The like charges that attract at close range

Like charges repel. Coulomb measured it with a torsion balance in 1785, and it is the first thing taught about electric charge. For two point charges it is exactly true at every distance. But charged objects are not points, and the bodies most often charged — metal spheres on insulating stands, water drops, dust grains, droplets in a printer — are conductors, on which charge is free to move. On a conductor, a charge’s position is not fixed by where it was put. It is fixed by the field of everything nearby, and a nearby charged body rearranges it.

That rearrangement can turn the sign of the force. Two metal spheres, both carrying positive charge, will attract each other when they are close enough, provided their charges are unequal. The effect was known in the nineteenth century to people who worked out the electrostatics of spheres, and it was set out in detail by John Lekner in 2012, who showed how general it is. It is not a small correction. Close enough, the attraction always wins.

The attraction that needs no charge #

The attraction that needs no charge began the story with a neutral conductor near a charge. The charge pulls opposite charge to the near side of the conductor and pushes like charge to the far side, and since the near side is closer, the attraction wins: a charged rod picks up scraps of foil. Now give the conductor a small positive charge of its own. Far away, the two repel, because the net charges dominate. But the induced separation does not care whether the conductor is neutral: it is driven by the stronger body’s field, and it grows sharply as the gap closes. At some distance the induced attraction, which grows faster, overtakes the repulsion between the net charges.

The curves come from an exact calculation, Kelvin’s method of successive images. A charged sphere at a fixed potential is equivalent, outside itself, to a point charge at its centre. Bring a second sphere near, and to keep the second at its potential an image charge has to be placed inside it, at the point where a sphere’s image of the first charge sits. That image disturbs the first sphere, which needs an image of the image, and so on. The images alternate between the spheres, get smaller and move towards the point of closest approach, and their sums give the charge each sphere holds at given potentials. From those come the energy at fixed charges, and from the energy, by differentiating with respect to separation, the force.

Far apart, the result is Coulomb’s law, checked in the calculation at sixty radii. Equal charges on equal spheres repel at every gap, because by symmetry the spheres are always at the same potential and no charge needs to rearrange towards the other: each sphere’s own charge simply crowds to its far side. But make the charges unequal and there is a gap below which the force changes sign. For a ratio of one half, the spheres have to be within a twentieth of a radius. For a ratio of one twentieth, they attract when more than a radius and a half apart.

Why the images crowd into the gap #

The method of images rests on one geometric fact about spheres. A point charge $q$ at a distance $s$ from the centre of a sphere of radius $R$ held at zero potential is mimicked, everywhere outside the sphere, by an image charge

$$q' = -\frac{qR}{s} \quad \text{at a distance} \quad \frac{R^2}{s} \text{ from the centre}.$$

The image sits inside the sphere, on the line to the real charge, and the closer the real charge, the closer to the surface its image lies and the nearer its size is to $-q$. That is the whole calculation. Each sphere receives a charge at its centre that sets its potential, then images of the other sphere’s charges, then images of those images, and every new image sits nearer the point of closest approach than the one before.

How fast the series converges depends on the gap. For two spheres far apart, each new image is smaller than the last by roughly the ratio of radius to separation, and three or four terms are enough. For spheres nearly touching, each image is only slightly smaller than the one it mirrors and only slightly closer to the gap, and the ratio between successive images approaches one. For two equal spheres with a gap $g$, the images shrink by a factor of about $e^{-\sqrt{g/a}}$ per reflection, so a gap of a ten-thousandth of a radius needs of order a hundred reflections to shrink the images by a factor of $e$, and thousands for the sum to settle. The crowding of images into the gap is the mathematical form of the physical fact in the next figure: as the spheres close, more and more of their charge concentrates into the narrow region between them, and the series has to work harder and harder to resolve it.

