Electrostatic Patch Effect in Cylindrical Geometry. II. Forces
aa r X i v : . [ g r- q c ] S e p Electrostatic Patch Effect in Cylindrical Geometry. II. Forces
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: November 3, 2018) Abstract
We continue our study of patch effect (PE) for two close cylindrical conductors with parallelaxes, slightly shifted against each other in the radial and by any length in the axial direction,started in [1], where the potential and energy in the gap were calculated to the second order inthe small transverse shift, and to lowest order in the gap to cylinder radius ratio. Based on theseresults, here we derive and analyze PE force. It consists of three parts: the usual capacitor forcedue to the uniform potential difference, the one from the interaction between the voltage patchesand the uniform voltage difference, and the force due to patch interaction, entirely independent ofthe uniform voltage. General formulas for these forces are found, and their general properties aredescribed. A convenient model of a localized patch is then suggested that allows us to calculate allthe forces in a closed elementary form. Using this, a detailed analysis of the patch interaction forone pair of patches is carried out, and the dependence of forces on the patch parameters (width andstrength) and their mutual position is examined. We also give various estimates of the axial patcheffect force important for the Satellite Test of the Equivalence Principle (STEP), and recommendintensive pre-flight simulations employing our results.
PACS numbers: 41.20Cv; 02.30Em; 02.30Jr; 04.80CcKeywords: Electrostatics - Patch effect - Cylindrical capacitor - Forces - Precision measurements - STEP ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Electrostatic patch effect [2] is a nonuniform potential distribution on the surface of ametal first examined theoretically in paper [3], where the PE force in a plane capacitor withvarying boundary voltages has been calculated. This analysis was particularly motivated bythe LISA space experiment to detect gravitational waves (see c.f. [4]). PE can similarly affectthe accuracy of any other precision measurement if its set-up includes conducting surfaces ina closed proximity to each other. PE torques turned out one of the two major difficulties [5–7] in the analysis of data from Gravity Probe B (GP-B) Relativity Science Mission (2004–2005 flight), that measured relativistic drift of a gyroscope predicted by Einstein’s generalrelativity [8]. This required theoretical calculation of PE torques [9] for the case of twoconcentric spherical conductors.Here we continue studying PE in cylindrical geometry that we started in paper [1], hence-forth referred to as CPEI, for ”‘Cylindrical Patch Effect”’. It is strongly motivated by theexperimental configuration of STEP [10–13], where each test mass and its superconductingmagnetic bearing is a pair of approximately coaxial conducting cylinders (see section VI Afor more details). 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 properly ac-counting for PE forces is evident. Notably, the axial and transverse force we find is inverselyproportional to the gap and its square, respectively, exactly as in the plane capacitor [3].We determine the PE forces between the two slightly shifted cylinders with parallel axesby the energy conservation argument: a small shift, ~r , of one of the conductors relative tothe other causes an electrostatic force given by (see, for instance, [14]) ~F ( ~r ) = − ∂W ( ~r ) ∂~r , ~F (0) = − ∂W ( ~r ) ∂~r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~r =0 , (1)where W ( ~r ) is the electrostatic energy as a function of the shift. The latter was found inCPEI to the second order in the small transverse shift, ρ ≪ d , where d is the gap betweenthe cylinders in the coaxial position, so the force to the first order in ρ is found below. Fortypical experimental conditions, such as the STEP configuration [11, 12], the gap is muchsmaller than either of the cylinders’ radii; thus two small parameters are actually involved inthe problem, ρ /d ≪ , d/a ∼ d/b ≪
1. While justifying the model of infinite cylinders,this also allows for a significant simplification of the results to l. o. in d/a .2n the next section we summarize the results from CPEI needed for the force calculation.Based on this, we calculate PE forces in section III, and illucidate their general properties.In section IV we introduce a convenient model of a localized patch potential allowing oneto find simple closed–form expressions for the forces. Section V contains a detailed analysisof PE forces when a single localized patch described by our model is sitting at each ofthe cylinder boundaries. In section VI estimates of the axial patch force for the STEPexperiment set–up are given. The details of calculations, in places rather complicated andcumbersome, are found in the appendix.
II. SUMMARY OF RESULTS FROM CPEI
We use Cartesian and cylindrical coordinates in two frames of the inner and outer cylin-ders 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 } . Frame origins are separated by ~r , hence the coordinates are related by ~r ′ = ~r + ~r ; x ′ = x + x , y ′ = y + y , z ′ = z + z . (2)We also use an alternative writing x ≡ x , y ≡ x , x ≡ x , y ′ ≡ x ′ , etc.The surfaces of the inner and outer cylinders are ρ ′ = a and ρ = b , respectively, with d = b − a . They carry arbitrary voltage distributions given by the conditionsΦ (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 ) , (3)where Φ is the electrostatic potential, and V − is the uniform potential difference: all thevoltages in the problem are counted from the uniform voltage of the inner cylinder takenas zero. The non–uniform potentials (patch patterns) are described by arbitrary smoothenough functions G ( ϕ ′ , z ′ ) and H ( ϕ, z ), whose local nature is emphasized by requiring || G || = π Z ∞ Z −∞ dϕ ′ dz ′ | G ( ϕ ′ , z ′ ) | < ∞ , || H || = π Z ∞ Z −∞ dϕdz | H ( ϕ, z ) | < ∞ . (4)For any function u ( ϕ, z ) satisfying the square integrability condition we denote its 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 −∞ dϕdz u ( ϕ, z ) e − i ( kz + nϕ ) ; (5)3n particular, Fourier coefficients of the patch voltages G ( ϕ ′ , z ′ ) and H ( ϕ, z ) are G n ( k ) and H n ( k ), respectively. Since these functions are real, their Fourier coefficients satisfy G n ( k ) = G ∗− n ( − k ) , H n ( k ) = H ∗− n ( − k ) ; (6)here and elsewhere the star denotes complex conjugation. For any two squarely integrablefunctions 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 ) . (7)In the case u = v the identity (7) shows that the squared norm, || u || , of a function u isequal to the squared norm of its Fourier coefficient u n ( k ).As shown in CPEI, section III, electrostatic energy in the gap between the two cylinders,as a function of their mutual shift ~r