index n by one, n ′ = n −
1, leads to the final more compact representation: T px = ǫ ad Z ∞−∞ dk ∞ X n = −∞ ℜ (" G ∗ n ( k ) − H ∗ n ( k ) e ı nγ ∂∂k " G n +1 ( k ) − H n +1 ( k ) e − ı ( n +1) γ . (34)The other component, T py , can be determined in a similar way starting with the expression(28) and combining it with the formulas (19), (A6), (A9), and (A11). The result is: T py = − ǫ ad Z ∞−∞ dk ∞ X n = −∞ ℑ (" G ∗ n ( k ) − H ∗ n ( k ) e ı nγ ∂∂k " G n +1 ( k ) − H n +1 ( k ) e − ı ( n +1) γ , (35)9ntirely similar to (34). The general analysis of PE torques is completed. D. General properties of the PE torque
Looking at the results of the calculation of the PE torques, one can come up with a fewgeneral conclusions regarding their properties, such as:1. Uniformly charged cylinders do not give rise to any torque.2. The axial torque is inversely proportional to the gap width, the transverse componentsare inversely proportional to its square.3. Patches need to be present on both the cylinders to generate an axial torque.4. A non-zero axial torque is generally found when the cylinders are not rotated againsteach other ( γ = 0), unless the patch voltage distributions on both cylinders are the same upto a factor.5. Just one patch is enough to generate a slanting torque.6. A non-zero slanting torque is generally found when γ = 0 unless the patch voltagedistributions on both cylinders are the same.7. The interaction between patches and uniform potentials involves only the first har-monics of the azimuthal angle of the patch distribution.So, the general formulas that we obtained enable one to make some significant conclusionsabout PE torques agreeing with the physical insights into their origin. V. SINGLE PATCH AT EACH OF THE ELECTRODES: A PICTURE OF PATCHINTERACTION
To study the features of PE torques we analyze them in the case when only one localizedpatch is found at each of the cylinders, as it was done with PE force in CPEII. To make theanalysis results transparent, one needs to have the torques in a simple enough closed form,which requires some special choice of the generic patch model, a rather delicate task. Thelocalized potential distribution that satisfies this very well has been suggested and developedin CPEII, section IV. We repeat basic facts about this model and use it to calculate thetorques and examine the patch interaction.10 . The Patch Model
The suggested model of the patch potential is: V ( ϕ − ϕ ∗ , z − z ∗ ) ≡ V ( ϕ − ϕ ∗ , λ, z − z ∗ , ∆ z ) = V ∗ f ( z − z ∗ ) u ( ϕ − ϕ ∗ ) , (36)where f ( z ) = exp − z √ z ! , u ( ϕ ) = u ( ϕ, λ ) = (1 − λ ) ϕ − λ cos ϕ + λ ; (37)[note (cid:12)(cid:12)(cid:12) f ( z ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) u ( ϕ ) (cid:12)(cid:12)(cid:12) ≤
1, and f (0) = u (0) = 1]. Here V ∗ = V (0 ,
