Capacitance coefficients of two spheres

Journal of Electrostatics 69 (2011) 11e14

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Capacitance coefficients of two spheres John Lekner The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Chemical and Physical Sciences, P O Box 600, Wellington, New Zealand

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 December 2009 Received in revised form 30 July 2010 Accepted 22 October 2010 Available online 10 November 2010

Compact and exact expressions are obtained for the capacitance coefficients Caa, Cbb and Cab of two conducting spheres of radii a and b, for any distance c between the sphere centres. The results are equivalent to those of Maxwell, Russell and Jeffery, but enable rapid calculation of the coefficients in the limit of close approach, which is otherwise computationally difficult. Erroneous results published by several authors are corrected. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Maxwell [1] defined and calculated the capacitance coefficients for two or more conductors. For spheres of radii a and b, at potentials Va and Vb in otherwise empty space, the charges on them are, by definition of the coefficients Cij,

Qa ¼ Caa Va þ Cab Vb ;

Qb ¼ Cab Va þ Cbb Vb

CðV; VÞ/a þ b;

(2)

is the capacitance of the two-sphere system when they are held at the same potential. When the spheres carry equal and opposite charges þQ and Q, the potentials Va and Vb given by (1) are

Va ¼

Q ðCab þ Cbb Þ ; 2 Caa Cbb  Cab

Vb ¼

 Q ðCab þ Caa Þ 2 Caa Cbb  Cab

(3)

In this case the capacitance of the two-sphere system is

CðQ ; Q Þ ¼

2 Caa Cbb  Cab Q ¼ Caa þ 2Cab þ Cbb Va  Vb

(4)

0304-3886/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2010.10.002

ab aþb

(5)

(The capacitance of an isolated sphere in vacuum is equal to its radius, in Gaussian units.) For two equal spheres, widely separated, C(V,V) is four times C(Q,Q). The electric field lines emerging from the spheres at the same potential go off to (the sphere at) infinity, whereas for the oppositely charged spheres the field lines emerge from one and terminate on the other, as illustrated in Fig. 1.

2. Closed-form expressions for the capacitance coefficients The capacitance coefficients Caa, Cab and Cbb, for the twosphere system have been given by Maxwell [1], Russell [2], Jeffery [3] and Smythe [4]. We shall just state the results, and go on to give more compact expressions than the infinite series which result from summing over electrical images [1,2,4] or over products of exponentials or hyperbolic functions and Legendre polynomials [3]. Let a and b be the radii of the two spheres, and c  a þ b be the separation of their centers. An auxiliary dimensionless positive parameter u is defined by

cosh u ¼ E-mail address: [email protected].

CðQ ; Q Þ/

(1)

All possible electrostatic configurations of the two spheres are allowed for by (1). When the spheres are at the same potential V, the charges on them are then Qa ¼ ðCaa þ Cab Þ V and Qb ¼ ðCab þ Cbb Þ V so the total charge on the pair is Q ¼ Qa þ Qb hCðV; VÞ V, where

CðV; VÞ ¼ Caa þ 2Cab þ Cbb

The capacitances C(V,V) and C(Q,Q) are physically and analytically completely different. At close approach C(V,V) smoothly tends to the value taken at contact, whereas C(Q,Q) diverges logarithmically. When the spheres are far apart, the results of Section 2 imply the limiting forms

c2  a2  b2 2ab

or

c2 ¼ a2 þ 2abcosh u þ b2

(6)

12

J. Lekner / Journal of Electrostatics 69 (2011) 11e14 n2 X

Z n2 f ðnÞ ¼

n1

dn f ðnÞ þ

n1

ZN þ2

1 1 f ðn1 Þ þ f ðn2 Þ 2 2

Imff ðn2 þ iyÞ  f ðn1 þ iyÞg dy e2p y  1

(9)

0

The simplest sum is Cab, so we shall start with that. The result of applying (9) is

Cab

8   u ab<1 sinh u e þ1 ¼ þ ln u c :2 u e 1 ZN þ 2sinh u cosh u 0

 1 9 = sinuy e2p y  1 dy 2 2 cosh u  cos uy ;

(10)

(Note that, in the limit of large u, the last term in the braces contributes 12, and hence Cab /  ab c , as given in (8)). Applying the AbelePlana formula to Caa and Cbb one finds

Fig. 1. Electric field lines for two oppositely charged spheres. Only those in one plane are shown here. The field lines are arcs of circles, appearing as ellipses because of the oblique view. In the notation of this paper, the spheres have radii a and b, and the distance between their centres is c ¼ a þ b þ s.

