Two positively charged conducting spheres have been shown to attract at close enough range, unless they have a charge ratio that would result from contact. We give analytical results for the charge ratio at which the cross-over between electrostatic attraction and repulsion occurs, as a function of the sphere separation.

The purpose of this note is to give a precise criterion for the existence of attraction between like-charged spheres. The criterion is that at a given sphere separation the charge ratio must either exceed a critical value or be smaller than a critical value. Outside of this region, for large enough or small enough values of the charge ratio, the spheres will attract. We give analytical results in the case of conducting spheres of equal radii.

The theory is based on Ref. 1, where attraction of like-charged conducting spheres at small separation was predicted by analytical means. Naturally, this prediction faced healthy scepticism. For example, Griffiths wrote,2 “Suppose I take two conducting spheres, of equal radius and equal charge. They repel, regardless of the separation distance. Paint a thin layer of insulation over their surfaces, and place them in contact (or, if you like, leave them uninsulated and place them a tiny distance apart). Add an infinitesimal charge (maybe a single electron) to one of them. Now they attract (by your theorem). Are we really to believe that this minute increment in charge switches a force that was large and repulsive to one that is (perhaps small but) attractive?”

The answer given below amounts roughly to the following: if a small charge is added to one of the spheres, the spheres will attract at a separation that is proportional to the square of the deviation of the charge ratio from unity. (A more precise criterion will be given later.) Thus, a charge deviation from unity of one part per thousand requires approach to within one millionth of the radius. As discussed in Ref. 1, surface roughness of the spheres becomes important at very small separations, and is likely to lead to an electrical short. Electron tunneling through a very thin layer of insulator (as in the Griffiths scenario above) is another way in which charges on the spheres can equalize.

The analytical results of Ref. 1 for the capacitance coefficients and for the charge distributions on the two spheres have been confirmed numerically.3 In addition, generalizations to pairs of non-spherical conductors have been explored.4,5 In each case the attraction between like-charged conductors, where it exists, can be traced to the redistribution of surface charge on the conductors. Figure 7 of Ref. 1 and Figs. 3 and 4 of Ref. 3 show the charge distribution on spheres of radii $a$ and $2a$ with positive total charges $2Q$ and $Q$, respectively. The separation between the spheres is $s=a$ ($s$ is the separation distance of the closest points of the two spheres), not close enough for net attraction to occur, but already there is an induced negative surface charge on the part of the larger sphere closest to the smaller one.

Figure 1 of this paper shows the equipotentials for this configuration, for which the potential of the smaller sphere is larger by a factor of about 2.17 than that of the larger sphere (zero potential at infinity being assumed). The variable charge density is due to very small displacements (relative to the nuclei) of a very large number of conduction electrons. The displacements can produce locally a negative surface charge on a positively charged sphere.

The electrostatic energy of two conductors $a,b$ at potentials $Va,Vb$ carrying charges $Qa, Qb$ is

$W=12QaVa+12QbVb.$
(1)

The charges are related to the potentials via Maxwell's capacitance coefficients (Ref. 6, Sec. 87)

$Qa=CaaVa+CabVb, Qb=CabVa+CbbVb.$
(2)

Inverting these equations gives the potentials in terms of the charges and the capacitance coefficients

$Va=QaCbb−QbCabCaaCbb−Cab2, Vb=QbCaa−QaCabCaaCbb−Cab2.$
(3)

Thus the potential energy of two conductors (of arbitrary shape) carrying charges $Qa, Qb$ is

$W=Qa2Cbb−2QaQbCab+Qb2Caa2(CaaCbb−Cab2).$
(4)

When the conductors are spheres of radii $a,b$, the capacitance coefficients depend only on $a,b$ and the distance between their centers $c=a+b+s$. Maxwell (Ref. 6, Sec. 173) was the first to find the capacitance coefficients for two spheres. They are infinite sums over ratios of hyperbolic functions of integral multiples of the variable $U$, defined by

$coshU=c2−a2−b22ab=1+sa+sb+s22ab.$
(5)

