Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres
EExact closed-form and asymptotic expressions for theelectrostatic force between two conducting spheres
Shubho Banerjee, Thomas Peters, Nolan Brown, Yi Song
Department of Physics, Rhodes College, Memphis, TN 38112
Abstract
We present exact closed-form expressions and complete asymptotic expansionsfor the electrostatic force between two charged conducting spheres of arbitrarysizes. Using asymptotic expansions of the force we confirm that even like-charged spheres attract each other at su ffi ciently small separation unless theirvoltages / charges are the same as they would be at contact. We show that forsu ffi ciently large size asymmetries, the repulsion between two spheres increases when they separate from contact if their voltages or their charges are held con-stant. Additionally, we show that in the constant voltage case, this like-voltagerepulsion can be further increased and maximised though an optimal lowering ofthe voltage on the larger sphere at an optimal sphere separation.
1. Introduction
The interaction of two charged conducting spheres is a fundamental prob-lem in electrostatics that has a history of more than one hundred years. LordKelvin [1], Poisson, Kircho ff , Maxwell [2], and Russell [3] are among thosewho have worked on this problem. Due to its nontrivial nature, the problem con-tinues to generate interest to this date [4, 5, 6, 7, 8, 9, 10, 11]. The simple andubiquitous geometry of spheres makes the analytical results relevant in a widevariety of applications where electrostatic forces play an important role such asthe interaction of raindrops in clouds [12], dust and powder interactions [13, 14],cellular and molecular interactions [15], atomic force microscopy [16, 17], etc.1 a r X i v : . [ phy s i c s . c l a ss - ph ] N ov igure 1: Two conducting spheres 1 and 2 are held at voltages V and V respectively. Thepermittivity of the surrounding medium is (cid:15) . The spheres acquire charges Q and Q to maintaintheir given voltages in the presence of each other. These charges vary with separation. If thebatteries are disconnected at some given charge, then the voltages of the two spheres vary withseparation. The calculations and analysis of the force becomes di ffi cult as the two spheresapproach each other due to charge polarisation on each sphere. In the classicalsolution using the method of images for example, an ever increasing number ofimages need to be included for convergence and the solutions fail completelywhen the two spheres touch each other. In the near limit asymptotic solutionsprovide an alternative series to calculate the force accurately with a short com-putation time. However, in this asymptotic limit, the force expansions are knownonly to the first order in the sphere surface-to-surface separation [5]. Due to thedi ffi culty of this analysis, only recently has it been shown by Lekner [6] thattwo like-charged spheres attract each other at su ffi ciently small separation unlessthey have the exact same charge ratio as they would at contact.In this paper we present a complete asymptotic analysis of the force in thenear limit to all orders in sphere separation. Additionally, we derive exact closed-form expressions for the force using the q -digamma function [18]. Our analysisconfirms the attraction between like-charged spheres and provides new results onhow the electrostatic force varies with size asymmetry. In particular, we find thatat large size asymmetries the behaviour of the force is radically di ff erent fromthe behaviour at small asymmetries. 2he setup of the problem is shown in Fig. 1. Two spheres 1 and 2 with radii R and R are at a distance S between their centres and are given the voltages V and V respectively. The charges Q and Q acquired by spheres are related totheir given voltages through the matrix equation Q Q ≡ C C C C V V , (1)where C and C are the self capacitances of the two spheres and C = C istheir mutual capacitance. These capacitance coe ffi cients are geometric propertiesof the interaction and depend only on R , R , and S [2, 19].In a recent paper [11] we presented exact closed-form and asymptotic so-lutions to all orders in sphere separation for these capacitance coe ffi cients. InSec. 2 we present a summary of these results along with the well known classi-cal solutions since the knowledge of the capacitance coe ffi cients is vital to theanalysis presented in this paper.The basic formalism for calculating the force between any two conductors(see Art. 93 in Ref. [2] for example), starts by writing the overall electrostaticenergy of the system as W V = (cid:104) C V + C V V + C V (cid:105) (2)when the voltages of the two spheres are held constant. The mutual electrostaticforce between the two spheres is related to the variation of this energy with dis-tance F V = dW V dS = (cid:34) dC dS V + dC dS V V + dC dS V (cid:35) . (3)In Sec. 3 we carry out the derivatives of all three forms of the capacitance co-e ffi cients discussed in Sec. 2. These derivatives are the essential componentsrequired for calculation of the force.In Sec. 4 we calculate and plot this force at constant voltage for some typical3phere size ratios. For spheres with su ffi ciently large size asymmetries, we showthat repulsion at contact initially increases with distance. Additionally, we showthat the maximum repulsion between two spheres happens not when both sphereshave the same voltage but when voltage of the larger sphere is optimally lower and the spheres are at an optimal non-zero separation. Our analysis reveals nu-merical values for the size asymmetry thresholds above which these anomalousbehaviours of the force are observed.If the spheres have a constant amount of charge instead, then inverting thecharge voltage relation gives V V = C C C C − Q Q ≡ P P P P Q Q , (4)where P i j are the coe ffi cients of potential [2]. The electrostatic energy of thesystem and the electrostatic force are given by (see Art. 93 in Ref. [2]) W Q = (cid:104) P Q + P Q Q + P Q (cid:105) , F Q = − dW Q dS . (5)By analysing the coe ffi cients P i j , Lekner showed that two charged spheres al-ways attract each other unless they have the exact same charge ratio as theywould at contact. In this paper we confirm this result by the explicit calculationof the asymptotic force to all orders in sphere separation. In addition, we showthat above a certain size asymmetry, similar to the constant voltage case, the re-pulsion between two charged spheres increases at first when they separate fromcontact even as their charges remain fixed.