0

 1 sinuy e2p y  1 ða þ bcosh uÞ2 c2 cos2 uy

  u Cbb 1 sinh u ae þ b þ c ¼ þ ln ab 2a cu aeu þ b  c ZN dy þ 2sinh uðacosh u þ bÞ

Then

Caa ¼ absinh u  Cab ¼

  Caa 1 sinh u a þ beu þ c þ ln ¼ cu a þ beu  c ab 2b ZN þ 2sinh uða þ bcosh uÞ dy

absinh c

Cbb ¼ absinh u

N P

½asinh nu þ bsinhðn þ 1Þu1

n¼0 N P

u

½sinh nu1

n¼1 N P

0

(7)

½bsinh nu þ asinhðn þ 1Þu1

n¼0

These infinite series converge rapidly provided u is not small, ie provided the spheres are not close together. When the spheres are far apart, u is large, and by inspection of (7)

 1 sinuy e2p y  1 ðacosh u þ bÞ2 c2 cos2 uy

(11)

(12)

(For large u the last terms in (11) and (12) contribute 1/2b and 1/2a, respectively, so Caa and Cbb take the values given in (8)). The integral forms of the capacitance coefficients given in (10)e (12) are exact, and may be used at any value of u. Their real advantage is ease of dealing with the physically interesting and numerically challenging close-approach configuration, discussed next. 3. Near approach of the two spheres

Caa /a;

ab Cab /  ; c

Cbb /b

(8)

When c >> a þ b, Cab is small compared to a or b and the results given in (5) follow. The number of terms required in the summations (7) depends on the accuracy required. When u is small compared to unity, the number of terms is proportional to 1/u. For example, if the remainder on terminating the series at n ¼ N is to be less than one part per million, N should be about 15/u, since sinh(14.5) is approximately a million. In terms of the separation distance s ¼ c  a  b, u is given by (23) if it is small. When a ¼ b this implies uz2ðs=aÞ1=2 , while when b >> a, uzð2s=aÞ1=2 . Thus when a/s ¼ 104, for example when a ¼ 1 mm, s ¼ 100 nm, about 1000 terms would be needed for part per million accuracy. The numerical difficulties associated with evaluation of the sums at small u can be avoided, as explained below. The infinite sums in (7) can be converted into closed-form expressions by using the AbelePlana formula [5], stated here in the form taken when the function f is real when its variable is real:

Let s be the separation distance between the spheres, with

c ¼ a þ b þ s;

coshu ¼ 1 þ

aþb s2 sþ 2ab ab

(13)

As the spheres get close together, s and u tend to zero. To find the behaviour of the capacitance coefficients in this limit, one can make use of the integral [6]

ZN 2

dy

 1 y e2p y 1 p 1 ¼ lnz  jðzÞ; jargðzÞj < 2z 2 z2 þy2

(14)

0

where j(z) is the logarithmic derivative of the G function, with the properties [6]

jð1Þ ¼ g;

1 z

jð1 þ zÞ ¼ jðzÞ þ ;

jð1  zÞ ¼ jðzÞ þ pcotpz (g ¼ 0.5772. is Euler’s constant.)

ð15Þ

J. Lekner / Journal of Electrostatics 69 (2011) 11e14

In calculating the close-approach limits of the capacitance coefficients, one needs to replace c in the formulae (10)e(12) by (6), and then expand in powers of u. The results follow directly, once the integral (14) is employed:

Cab

Caa

   ab 2 ¼ ln þ g þ O u2 aþb u

(16)

(17)

  a    ab 2 ln  j þ O u2 aþb u aþb

(18)

   a 4 ln þ g þ O u2 2 u

(22)

The close-approach results have been given in terms of the dimensionless parameter u. These expressions can be converted to expansions in the sphere separation distance s ¼ c  a  b by means of (13), from which one finds,

u2 ¼

  2ða þ bÞ s þ O s2 ab

(23)