0) is the maximummagnitude of the potential (positive or negative); the center of the patch is at ϕ = ϕ ∗ , z = z ∗ ,∆ z denotes the axial size of the patch, and λ controls its angular size. Indeed, λ is relatedto ∆ ϕ , the angular half–width of the patch [defined as the angle for which u is equal to itsmean value, u (∆ ϕ ) = u av ], by the equalities:cos ∆ ϕ = λ , ∆ ϕ = arccos λ . (38)Fourier coefficients of the patch model (36), (37) are: V n ( k ) ≡ V n ( k, λ, ∆ z ) = V ∗ ˜ f ( k ) e − ikz ∗ u n e − inϕ ∗ , ˜ f ( k ) = ∆ z exp − k ∆ z √ ! ; (39) u n ≡ u n ( λ ) = √ π − λ λ λ | n | , n = 0; u ≡ u ( λ ) = √ π − λ . (40)For a single patch at each of the two cylinders, the boundary functions G ( ϕ, z ) and H ( ϕ − γ, z ) are given by the formula (36) as: G ( ϕ, z ) = V ( ϕ − ϕ , λ , z − z , ∆ z ) ; H ( ϕ − γ, z ) = V ( ϕ − ϕ − γ, λ , z − z , ∆ z ) . (41)The torques corresponding to these distributions are calculated in Appendix B in a closedform. Here we study the patch interaction for a particular case when the sizes of the patchesare identical, ∆ z = ∆ z = ∆ z , ∆ ϕ = ∆ ϕ = ∆ ϕ , and V = ± V = V . B. Axial Torque
As shown in our general analysis, sections III and IV, the axial torque occurs only whenboth cylinders carry non-uniform voltages. For the case of two patches of equal sizes and11agnitude, its expression is found by the formula (B15) [recall that we have set γ = 0]: T pz = ∓ √ π ǫ ad V ∆ z sin (∆ ϕ ) µ sin ( ϕ − ϕ ) exp [ − ˜ z ] , (42) µ ≡ µ ( λ, ϕ − ϕ ) ≡ λ [1 − λ cos ( ϕ − ϕ ) + λ ] ; ˜ z ≡ z − z z . The signs ∓ in formula (42) stand for patches of equal or opposite potential, respectively.This torque is proportional to the inverse of the relative gap, d/a . Its dependence on theaxial patch distance is driven by the Gaussian exponent: the axial torque monotonicallyand rapidly decreases toward zero with increasing values of (cid:12)(cid:12)(cid:12) z − z (cid:12)(cid:12)(cid:12) . The dependence onthe angular patch distance, 0 ≤ ϕ − ϕ ≤ π is shown in fig.2 for various values of ∆ ϕ : thetorque goes down when the patch angular width gets smaller.The axial and azimuthal sizes of the patch play a very distinctive role in the expression(42). As ∆ z grows (and the patch thus becomes strip–like, for fixed ∆ ϕ ), the torque mag-nitude increases and goes linearly to infinity in ∆ z → ∞ . On the other hand, the influenceof the azimuthal patch width is represented predominantly by the sixth power of the sine of∆ ϕ : the torque vanishes for belt–like patches, and goes to zero as (∆ ϕ ) for ∆ ϕ → C. Slanting Torques
1. Uniform and patch potential interaction
In our case of identical patches expressions (B3) for the interaction torque simplify to T Intx = −√ π ǫ ad V V − ∆ z sin (∆ ϕ ) (cid:16) z sin ϕ ∓ z sin ϕ (cid:17) ; (43) T Inty = √ π ǫ ad V V − ∆ z sin (∆ ϕ ) (cid:16) z cos ϕ ∓ z cos ϕ (cid:17) . Recall that V − is the difference between the uniform potentials at the boundaries, seeformula (10). The minus or plus sign above is taken when the patches have the same or theopposite voltages, respectively. Expressions (43) show that this torque is a superposition oftwo contributions each coming from a single patch interacting with the uniform voltage V − .Each of these torques can be expressed as the product of a force acting at the center of thepatch and the respective arm, z or z . It is interesting that these forces are equal to thezero order interaction forces obtained in CPEII, section VA. For 0 ≤ ∆ ϕ ≤ π/ ϕ →
0, it goes to zero as(∆ ϕ ) . The torque is again proportional to ∆ z . However, as the patch becomes strip–like,the torque goes to zero, by symmetry. In fact, the arm of the force is zero in this latter case.