consists of three parts, W ( ~r ) = W u ( ~r ) + W int ( ~r ) + W p ( ~r ) , (8)where the first one is due to the uniform potential difference, the second results from theinteraction between the uniform and patch voltages, and the third one is the energy ofthe patch interaction. Each of these contributions was found in CPEI in the form of anexpansion in the small transverse shift ρ to quadratic order in the small ρ /d ratio, withthe coefficients depending generally on the axial shift z : W A ( ~r ) = W A ( z ) + W A µ ( z ) ( x µ /d ) + W A µν ( z ) ( x µ /d ) ( x ν /d ) + O h ( ρ /d ) i ; A = u, int, p . (9)The explicit coefficients for each kind of energy follow immediately.The uniform potential energy is, naturally, proportional to the length of the capacitor,same as its expansion coefficients which, for | z | ≤ L , are W u ( L ) = 2 πLǫ ad (cid:16) V − (cid:17) , W uµ ( L ) = 0 , W uµν ( L ) = πLǫ ad (cid:16) V − (cid:17) δ µν . (10)Here and everywhere else we adopt the summation rule over repeated Greek indeces µ, ν ,etc.: the summation over them runs from 1 to 2, corresponding to transverse coordinates inthe cylinder cross–section. The first order term vanishes as it should be due to symmetry(otherwise there would be a non–zero transverse force in a perfectly symmetric configurationof coaxial conductors under uniform potentials).4he expansion coefficients for the interaction energy are: W int = − πǫ ad V − [( G (0) − H (0))] ,W intµ = +4 πǫ ad V − ℜ h c + µ ( G (0) − H (0)) i , (11) 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 , where c +1 = 0 . , c +2 = 0 . i , ℜ ( · ) denotes the real part of ( · ); recall also that G n ( k ) and H n ( k )are the Fourier coefficients of the patch voltages (3). Same as the uniform part of energy,the interaction one does not depend on the axial shift, because the longitudinal shifting ofan electorde with the uniform potential does not change the actual charge configuration.Finally, the patch energy coefficients are found to be: 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 ; (12) 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 . These quantities do depend on the axial shift, z . Combining expressions (8) and (9) we canalso write for the total energy: W ( ~r ) = W ( z ) + W µ ( z ) ( x µ /d ) + W µν ( z ) ( x µ /d ) ( x ν /d ) + O h ( ρ /d ) i ; (13) W ξ ( z ) = W uξ + W intξ + W pξ ( z ) , ξ = 0 , µ, µν . (14)Note that all parts of energy are given to l. o. in d/a . III. PATCH EFFECT FORCES
As explained in the introduction, the force is found by the formulas (1) and (13): F µ = − ∂W ( ~r ) ∂x µ = − h W µ ( z ) + 2 W µν ( z ) ( x ν /d ) + O (cid:16) ( ρ /d ) (cid:17)i = − h W µ (0) + W ′ µ (0) z + 2 W µν (0) ( x ν /d ) + O (cid:16) ( r /d ) (cid:17)i , µ = 1 , r = | ~r | ≡ q ( x ) + ( y ) + ( z ) , F z = − ∂W ( ~r ) ∂z = − h W ′ ( z ) + W ′ µ ( z ) ( x µ /d ) + O (cid:16) ( ρ /d ) (cid:17)i = − h W ′ (0) + W ′′ (0) z + W ′ µ (0) ( x µ /d ) + O (cid:16) ( r /d ) (cid:17)i , (16)for the axial force (here and everywhere else primes denote the derivatives in z ). Same asenergy, the force consists, of course of three parts, which we study below one by one. A. The Force due to the Uniform Potential Difference
Using the formula (15) for the transverse force and expansion (9) with the coefficients(10) we obtain (as usual, F u = F ux , F u = F uy ): F ux = − πLǫ ad (cid:16) V − (cid:17) (cid:16) x /d (cid:17) , F uy = − πLǫ ad (cid:16) V − (cid:17) (cid:16) y /d (cid:17) , F uz = 0 . (17)Zero axial force, obvious by symmetry, is formally due to the independence of the energy onthe axial shift, z . Recall that 2 L is the cylinder length, so the force per unit length of thecylinders that is finite. B. The Patch and Uniform Potential Interaction Force
To get it, we combine formulas (15), (9) and (11); after some simplifying transformationsthe result becomes [ ℑ ( · ) is the imaginary part of ( · )]: F intx = − πǫ ad V − {ℜ [ G (0) − H (0)] −ℜ [( G (0) − H (0)) + ( G (0) − H (0))] (cid:16) x /d (cid:17) + ℑ [ G (0) − H (0)] (cid:16) y /d (cid:17)o ; (18) F inty = − πǫ ad V − n −ℑ [ G (0) − H (0)] + ℑ [ G (0) − H (0)] (cid:16) x /d (cid:17) + ℜ [( G (0) − H (0)) − ( G (0) − H (0))] (cid:16) y /d (cid:17)o ; F intz = 0 . The axial force vanishes again, by symmetry.
C. Forces due to the Patch Interaction
By the formula (15) and the energy expansion (9) with the coefficients (12) we find: F px = ǫ a d Z ∞−∞ dk ∞ X n = −∞ n ℜ h(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 − | G n ( k ) e ıkz − H n ( k ) | + ℜ h(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)ii (cid:16) x /d (cid:17) + ℑ h(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)i (cid:16) y /d (cid:17)o ; (19) F py = − ǫ a d Z ∞−∞ dk ∞ X n = −∞ n ℑ h(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 −ℑ h(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)i (cid:16) x /d (cid:17) + h | G n ( k ) e ıkz − H n ( k ) | − ℜ h(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)ii (cid:16) y /d (cid:17)o . To get the axial force, we use formulas (9), (12), and (16): F pz = − ǫ ad Z ∞−∞ dk ∞ X n = −∞ n ℑ h kG n ( k ) H ∗ n ( k ) e ıkz i + (20) ℑ h G ∗ n ( k ) H n +1 ( k ) e − ıkz − H ∗ n ( k ) G n +1 ( k ) e ıkz i k (cid:16) x /d (cid:17) + ℜ h G ∗ n ( k ) H n +1 ( k ) e − ıkz − H ∗ n ( k ) G n +1 ( k ) e ıkz i k (cid:16) y /d (cid:17)) . Formulas (19) and (20) provide the force due to patches to linear order in a small trans-verse shift for an arbitrary axial displacement. In many cases, such as the STEP set–updescribed in sec. VI, the axial shift is also small, and PE forces to linear order in all theshifts are only needed. We obtain these expressions by replacing exp ( ± ıkz ) with the twoterms of its Maclaurin expansion, assuming that all the arising integrals in k converge: F px = ǫ a d Z ∞−∞ dk ∞ X n = −∞ {ℜ [( G ∗ n ( k ) − H ∗ n ( k )) ( G n +1 ( k ) − H n +1 ( k ))] − h | G n ( k ) − H n ( k ) | + ℜ [( G ∗ n ( k ) − H ∗ n ( k )) ( G n +2 ( k ) − H n +2 ( k ))] i (cid:16) x /d (cid:17) + ℑ [( G ∗ n ( k ) − H ∗ n ( k )) ( G n +2 ( k ) − H n +2 ( k ))] (cid:16) y /d (cid:17) + ℑ ( G n ( k ) H ∗ n +1 ( k ) − H n ( k ) G ∗ n +1 ( k )) (cid:16) kz (cid:17)o ; F py = − ǫ a d Z ∞−∞ dk ∞ X n = −∞ {ℑ [( G ∗ n ( k ) − H ∗ n ( k )) ( G n +1 ( k ) − H n +1 ( k ))] − (21) ℑ [( G ∗ n ( k ) − H ∗ n ( k )) ( G n +2 ( k ) − H n +2 ( k ))] (cid:16) x /d (cid:17) + h | G n ( k ) − H n ( k ) | − ℜ [( G ∗ n ( k ) − H ∗ n ( k )) ( G n +2 ( k ) − H n +2 ( k ))] i (cid:16) y /d (cid:17) + ℜ ( G n ( k ) H ∗ n +1 ( k ) − H n ( k ) G ∗ n +1 ( k )) (cid:16) kz (cid:17)o ; F pz = − ǫ ad Z ∞−∞ dk ∞ X n = −∞ {ℑ [ kG n ( k ) H ∗ n ( k )] + ℑ G ∗ n ( k ) H n +1 ( k ) − H ∗ n ( k ) G n +1 ( k )] k (cid:16) x /d (cid:17) + ℜ G ∗ n ( k ) H n +1 ( k ) − H ∗ n ( k ) G n +1 ( k )] k (cid:16) y /d (cid:17) + ℜ ( G n ( k ) H ∗ n ( k )) k (cid:16) kz (cid:17)) . x and y , the axial shift enters here not in the ratio to the gap, d , but in the product with k , which is the inverse characteristic length in the axial direction.Note that all the forces are derived as acting on the inner cylinder, the forces on the outerone have the opposite sign. D. General Properties of Electrostatic Patch Effect Forces