That concentration has its own mechanical consequence. A conductor’s surface charge feels an outward pull of $\sigma^2/2\varepsilon_0$ per unit area, the pressure that the pressure a charge puts on its own metal found on any charged surface, whatever the sign of the charge. In the gap the two facing surfaces carry dense layers of opposite sign, each pulled towards the other by that pressure, and integrating it over the facing surfaces gives the same attraction as differentiating the energy. Two very different computations — the stress on the surfaces and the change in stored energy — have to agree, and in the calculation behind these figures they do.

A positive sphere, negative where it faces #

The surface charge shows what is happening.

The weaker sphere is positive overall, but the side facing its neighbour is strongly negative. The stronger sphere’s field is large in the narrow gap and drives electrons into the facing surface, and the positive charge they leave behind is pushed round to the back. On the stronger sphere the same thing happens in reverse, less dramatically: its facing surface becomes more positive. So the gap is bordered by a positive surface on one side and a negative one on the other, a few hundredths of a radius apart, and the attraction between these two concentrated layers overwhelms the repulsion between the net charges, which act as if from much further away.

The inside of a conductor explained why a conductor’s charge sits on its surface wherever it has to in order to make the field inside zero. The negative patch is that rule at work. It is not a separate phenomenon added to Coulomb’s law; it is Coulomb’s law applied to charges that are allowed to move.

Where the pull begins #

How far out does the attraction start? It depends on how unequal the charges are, and on the sizes of the spheres.

The curves dive towards zero at one ratio and rise on either side of it. That special ratio is the one at which the two spheres, brought close, would already be at the same potential, so that no charge has to move towards the gap. For equal spheres, it is equal charges. For unequal spheres it is the ratio in which they would share a charge if they touched, which is not simply the ratio of their radii. Away from that ratio, in either direction, there is always a gap below which the spheres attract — and it reaches a sizeable fraction of a radius when the charges differ by a factor of a hundred.

The asymmetry between the curves says something about which sphere does the pulling. A small sphere near a large one with much more charge is attracted from far out, because the large sphere’s field is strong and the small one is easily polarised relative to its own charge. A large sphere near a small one with much more charge is a stranger case: the small sphere, highly charged, induces a large patch on the big one, and the attraction can also set in well before contact.

A hill before the fall #

The energy of the arrangement makes the behaviour intuitive.

Bringing the spheres together from a distance costs energy at first: they repel, and work has to be done against that. At the gap where the force changes sign, the energy reaches a maximum, a barrier; inside it, the attraction does work and the energy falls. The force near contact grows without bound, because the field in a vanishing gap at a finite potential difference grows without bound. So two like-charged conductors released inside the barrier will snap together, and released outside it will fly apart.

That is the shape of a chemical reaction’s energy, with an activation barrier between reactants and products, and it plays the same role. Two like-charged droplets in a cloud will usually bounce away from each other, but if turbulence drives them past the barrier, they coalesce. The same consideration arises in electrospraying, in electrostatic printing, and in the coagulation of charged dust in planetary rings and young planetary systems, where it decides whether grains that carry charges of the same sign can stick. The dust a plasma freezes into a crystal met grains in a plasma that all carry the same negative charge and arrange themselves in lattices because they repel; their charges are nearly equal, which is the case in which induced attraction is weakest, and that is part of why the lattices survive.

The one ratio that never attracts #

The special ratio is worth computing directly.

When two spheres touch, they become one conductor at one potential, and the charge divides itself between them in a definite ratio. For equal spheres it is equal. For a sphere half the size of the other, the small one takes 0.29 of the large one’s charge. That is more than the ratio of areas, a quarter, and less than the ratio of radii, a half, which is what isolated spheres at the same potential would hold: the small sphere is partly shielded by the large one, its near side lying in the large one’s shadow. How much charge a shape will hold found the same kind of shielding on any conductor with a sharp feature or a hollow, where the geometry decides which parts carry charge.