2. Capacitance coe ffi cients As discussed in Sec. 1, the key to understanding the force between the twospheres lies in examining the geometrical dependence of the capacitance coe ffi -cients. In this section we present a recap of the classical solutions [2, 19] andsome recently published asymptotic and closed-form capacitance results [11]that are required for accurate force calculations.4 .1. Preliminary definitions In writing down expressions for the capacitance coe ffi cients, it is convenientto write the following two geometric definitions: s ≡ SR + R (6)is the dimensionless distance between the spheres. When the two spheres areabout to touch, s →
1. The dimensionless surface separation of the two spheresis s − R + R . In addition to s we define a second parameter thatmeasures the asymmetry in the radii of the two spheres r ≡ R − R R + R . (7)For equal sized spheres, r =
0, and when the second sphere is much smaller insize, r →
1. Interchanging the two spheres takes r → − r .The quantities s and r are a complete description of the geometry of theproblem. In terms of s and r it is convenient to define α = √ s − r − √ s − √ − r , (8)which scales the entire distance range between the two spheres to the unit inter-val. When the two spheres are about to touch α →
1, and when the spheres movevery far away α → s and α in Eq. (8) can be inverted togive s = (cid:112) (1 + α ) − r (1 − α ) √ α . (9)To write the capacitances in compact form it is useful to define λ = (1 − r )2 s (cid:32) − α α (cid:33) = − r ) sinh µ (cid:113) − r tanh µ (10)5nd 12 − x = y − = tanh − ( r tanh log √ α )log α = tanh − ( r tanh µ )2 µ , (11)where µ = − log √ α or we may write α = exp( − µ ). Similar to s − µ → u = − log α is used by Lekner to express the capacitance coe ffi cients using the Abel-Plana formula [4]. Maxwell first defined this u variable (see (cid:36) in Chapter XI ofRef. [2]) and Russell [3] uses this variable in his papers as well.The size asymmetry r , along with one distance variable α , µ , or s , completelyspecifies the geometry of the problem. For our numerical calculations we use r and α as our main quantities, and express µ , s , λ , x and y in terms of thesetwo quantities. We choose α as the main distance variable in our numericalcalculations because it is easily converted to s or µ through the simple relationsdiscussed above; this choice optimises the computation time for the force plotspresented in this paper.To gain more insight into the interdependence of these quantities it is usefulto look at their behaviour when the spheres are about to touch each other. Forsmall µ we have s − = (1 − r )2 µ + (1 + r − r )24 µ + O ( µ ) , (12) λ = − r ) µ + (1 + r − r )3 µ + O ( µ ) , (13)and 12 − x = y − = r − r (1 − r )6 µ + r (2 − r + r )30 µ + O ( µ ) . (14)For spheres of equal size, both x and y are equal to 1 / λ is the derivative of 2 s with respect to µ (see Sec. 3). It is useful to define the dimensionless capacitance coe ffi cients for the twospheres as c i j ≡ C i j / π(cid:15) ( R + R ). With the definitions above, the classical6olutions using the method of images can be written in the following infiniteLambert series [20] form c = λ ∞ (cid:88) n = α n + x − α n + x , c = λ ∞ (cid:88) n = α n + y − α n + y , (15)and c = − λ ∞ (cid:88) n = α n − α n . (16)The series above (with minor di ff erences) were first given by Kirchho ff [2] anddiscussed in Ref. [11] in their current form. We include terms through n =
10 andlimit the range to α ≤ .
15 with errors of order 10 − or less for the calculationspresented in this paper. Note that these series are defined for all α (cid:44) q -digamma function [18,20] the capacitance coe ffi cients can be written in closed-form as [11] c = λ µ (cid:104) ψ α ( x ) − ψ α ( x ) + log + α − α (cid:105) , (17) c = λ µ (cid:104) ψ α ( y ) − ψ α ( y ) + log + α − α (cid:105) , (18)and c = λ µ (cid:104) ψ α ( ) + log(1 − α ) (cid:105) , (19)where ψ α ( x ) is the q -digamma function of x with q = α . These solutions arevalid for 0 < α <
1. They are not defined at α = α = ffi cients by utilising the existing special function library ofnumerical software such as Mathematica [21]. Built-in convergence tests in thesoftware can optimise the number of terms required for accuracy in calculatingthe q -digamma function. We use the closed-form solutions for calculating rel-ative errors when deciding on the number of terms and the range of use of theclassical solutions above and the asymptotic solutions discussed below.7oth classical and closed-form solutions show a slow down in convergencewhen the spheres are near each other, i.e. µ <<
1. In the near regime ( µ (cid:46)
1) thefollowing asymptotic solutions provide fast and accurate computation [11]: c ≈ λ µ (cid:34) log 1 µ − ψ ( x ) − K (cid:88) k = k − B k ( x ) B k (cid:16) (cid:17) (2 k )! k µ k − π sin(2 π x ) e − π µ (cid:35) , (20) c ≈ λ µ (cid:34) log 1 µ − ψ ( y ) − K (cid:88) k = k − B k ( y ) B k (cid:16) (cid:17) (2 k )! k µ k − π sin(2 π y ) e − π µ (cid:35) , (21)and c ≈ − λ µ (cid:34) log 1 µ + γ − K (cid:88) k = k − B k B k (cid:16) (cid:17) (2 k )! k µ k (cid:35) , (22)where ψ = ψ α = is the digamma function, γ = . ... is the Euler’s constant,and B k , B k ( x ) are the 2 k th Bernoulli number and Bernoulli polynomial [22].Using the cut-o ff K (cid:39) π / µ leads to optimally low errors in capacitances atany given µ [11]. Using K = ff for µ ≤ .
35 (or α ≥ . − or less. Note that using too large a K value makes theerrors worse. For µ = .
35, the optimal K is π / . (cid:39)
14. The exp ( − π /µ ) termin c , c matters only if the optimum K is used. In this paper, with K = ff , we omit that exp ( − π /µ ) term as it is smaller than the error.