The above results allow for some general conclusions regarding the patch interaction.1. To lowest order, the axial patch force is inversely proportional to the gap width, thetransverse force components go as its inverse square.2. Forces and shifts are directionally coupled: a transverse shift causes generally someaxial force, and vice versa, a transverse force appears due to an axial shift.3. The axial force vanishes when the patches are present on one of the cylinders only,i.e., when either G n ( k ) = 0, or H n ( k ) = 0.4. The transverse force does not vanish when the patches are on one of the cylinders only.Moreover, expressions (19) for its components can be given in terms of the non-zero patchpotential (and not its Fourier coefficient!) using the Parceval identity (7). For example, inthe case when there are no patches on the inner cylinder [ V a ( ϕ ′ , z ′ ) ≡ , G n ( k ) ≡
0] thetransverse components are: F px = ǫ a d Z ∞−∞ dz Z π dϕ V b ( ϕ, z ) h cos ϕ − (1 + cos 2 ϕ ) (cid:16) x /d (cid:17) − sin 2 ϕ (cid:16) y /d (cid:17)i ; F py = ǫ a d Z ∞−∞ dz Z π dϕ V b ( ϕ, z ) h sin ϕ − (1 − cos 2 ϕ ) (cid:16) y /d (cid:17) − sin 2 ϕ (cid:16) x /d (cid:17)i . The transverse interaction force also does not vanish in such a case, by the formulas (18).5. Uniformly charged cylinders give rise to a mutual restoring (elastic) force, so that thecoaxial condensor is at a neutrally stable equilibrium.6. The interaction between patch and uniform potentials involves only the first andsecond polar angle harmonics of the patch distribution if the force is taken, as above, tolinear order in the transverse shift.Some of the above conclusions might be rather obviuos or intuitively clear, however, allof them are now accurately established by our analysis.8
V. THE PATCH MODEL
To understand better patch interaction, it is natural to examine the case when just acouple of patches are present, and the PE forces are described by as simple expressions aspossible. One thus needs some convenient model of a patch as a localized deviation from theuniform potential described by some particular functions with just few parameters involved.One of them should control the patch potential, two more have to govern the spot width inthe axial and azimuthal directions, and two more parameters specify the patch position.Such a patch model is desired for another reason as well. In the experiments there isusually no way to directly measure the patch distribution at the electrode surfaces. In-stead, one should infer it from some other signals, like the patch forces. However, our forceformulas are not fit for immediate modeling, since the unknowns in them are the Fouriercoefficients G n ( k ) and H n ( k ), with no means to estimate these functions unless properlyparameterized. The existing experience of such parameterizations, along with the commonsense, demonstrate clearly that only the models based on the underlying physics, ratherthan ad hoc ones, turn out efficient and work successfully.So, the goal of an effective patch model is to find such functions that: a) both Fouriercoefficients G n ( k ) and H n ( k ) are found in a closed form, and b) all the series and integrals inthe formulas (19) and (20) for the forces are computed analytically in a closed form. Fromthis standpoint, separation of variables is the simplest representation: V ( ϕ − ϕ ∗ , z − z ∗ ) = V ∗ f ( z − z ∗ ) u ( ϕ − ϕ ∗ ) . (22)Here the normalizing constant V ∗ has the dimension of a potential, and the dimensionlessfunctions f ( z ) and u ( ϕ ) are chosen so that | f ( z ) | ≤ , | u ( ϕ ) | ≤ f (0) = 1 , u (0) = 1.The center of the patch is at ϕ = ϕ ∗ , z = z ∗ , where the potential achieves the maximummagnitude V ∗ (positive or negative). The Fourier coefficient of the function (22) is: V n ( k ) = V ∗ ˜ f ( k ) e − ikz ∗ u n e − inϕ ∗ = V ∗ ˜ f ( k ) u n e − i ( kz ∗ + nϕ ∗ ) ; (23)˜ f ( k ) = 1 √ π Z ∞−∞ dz f ( z ) e − ikz , u n = 1 √ π Z π dϕ u ( ϕ ) e − inϕ . Successful implementation of the above requirements a) and b) is in the choice of functions f ( z ) and u ( ϕ ). The first of them is natural to choose as the Gaussian exponent in z , f ( z ) = exp − z √ z ! , ˜ f ( k ) = ∆ z exp − k ∆ z √ ! , (24)9ith the Fourier coefficient a Gaussian exponent, too, and the parameter ∆ z giving the axialhalf–width of the spot. Since the product of two Gaussians entering the force expressions(19) and (20) is again a Gaussian, the integrals in k there combinations of the elementaryfunctions, as desired. A good choice of the second function, u ( ϕ ), which is 2 π –periodic,turns out much more difficult. Nevertheless, eventually one can come up with the following: u ( ϕ ) = u ( ϕ, λ ) = (1 − λ ) ϕ − λ cos ϕ + λ , (25)where − ≤ λ < ϕ = 0 (see fig. 2 below).Indeed, when λ = − u ( ϕ, − ≡
1, for λ = 0, u ( ϕ,
0) = 0 . ϕ ), and finally, when λ → −
0, the function demonstrates a ‘bounded delta–like’ behavior by going to zeroeverywhere except ϕ = 0, where the limit is unity (zero width peak). We introduce theazimuthal patch half–width, ∆ ϕ , in an accurate way as the abscissa at the point where u coincides with its mean value over the whole interval, u (∆ ϕ ) = u av , which givescos ∆ ϕ = λ, ∆ ϕ = arccos λ . (26)In a complete agreement with the above, when λ → −
0, the width shrinks according as∆ ϕ ≈ q − λ ) →
0; in the opposite case λ = − ϕ = 2 π .What makes the choice (25) really invaluable for calculations is its Fourier coefficientsgotten by simply expanding the function in powers of λ exp ( − iϕ ): u n = u n ( λ ) = √ π − λ λ λ | n | , n = 0; u = u ( λ ) = √ π − λ , (27)which are essentially just exponents of | n | , as in the geometric progression. Apparently,expressions (25) and (27) satisfy our requirements a) and b), perhaps even in the simplestpossible way. The profiles of u ( ϕ ) are plotted in fig. 2 for various width values.With u ( ϕ ) and f ( z ) defined by formulas (24) and (25), our patch model (22) becomes: V ( ϕ − ϕ ∗ , z − z ∗ ) = V ( ϕ − ϕ ∗ , z − z ∗ ; ∆ z, ∆ ϕ ) ≡ V ∗ v ( ϕ − ϕ ∗ , z − z ∗ ; ∆ z, ∆ ϕ ) = (28) V ∗ (1 − λ ∗ ) ϕ − ϕ ∗ )1 − λ ∗ cos ( ϕ − ϕ ∗ ) + λ ∗ exp − z − z ∗ √ z ∗ ! . The corresponding Fourier coefficients are found by the expressions (23), (24), and (27) as V n ( k ) = √ π ∆ z ∗ V ∗ − λ ∗ λ ∗ λ | n |∗ exp − k ∆ z ∗ √ ! e − i ( nϕ ∗ + kz ∗ ) , n = ± , ± . . . , V ( k ) = √ π ∆ z ∗ V ∗ − λ ∗ − k ∆ z ∗ √ ! e − ikz ∗ . (29)10ertainly, these functions are smooth enough to satisfy conditions (4), in fact, the function(28) and all its derivatives in ϕ and z are squarely integrable, or else V belongs to theSobolev space H p for any p ≥