The charge that has to be somewhere else made the general point that a conductor’s charge goes wherever the whole arrangement requires, not where it was placed, and the touching spheres are the cleanest example: no amount of reasoning about either sphere alone gives 0.29. So there is a single physical meaning to the special ratio: it is where the spheres would end up if they touched. Every other pair of like charges is out of that equilibrium, and close approach drives the system towards it — first by an attraction, then, at contact or through a spark across the narrowing gap, by a transfer of charge. A spark between two like-charged bodies is therefore not a contradiction. It happens because they are at different potentials, which unequal charges usually are, and the same potential difference that drives the spark is what makes the close-range attraction. The half a wire always takes followed the energy cost of such sharing between two capacitors, and found that half the energy is lost whatever does the sharing.

Point charges, and what the rule really says #

None of this contradicts Coulomb’s law. Coulomb’s law is a statement about point charges, which cannot be polarised, and it holds for the fixed charges and their images at every step of the calculation. What fails is the extension of the rule from point charges to bodies, the assumption that a body’s charge can be summarised by its total. For a conductor far from anything, it can. Close to another body, the distribution matters as much as the total, and a body with positive total charge can present a negative face.

It matters in practice wherever conducting droplets of one sign are expected to keep apart. Electrostatic spray painting and charged agricultural sprays give every droplet a charge of the same sign, so that the droplets spread out rather than clump and are drawn to the earthed target. But droplets never carry identical charges, and two that are brought close by turbulence can pass inside their barrier and merge, which coarsens the spray. The design aim of an even charge is, by this argument, an aim of nearly equal charges, since equal charges on equal drops are the one pair that repels all the way in. A uniform charge keeps droplets apart; a spread of charges, at close range, can make them stick. The same subtlety turns up in places where it is easy to miss. Field lines are a choice described the attraction of opposite charges as lines in tension and the repulsion of like charges as lines pressing sideways. For two like-charged conductors that picture shows the transition directly: far apart, no line joins them; close together, lines run from the stronger sphere’s facing surface to the negative patch on the weaker, and the tension in those short lines is the attraction. The lines did not change their rules. The charges moved.

What the picture cannot show #

The spheres are perfect conductors, the charges fixed in total, and everything is static. Real conductors are near-perfect for this purpose, but water drops conduct well only on time scales longer than their charge relaxation time, a microsecond or so for pure water and much less for salty water, so for slowly approaching cloud drops the calculation applies while for fast collisions it may not. Dust grains of insulating material are not conductors at all; their charges are stuck where they landed, and they show only the polarisation of a dielectric, which is weaker and gives a different crossover.

The image series converges slowly when the spheres are very close, and the figures stop at a gap of a few thousandths of a radius, or a ten-thousandth for the contact ratio, where many thousands of images are summed. At smaller gaps, real surfaces are not smooth and spheres are not perfect; the air in the gap breaks down when the field exceeds about three megavolts per metre, and the induced attraction then ends in a spark rather than in contact. Quantum effects and the work function of the metal enter only at gaps of nanometres, far below anything drawn.

The force is computed at fixed total charges, which is right for isolated spheres. Spheres connected to batteries at fixed potentials have different forces, given by the same capacitance coefficients with a change of sign in the energy, and their behaviour near contact is different again.

Still open: what like-charged drops do in a thundercloud #

The force between charged drops matters most in clouds, where it affects how fast drops grow by colliding and so how fast rain forms. Laboratory experiments with charged drops show the expected attraction at close range and also show that strong fields and large charges change how drops collide, sometimes making them bounce, sometimes merge, sometimes break up. How much charge drops in a developing thunderstorm actually carry, how unequal the charges are between neighbours, and how much the close-range attraction speeds up rain formation compared with the turbulence and gravity that bring drops together, are still being worked out from field measurements and simulations. The electrostatics is exact; the drops’ charges are the unknown.

Two conducting spheres carrying charge of the same sign repel far apart, but unless their charges stand in the one ratio they would share on touching, the stronger induces a patch of opposite charge on the facing side of the weaker, and below a gap set by how unequal they are — 0.40 radii for equal spheres at a ratio of five, 1.56 radii at twenty — they attract, past an energy barrier and with a force that grows without limit as the gap closes. The habit worth carrying is to ask whether a charge is free to move before applying a rule written for charges that cannot.