In the above formulae one can thus make the replacement

  2 1 2ab 1 ln / ln u 2 aþb s

(24)

For example, Eq. (21) for the equal-sphere capacitance becomes

CðV; VÞ ¼ Caa þ 2Cab þ Cbb (19)

This result (without the O(u2)) was obtained by Russell [2] in 1909, and again by Moussiaux and Ronveaux [7] in 1979. When b ¼ a one finds, on using jð12Þ ¼ g  2ln2 [6], that

  CðV; VÞ ¼ 2a ln2 þ O u2

(21)

aþb

When b ¼ a this takes the value

CðQ ; Q Þ ¼

These results give useful estimates even when u is not small, as shown in Fig. 2. From the formulae (16)e(18) one can immediately obtain the form of the capacitances C(V,V) and C(Q,Q) when the spheres are close. The capacitance of the spheres held at the same potential is

      a  ab b 2jð1Þ  j j þ O u2 ¼ aþb aþb aþb

2 Caa Cbb Cab Caa þ2Cab þCbb     ) ( 2 a b   ab 2 g  j aþb j aþb 2     þO u ¼ ln þ aþb u 2g þ j a þ j b aþb

     ab 2 b ln  j þ O u2 ¼ aþb u aþb

Cbb ¼

CðQ ;Q Þ ¼

13

(20)

CðQ ; Q Þ ¼

  s  a 1 a ln þ ln2 þ g þ O 2 2 s a

(25)

Fig. 3 shows the capacitances C(V,V) and C(Q,Q)as functions of the relative size of the spheres. 4. Sphere in close approach to a plane The above results include, as a limiting case, the problem of a conducting sphere and a conducting plane. Formally, one takes

When b ¼ 2a and 3a the lead terms in (19) are 2a ln 3 and 92 a ln2, respectively. The capacity of the two-sphere system when they carry equal and opposite charge is

Fig. 2. Comparison of the exact and close-approach values of the capacitance coefficients. Shown are the dimensionless quantities Caa/a and Cab/a, drawn for b ¼ a. The exact values are full curves, the approximations (16) and (17) are dashed curves. At u ¼ 0.2 both close-approach forms are within 2 parts per thousand of the exact values. At u ¼ 1 they are too low by 5% and 4%, respectively.

Fig. 3. The equal-potential capacitance C(V,V), and oppositely charged capacitance b , in near approach. C(V,V) is independent of the sphere C(Q,Q), as function of b ¼ aþb separation s to lowest order, so only one curve is shown, of C(V,V)/a. Three curves show C(Q,Q)/a for s/a ratios of 0.05, 0.1 and 0.2. As b/0 the capacitance C(V,V) becomes that of a single sphere of radius a, hence the curve begins at 1. For oppositely charged spheres the system capacitance goes to zero with b, since then the vanishingly small sphere of radius b can hold no charge.

14

J. Lekner / Journal of Electrostatics 69 (2011) 11e14

the limit of b/N in the Eq. (7). This is done in [4]. More physically, one notes that the sphere-plane problem is related to the electrostatics of two equal spheres. For example, when two equal spheres carry equal and opposite charges, the plane normal to and bisecting the line joining their centres is at potential zero. One can replace this zero equipotential surface by a conducting plane. The field lines above this plane remain unchanged, and now correspond to the sphere-plane problem. If the original spheres were at potentials V and V, the sphere and plane potentials are now V and zero. If the original spheres were separated by distance s, the sphere-plane separation is s/2. Thus, for close approach of sphere and plane, one can either take the b/N limit of Eq. (21), or take twice the value of the result of substituting s/s=2 in (22). [The capacitance is twice the twosphere value because the potential difference is V rather than 2V]. Either method gives the capacitance of a sphere of radius a in close approach to a plane, with charges Q and eQ on them, as

CðQ ; Q Þ ¼

 s o an a ln þ ln2 þ 2g þ O 2 s a

(26)