0. The picture of equipotentials of the patch (28) normalizedby the maximum voltage is in fig. 3.
V. SINGLE PATCH AT EACH OF THE ELECTRODES: A PICTURE OF PATCHINTERACTION
We consider now two patches of the form (28), one at the inner, the other at the outerboundary. Patch voltages in the boundary conditions (3) become ( i = 1 , G ( ϕ ′ , z ′ ) = V ( ϕ ′ − ϕ , z ′ − z ) , H ( ϕ, z ) = V ( ϕ − ϕ , z − z ) ; (30) V ( ϕ − ϕ i , z − z i ) = V i (1 − λ i ) ϕ − ϕ i )1 − λ i cos ( ϕ − ϕ i ) + λ i exp − z − z i √ z i ! , where, according to the relation (26) between λ and the angular width, ∆ ϕ ,0 < ∆ ϕ i ≤ π, ≤ ∆ z i < ∞ , − π < ϕ i ≤ π, −∞ < z i < ∞ , i = 1 ,
2. The forces cor-responding to these distributions are derived in the appendix. We study here the interactionof two identical patches, ∆ z = ∆ z = ∆ z , ∆ ϕ = ∆ ϕ = ∆ ϕ , and V = ± V = V . A. Transverse Force
1. Transverse force due to patch and uniform potential interaction
By formula (A1) of the appendix, the transverse force due to the interaction between theuniform potential and patches reduces to the following expressions: F intx F = − π r π za sin ∆ ϕ ( (cos ϕ ∓ cos ϕ ) − x d " (cos 2 ϕ ∓ cos 2 ϕ ) cos ∆ ϕ + 2(1 ∓ ϕ − y d [(sin 2 ϕ ∓ sin 2 ϕ ) cos ∆ ϕ ] ) ; F inty F = − π r π za sin ∆ ϕ ( (sin ϕ ∓ sin ϕ ) − (31) x d [(sin 2 ϕ ∓ sin 2 ϕ ) cos ∆ ϕ ] − y d " ∓ ϕ − (cos 2 ϕ ∓ cos 2 ϕ ) cos ∆ ϕ . Here F is the characteristic force defined as F = ǫ V V − ( a/d ) , (32)11nd V − is the uniform voltage difference (3). The minus or plus sign is taken for the patchvoltages of the same or opposite sign, respectively; the signs of the charges induced by patcheson the cylinders are opposite in the first case, and same in the second. As expected, themagnitude of the transverse force is inversely proportional to ( a/d ) . It is also proportionalto the relative axial width, ∆ z/a , and entirely independent of the axial positions z , of thepatches. The dependence on the angular width is more complicated: the maximum force isat ∆ ϕ = π/
2, as prompted by geometry; for a small width the force goes to zero as (∆ ϕ ) .When ∆ ϕ = π , i.e., the patches are the ‘belts’ of voltage uniform in ϕ , the zeroth orderforce vanishes, and the total becomes proportional to the shift and directed along it: ~F int = (2 π ) / (1 ∓ F (cid:18) ∆ za (cid:19) ~ρ d ! . (33)This is zero when the patch voltages are equal: the forces from each of them have the samemagnitude and opposite directions (see more on this below).The main contribution, i.e., the force in the centered position is best characterized by itspolar components, namely: F intρ F = − π r π za sin ∆ ϕ [cos( ϕ − ϕ ) ∓ cos( ϕ − ϕ )] ; F intϕ F = π r π za sin ∆ ϕ [sin( ϕ − ϕ ) ∓ sin( ϕ − ϕ )] . The total force is a superposition of the two forces from each of the patches. They act alongthe radial direction to the corresponding patch center, and the total is a vector sum of thesetwo radial vectors, see figs. 4. That is why the interaction force vanishes when the patchesare one opposite the other ( ϕ = ϕ ): their contributions, aligned and of the opposite signs,exactly cancel each other. So here a patch behaves as an effective point charge, q eff , in theuniform radial field E u = − V − /d : the force due to it is just q eff E u . The effective chargesare readily found by comparison with the above expressions of the force.The forces of the first order are directionally coupled to the shifts, meaning an x –forcedepends on the y –shift, and vice versa. The forces consist of a constant term and the secondharmonics of the patch angular position. 12 . Transverse force due to patch interaction Its general expressions (A8) and (A10) simplify for the same size patches: F px F tr = 2( π ) / ∆ za sin (cid:18) ∆ ϕ (cid:19) (" N ∓ M exp " − (cid:18) z − z z (cid:19) (cos ϕ + cos ϕ ) − x d " N + N (cos 2 ϕ + cos 2 ϕ ) ∓ M + M cos ( ϕ + ϕ ) ! exp " − (cid:18) z − z z (cid:19) − y d " N (sin 2 ϕ + sin 2 ϕ ) ∓ M sin ( ϕ + ϕ ) exp " − (cid:18) z − z z (cid:19) ∓ z ∆ z " M (cos ϕ + cos ϕ ) z − z ∆ z exp " − (cid:18) z − z z (cid:19) ; F py F tr = 2( π ) / ∆ za sin (cid:18) ∆ ϕ (cid:19) (" N ∓ M exp " − (cid:18) z − z z (cid:19) (sin ϕ + sin ϕ ) − (34) x d " N (sin 2 ϕ + sin 2 ϕ ) ∓ M sin ( ϕ + ϕ ) exp " − (cid:18) z − z z (cid:19) − y d " N − N (cos 2 ϕ + cos 2 ϕ ) ∓ M − M cos ( ϕ + ϕ ) ! exp " − (cid:18) z − z z (cid:19) ∓ z ∆ z " M (sin ϕ + sin ϕ ) z − z ∆ z exp " − (cid:18) z − z z (cid:19) . Here the characteristic transverse force, F tr , is almost as in the previous case [formula (32)], F tr = ǫ V a /d , (35)with just a natural replacement of V − with V . The coefficients involved are found bycombining the formulas (A5), (A6) and (A7) of the appendix with the representation (A12): N = 3 − λ N = 1 + λ − λ ) ; N = 1 + λ
16 (1 + 4 λ − λ ) ; M = 1 − λ " λ ) cos( ϕ − ϕ ) − λ D ; D = 1 − λ cos ( ϕ − ϕ ) + λ ; (36) M = 1 − λ " − λ (1 + λ ) 1 + λ − ϕ − ϕ )2 D ; M = 1 − λ ( λ λ " ϕ − ϕ ) + λ (1 + λ ) cos 2( ϕ − ϕ ) − λ cos( ϕ − ϕ ) D .