3. Capacitance derivatives
The electrostatic force in Eq. (3) is a linear combination of the derivatives ofthe capacitance coe ffi cients with respect to the sphere separation. In this sectionwe calculate these derivatives of the capacitance coe ffi cients discussed in Sec. 2. It is convenient to first calculate the derivatives of the main quantities usedin expressing the capacitance coe ffi cients. These derivatives allow us to expressthe derivatives of the capacitance coe ffi cients in a more compact manner.8he variables α and µ are mathematical measures of the distance between thespheres. Their derivatives with respect to s are d α ds = − αλ and d µ ds = λ . (23)The capacitance coe ffi cients in the previous section are expressed in terms of µ and α . The derivatives above allow us to relate the µ and α derivatives of thecapacitance to the derivatives with respect to s . The derivative of λ with respectto s is d λ ds = (cid:32) + α − α (cid:33) − λ s = µ − λ s . (24)The variation of x , y with respect to α and µ are x (cid:48) ( α ) = − (1 − x )4 αµ + r cosh[(1 − x ) µ ] (1 + α ) µ , x (cid:48) ( µ ) = − α x (cid:48) ( α ) , (25) y (cid:48) ( α ) = − x (cid:48) ( α ) , y (cid:48) ( µ ) = − x (cid:48) ( µ ) . (26)These derivatives of x and y go to zero as α → µ → Di ff erentiating the classical result for capacitance c in Eq. (15) with respectto s gives dc ds = c λ d λ ds − ∞ (cid:88) n = (cid:16) α n + x + α n + x (cid:17) (cid:2) n + x + α x (cid:48) ( α ) log α (cid:3)(cid:0) α n + x − (cid:1) , (27)and for capacitance c in Eq. (16) we get dc ds = c λ d λ ds + ∞ (cid:88) n = (cid:16) α n + α n (cid:17) n (cid:0) α n − (cid:1) . (28)The derivative of c can be obtained by replacing x with y and c with c in Eq. (27). These derivatives of the classical results converge well when the9pheres are relatively far away. Only terms through n =
10 are needed for anumerical accuracy of order 10 − for α ≤ .
15 (or µ ≥ . Di ff erentiating the closed-form expression for c in Eq. (17) with respect to s we have dc ds = − αµ (cid:34) α ψ (1 , α ( x ) − ψ (1 , α ( x ) + − α (cid:35) − c (cid:32) λµ + s − µλ (cid:33) − α x (cid:48) ( α ) µ (cid:104) ψ (0 , α ( x ) − ψ (0 , α ( x ) (cid:105) , (29)where the superscript (0 ,
1) indicates the derivative with respect to x and (1 , α or α depending on the case. Similarly, dc ds = − α µ (cid:34) ψ (1 , α (cid:16) (cid:17) − − α (cid:35) − c (cid:32) λµ + s − µλ (cid:33) . (30)The derivative of c can be obtained by replacing x with y and c with c inEq. (29). These derivatives are defined at all points except α = α = α derivative of ψ α ( x ) is numerically unstable in Mathemat-ica [21] for some parts of the range 0 < α <
1. To avoid inaccuracies in calcu-lating the capacitance derivatives above, we use the inversion symmetry [18] ψ α ( x ) = (cid:16) x − (cid:17) log α + ψ /α ( x ) (31)as an alternate definition for ψ α ( x ) within the software.In principle, these closed-form expressions for the capacitance derivativescan be used to calculate the electrostatic force between the spheres for all 0 <α <
1. However, derivative calculations of the q -digamma function encounternumerical errors near α = α = . ≤ α ≤ .
5, which corresponds to 0 . ≥ µ ≥ . .4. Asymptotic capacitance derivatives Di ff erentiating the asymptotic expression for c in Eq. (20) gives dc ds ≈ − µ + c λ (cid:32) d λ ds − µ (cid:33) − x (cid:48) ( µ )2 µ (cid:34) ψ (cid:48) ( x ) + K (cid:88) k = k B k − ( x ) B k (cid:16) (cid:17) (2 k )! µ k (cid:35) − K (cid:88) k = k − B k ( x ) B k (cid:16) (cid:17) (2 k )! µ k − − π e − π µ µ (cid:104) π sin(2 π x ) + µ cos(2 π x ) x (cid:48) ( µ ) (cid:105) , (32)and di ff erentiating asymptotic c in Eq. (22) we get dc ds ≈ µ + c λ (cid:32) d λ ds − µ (cid:33) + K (cid:88) k = k − B k B k (cid:16) (cid:17) (2 k )! µ k − . (33)The asymptotic derivative of c are obtained by replacing x with y and c with c in Eq. (32).The asymptotic derivatives are be useful for fast and accurate calculationsin the near range ( µ (cid:46)
1) provided the cut-o ff K is carefully chosen [11]. Forpractical considerations we choose K = − for µ ≤ .
35 (or α ≥ . − π /µ ) term Eq. (32) isomitted in this approximation as discussed in Sec. 2.
4. Energy and force at constant voltage
In this section we analyse the electrostatic force when both spheres are heldat constant voltages. The force in this case is a linear combination of the capac-itance derivatives discussed in Sec. 3. We combine the classical, closed-form,and asymptotic expressions to create an accurate description of the force at allseparations between the spheres.