2. Patch–patch interaction
The slanting patch torque expressions (B9) and (B11) become T px = − √ π ǫ ad V ∆ z sin (∆ ϕ ) × (44) " N ( z sin ϕ + z sin ϕ ) ∓ ( z + z ) M (sin ϕ + sin ϕ ) e − ˜ z ; T py = 2 √ π ǫ ad V ∆ z sin (∆ ϕ ) × " N ( z cos ϕ + z cos ϕ ) ∓ ( z + z ) M (cos ϕ + cos ϕ ) e − ˜ z ;˜ z = z − z z ; N = 2 − λ M = 1 − λ h − λ λ ) 1 + λ − ϕ − ϕ )1 − λ cos ( ϕ − ϕ ) + λ i , in our case of identical patches. Above we have used a slightly different notation for M and N as compared to Appendix B, formulas (B7). The first term in the square brackets carriesthe contributions of each single patch independent of their signs. The second term representsinteraction between the patches. It decreases as the distance between them grows, due tothe presence of the coefficient M and the Gaussian exponent. The decay is faster, thesmaller the widths ∆ z and ∆ ϕ are. The plus or minus sign of this term is taken dependingon whether the patch potentials have the same or opposite signs. Similarly to the previouscase, the patch torque can be expressed as the sum of the products of a force acting oneach patch and the arm z or z , plus a mutual force between the patches times the arm z + z [see fig.3 giving torque versus z for different values of z ]. The slanting patch torqueis essentially proportional to the square sine of ∆ ϕ , and to ∆ z . However, by symmetry, as∆ z → ∞ , the torque vanishes, just like the interaction torque above. VI. AXIAL PATCH EFFECT TORQUES FOR STEP
In this section we use the obtained results to evaluate the effect of non–uniform potentialson the performance of the instrument that will be used in STEP. The pertinent information13bout this experiment is found in CPEII, section VIA; for more details, see [5–7]. Here werecall the basic design of its core system, the differential accelerometers (DACs). As shownin fig.4, each DAC consists of two test masses (TMs) shaped as coaxial cylindrical shells, andof a system of electrodes and magnetic bearings. It essentially constrains the TMs in fourdegrees of freedom, with the translation along their symmetry axis and the spin about it leftfree. The science signal is read by a magnetic SQUID readout system from the differentialaxial motion of the TMs.In CPEII we analyzed the patch effect on the axial translation of the TMs. Here weconcentrate on the axial PE torque causing the spin motion, i. e., relative rotation of thecylinders about their common axis. The main reason to examine it is that this motionmakes PE forces change with the time. Just like in CPEII, we consider a TM and itsmagnetic bearing as a reference case of our pair of cylinders, since the gap between them isat least 3 times smaller - and the axial torque thus 3 times larger - than the gap betweenthe TM and the electrodes. As was assumed in section V, we consider each of the cylindershaving just one patch, both patches of identical sizes and magnitudes. Even in this case theequation of motion proves to be a nonlinear one with the periodic potential such that twoequilibrium points exist within a single period, one stable and one unstable. We describe thegeneral picture of motion pointing out the regimes expected as typical in the real situationof the STEP experiment, namely, the regimes of oscillations. We give the expressions for thefrequency of small oscillations near stable equilibria and the runaway time from the unstableones, and then estimate these quantities under the STEP conditions. An estimate of thetorque is also provided. Finally, we show that spin motion for any patch voltage distributionwill be qualitatively the same as in the case of two patches only.In compliance with CPEII, we use the following parameters: patch voltage V = 10 mV ,TM radius a = 2 . cm , TM to magnetic bearing gap d = 0 . mm , TM length 2 L = 0 . cm .The moments of inertia for some flight–like TMs vary from 2 . × − Kg m to 1 . × − Kg m for different test masses, but