When $c$ is large compared to the sum $a+b$ of their radii, the leading terms of the expansion of the energy in inverse powers of $c$ (which in turn follows from the expansion of the capacitance coefficients in inverse powers of $c$) are

$W=Qa22a+Qb22b+QaQbc−Qa2b3+Qb2a32c4−Qa2b5+Qb2a52c6+⋯.$
(6)

The first two terms are the self-energies of the two charged spheres, then comes the Coulomb energy, followed by an attractive term that originates in the mutual polarization of the spheres, as explained in Ref. 1. The next term is also attractive. Remarkably, Maxwell6 carried out an equivalent expansion in reciprocal powers of the sphere separation to order $c−22$! (Maxwell's calculation is discussed in Ref. 7.)

The force between two widely separated spheres is therefore

$F=−∂cW=QaQbc2−2Qa2b3+Qb2a3c5+⋯.$
(7)

Assuming $Qa$ and $Qb$ have the same sign, to this approximation the force will be attractive (negative) for

$c3<2(QaQbb3+QbQaa3).$
(8)

Attraction between two widely separated like-charged spheres is thus possible only for a charge ratio very different from unity. At a given (large) separation, the required charge ratios are

$QbQa≈12(ca)3 or QbQa≈2(bc)3.$
(9)

The close-approach situation is more interesting. The capacitance coefficients have terms logarithmic in the sphere separation, and lead to the prediction that attraction will occur when the spheres are close enough to each other, unless they have the charge ratio that they would attain on contact. The question “how close is close enough for attraction?” is answered in general as follows.

To find the electrostatic force between the spheres, we differentiate the energy in Eq. (4) with respect to the sphere separation. We find that $F=−∂cW=−∂sW$ is zero when the charge ratio $q=Qb/Qa$ satisfies the quadratic

$A0+A1q+A2q2=0,$
(10)

where

$A0=2CbbCabDab−Cbb2Daa−Cab2Dbb,$
(11)
$A1=2Cab(CaaDbb+CbbDaa)−2(CaaCbb+Cab2)Dab,$
(12)

and

$A2=2CaaCabDab−Caa2Dbb−Cab2Daa.$
(13)

The coefficients $D$ are derivatives of the capacitance coefficients with respect to separation, $Dab=∂cCab=∂sCab$, etc. Equivalently, we can replace derivatives with respect to $c$ or $s$ by derivatives with respect to the dimensionless coordinate $U$ (the sum of the bispherical coordinates of the two spheres) defined in Eq. (5). ($U$ increases monotonically with $c$ or $s$, so the zero-force quadratic is the same in either case.) Equation (10) is simultaneously a transcendental equation for the zero-force sphere separation in terms of the charge ratio, but it is simpler to treat it as a quadratic in $q=Qb/Qa$.

The large-separation capacitance coefficients of Ref. 1 inserted into Eq. (10) lead to Eqs. (8) and (9). The discriminant of the quadratic (10) is

$A12−4A0A2=4(CaaCbb−Cab2)2(Dab2−DaaDbb).$
(14)

For large separation, we find that this is positive, with leading term $4a4b4/c4$, decreasing with sphere separation. At close approach, the discriminant is positive and increasing with separation. Thus the quadratic for $q$ has two real roots in each limiting case. That two real charge-ratio solutions exist at any separation would follow from an analytic proof of the inequality $Dab2−DaaDbb>0$, which seems difficult. A physical argument for the existence of cross-over between repulsion and attraction at any separation applicable to spheres of equal radii, based on a suggestion of Griffiths, goes as follows. For any value of the separation distance, if one of the charges is zero the force is obviously attractive, and for equal charges it is by symmetry repulsive, so there must be some cross-over charge ratio where it goes from attractive to repulsive.