We define a dimensionless form of the electrostatic energy as w V ≡ W V π(cid:15) ( R + R ) V = c + c v + c v , (34)11here v = V / V is the voltage of the second sphere relative to the first sphere.We assume that V is the larger of the two voltages so that − ≤ v ≤
1. Sinceone of the two spheres has to be at a non-zero voltage for the problem to bemeaningful, normalising the energy using the larger voltage ensures that there isno division by zero.We now define a dimensionless version of the force in Eq. (3) as f V ≡ F V π(cid:15) V = dc ds + v dc ds + v dc ds , (35)for which the capacitance derivatives are calculated in Sec. 3. As defined, thedimensionless force at constant voltage satisfies the following symmetry underthe interchanging of the spheres and their voltages f V ( r , v ) v = (cid:34) f V ( r , / v )1 / v (cid:35) r = − r . (36)All possible voltage and size scenarios can be covered by analysing the full rangeof the size asymmetry ratio − < r <
1. The equations are valid for | v | > µ in the asymptotic limit so that the coe ffi cients only dependupon the size asymmetry and the voltage ratio. Using the asymptotic derivativesin Sec. 3 we get f V = − (1 − v ) µ + (1 − v ) (1 + r )6 log 1 µ − (1 − r )(1 + v ) + v − (1 + r )3 φ v ( y ) + ( r − r )3 φ (cid:48) v ( y ) − (1 − v ) + r − r µ log 1 µ − (cid:20) (33 + r − r )(1 + v ) + + r ) v − + r − r φ v ( y ) + (19 r − r + r )90 φ (cid:48) v ( y ) + ( r − r ) φ (cid:48)(cid:48) v ( y ) (cid:21) µ + O (cid:16) µ log µ (cid:17) , (37)where φ v ( x ) ≡ [ v ψ ( x ) + ψ (1 − x ) + v γ ] / r → − r , andincreases with size asymmetry. The O (1) term changes under r → − r due to thepresence of φ (cid:48) v which involves the trigamma function.For the numerical calculations presented in this paper (see Fig. 2 for example)we use the closed-form expressions for the middle region 0 . ≤ µ ≤ .
95 whichcorresponds to 0 . ≥ α ≥ .
15. In the near region µ ≤ .
35 (or α ≥ .
5) weuse the asymptotic capacitance and derivatives with a cut-o ff of K =
5, and inthe far region µ ≥ .
95 (or α ≤ .
15) we use the classical capacitances and theirderivatives with terms through n = q -digamma function is unavailable then the asymptotic solutions (with K =
5) can be used for µ ≤ . α ≥ .
3) and the classical solutions (with n = − or less. The regions of strong overlap inthe three forms of solutions lets us accurately calculate the force between thespheres at any distance with a high degree of confidence. V when R > R We first analyse the case where the first sphere, with the higher voltage, islarger in size as well. In Fig. 2 we plot the force between the two spheres for r = which corresponds to the case where R = R . As expected, the force isattractive at su ffi ciently small sphere separation when the two spheres are not atthe exact same voltage. The force is repulsive at all distances only when the twospheres have the exact same voltage, i.e., v =
1. This attraction at su ffi cientlysmall separation when v (cid:44) igure 2: The dimensionless force at constant voltage, f V , is plotted versus the relative surfaceseparation, s −
1, of the two spheres for r = (or when R = R ). The force is attractive(negative) at su ffi ciently small distances for all voltage ratios v (cid:44)
1. The force is always repulsive(positive) when the two spheres have the exact same voltage, v =
1. This repulsion at equalvoltages decreases monotonically with increasing sphere separation. second sphere is at a lower voltage. At the end of this field line there must benegative charge even though the second sphere has a positive voltage. The dis-tance between this negative charge and the positive charge at the beginning ofthe field line goes to zero as the two spheres are about to touch. This attractiveforce eventually overcomes any repulsion between the spheres. There is no suchfield line only when the two spheres are at the exact same voltage. In this casethe force is finite even at zero separation because the like charges move awayfrom each other.In Fig. 3 we plot the force between the two spheres for r = which corre-sponds to the highly asymmetric case where R = R . Even in this case theforce is attractive at su ffi ciently small sphere separation when v (cid:44) v =
1. This repulsive force, however, shows non-monotonic behaviour versussphere separation. The repulsion initially increases as the spheres move awayfrom each other at contact and reaches a maximum of 1 .
65 times the value at14 igure 3: The dimensionless force at constant voltage, f V , is plotted versus the relative spheresurface separation, s −
1, for r = (when R = R ). Similar to the r = case, the force isattractive at su ffi ciently small distances for all voltage ratios v (cid:44) v =
1. However, unlike the r = case, here the repulsion at v = increases with separationfor small surface separations. contact at s − = .
353 before eventually decreasing.We may qualitatively understand this increase in the repulsion as the spheresmove away from contact by noting that the smaller sphere gains 2 .
45 times itscharge at contact when separation increases to s − = . .
9% of its original value. The product of thetwo charges therefore increases by a factor of 2 .
42. However, since the separationbetween the charges increases as well, the force increases only by a factor of1 .
65. Note that the first two leading order terms in Eq. (37) go to zero at v = V when R < R We now examine the case where sphere 1 with the higher voltage has asmaller radius than sphere 2. In Fig. 4 we plot the force between the two spheresfor r = − which corresponds to the case when R = R . The force is attrac-tive at su ffi ciently small sphere separation when the two spheres are not at theexact same voltage. The force is repulsive at all distances only when the two15 igure 4: The dimensionless force, f V , is plotted versus the relative surface separation, s −
1, for r = − (solid lines), i.e., when R = R . The behaviour is similar to the r = case (dottedlines), except that the polarisation e ff ect is weaker for the solid lines. This e ff ect can be seenby comparing the switch to attraction which happens at comparatively smaller separation for thesame voltage ratio. spheres have the exact same voltage, i.e., v =
1. The polarisation e ff ect, whichcauses the attraction at small separation, is weaker in this case than for r = .This makes sense if we note that the smaller sphere at lower voltage loses chargerapidly with decreasing distance to maintain its given voltage in the presence ofthe larger sphere. In comparison, the charge on the lower voltage larger spheredoes not change much in the vicinity of the smaller sphere at a higher voltage.In Fig. 5 we plot the force between the spheres for r = − which correspondsto case where R = R . The force shows similar behaviour to the r = case inFig. 3. The v = r = case, at certain distances, it is possible to increase the repulsive forcebetween the two spheres by decreasing the voltage of the larger sphere! Thevalue v = . ... is calculated numerically to maximise the relative verticaljump from the v = igure 5: The dimensionless force, f V , is plotted versus the relative surface separation, s − r = − (i.e., R = R ). The force plots are similar to the r = case, with one notableexception: for some separations, decreasing the voltage on the larger sphere can increase therepulsion. The “optimal” v = . ... maximises the relative vertical “jump” from the v = show that decreasing the voltage to 86 .