13s before, the signs ∓ correspond to the case of the same or opposite signs of the patchvoltages. The force (34) is again proportional to ( a/d ) and ∆ z/a . The dependence on theangular width here is even more complicated than in the previous case; still, the force is ∝ (∆ ϕ ) in the narrow patch limit. For ∆ ϕ = π , an analog of the formula (33) holds: ~F p = 2( π ) / F tr (cid:18) ∆ za (cid:19) ( ∓ exp " − (cid:18) z − z z (cid:19) ~ρ d ! . This force, however, does not vanish for identical voltages, unless the patches are one rightagainst the other.The nature of the force due to the patch interaction is most clearly seen in the main termcorresponding to coaxial cylinders with no axial shift:a) V = V = V F p ⊥ F tr = 4( π ) / ∆ za sin (cid:18) ∆ ϕ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos ϕ − ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( N − M exp " − (cid:18) z − z z (cid:19) , b) V = − V = V F p ⊥ F tr = 4( π ) / ∆ za sin (cid:18) ∆ ϕ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos ϕ − ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( N + 2 M exp " − (cid:18) z − z z (cid:19) . The direction in the cylinder cross–section is along the bisectrix of the angle subtendedby the patches, no matter what the voltage signs are:tan θ p = tan[0 . ϕ + ϕ )]. Thus for ϕ = ϕ the total force is maximum because the contribution of each patch doubles, whilefor ϕ = ϕ + π the total force is apparently zero. The force is a combination of two terms:one is proportional to N depending only on the angular patch width and non–negative, bythe formulas (36). The other is the exponent of ( z − z ) with the coefficient M dependingon | ϕ − ϕ | . The coefficient M , along with M , and M , are shown in fig. 5 versus theangular distance. They have a sharp maximum at ϕ = ϕ , and drop quickly away from it;the maximum is the sharper, and the drop the faster, the smaller the width of the patch is.This is a strong manifestation of screening of patch charges: the patches interact strongestof all when their centers are right opposite each other; the interaction drops when one chargestops ‘seeing’ the other due to the obstruction by the inner cylinder.In fig. 6 F p ⊥ is plotted as a function of the angular distance | ϕ − ϕ | for both cases, a)and b). The two curves differ significantly for the moderate angular separations, and tendto zero when ϕ − ϕ → π . The dependence of F p ⊥ on the axial distance | z − z | is shown14n fig. 7. The force does not vanish at infinity, but goes instead to the asymptotic valuecommon for the two cases.As prompted by the similarity between the coefficients M — M and N — N , being justsome bounded functions of ∆ ϕ , the terms in the force (34) proportional to the transverseshifts have the structure, and thus the behavior, similar to that of the zeroth order expres-sions. The term with the axial shift is rather different, first of all because of an additionalfactor, ( d/ ∆ z ). Thus z no longer compares to the gap, d : quite naturally, it is the ratio( z / ∆ z ) that stands as a small parameter at this part of the force. The other peculiarity isthe additional factor proportional to z − z . This part of the force vanishes in both limits,( z − z ) / ∆ z → ∞ and ( z − z ) / ∆ z = 0, with the maximum magnitude at | z − z | = √ z as seen in fig.8. Finally, it is proportional to the first harmonics of the angular patch posi-tions and the coefficient M . The coefficient, and thus the force, is maximum when ϕ = ϕ ,then it decreases monotonically as | ϕ − ϕ | increases, fig. 5. B. Axial Force
As demonstrated in section III, the axial PE force is only due to the patch interaction.Its general expression (A11) reduces in our case to: F pz F ax = ± π / sin (cid:18) ∆ ϕ (cid:19) z − z ∆ z exp " − (cid:18) z − z z (cid:19) M − (37) x d M (cos ϕ + cos ϕ ) − y d M (sin ϕ + sin ϕ ) − z z − z M − z − z √ z ! ) . The coefficients M , M are found in the formulas (36), and the characteristic force isinversely proportional to the relative gap, d/a , and not to its square, as before: F ax = ǫ V a/d . (38)The sign of the force (37) switches from plus to minus for patch voltages of the sameor opposite signs, respectively. The most striking feature of the axial force is its overalldependence on the ratio ( z − z ) / ∆ z . As one expects intuitively, the axial force tends tozero in both limits ∆ z → z → ∞ , due to the Gaussian exponent of the aboveargument multiplied by this same argument. It has the maximum π / e − sin (∆ ϕ/ F ax at | z − z | = √ z , same as the axial shift part of the transverse force from fig. 8.15he overall dependence of the axial force on ∆ ϕ is just like that of the transverse force;particularly, for belt–like patches, ∆ ϕ = π , the force is: F pz F ax = ± π / z − z ∆ z exp " − (cid:18) z − z z (cid:19) − z z − z − z − z √ z ! ) . Here the main term vanishes when the two patches are in the same cross–section, z = z , butthe ‘correction’ proportional to z does not: F pz /F ax = ∓ π / ( z / ∆ z ) , ∆ ϕ = π, z = z .The first order force proportional to the axial shift has a factor ( z − z ) , instead of( z − z ), in front of the Gaussian. This contribution changes sign at z − z = √ z , andis maximum when the patches are in the same z plane, and it as illustrated by fig. 9.The axial force decreases with the angular distance between the patches. The zeroth ordercomponent of the axial force together with the first order term proportional to the axial shiftdepend on | ϕ − ϕ | through the coefficient M only. The two other components due tothe transverse shifts are proportional to the harmonics of the patches angular positions andthe coefficient M ; they both vanish in just one case, when ϕ − ϕ = π . Despite theseminor differences, each of these terms follows basically the characteristic behavior of thecoefficients M or M versus the distance | ϕ − ϕ | , as presented in fig. 5. VI. ESTIMATES OF AXIAL PATCH EFFECT FORCE FOR STEP.CONCLUDING REMARKSA. Basics of the STEP Experimental Set-up and Some Requirements
STEP, a medium size scientific satellite ( < Kg ), will be put into a drag–free earthorbit at the altitude of ∼ Km . It is to measure the relative free fall acceleration of pairsof test masses (TM) of different materials with an accuracy of 10 − m/sec , to determine theequivalence of the inertial and gravitational mass with an uncertainty 6 orders of magnitudesmaller than the existing results [15], or find violations of the Equivalence Principle (EP; inother words, Universal Free Fall) somewhere between 1 part in 10 and 1 part in 10 .STEP will fly four differential accelerometers (DAC), each with a pair of TMs shaped ascoaxial cylindrical shells. The cross–section of the DAC is shown in fig. 10. An electromag-netic system of magnetic bearings and capacitors keeps TMs alligned and centred to within < nm . The transverse degrees of freedom are constrained, the two axial ones are left free,16o the axial motion and the rotation about the TM axis are most important. So we discussonly the axial PE force; the axial torque will be examined in the final part of this paper.The readout of the differential longitudinal displacement for each TM pair is provided bya Superconducting Quantum Interference Device (SQUID), sensitive to an acceleration of3 × − m/sec in an accumulation time of at least 20 orbits, i.e., ∼ days . Nominally theDACs are kept inertially fixed and the EP violation signal will be at the orbital frequency, f orb = 1 . × − Hz . However, changing the signal frequency from one measurementsession to the other helps to discover and remove systematic readout errors. So modulationof the frequency is planned by rolling the satellite (and the DACs) about the normal to theorbital plane at some frequency f roll ≥ f orb (the DAC axes lie in the orbital plane). Withthis procedure, the science signal will be at the frequency f s = f roll ± f orb bounded frombelow as f s ≥ . × − Hz . (39)The STEP design rejects all perturbations able to mimic or mask the signal from theTM free fall at the level of the target accuracy. Thus the magnitude of any perturbing axialacceleration at the signal frequency should be at least less than 10 − m/sec . In addition,axial motion of TMs is allowed only within certain range, as required by drag–free controland SQUID readout limits. The total axial shift must not exceed some limit, z max , whosevalue may be adjusted while developing the instrument; currently it is z max = 1 µm .The SQUID sensor will keep the TM position in the body by applying a restoring (spring)force to counteract any D.C. axial force, such as the one due to patches. Its control authorityin terms of the range of the corresponding oscillation period is 300 − s . For a TM of 1 Kg ,this converts into the spring constant, k , within the range 3 . × − − . × − N/m . Itsmaximum value k max = 4 . × − N/m provides the maximum force that can be controlledaccording to F max = k max z max = 4 . × − N , or maximum acceleration a max = F max / Kg = 4 . × − m/s . (40)The related criterion is: any distribution of patches is acceptable as soon as the axial D.C.acceleration due to them does not exceed a max .We consider a TM and its magnetic bearing as a reference case of our pair of cylinders(the gap between them is at least 3 times smaller - and the force equally larger - than thegap between the TM and electrodes, see fig. 10). First we assume each cylinder to carry17ust one patch of the same sizes and magnitudes; then we consider an arbitrary number ofsuch patch pairs. Using slightly different notations the formula (37) for the axial force with x = y = 0 becomes: F z = ± π / F ax sin (cid:18) ∆ ϕ (cid:19) µ exp " − (cid:18) z − z z (cid:19) z − z ∆ z − z ∆ z − z − z √ z ! ) ; (41) µ = µ ( λ, | ϕ − ϕ | ) = 1 + (1 + λ ) ϕ − ϕ ) − λ − λ cos ( ϕ − ϕ ) + λ , λ = cos ∆ ϕ ;the characteristic force F ax is defined by the expression (38). To l. o. the force (41) isconstant (unless the patches move on the surfaces, which has not been observed so far, orthe cylinders rotate); the correction proportional to the axial shift, z , can produce harmonicoscillations near a stable equilibrium, or exponential runaway from an unstable one. We takethe patch voltage V = 10 mV for our estimates. This seems a plausible number for patchesat low temperatures (see some relevant result for the GP-B experiment in [7]). The TMparameters used are: TM radius a = 2 . cm (outer TM; the inner radius, and hence theforce, is about 5 times smaller), TM height 2 L = 14 cm , the TM to magnetic bearing gap d = 0 . mm ( d/a ≈ − ). The TM masses vary from 0 . Kg to 2 . Kg , so to make therescaling easy, we give all the accelerations (or specific forces) per 1 Kg of mass . B. Constant Axial Acceleration due to Patches
By the expression (41), the constant part of the axial patch effect force is F z = ± π / F ax sin (cid:18) ∆ ϕ (cid:19) µ ˜ z exp (cid:16) − ˜ z (cid:17) , ˜ z ≡ ( z − z ) / z ; (42)the ballpark number for the resulting acceleration comes from the characteristic force (38): a ax = F ax / Kg = ǫ V ( a/d ) / Kg ≈ . × − m/s . (43)It is almost 4 orders of magnitude smaller than the one required by the formula (40). Next isan upper bound on the acceleration from the force (42) with two last factors set at their max-imum. Since the maximum of µ with regards to | ϕ − ϕ | is h (∆ ϕ/ i / (∆ ϕ/ / √ e when ˜ z = 1 / √
2, the bound is | a z | < s π e a ax sin (cid:18) ∆ ϕ (cid:19) ≈ . × − sin (cid:18) ∆ ϕ (cid:19) m/s . (44)18t is still more than 3 orders of magnitude smaller than the required value (40) for any ∆ ϕ ,and drops as its square when it becomes small.We now consider the mean value of the force (42) over all angular | ϕ − ϕ | and the axial,˜ z patch distances ( l ≡ L/ z ). The average in the angle is unity while that in the distanceis ( l ) − R l ˜ z exp( − ˜ z ) d ˜ z = ( l ) − [1 − exp( − l ) ], so¯ a z = 2 π / a ax ∆ zL h − e − ( L/ z ) i sin (cid:18) ∆ ϕ (cid:19) < . × − ∆ zL sin (cid:18) ∆ ϕ (cid:19) m/s . (45)Two new pleasant features here as compared to the estimate (44) are: the fourth powerinstead of the square of sine, and a factor (∆ z/L ), which is less than unity in any case, andis expected to be essentially smaller. The upper bound (45) satisfies the condition (40) withthe margin of 3 orders of magnitude for any patch sizes.However, this is hardly the case with some number, N , of patches pairs identical in sizesand magnitude, since N may be quite large. Those pairs generate N interactions eachgiving the force (42), with either sign. Assuming the signs of the patches random, singlecontributions do not cancel each other completely, but sum up to a factor √ N = N for N ≫
1. Thus the total force is reasonably estimated as N times the average of the force(42). The resulting acceleration satisfies the condition (40) if the following inequality is true: N ∆ zL sin (cid:18) ∆ ϕ (cid:19) < . × . (46)For N up to 600 the condition is satisfied by the patches of any sizes. Limitations on thelatter start with N ∼ N ∼ , ϕ = 60 ◦ , any axial size of the patch is still acceptable.Moreover, the upper bound for N is available through the patch sizes, if the patches do notoverlap within their nominal widths 2∆ z , 2∆ ϕ . The effective patch area is (2∆ z ) × (2 a ∆ ϕ ),the total surface area of the cylinder is (2 πa )(2 L ), so N is bounded by the ratio N ≤ πaL a ∆ ϕ )∆ z = π ∆ ϕ L ∆ z . (47)Introducing this to the inequality (46), we obtain a universal estimate in terms of ∆ ϕ only:1∆ ϕ sin (cid:18) ∆ ϕ (cid:19) < , (48)which, of course, is always true. So, non-overlapping patches covering the surfaces completelysatisfy the STEP limitation on the DC acceleration, if the averaged force is used and theassumption of random signs of the patch potentials is valid.19 . Harmonic Oscillations and Exponential Runaway due to Patches The zero order force (41) vanishes at z = z . The first correction to it, δF z , is: δF z = ∓ π / F ax ∆ z sin (cid:18) ∆ ϕ (cid:19) µ (cid:16) − z (cid:17) exp (cid:16) − ˜ z (cid:17) , ˜ z ≡ ( z − z ) / z . (49)As explained in sec. V B, its overall sign depends on the signs of the patches and the distancebetween them, because the factor in brackets is positive or negative depending on whether˜ z is smaller or larger than 1 / √ . So we denote ω = | δF z | / Kg , then the equation of TMmotion in the axial direction near the equilibrium becomes:¨ z = ∓ ω z , (50)and describes small oscillations for the minus sign (restoring force), and exponential runawayotherwise. In the first case, the oscillation frequency is f = ω/ π >
0, in the second theTM drifts away exponentially with the characteristic time τ = 1 /ω .The maximum of ω , needed for an upper bound for the frequency f or a lower bound forthe time constant τ , is clearly attained when ˜ z = ϕ − ϕ = 0, so: ω ≤ s π / a ax ∆ z sin (cid:18) ∆ ϕ (cid:19) = 6 . × − √ ∆ z sin (cid:18) ∆ ϕ (cid:19) rad/s , (51)with ∆ z in meters. Remarkably, this estimate of ω decreases when ∆ ϕ goes down, butincreases with the decrease of ∆ z . The frequency of TM oscillations should be below theminimum science signal frequency (39), f < . × − Hz , (52)to avoid the signal corruption. Based on the estimate (51), this condition holds when∆ z > . × − sin (cid:18) ∆ ϕ (cid:19) m . (53)Even if sin (∆ ϕ/
2) = 1 (circular patches), the axial width should be only about two hundredsnanometers to satisfy this.In the runaway case the characteristic time must satisfy τ ≫ T obs = 1 . × s , (54)20o avoid saturation: if it is true, then any initial shift smaller than z max = 1 µm will notgrow critically during a science session. This criterion and the estimate (51) lead to∆ z ≫ . × − sin (cid:18) ∆ ϕ (cid:19) m , (55)five orders of magnitude more restrictive than the bound (53). However, a small enoughangular size helps to meet the requirement (54). The maximum acceleration allowed in therunaway case when the condition (54) holds is implied by the estimate (51) with ∆ z min taken from the r.h.s of the estimate (55) and the maximum shift z max = 1 µm : a max = ω z max = 3 . × − sin (∆ ϕ/ z min m/s = 3 . × − m/s . (56)Its value is 7 orders of magnitude smaller than the STEP requirement (40).Other estimates for the patch parameters are produced by the approach of the previoussection using the expression of | δF z | averaged over the axial and the angular distancesbetween the patches. As we know, the average of µ in the formula (49) is unity, andthe average over the axial distance is l − R l d ˜ z (1 − z ) exp [ − ˜ z ] = exp( − l ) (as before, l = L/ z ). Thus the mean angular frequency is found to be¯ ω = vuut π / a ax ∆ z sin (cid:18) ∆ ϕ (cid:19) exp " − (cid:18) L z (cid:19) =6 . × − √ ∆ z sin (cid:18) ∆ ϕ (cid:19) exp − L √ z ! rad/s , (57)with ∆ z on the far right in meters, as usual. As compared to the estimate (51), here isan extra power of the sine of ∆ ϕ , and, most important, the Gaussian exponent that dropssharply with L/ ∆ z growing. Due to this, the conditions (52) and (54) on the oscillationfrequency and runaway time are satisfied for the patches of any sizes, by the estimate (57).Something close to this happens if we consider N pairs of patches instead of one. Let usexamine the condition (54), which is more stringent than (52). We can again estimate theoverall contribution of N pairs of patches with random voltage signs as √ N δF z , and takethe mimimum √ L ( e/ / for ∆ z = L/ √
2. This gives N / sin (∆ ϕ/ ≪ . √ L ( e/ / = 2 . L = 7 cm cited above. The inequality does not hold always, somewhat limiting theangular patch size to be small enough given N . However, assuming the patches do not21verlap, we can use the upper bound (47) of N with the above value ∆ z = L/ √ (∆ ϕ/ ϕ/ − / ≪ .