a single TM is fabricated in such a way that theprincipal moments of inertia are all equal to a high accuracy [12], I x = I y = I z = I . Wehere need, in fact, only I = I z , and we use the smallest value, I = 2 . × − Kg m .14 . Spin Motion in the Case of Two Patches We take the torque expression (42), and write the motion equation, I ¨ γ = T z = ∓ T ∗ (˜ z ) ∆ z sin (∆ ϕ ) µ ( γ, λ ) sin γ , (45)for the rotation of the outer cylinder about the symmetry axis; here the dimesionless function µ = µ ( γ, λ ) = 1 + λ (1 − λ cos γ + λ ) > , (46)and the characteristic value of the torque is T ∗ (˜ z ) = π / ǫ V ad e − ˜ z , ˜ z = z − z z . (47)Without any loss of generality, we count here the rotation angle γ from the position wheretwo patches are right one against the other, ϕ − ϕ = 0. The torque has the minus (plus) signwhen the patch voltages have the same (opposite) sign. Equation (45) strongly resemblesthe classical motion equation of the pendulum, with just one additional coefficient µ ( γ, λ )being a strictly positive non-singular function of γ . Extending this similarity, we note that,by the expression (22) for the axial torque through the patch energy W p ( γ ), equation (45)has the potential W p ( γ ) / I : it can be equivalently written as I ¨ γ = − ∂W p ( γ ) ∂γ . (48)Since the potential is periodic in γ , the complete qualitative picture of motion is well known(see, for instance, [13], Ch.1). It follows from the energy integral of the equation (48), I ˙ γ W p ( γ ) = E , (49)where E is the total energy determined by the initial conditions: E = I ˙ γ W p ( γ ) , γ = γ ( t ) , ˙ γ = ˙ γ ( t ) . Potential energy W p ( γ ) is bounded, with the bounds denoted as − ∞ < W − = min ≤ γ< π W p ( γ ) < W + = max ≤ γ< π W p ( γ ) < ∞ . (50)The minimum potential energy W − = W p ( γ − ) corresponds, of course, to a stable equilibrium γ ( t ) ≡ γ − = const, while the maximum one, W + = W p ( γ + ), is achieved at an unstableequilibrium point, γ ( t ) ≡ γ + = const. This is enough to qualitatively describe the motion.15ndeed, from the energy conservation (49) it is clear that E ≥ W − . If E = W − , then thesystem stays at the stable equilibrium, γ ( t ) ≡ γ = γ − , ˙ γ ( t ) ≡ , t ≥ t . If W − < E < W + ,then the system can never reach the peak of the potential, the rotation angle is boundedat all times, γ − ≤ γ ( t ) < γ max < γ + , the motion is finite, which means that the cylinderoscillates about the stable equilibrium. If, next, E > W + , then the system always remainsabove all the potential wells, the motion is infinite, the cylinder rotates indefinitely andnon-uniformly in one direction depending on the sign of the initial velocity. What remainsis the exceptional case E = W + , when, in purely mathematical view, the system stays atthe equilibrium γ + ; however, this rest point is unstable, so in reality any small perturbationin this or that direction leads, again, either to the oscillational, or to the rotational motion.To make all this even more particular in our case of just two patches, we give the patch en-ergy explicitly, as easily found either by the general expression (21) or, up to an insignificantconstant, by the direct integration of the torque in the r.h.s. of the equation (45): W p ( γ ) = ∓ T ∗ (˜ z )∆ z sin (cid:18) ∆ ϕ (cid:19) w ( γ ) + const , w ( γ ) = (1 + λ ) λ − λ cos γ + λ . (51)There is no essential difference between the two cases with the opposite signs, so we discussonly the case of the minus sign below.As seen yet from the equation (45), in this case we have just two equilibria at each periodof the potential, the stable one, γ − = 0 (mod 2 π ), and the unstable γ + = π (mod 2 π ). Thisis also clear from the plot of the patch energy given in fig.5; the horizontal line through thepeaks shows the critical energy, W + , that separates the finite motions (oscillations) fromthe infinite ones (rotations). The period of oscillations is the