We now look in more detail at the close-approach behaviour. Reference 1 gives the capacitance coefficients to order $U2$ for general $a, b$. The equal-radii coefficients are simpler

$Caa=a2{ln2U+γ+2ln2+U224[ln(2U)+γ+2ln2−16]}=Cbb,$
(15)
$Cab=−a2{ln(2U)+γ+U224[ln(2U)+γ+13]},$
(16)

($γ≈0.5772$ is Euler's constant). When $a=b$ there is extra symmetry in the quadratic (10) for the charge ratio $q=Qb/Qa$: $A0=A2$, so the quadratic is unchanged on the replacement of $q$ by $q−1$, as expected. Equation (10) now gives the cross-over charge ratio as

$q+q−1=−A0A1=2+O(U2).$
(17)

The term of order $U2$ is positive, so the quadratic has real solutions, as noted above. The solutions are also positive, and reciprocals of each other: $q±=(1+δ)±1$, where $δ2$ is equal to the $O(U2)$ term in Eq. (17), which follows from Eqs. (10), (15), and (16):

$δ2≈4ln2−112(ln4U+γ)2U2.$
(18)

For spheres of equal radii, $U≈2(s/a)1/2$ at close approach, and the cross-over charge ratios differ from unity by

$δ≈(4ln2−112)1/2[ln(4as)+2γ](sa)1/2.$
(19)

Figure 2 shows the degree of repulsion or attraction between two equal spheres, as a function of their separation and of the charge ratio $q=Qb/Qa$. The contours give the ratio of the force between the spheres to the Coulomb force between point charges $Qa, Qb$ separated by the center-to-center distance $c$ between the spheres, namely $FC=QaQb/c2$. The thick contours give the charge ratio at which the force is zero, and thus bound the region of repulsion.

From Eq. (19) we see that the dominant variation in the charge ratio for which the force is zero is (at close approach) determined by the square root of separation to radius ratio. Conversely, the separation at which cross-over to attraction occurs varies as the square of the deviation of the charge ratio from unity, $s/a∼ δ2=(Qb/Qa−1)2$.

The theoretical $F/FC$ contours in Fig. 2 all meet at the singular contact point (0, 1) in the (separation/sphere diameter, charge ratio) plane. The neighbourhood of the contact point is interesting. We shall discuss the equal-sphere case; for the general case see Sec. 4 of Ref. 1. Once the spheres touch and the charges are equalized [$Q=(Qa+Qb)/2$ is the charge on each sphere after contact], the force is repulsive and given by the Kelvin7,8 expression at contact, namely, by

$F0=4ln2−16(ln2)2Q2(2a)2≈0.6149Q2(2a)2.$
(20)

The force remains repulsive at all distances, by symmetry. It is almost constant at small separations.1 Before the charges are equalized the leading close-approach term of the force is attractive. It increases with decreasing separation and is proportional to the charge difference squared1

$F=−(Qa−Qb)22as[ln(4as)+2γ]+O(1).$
(21)

If the charge difference is very small, the next (repulsive) term can dominate at finite separation, but the repulsion does not exceed the Kelvin contact force. So, near the contact point the force can be attractive or repulsive, depending on the charge difference and the separation. The repulsion is bounded by the Kelvin contact force; the attraction is in theory unbounded but in practice eventually converted into repulsion by an electrical short at close approach.

The same square root variation of the bounding charge ratio curves persists in the general case of unequal radii. The charge ratio that is attained on contact is, from Eqs. (4.1) and (4.2) of Ref. 1,

$QbQa=γ+ψ(aa+b)γ+ψ(ba+b)≈(ba)2(π26)a−ba+b, ψ(z)=ddzlnΓ(z).$
(22)

For example, when $b=2a$ the contact charge ratio (for which there is repulsion at all separations) is

$QbQa=γ+ψ(1/3)γ+ψ(2/3)=9ln3+π39ln3−π3≈3.4477.$
(23)

[The value in Eq. (23) is 1.7% larger than is given by the approximate expression in Eq. (22).]

For unequal radii, the bounding curves of the repulsion region are no longer reciprocal, as they are in the equal radii case. They meet at zero separation at the charge ratio given in Eq. (22). Figure 3 shows the contours of the force ratio $F/FC$ and the regions of repulsion and attraction in the case $b=2a$.

The author is grateful to David Griffiths for permission to quote his private communication, and for comments on a draft manuscript. Helpful suggestions of the reviewers are also gratefully acknowledged.

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