7% on the larger sphere at s − = . ff ective distance between thetwo charge distributions increases. Additionally, there is now some negativecharge on the larger sphere as well which causes some attraction. Thus, there isan overall increase of 18% in the repulsive force.In Table 1 we list such force anomalies for several di ff erent sizes of the twospheres. For every r value we numerically calculate the optimum lowering of thevoltage from v = s −
1, that causes the maximumrelative increase in the force. Note that when the two spheres are not at equalvoltage, there are some field lines from sphere 1 that end up on sphere 2 andcause some attraction between charges of opposite polarity. Even then, the lowervoltage case has overall more repulsion than the equal voltage case.17 R : R optimal v s − f V ( v = v opt ) f V ( v = f V ( v opt ) / f V (1)– / / / / / / / / Table 1: The data shows that the repulsive force is greater when the larger sphere is at an optimal lower voltage f V ( v = v opt ) than when both spheres are at the same voltage, f V ( v = To calculate the minimum size asymmetry where lowering the voltage canincrease the force, let the voltage ratio be v = − ε , where ε <<
1. Substitutingthis voltage ratio in the expression for force we get f V = dc ds + − ε ) dc ds + (1 − ε ) dc ds = f V ( v = − ε (cid:32) dc ds + dc ds (cid:33) + O ( ε ) . (38)So the criterion for the force to increase when the voltage is reduced is that thecoe ffi cient of ε has to be positive at some separation. The conditions for thecritical case are given by dds ( c + c ) = d ds ( c + c ) = . (39)Solving for the two conditions numerically gives r = − . ... and s − → R (cid:39) R are about to toucheach other. Note that the data in Table 1 alludes to this critical point as well.18nalysing the capacitance coe ffi cients near µ = c + c − r = − (cid:2) γ + ψ ( y ) (cid:3) + µ × (40) (cid:104) − + r − (2 + r )[ γ + ψ ( y )] + r (1 − r ) ψ (cid:48) ( y ) (cid:105) + O ( µ ) , where y = (1 + r ) / y as µ →
0. Plotting the coe ffi cient of µ shows that it is negative for r < − . ... which corresponds to the size ratio R (cid:39) R . That is, the repulsion can be higher at lower voltage if the first sphereis smaller than about half the size of the second sphere. Note that µ ∼ ( s −
1) inthis limit.
5. Energy and force at constant charge
In this section we analyse the electrostatic force when the charges on thespheres are held constant. In addition to the standard definition discussed inSec. 1, we develop another alternative formulation for the force which allowsus to use our results from the constant voltage case. We combine the classical,closed-form, and asymptotic expressions to create an accurate description of theforce at all separations between the spheres.
We define a dimensionless form of the electrostatic energy as w Q ≡ π(cid:15) ( R + R ) W Q q Q = q (cid:16) p + p q + p q (cid:17) , (41)where p i j ≡ π(cid:15) ( R + R ) P i j are dimensionless, q = Q / Q is the charge ratiobetween the spheres, and q = γ + ψ ( y ) γ + ψ ( x ) = φ ( y ) − π cot π y φ ( y ) + π cot π y (42)is the charge ratio at contact with φ = φ v = . In the ratio q above, x = (1 − r ) / x as µ →
0. Note that y = − x .19his normalisation of W Q ensures that the dimensionless energy and force atcontact are the same regardless of which sphere is designated as 1. Sphere 1 ischosen such that q | Q | ≥ | Q | to ensure that | V | ≥ | V | near contact. Addition-ally, sphere 1 is chosen to be of positive polarity without loss of generality sincethe energy and the force remain the same if we switch polarities of both spheres.We can now calculate the dimensionless force at constant charge as f Q ≡ π(cid:15) ( R + R ) q Q F Q = − π(cid:15) ( R + R ) q Q dW Q dS = − dw Q ds = − q (cid:32) d p ds + q d p ds + q d p ds (cid:33) . (43)As defined, the dimensionless force at constant charge satisfies the symmetry f Q ( r , q ) = (cid:2) f Q ( r , q ) (cid:3) r = − r (44)at q = q and more generally for any q , q f Q ( r , q ) q = (cid:34) q f Q ( r , / q )1 / q (cid:35) r = − r . (45)All possible charge and size scenarios are covered by analysing the parameterspace − q ≤ q ≤ q along with − < r <
1. Although we restrict ourselves to | q | ≤ q , the force expressions hold for | q | > q as well. However, those cases donot provide any additional information than that already discussed in the paper.Any | q | > q is equivalent to a case with ratio q → / q (and thus | q | < q ) and r → − r in our parameter space as shown by the symmetry above. The derivatives of the coe ffi cients of potential p i j are nontrivial to calculateand simplify. Here, we rewrite f Q in terms of the force at constant voltage dis-cussed earlier in Sec. 4.1 in the manner below.Imagine disconnecting the batteries at a certain distance s . At this point thespheres now are at fixed charge. But the force before and after disconnecting the20atteries should not change at the same s value since the charge on the spherestays the same. Setting F V = F Q and replacing V in terms of its correspondingcharge values gives f V = F V π(cid:15) V = F Q π(cid:15) ( P Q + P Q ) = π(cid:15) ( R + R ) F Q Q ( p + p q ) . (46)Therefore 4 π(cid:15) ( R + R ) F Q q Q = f Q = ( p + p q ) q f V . (47)Note that f V is in terms of v which should be replaced by its equivalent in termsof q , v = V V = p q + p p + p q = c q − c c − c q , (48)since the charges are now fixed whereas the voltage ratio v varies with distance.After expressing all the p i j in terms of c i j and comparing the coe ffi cients of thederivatives of c i j , it is easily verified that the f Q expressions in Eqs. (43) and (47)are equal to each other.To understand the plots presented below it is useful to expand this force atconstant charge in the limit µ → f Q = − (cid:16) − qq (cid:17) (1 − ξ )(1 − r ) µ (cid:104) µ − φ − ( y ) + ξ φ ( y ) (cid:105) (49) + (cid:104)(cid:16) + qq (cid:17) + ξ (cid:16) − qq (cid:17)(cid:105) (cid:104) (1 − r )(2 r φ (cid:48) ( y ) − − (2 + r ) φ ( y ) (cid:105) (cid:0) − ξ (cid:1) (cid:0) − r (cid:1) φ ( y ) + O (cid:32) µ (cid:33) , with ξ = π cot( π y ) / φ ( y ) where φ ± = φ v at v = ±