larger, the higher energy E is, i.e., the larger the oscillation amplitude (the limit of small oscillations, when the periodis independent of the amplitude, is described in the next section).The energy integral (49) allows also for the representation of motion in the phase plane γ, ˙ γ . The corresponding plot is given in fig.6; closed orbits in it correspond to oscillatory(finite) motions (the size of these ovals grows with E ). They are separated from the infinitetrajectories (rotations) by the so called heteroclinic curves, which go from one unstableequilibrium to another nearest to it. The total time of motion along these separatrices fromone rest point to the other is infinite.Note that under the STEP conditions one expects only oscillatory spinning of the testmass, rather than its rotation. The reason is that the TM will be caged (fixed) during16he satellite launch, and then rather accurately released, practically with no initial velocity,which leads to pure oscillations. B. Estimates for Small TM Motions Near Its Equilibria in the Case of Two Patches
Here we describe small motions of the cylinder near its rest points. Accordingly, welinearize the equation (45) by setting γ ( t ) = γ ∓ + δγ ( t ) , | δγ ( t ) | ≪ , which results in ¨ δγ = ∓ ω ∓ δγ, ω ∓ = T ∗ (˜ z ) I ∆ z sin (∆ ϕ ) (1 + λ )(1 ∓ λ ) . (52)The cylinder thus oscillates about the stable equilibirium position γ − = 0 with the frequency f P E = ω − π = π − / V r ǫ a I d √ ∆ z √ ∆ ϕ sin ∆ ϕ e − . z ≤ − √ ∆ z √ ∆ ϕ sin ∆ ϕ Hz . (53)Accordingly, it rotates exponentially away from the unstable position γ + = π with thecharacteristic time τ P E = 1 ω + = 2 π / V s I dǫ a √ ∆ z (1 + cos ∆ ϕ ) / sin ∆ ϕ e . z ≥ . × (1 + cos ∆ ϕ ) / √ ∆ z sin ∆ ϕ s . (54)Numerical estimates (53) and (54) hold for the above set of STEP parameters, with ∆ z inmeters. To get the feeling of what the numbers are in reality, let us consider some examples.For instance, the longitudinal patch size cannot be larger than the size of the TM, ∆ z ≤ L ;in the case of the maximum size, the upper bound for the frequency of small oscillationsbecomes: f P E ≤ . × − √ ∆ ϕ sin ∆ ϕ Hz . This value remains below the STEP signal frequency range1 . × − Hz < f < . × − Hz ;(see CPEII, section VIA) for the patches as small in the azimuthal direction as ∆ ϕ ∼ deg ,or larger. The frequency tends to infinity when the angular size tends to zero, because thepotential well becomes infinitely deep. The smallest runaway time in the case ∆ z = L is τ P E ≥ . × (1 + cos ∆ ϕ ) / sin ∆ ϕ s , ∼ hr , for∆ ϕ ≤ deg . So if in such case the science session starts with the TM close to an unstableequilibrium, then it will be practically stay in this position for its whole duration. C. Estimate of the Axial Torque
By the equation (45), we got the expression of the axial torque, | T z | = T ∗ (˜ z ) ∆ z sin (∆ ϕ ) µ ( γ, λ ) sin γe − ˜ z , µ ( γ, λ ) = 1 + λ (1 − λ cos γ + λ ) . (55)The ballpark number for the torque is obtained by the maximum of the expression (47): T ∗ (0) = π / ǫ V ad ≈ . × − N ;of the dimension of a torque per unit length. Contributions of the patch widths, and thepatch azimuthal distribution were not taken into account. A more meaningful estimateis obtained by computing the average torque for all the relative positions of the patches.Introducing the normalized length l = L/ z , we write, using expression (55): | ¯ T z | = T ∗ ∆ z sin (∆ ϕ ) 1 l Z l d ˜ ze − ˜ z π Z π dγ µ ( γ, λ ) sin γ , we calculate the two integrals Z l d ˜ ze − ˜ z = √ π erf ( l ) , Z π dγ µ ( γ, λ ) sin γ = 2sin ∆ ϕ (1 + cos ∆ ϕ ) , and substitute them in the expression, to find | ¯ T z | = 2 √ π T ∗ (∆ z ) L erf ( l ) sin (∆ ϕ )1 + cos ∆ ϕ ≈ . × − (∆ z ) erf ( l ) sin (∆ ϕ )1 + cos ∆ ϕ N m , with ∆ z in meters. For the typical case l ≫