An approximate version of (26) been published [12], namely (in our notation),

   q a a 23 þ ln2 þ CðQ ; Q Þz ln þ 2 s 20 63

(27)

where q is an unknown parameter that lies between 0 and 1. Since 23/20 ¼ 1.15 and 2gz1:1544, the agreement with (26) is close. 5. Comment on some incorrect published results The close-approach limit of the capacitance of equal and oppositely charged spheres given in (22) is not quoted by Russell [2], but is in agreement with his results, which are equivalent to (16)e(18). However, several writers have obtained different values. Keller [8] gives the value 2p aln2u. He quotes Weber’s [9] Eqs. (31) and (32) on page 232, the first of which gives C(Q,Q) as 2 (Caa  Cab), which is too large by a factor of 4. [Eq. (4), for equal radii, gives CðQ ; Q Þ ¼ 12ðCaa  Cab Þ.] Keller also adds a factor of p to Weber’s equation, hence the disagreement by 4p. Love [10] also quotes Weber [9] for the incorrect capacitance formula, and obtains the value 2a½ln4u þ g. Apart from the factor 4, this agrees with (22). Rawlins [11] quotes Maxwell [1] for the capacitance, but obtains 2a½ln4u þ g, the same as Love [10]. Weber, Keller, Love and Rawlins all use a parameter b, with u¼2b when a ¼ b. The factor 4 common to these authors can be traced to Weber’s error, since no such error appears in Maxwell’s treatise. Keller, Love and Rawlins do not refer to the work of Russell or of Jeffery. Apart from the factor of 4, the result given by Love and Rawlins is correct:

their expression, which contains an infinite sum over powers of b involving the Bernoulli numbers in the coefficients, is equivalent (when a ¼ b) to the capacitance derived from our integral expressions (10)e(12). 6. Summary and discussion Exact and compact formulae have been obtained for the capacitance coefficients of two spheres, of arbitrary radii and at any separation. The physically interesting case of close approach was examined in detail, and simple results were given for the logarithmic dependence on the separation. When two uncharged conducting spheres are placed in an external field E0, the field between the spheres is larger than E0 because of their polarization. This enhancement of the field has been explored in [13]. The ratio of the average field between two equal spheres (of radius a) to the imposed field grows without limit as the distance s between the spheres decreases:

p2 =3 Eave  z

s 1 a E0 ln þ ln2 þ g a 2 s

(28)

Four published erroneous results are noted, the first error (in the Weber text [9]) being the likely source of the others. While it is strange that an error of this magnitude has arisen and has propagated, the erroneous result does at first sight seem plausible. This is because the incorrect Weber formula makes the capacitance of two widely separated oppositely charged spheres equal to the sum of the capacitances of these spheres, if each were acting as an isolated conductor. However, as pointed out in the Introduction, oppositely charged conductors are linked by field lines, no matter how widely separated, and so cannot be considered independent. References [1] J.C. Maxwell, A Treatise on Electricity and Magnetism, third ed., vol. 1, Dover, New York, 1891/1954, Section 173. [2] A. Russell, Proc. R. Soc. A82 (1909) 524e531. [3] G.B. Jeffery, Proc. R. Soc. A87 (1912) 109e120. [4] W.R. Smythe, Static and Dynamic Electricity. McGraw-Hill, New York, 1950, Section 5.08. [5] F.W.J. Olver, Asymptotics and Special Functions. Academic Press, New York, 1974, (Chapter 8), Section 3. [6] P.J. Davis, Chapter 6 of Handbook of Mathematical Functions, NBS, Washington 1972, M. Abramowitz and I.A. Stegun, eds. [7] A. Moussiaux, A. Ronveaux, J. Phys. A Math. Gen. 12 (1979) 423e428. [8] J.B. Keller, J. Appl. Phys. 34 (1963) 991e993. [9] E. Weber, Electromagnetic Fields, vol 1, Wiley, New York, 1950, p. 232. [10] J.D. Love, J. Inst. Math. Applics. 24 (1979) 255e257. [11] A.D. Rawlins, IMA J. Appl. Math. 34 (1985) 119e120. [12] L. Boyer, F. House, A. Tonck, J.-L. Loubet, J.-M. Georges, J. Phys. D Appl. Phys. 27 (1994) 1504e1508. [13] J. Lekner, J. Electrostat. 68 (2010) 299e304.