1. Note that the leadingterm in Eq. (49) is always negative. Thus for q (cid:44) q this leading term alwaysdominates the higher order terms and causes an overall attractive (negative) forcefor su ffi ciently small µ , which agrees with Lekner’s result [5]. The second term,dominant when q = q , is always repulsive and approaches a finite limit as µ →
0. All higher order terms go to zero in this limit.21 igure 6: The dimensionless force, f Q , is plotted versus the relative surface separation, s −
1, for r = − (solid lines), i.e., R = R . Also plotted is the r = case (dotted lines) for comparison.The force is attractive at su ffi ciently small distances for all charge ratios when q < q and alwaysrepulsive only when the two spheres have the charge ratio at contact, q = q . The polarisatione ff ect that causes attraction at small separation is stronger when the smaller sphere is the more“positive” of the two. Q We now analyse the electrostatic force between the two spheres at constantcharge using the expressions developed above. The sphere designated as 1 is themore “positive” of the two and does not develop any negative charge density evenupon polarisation. The second sphere develops some negative charge densitybefore contact (because of its lower voltage) even if it has an overall positivecharge. To cover all possible charge and size scenarios, we examine both r > R > R ) and r < R < R ) cases since even the smaller sphere can be themore positive of the two.In Fig. 6 we plot the force when one sphere is twice the size of the other,i.e., r = ± . In both cases the spheres attract each other at su ffi ciently smallseparations unless their charge ratio is exactly the same as the charge ratio atcontact, i.e., q = q . This attraction due to charge polarisation is stronger for r = − (solid lines), where the smaller sphere is the more positive of the two.22 igure 7: The dimensionless force, f Q , is plotted versus the relative surface separation, s − r = (dotted lines) and r = − (solid lines). The force is attractive at su ffi ciently smalldistances for all charge ratios q < q and repulsive at all distances only when the two sphereshave q = q , the charge ratio at contact. This plot is similar to that in Fig. 6 with one exception:the repulsion at q = q increases as the two spheres separate away from contact before eventuallydecreasing. The behaviour of the force versus separation in Fig. 6 is qualitatively similar tothat of equal-sized spheres with fixed charges [9].The “perfect” charge ratio, q , ensures that both spheres are at the same volt-age right before contact and there are no electric field lines between them. Thus,there is no negative charge density on either sphere which can cause any attrac-tion. When one of the spheres has less charge than this ideal ratio, there is at leastone field line from the sphere with the greater (than perfect) charge to the spherewith less (than perfect) charge along the line joining the two. The opposite endsof this field line have opposite charge density. When regions of opposite chargedensity get su ffi ciently close before contact, their attraction overcomes the repul-sion between the like charges to cause an overall attraction between the spheres.In Fig. 7 we plot the force when one sphere is ten times the other, i.e., r = ± . In both cases the spheres attract each other at su ffi ciently small separationsunless their charge ratio is exactly the same as the charge ratio at contact, q = q .23he features of this plot are similar to those for the r = ± case in Fig. 6 withone exception: similar to v = q = q is not maximum when the two spheres are in contact and increases when the two spheres separate.To understand how two spheres can repel more at a finite separation thanwhen at contact, we hypothesise that the charge on the larger sphere moves backtowards the point of contact when the two spheres separate. This charge rear-rangement must be the main mechanism that causes a stronger horizontal pushalong the axis of symmetry since neither sphere gains any charge.Comparing the force plots for constant charge with the force plots at constantvoltage in Sec. 4, we see that polarisation e ff ects in constant charge and constantvoltage are in the opposite directions; that is, the dashed curves lie on oppositesides of the solid curves where the force switches to attraction. This contrastingbehaviour is due to the di ff erence in main mechanism of charge polarisation inthe two cases.In the constant voltage case the main mechanism that drives the force char-acteristics is the gaining / losing of charge by the smaller sphere. This mecha-nism is stronger when the larger sphere has more voltage. In comparison, themain mechanism of polarisation in the constant charge case is the redistributionof charge on the larger sphere. This charge redistribution is stronger when thesmaller sphere has a greater than its fair share of the charge and is able to causemore charge separation on the larger sphere.Another contrast between the constant charge and the constant voltage casesis the value of the separation at which the repulsion switches to attraction. Thisseparation is much smaller in the charge case for similar numerical value of thevoltage and charge ratios. By expanding the variable voltage ratio, v , for a fixedcharge ratio q we see that v = − q ψ (cid:16) − r (cid:17) − γ (1 − q ) − ψ (cid:16) + r (cid:17) (1 + q ) log µ + γ q − ψ (cid:16) + r (cid:17) + O (cid:32) µ log µ (cid:33) . (50)24ote that the voltage ratio v goes to 1 as the separation µ → q . Hence, much smaller separations are needed to overcome the v =