1, the above expression becomes | ¯ T z | ≈ . × − (∆ z ) sin ∆ ϕ ∆ ϕ N m , (56)which is proportional to the square of the axial patch width; for small angular widths themean value goes to zero with the square of ∆ ϕ , too.18 . Spin Motion for a General Patch Distribution Here we make an important concluding remark. It is easy to see that the picture ofthe spin motion due to the patch effect torque given in section VI A for the case of twopatches is valid, in fact, for any boundary voltage distributions well. Indeed, in any casethe electrostatic patch energy is bounded and is a 2 π –periodic function of the rotation angle γ , therefore equations (48)—(50) hold, along with the whole following argument aboutthe picture of motion based on them. Thus in the most general case the cylinder eitherrotates all the way in the same direction, or oscillates about a stable rest point, dependingon whether the total energy is above or below the critical value, the global maximum ofpotential energy. The only significant difference is that for a general patch distribution thenumber of equilibria can be larger than two (but always even, with the equal number ofstable and unstable rest points, since the torque is continuous).The increase of the number of equilibrium positions leads to a trend of decreasing theamplitude of the typical oscillatory motion. The remark at the end of section VI A is alsovalid in the general case: rotational regimes are not anticipated under the STEP set–upat all. The experiment conditions are such that any friction or other dissipation should beextremely low, so the typical spin oscillations of a TM should be damped but very slowly, ascompared even with the duration of a single science session. (One should note, however, thatif some resistive energy losses are present, as it appears to have happened with GP-B [14],then this picture might change significantly). These oscillations will make the axial forcefound in CPEII change periodically with the time. If the basic frequency or some of itsharmonics is close to the frequency of the STEP science signal, then it might introduce asystematic error in the experiment, provided that the PE force and the acceleration due toit is large enough. For this reason, as well perhaps for many other ones, careful pre- andpost-mission calibrations on orbit are recommended, along with the extensive simulationsbefore the flight as described in CPEII, section VID. 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 Everitt19or their permanent interest in and support of this work, and for some valuable remarks.Many our colleagues at GP-B and STEP made valuable remarks and comments on thepaper and provided some valuable information. We are greatful to all of them for this, inparticular, to Dan DeBra, Sasha Buchman, David Hipkins, John Mester, and Paul Worden.
Appendix A: Calculation of the Slanting Torques
We provide here intermediate results needed for computing of the slanting torques. Par-ticular terms that appear under the integrals in the formulas (25)—(28) are: z ∂ Φ p ∂ρ cos ϕ, z ∂ Φ p ∂ρ sin ϕ, z ∂ Φ p ∂ϕ cos ϕ, z ∂ Φ p ∂ϕ sin ϕ, ∂ Φ p ∂z cos ϕ, ∂ Φ p ∂z sin ϕ ; (A1)we need to determine the Fourier coefficients of these functions evaluated at the boundary ρ = b . The radial and the angular derivatives of the potential involved here are: ∂ Φ p ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = − d Z ∞−∞ dk ∞ X n = −∞ e ı ( kz + nϕ ) h G n ( k ) − H n ( k ) e − ı nγ i ; (A2) ∂ Φ p ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = 12 π Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) h in H n ( k ) e − inγ i ; (A3)they come from the formulas (19) and (31), respectively. The z derivative is computed fromthe second of the boundary conditions (11): ∂ Φ p ∂z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = i π Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) h k H n ( k ) e − inγ i . (A4)For any function u ( ϕ, z ) such that z u ( ϕ, z ) is squarely integrable, by the definition of theFourier transform (7), the following equalities