8. The maximum of the l.h.s hereis always less than unity, so the condition (58) holds always for non-overlapping patches.Concluding this section we remind that all the numbers in the estimates were obtainedusing the patch voltage V = 10 mV , the TM mass 1 Kg , and the TM radius–to–gap ra-tio a/d = 77. All the accelerations scale as voltage square, proportional to this ratio, andinversely proportional to the mass; the frequencies and inverse runaway times scale propor-tional to the voltage, q a/d , and the inverse of the square root of the mass. D. Concluding Remarks. Perspectives of Patch Force Modeling
The main message of the above estimates is that the STEP requirements (40), (52) and(54) can be met by patches whose sizes and number are appropriately limited; the limitsseem not too restrictive and practically achievable. However, those estimates were derivedunder a number of simplifying assumptions. Perhaps the strongest of them was that all thepatches differ at most by the sign of the voltage, having the same voltage magnitude andspot sizes. The results obtained in this paper allow, in fact, for an essentially more realisticpatch force modeling, which can be done in the following way.One represents both patch potentials V a ( ϕ ′ , z ′ ) and V b ( ϕ, z ) as a superposition of somenumber, N a, b , of model patches (28), V µ ( ϕ, z ) = N µ X n =0 V ( ϕ − ϕ µn , z − z µn ; ∆ z µn , ∆ ϕ µn ) = N µ X n =0 V µn v ( ϕ − ϕ µn , z − z µn ; ∆ z µn , ∆ ϕ µn ) , µ = a, b , (59)with different voltages, sizes, and positions. By the formulas of section III the patch effectforces corresponding to the distributions (59) can be explicitly calculated, as it is done herefor a single patch at each of the boundaries. Being cumbersome, this general calculationis otherwise straightforward, without any new technical difficulties. It leads to the forceexpression as a quadratic form of the patch voltages V µn , with the coefficients depending onall other parameters in a known way.Having these formulas at hand, one then carries out simulations by specifying parametersets in various ways and computing the patch forces. One can pick the parameters randomly,22nd eventually come up with the patch force statistics. One can also use any lab informationon the patch distributions, arranging for a semi-random patch sets, as was done, for instance,when simulating magnetic trapped flux distribution on GP-B rotors [16]. Such exhaustiveanalysis can be strongly recommended before the STEP flight. On the other hand, the samegeneral formulas for the transverse forces can be used for fitting control effort data obtainedduring the experiment, for restoring the voltage patch patterns on the proof masses andbearings. Once the latter are known, the axial forces can be computed, and the systematicexperimental error due to them can thus be bounded. Of course, all this is applicable toany experimental set-up with cylindrical geometry. 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. Wealso thank our colleagues at GP-B and STEP, particularly, Dan DeBra, Sasha Buchman,David Hipkins, John Mester and Paul Worden for valuable discussions and remarks.
Appendix A: Calculation of the Transverse and Axial Force for a Single Patch atEach of the Cylinders
Here we consider one patch at each boundary described by our patch model (28), i.e., forthe voltage distributions (30). The force (18) due to the interaction between the uniformpotential and patches is a linear function of the Fourier coefficients of the boundary patchvoltages computed for k = 0 and n = 0 , ,
2. Thus formulas (29) allow one to obtain( l i = √ z i , i = 1 , F intx = π √ π ǫ ad V − ( − V l − λ ϕ + V l − λ ϕ + x d " V l (1 − λ ) λ λ cos 2 ϕ ! − V l (1 − λ ) λ λ cos 2 ϕ ! + y d " V l − λ λ sin 2 ϕ − V l − λ λ sin 2 ϕ ; (A1) F inty = π √ π ǫ ad V − ( − V l − λ ϕ + V l − λ ϕ +23 d " V l − λ λ sin 2 ϕ − V l − λ λ sin 2 ϕ + y d " V l (1 − λ ) − λ λ cos 2 ϕ ! − V l (1 − λ ) − λ λ cos 2 ϕ ! . The transverse force due to the patch interaction is essentially more cumbersome to derive,with the additional difficulty of computing some integrals in k and sums over n containingproducts of Fourier coefficients. We start with F px from the first of the formulas (21). Usingthe expressions (23)and (30), combined with the equality (24), we find: F px = − π ǫ ad Z ∞−∞ dk ( −ℜ " V l e − k l N ( λ ) e ıϕ + V l e − k l N ( λ ) e ıϕ − (A2) V V l l M e − k ( l + l ) (cid:16) e − ik ( z − z ) + e ik ( z − z ) (cid:17)(cid:21) + x d ℜ " V l e − k l (cid:18) N ( λ ) + N ( λ ) e ıϕ (cid:19) + V l e − k l (cid:18) N ( λ ) + N ( λ ) e ıϕ (cid:19) − V V l l e − k ( l + l ) (cid:16) (2 M + M ) e − ik ( z − z ) + M e ik ( z − z ) (cid:17)(cid:21) + y d ℑ " V l e − k l N ( λ ) e ıϕ + V l e − k l N ( λ ) e ıϕ − V V l l M e − k ( l + l ) (cid:16) e − ik ( z − z ) + e ik ( z − z ) (cid:17)(cid:21) − kz ℑ (cid:20) V V l l M e − k ( l + l ) (cid:16) e − ik ( z − z ) − e ik ( z − z ) (cid:17)(cid:21)(cid:27) . Here we have introduced the following notations for the coefficients: M q = M q ( ϕ , λ , ϕ , λ ) ≡ π ∞ X n = −∞ u n ( λ ) e − ınϕ u ∗ n + q ( λ ) e ı ( n + q ) ϕ ; (A3) N q ( λ ) ≡ π ∞ X n = −∞ u n ( λ ) u ∗ n + q ( λ ) = M (0 , λ ; 0 , λ ) ; q = 0 , , u n ( λ ) defined by the formulas (27). Coefficients (A3) are clearly symmetric in the pairs ofarguments ϕ, λ corresponding to each of the patches, M q ( ϕ , λ , ϕ , λ ) = M q ( ϕ , λ , ϕ , λ ) , which we have already used in the above expressions. Now, formulas (27) lead to the explicitsums of the series (A3), since they reduce to geometric progressions: M ( ϕ , λ , ϕ , λ ) = (1 − λ ) (1 − λ )4 " λ ) (1 + λ ) cos( ϕ − ϕ ) − λ λ D ; M ( ϕ , λ , ϕ , λ ) = (1 − λ ) (1 − λ )8 ( e ıϕ (1 + λ ) " λ λ ) e ı ( ϕ − ϕ ) − λ λ D + e ıϕ (1 + λ ) " λ λ ) e − ı ( ϕ − ϕ ) − λ λ D ; (A5)24 ( ϕ , λ , ϕ , λ ) = (1 − λ ) (1 − λ )8 ( e ıϕ (1 + λ ) λ " λ λ ) e ı ( ϕ − ϕ ) − λ λ D + e ıϕ (1 + λ ) λ " λ λ ) e − ı ( ϕ − ϕ ) − λ λ D + (1 + λ ) (1 + λ )2 e ı ( ϕ + ϕ ) ) ; D = 1 − λ λ cos( ϕ − ϕ ) + ( λ λ ) . (A6)By the definition (A4), coefficients N q ( λ ) are found as a particular case of the expressions(A5) when λ = λ = λ and ϕ = ϕ = 0: N ( λ ) = 1 − λ − λ ); N ( λ ) = 1 − λ − λ ) ; N ( λ ) = 1 − λ