6. Repulsion increase with separation
The anomalous behaviour of the electrostatic force at contact, where the re-pulsion increases as the spheres separate, happens only for su ffi ciently large sizeasymmetries. Similar behaviour is observed in both constant voltage and con-stant charge cases although the force increase is much larger in the constantvoltage case. In this section we analyse this repulsion increase with separationbetween the two spheres as a function of their size asymmetry.For the constant voltage case, setting v = f V for small µ gives f V ( v = = − (cid:32) + r (cid:33) φ ( y ) + (cid:32) − r (cid:33) (cid:104) r φ (cid:48) ( y ) − (cid:105) + µ × (cid:20) − (cid:16) + r − r (cid:17) + (cid:16) + r − r (cid:17) φ ( y ) (51) − (cid:16) r − r + r (cid:17) φ (cid:48) ( y ) − (cid:16) r − r (cid:17) φ (cid:48)(cid:48) ( y ) (cid:21) + O (cid:16) µ (cid:17) . Setting the coe ffi cient of µ to zero and numerically solving for the critical asym-metry ratio yields r c = . ... ( R (cid:39) . R ). For | r | > r c , the coe ffi cient of µ ispositive and the force increases with separation starting at contact! As discussedearlier in Sec. 4.2, this increase in force is mainly due to the increase in chargeon the smaller sphere.In Fig. 8 we highlight this non-monotonic behaviour of the repulsive forcewith sphere separation as a function of the size asymmetry. To compare all thecases we normalise the force by its value at contact. For size ratios greater thanabout 5:2, the repulsion increases at first as the spheres move away from eachother from contact. We observe that the peak of the normalised force increasesmonotonically with r . 25 igure 8: The force at constant voltage, f V , normalised by the force at contact, f V , is plottedagainst the surface separation, s −
1, for the voltage ratio v =
1. Above a critical size asymmetry, r c = . ... ( R (cid:39) . R ), the force increases as the two spheres separate from contact. For r = . s − = .
41 (see Table. 2).
By using f V ( v =
1) in Eq. (51) and setting q = q the force at constant chargefor small µ yields f Q ( q = q ) = f V ( v = − r ) [ φ ( y ) − ( π /
4) cot ( π y )] × (52) (cid:34) − µ − r + (2 + r ) φ ( y ) − r − r ) φ (cid:48) ( y )6 φ ( y ) (cid:35) + O (cid:32) µ log µ (cid:33) . Isolating the coe ffi cient of µ above and setting it to zero gives r c = . ... (i.e., R (cid:39) R ). The force at contact, f Q , in its dimensionless form is the same as theKelvin factor calculated in Ref. [6].In Fig. 9 we plot f Q normalised by its value at contact for several di ff erentvalues of r . Compared to the constant voltage case in Fig. 8, the normalisedforce shows a much smaller increase and the peak happens at much smaller sep-arations. Since neither sphere gets any additional charge, the only mechanismby which the force can increase is the redistribution of the charge which createsa greater horizontal component of the force as separation increases.26 igure 9: The force f Q normalised by the force at contact, f Q , is plotted against separation s − q . Above a critical size asymmetry, r c = . ... ( R (cid:39) R ), the force increases as the two spheres move away from contact. For r = . r = .
99) are needed for a significant (14%) force increase.
In Table 2 we list values of the maximum normalised repulsion between thespheres and the corresponding separations at which this maximum happens. Thisdata can be tested against computer simulations or experimental measurements.The constant voltage case appears to be a more likely candidate for experimentalverification since the normalised force values are much larger than in the constantcharge case. Also, it is easier to maintain two spheres at constant voltage thanconstant charge due to the variety of mechanisms through which the charge candissipate from objects that are not perfectly isolated.
7. Summary
The main new results presented in this paper are the asymptotic expansionsand closed-form expressions for the electrostatic force between two conductingspheres. We supplement these results with the well known classical method-of-images series in the far region for best convergence and computational speed atall distances while maintaining high accuracy.27 R : R Max f V / f V s − f Q / f Q s − / .
014 0.100 – – / .
077 0.194 – – / .
159 0.244 – – / .
250 0.279 1.003 0.00890 / .
346 0.304 1.006 0.0127 / .
445 0.324 1.009 0.0157 / .
546 0.340 1.012 0.0180 /
10 : 1 1 .
648 0.353 1.015 0.0198 /
13 : 1 1 .
961 0.382 1.025 0.0233 /
16 : 1 2 .
278 0.401 1.034 0.0249 /
19 : 1 2 .
597 0.415 1.042 0.0257
Table 2: The data shows that the repulsion between the two spheres increases as they moveaway from contact. The value of the force is listed as a ratio to the value of the force at contact, f V / f V for constant voltage, and f Q / f Q for constant charge. Also listed is the separation s − f V and f Q are given by the µ → Experiment 1.
Let one sphere be 19 times larger than the other (see r = /
10 curve in Fig. 8). Hold both spheres at equal voltages and measure theelectrostatic force at contact. Now move the spheres to 1.42 times this distanceof closest approach while keeping their voltages the same. The measured forceshould now be 2.60 times larger.
Experiment 2.
Let one sphere be 10 times larger than the other (see Fig. 5).Hold both spheres at equal voltages and measure the electrostatic force at acentre-to-centre distance of 1 . R + R ). Now decrease the voltage of thelarger sphere to 86 .
7% of the original value while keeping the separation fixed.The measured force should now be 18 .