hold: u ( ϕ, z ) z = i π Z ∞−∞ ∞ X n = −∞ e i ( kz + nϕ ) ∂u n ( k ) ∂k ; (A5) u ( ϕ, z ) cos ϕ = 14 π Z ∞−∞ ∞ X n = −∞ e i ( kz + nϕ ) [ u n − ( k ) + u n +1 ( k )] ; u ( ϕ, z ) sin ϕ = − i π Z ∞−∞ ∞ X n = −∞ e i ( kz + nϕ ) [ u n − ( k ) − u n +1 ( k )] . Using formulas (A5) and (A2) with the radial derivative playing the role of u ( ϕ, z ), the firsttwo functions (A1) are represented in the following way: z ∂ Φ p ∂ρ cos ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = i πd Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) ∂∂k h(cid:16) G n − ( k ) − H n − ( k ) e − i ( n − γ (cid:17) + (A6)20 G n +1 ( k ) − H n +1 ( k ) e − i ( n +1) γ (cid:17)i ; z ∂ Φ p ∂ρ sin ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = − πd Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) ∂∂k h(cid:16) G n − ( k ) − H n − ( k ) e − i ( n − γ (cid:17) − (A7) (cid:16) G n +1 ( k ) − H n +1 ( k ) e − i ( n +1) γ (cid:17)i . By the same token, using formula (A3) instead of (A2), for the two terms proportionalto the potential derivative in ϕ we obtain: z ∂ Φ p ∂ϕ cos ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = − π Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) ∂∂k h ( n − H n − ( k ) e − i ( n − γ + (A8)( n + 1) H n +1 ( k ) e − i ( n +1) γ i ; z ∂ Φ p ∂ϕ sin ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = i π Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) ∂∂k h ( n − H n − ( k ) e − i ( n − γ − (A9)( n + 1) H n +1 ( k ) e − i ( n +1) γ i . Note that expressions (A6)—(A9) here are valid under the additional conditions Z ∞−∞ dk ∞ X n = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂k G n ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ , Z ∞−∞ dk ∞ X n = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂k H n ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ , which are equivalent to: Z ∞−∞ dz Z π dϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) zG ( ϕ, z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ ; Z ∞−∞ dz Z π dϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) zH ( ϕ, z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ . (A10)To determine the remaining two terms in (A1) containing the derivative of the potentialwith respect to z , we just need the last two formulas from (A5) and the expression (A4) inplace of u ( ϕ, z ). This results in: ∂ Φ p ∂z cos ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = − i π Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) k h H n − ( k ) e − i ( n − γ + H n +1 ( k ) e − i ( n +1) γ i ; (A11) ∂ Φ p ∂z sin ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = b = 14 π Z ∞−∞ dk ∞ X n = −∞ e i ( kz + nϕ ) k h H n − ( k ) e − i ( n − γ − H n +1 ( k ) e − i ( n +1) γ i . (A12)Formulas (A6)—(A12) provide the expressions and the needed conditions (A10) for cal-culating the integrals in the formulas (25)—(28) for the slanting torque. Appendix B: Calculation of the Torque for a Single Patch at each of the Cylinders
To get the torque we first need to find the Fourier coefficients of the boundary distributions G ( ϕ, z ) and H ( ϕ − γ, z ) from the formulas (41) and their derivative with respect to k . The21ormer can be represented by the Fourier coefficients (39), (40) as: G n ( k ) = V n ( k, λ , ∆ z ) ; H n ( k ) = V n ( k, λ , ∆ z ) . (B1)The latter are the derivatives in k of these expressions, and they are given by the generalformula: ∂ V jn ( k ) ∂k = − V j ∆ z (cid:16) k (∆ z ) + iz j (cid:17) exp − k ∆ z j √ ! u n ( λ j ) e − i ( kz j + nϕ j ) ; j = 1 , , (B2)with V n ( k ) = G n ( k ) and V n ( k ) = H n ( k ).The torque due to the interaction between the patches and uniform potential differenceis a linear function of the derivatives in k of G n ( k ) and H n ( k ) calculated at k = 0 and n = 1.These are determined by the formulas (B2) as: ∂G ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = −√ πV ∆ z ( iz ) 1 − λ e − iϕ ; ∂G ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = −√ πV ∆ z ( iz ) 1 − λ e − iϕ , where we used equality (40) to express the coefficient u n . In order to calculate the torquewe need just to substitute the above expressions in the formulas (32) and (33). This leadsto the following result: T Intx = −√ π ǫ ad V − " V ∆ z (sin ∆ ϕ ) z sin ϕ − V ∆ z (sin ∆ ϕ ) z sin ( ϕ + γ ) ; (B3) T Inty = √ π ǫ ad V − " V ∆ z (sin ∆ ϕ ) z cos ϕ − V ∆ z (sin ∆ ϕ ) z cos ( ϕ + γ ) . Note that here we used the first of the relations (38) to replace λ with the more meaningfulparameter ∆ ϕ .The expressions for the slanting torque due to the interaction between the patches aremore cumbersome to find, since one needs to calculate the sum over n and the integral over k of the product of the Fourier coefficients of the boundary distributions and their derivativesin k . For the x component, we combine formula (34) with (B2) for the derivatives, and (B1)for the boundary functions, to obtain the expression: T px = 2 π ǫ ad Z ∞−∞ dk ℜ ( − V ∆ z N ( λ ) h k ∆ z + iz i exp h − k ∆ z i e − iϕ + (B4) V V ∆ z ∆ z M "(cid:16) k ∆ z + iz (cid:17) e ik ( z − z ) + (cid:16) k ∆ z + iz (cid:17) e − ik ( z − z ) exp " − k (cid:16) ∆ z + ∆ z (cid:17) − ∆ z N ( λ ) h k ∆ z + iz i exp h − k ∆ z i e − iϕ ) ;for the matter of space we set γ = 0, without any loss of generality, and denoted [comparewith CPEII, Appendix A]: M = M ( λ , ϕ , λ , ϕ ) ≡ π ∞ X n = −∞ u n ( λ ) e ınϕ u n +1 ( λ ) e − ı ( n +1) ϕ ; (B5) N ( λ ) ≡ π ∞ X n = −∞ u n ( λ ) u n +1 ( λ ) = M (0 , λ ; 0 , λ ; ) . (B6)The values of these coefficients are obtained by summing up geometrical progressions [seecoefficient u n in formula (40)]: M = (1 − λ ) (1 − λ )8 ( e − ıϕ (1 + λ ) " λ λ ) e − ı ( ϕ − ϕ ) − λ λ D + (B7) e − ıϕ (1 + λ ) " λ λ ) e ı ( ϕ − ϕ ) − λ λ D ; D = 1 − λ λ cos( ϕ − ϕ ) + ( λ λ ) ; N ( λ ) = 1 − λ − λ ) . (B8)All we need now to get T px is the two integrals in the formula (B4), which are well known: Z ∞−∞ dk exp " − k (∆ z + ∆ z )2 e ± ık ( z − z ) = √ π q ∆ z + ∆ z exp " − ( z − z ) z + ∆ z ) ; Z ∞−∞ dk exp " − k (∆ z + ∆ z )2 k e ± ık ( z − z ) = ± ı √ π z − z (∆ z + ∆ z ) / exp " − ( z − z ) z + ∆ z ) . In the case z = z and ∆ z = ∆ z = ∆ z they becomes: Z ∞−∞ dk exp h − k ∆ z i = √ π/ ∆ z ; Z ∞−∞ dk exp h − k ∆ z i k = 0 . With these results we are now able to rewrite formula (B4) for the torque as: T px = − π / ǫ ad ( V z ∆ z N ( λ ) sin ϕ + V z ∆ z N ( λ ) sin ϕ − (B9) V V ¯ l ℑ ( M ) h √ z (cid:16) ∆ z − ∆ z (cid:17) + ( z + z ) i exp [ − ˜ z ] ) , with the new notations¯ l ≡ √ z ∆ z q ∆ z + ∆ z ; ˜ z ≡ z − z q z + ∆ z ) . (B10)23he calculation of the y component of the patch torque does not present any additionaldifficulties. In the same way as we derived formula (B9), one can find for T py : T py = 2 π / ǫ ad ( V z ∆ z N ( λ ) cos ϕ + V z ∆ z N ( λ ) cos ϕ − (B11) V V ¯ l ℜ ( M ) h √ z (cid:16) ∆ z − ∆ z (cid:17) + ( z + z ) i exp [ − ˜ z ] ) , [again, the notations (B7) and (B10) are used].To obtain the closed form representation of the axial torque we combine the formula (31)with the expressions (B1). The integral in k there is found above, so the result is: T pz = 4 π / ǫ ad V V ¯ l ℑ ( M ) exp [ − ˜ z ] , (B12)with the coefficient M given by M = M ( λ , ϕ , λ , ϕ ) ≡ π ∞ X n = −∞ n u n ( λ ) e − ınϕ u n ( λ ) e ınϕ . (B13)Formula (40) for u n leads to an explicit sum of this series reduced to the derivative of ageometric progression: M = − i " (1 − λ ) (1 − λ )8 (1 − λ λ ) D sin ( ϕ − ϕ ) . (B14)Using this we write formula (B12) in the final explicit form: T pz = − π / ǫ ad V V ¯ l (cid:16) − λ (cid:17) (cid:16) − λ (cid:17) sin ( ϕ − ϕ ) − λ λ D ! exp [ − ˜ z ] . (B15) [1] Ferroni V., A.S. Silbergleit Electrostatic Patch Effect in Cylindrical Geometry I. Potential andEnergy between Slightly Non-Coaxial Cylinders (submitted to this journ).[2] Ferroni V., A.S. Silbergleit
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