16 (1+4 λ − λ ) . (A7)For the closed form of F px , two well known integrals in k are used: Z ∞−∞ dk exp " − k ( l + l )4 e ± ık ( z − z ) = 2 √ π q l + l exp " − ( z − z ) l + l ; Z ∞−∞ dk exp " − k ( l + l )4 k e ± ık ( z − z ) = ± ı √ π z − z ( l + l ) / exp " − ( z − z ) l + l . In the case z = z and l = l = l the first integral becomes: R ∞−∞ dk exp h − k l i = √ π/l . Thanks to all the above results for the series and integrals, we obtain the expression for F px ,to l. o. in the parameter r /d ≪ F px = π / √ ǫ ad ((cid:20) V l N ( λ ) cos ϕ + V l N ( λ ) cos ϕ − V V ¯ l ℜ (cid:18) M (cid:19) e − ˜ z (cid:21) − x d (cid:20) V l (cid:18) N ( λ ) + N ( λ ) cos 2 ϕ (cid:19) + V l (cid:18) N ( λ ) + N ( λ ) cos 2 ϕ (cid:19) − V V ¯ l ℜ (cid:18) M + M (cid:19) e − ˜ z (cid:21) − (A8) y d (cid:20) V l N ( λ ) sin 2 ϕ + V l N ( λ ) sin 2 ϕ − V V ¯ l ℑ (cid:18) M (cid:19) e − ˜ z (cid:21) − z ¯ l ¯ l / l l (cid:20) V V ¯ l ℜ (cid:18) M (cid:19) ˜ ze − ˜ z (cid:21)) ;˜ z = z − z q l + l ; ¯ l = 2 / l l q l + l . (A9)Calculation of F py is pretty similar, and its result, to linear order in r /d ≪
1, is: F py = π / √ ǫ ad ((cid:20) V l N ( λ ) sin ϕ + V l N ( λ ) sin ϕ − V V ¯ l ℑ (cid:18) M (cid:19) e − ˜ z (cid:21) − (A10) x d (cid:20) V l N ( λ ) sin 2 ϕ + V l N ( λ ) sin 2 ϕ − V V ¯ l ℑ (cid:18) M (cid:19) e − ˜ z (cid:21) − d (cid:20) V l (cid:18) N ( λ ) − N ( λ ) cos 2 ϕ (cid:19) + V l (cid:18) N ( λ ) − N ( λ ) cos 2 ϕ (cid:19) − V V ¯ l ℜ (cid:18) M − M (cid:19) e − ˜ z (cid:21) − z ¯ l ¯ l / l l (cid:20) V V ¯ l ℑ (cid:18) M (cid:19) ˜ ze − ˜ z (cid:21)) . Finally, we find a closed–form expression for the axial force. Starting from the last ofthe formulas (21), we just need to repeat the same steps as with F px . The only significantdifference is that here we need a slightly different integral, Z ∞−∞ dk exp " − k ( l + l )4 k e ± ık ( z − z ) = 4 √ π ( l + l ) / " − z − z ) l + l exp " − ( z − z ) l + l . In this way, using definitions (A9) and (A5), we arrive at the final expression for F pz : F pz = π ! / ǫ ad V V ¯ l l l e − ˜ z ( ˜ z ℜ (cid:18) M (cid:19) − (A11) x d (cid:20) ˜ z ℜ (cid:18) M (cid:19)(cid:21) − y d (cid:20) ˜ z ℑ (cid:18) M (cid:19)(cid:21) − z ¯ l ¯ l / l l (cid:20)(cid:16) − z (cid:17) ℜ (cid:18) M (cid:19)(cid:21)) . The calculation of the patch effect forces for the case when a single patch is placed oneach of the cylinders is now completed. The expressions (A8), (A10) and (A11) are for thepatches with different sizes and magnitudes. When those are identical, the results simplifyessentially due to λ = λ = λ , l = l = l , | V | = | V | , in particular: N ( λ ) = (1 − λ ) N ; N ( λ ) = (1 − λ ) N ; N ( λ ) = (1 − λ ) N ; M ( λ, ϕ , ϕ ) = (1 − λ ) M ; (A12) M ( λ, ϕ , ϕ ) = (1 − λ ) M ( λ, ϕ − ϕ ) [(cos ϕ + cos ϕ ) + ı (sin ϕ + sin ϕ )] ; M ( λ, ϕ , ϕ ) = (1 − λ ) M ( λ, ϕ − ϕ ) [cos ( ϕ + ϕ ) + ı sin ( ϕ + ϕ )] . The coefficients N i and M i here are given explicitly by the formulas (36). [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] Darling, T.W.
Electric Fields on Metal Surfaces at Low Temperatures , in: ‘School of Physics”’,University of Melbourne, Parkville, 1989, p.88.[3] Speake, C.C.
Forces and Force Gradients due to Patch Fields and Contact–Potential Differ-ences . Class. Quantum. Gravity, , A291–297, 1996.
4] Prince, T., J. Baker, P. Bender, et al. LISA-LIST-RP-436.Version 1.2. A revision of a documentoriginally prepared for the National Research Council ‘Beyond Einstein Program AssessmentCommittee’ (BEPAC) in 2007. 10 March 2009. http://lisa.nasa.gov/[5] Everitt C.W.F., M. Adams, W. Bencze, et al.
Gravity Probe B Data Analysis. Status andPotential for Improved Accuracy of Scientific Results . Space Science Reviews, (1–4), 53–70 (2009).[6] Heifetz M.I., W. Bencze, T. Holmes, A.S. Silbergleit, V. Solomonik.
The Gravity Probe BData Analysis Filtering Approach . Space Science Reviews, (1–4), 410–428 (2009).[7] Buchman S., J. Turneaure, E. Fei, D. Gill, J.A. Lipa.
The Effect of Patch Potentials on thePerformance of the Gravity Probe B Gyroscopes , to be submitted to JAP
Where is it??? .[8] Keiser G.M., M. Adams, W.J. Bencze, et al.
Gravity Probe B . Rivista del Nuovo Cimento (11), 555 - 589 (2009).[9] Keiser G.M., J. Kolodziejczak, A.S. Silbergleit. Misalignment and Resonance Torques andTheir Treatment in the GP-B Data Analysis . Space Science Reviews, (1–4), 383–396(2009).[10] Worden Jr., P.W.,
Almost Exactly Zero: The Equivalence Principle , Near Zero, 766-782,(1988).[11] Mester, J., et al.
The STEP mission: principles and baseline design . Class. Quant. Grav., ,2475–2486 (2001).[12] Overduin, J., C.W.F. Everitt, J. Mester, P.W. Worden. The Science Case for STEP . Adv. inSpace Res., , 1532–1537 (2009).[13] Worden Jr., P.W., J. Mester Satellite Test of the Equivalence Principle Uncertainty Analysis .Space Science Reviews, (1–4), 489–499 (2009).[14] Smythe, W.R.
Static and Dynamic Electricity , 3rd ed. Hemisphere Publ. Corp.,, New York–Washington–Philadelphia–London, 1989.[15] Will C.
Progress in Lunar Laser Ranging Tests of Relativistic Gravity . Phys. Rev. Lett. Explicit Green’s function of a boundary value problem fora sphere and trapped flux analysis in Gravity Probe B experiment . J. Appl. Phys., (1),614—24, 1999. IG. 1: Geometry of the problem and coordinate systemsFIG. 2: Azimuthal profile of the patch for various ∆ ϕ = 1 ◦ , ◦ , ◦ , ◦ , ◦ IG. 3: Equipotentials of the patch model for ∆ ϕ = π/ IG. 5: Force coefficients M n as functions of the angular separation of the patches for ∆ ϕ = π/ ϕ = π/ IG. 7: Normalized transverse force vs. the axial separation of the patchesFIG. 8: Normalized x –component of the force due to the axial shift as a function of the axialdistance between the patches ( ϕ = 0 is taken) IG. 9: Normalized z –component of the force due to the axial shift as a function of the axialdistance between the patchesFIG. 10: Cross-section of the STEP differential accelerometer–component of the force due to the axial shift as a function of the axialdistance between the patchesFIG. 10: Cross-section of the STEP differential accelerometer