4% larger.
Experiment 3.
Let one sphere be 19 times larger than the other (see r = / ff ectivedistance between the spheres even as their centre-to-centre distance increases.The three experiments listed above highlight the anomalous behaviour ofthe electrostatic force for high sphere size asymmetries. Additional details ofsuch behaviour are provided in Tables 1 and 2. For convenience we work withdimensionless form of the electrostatic force. If needed, the dimensionless forms29an be converted back to SI units through the relations F V = π(cid:15) V f V and F Q = q Q π(cid:15) ( R + R ) f Q . (53)In the paper we designate sphere 1 to be stronger than sphere 2 in order to put abound on the voltage and charge ratios and avoid repetition of cases. However,there is no such requirement; all relations hold as long as V and Q are nonzero. Acknowledgments
We thank the Mac Armour Fellowship for support.
References [1] William Thomson. On the mutual attraction or repulsion between two electrified spheri-cal conductors.
Reprint of Papers on Electrostatics and Magnetism , pages 86–97, 1872.doi:10.1017 / cbo9780511997259.007.[2] James Clerk Maxwell. IX and XI. In A Treatise on Electricity and Magnetism , volume 1.Dover Publishing, 3 edition, 1954.[3] A Russell. The electrostatic problem of two conducting spheres.
J. Inst. Electr. Eng. , 65:517–535(18), May 1927. ISSN 0099-2887. doi:10.1049 / jiee-1.1927.0053.[4] John Lekner. Capacitance coe ffi cients of two spheres. J. Electrostat. , 69:11–14, February2011. doi:10.1016 / j.elstat.2010.10.002.[5] John Lekner. Electrostatic force between two conducting spheres at constant po-tential di ff erence. J. Appl. Phys. , 111(7):076102, April 2012. ISSN 0021-8979.doi:10.1063 / Proc. R. Soc. A , 468(2145):2829–2848, 2012. doi:10.1098 / rspa.2012.0133.[7] Kiril Kolikov, Dragia Ivanov, Georgi Krastev, Yordan Epitropov, and Stefan Bozhkov. Elec-trostatic interaction between two conducting spheres. J. Electrostat. , 70(1):91–96, February2012. ISSN 0304-3886. doi:10.1016 / j.elstat.2011.10.008.[8] Matthias Meyer. Numerical and analytical verifications of the electrostatic attraction be-tween two like-charged conducting spheres. J. Electrostat. , 77:153–156, October 2015.ISSN 0304-3886. doi:10.1016 / j.elstat.2015.08.007.[9] Shubho Banerjee, Mason Levy, Mckenna Davis, and Blake Wilkerson. Exact and ap-proximate capacitance and force expressions for the electrostatic interaction between two qual-sized charged conducting spheres. IEEE Trans. Ind. Appl. , PP:1–1, February 2017.ISSN 0093-9994. doi:10.1109 / TIA.2017.2672744.[10] John Lekner. Regions of attraction between like-charged conducting spheres.
Am. J. Phys. ,84(6):474–477, 2016. ISSN 0002-9505. doi:10.1119 / ffi cients for the electrostatics of two spheres. J. Electrostat. , 101:103369, September 2019. ISSN 0304-3886. doi:10.1016 / j.elstat.2019.103369.[12] R Harrison, Keri A Nicoll, Maarten Ambaum, Graeme Marlton, Karen L Aplin, andMichael Lockwood. Precipitation modification by ionization. Phys. Rev. Lett. , 124, May2020. ISSN 0031-9007. doi:10.1103 / PhysRevLett.124.198701.[13] James Q Feng and Dan A Hays. Relative importance of electrostatic forces on pow-der particles.
Powder Technol. , 135-136:65–75, October 2003. ISSN 0032-5910.doi:10.1016 / j.powtec.2003.08.005.[14] Zachary C Cordero, Harry M Meyer, Peeyush Nandwana, and Ryan R Deho ff . Powderbed charging during electron-beam additive manufacturing. Acta Mater. , 124:437–445,February 2017. ISSN 1359-6454. doi:10.1016 / j.actamat.2016.11.012.[15] Arpita Varadwaj, Pradeep Varadwaj, and Koichi Yamashita. Do surfaces of positive elec-trostatic potential on di ff erent halogen derivatives in molecules attract? like attracting like! J. Comput. Chem. , 39, December 2017. ISSN 0192-8651. doi:10.1002 / jcc.25125.[16] S Hudlet, M Saint Jean, C Guthmann, and J Berger. Evaluation of the capacitive forcebetween an atomic force microscopy tip and a metallic surface. Eur. Phys. J. B , 2(1):5–10,March 1998. ISSN 1434-6036. doi:10.1007 / s100510050219.[17] Bruce Law and Francois Rieutord. Electrostatic forces in atomic force mi-croscopy. Phys. Rev. B Condens. Matter , 66, June 2002. ISSN 0163-1829.doi:10.1103 / PhysRevB.66.035402.[18] C Krattenthaler and H M Srivastava. Summations for basic hypergeometric series involvinga q -analogue of the digamma function. Comput. Math. Appl. , 32(3):73–91, 1996. ISSN0898-1221. doi:10.1016 / Static and Dynamic Electricity . McGraw-Hill, 3 edition, 1968.[20] Shubho Banerjee and Blake Wilkerson. Asymptotic expansions of Lambert series andrelated q -series. Int. J. Number Theory , 13(08):2097–2113, 2017. ISSN 1793-0421.doi:10.1142 / S1793042117501135.[21] Wolfram Research, Inc. Mathematica, Version 12.0. Champaign, IL, 2019.[22] Milton Abramowitz and Irene A Stegun. Chapter 23. In
Handbook of Mathematical Func-tions with Formulas, Graphs, and Mathematical Tables . Dover, 1972.. Dover, 1972.