The charge that steers a current round a bend

The inside of a conductor established the first thing anyone learns about metal in a field: at equilibrium, inside a conductor the field is exactly zero. That rule has an exception people learn in the same week, and the two are seldom put next to each other. In a wire carrying a steady current, the field inside is not zero. It has to be there, pointing along the wire, to push the electrons against the resistance. In a copper wire with 1.5 V across a metre of it, the field is 1.5 V per metre. What is more, it stays pointing along the wire when the wire curves, doubles back or ties itself in a knot.

Which leaves a question that a surprising number of textbooks do not ask. What makes it do that? A battery has two terminals and it sits in one place. Its own charges produce a field that falls off with distance and points in directions fixed by where the battery is. A loop of wire on the far side of a bench has no reason to feel a field along its length, from the battery alone, and the field from the battery alone would in most places be pointing across the wire rather than along it. Yet the current in a thin wire across the room behaves as if the wire had told it where to go.

The electrons are not bouncing along off the walls as water flows down a pipe. A wall can stop a flow reaching it, and that is the whole of what it can do. The thing that turns the current has to be a field, and a field has to have a source. The source is charge on the surface of the wire, a very small amount of it, arranged in a pattern that a circuit sets up for itself within nanoseconds of being connected. It was argued for in the nineteenth century, worked out by Oleg Jefimenko and others in the twentieth, and shown experimentally by Jefimenko’s photographs of grass seed lining up beside painted circuits on glass.

A wire with a voltage on it is a charged object #

Start from what a steady current demands. Along a uniform wire the potential falls linearly with distance, from one terminal to the other. So each point along the wire is at its own potential, with respect to the room. Treat that as a boundary condition. The space outside the wire has no charge in it, so outside it the potential obeys Laplace’s equation, and with the wire’s potential given and the distant room at a fixed value, it has one solution.

The whole of the picture is what a circuit does to the room around it. The field does not live only in the wire. It fills the space, and it is strongest near the battery, where the two ends at opposite potentials are close together. Each equipotential comes out of the surface of the wire at the point on it with that voltage. So just outside the surface, a contour runs away from the wire at some angle, and the field outside has a component along the wire as well as one straight out.

The component along the wire is the one that matters here. The tangential field is continuous across a surface, so what is true just outside is true just inside. It follows, without any assumption about what electrons do, that the field just outside a wire carrying a current has a component along the wire, equal to the one inside. A static charge on a resting conductor could never produce that, because a resting conductor is one potential.

The field straight out, on the other hand, ends on charge. Gauss’s law, as counting what comes out put it, turns the jump in the normal field across any surface into a surface charge. Laplace’s equation has no sources in the space outside, but on the wire there is a residual: a discrete Laplacian at each cell along the wire, and that number is the charge at that place along it, in this two-dimensional calculation.

There it is, the charge on a wire that has no net charge. The half near the positive terminal is positive. The half near the negative terminal is negative. A point at the far side of the loop, halfway round, is uncharged. Nothing in this calculation has been told what a current is. It needs only the falling potential, and any conductor that has one, under any cause, will have this distribution, or the corresponding one for its own geometry.

A gradient in surface charge is the source of the field along the wire. Each short length of wire has more charge on one side of it than on the other, and the imbalance points from higher to lower, along the wire. That is the same statement as the falling potential. Put a kink in the wire, and the pattern of charge rearranges until the field inside has been turned to match. It has to, since a field across the wire would otherwise drive electrons to one side, where they would add to the surface charge and cancel the sideways component, in about the time it takes light to cross. A surface charge that adjusts itself until no current crosses the surface is exactly what the charge that has to be somewhere else described for a conductor in a static field, doing a different job.

What actually happens at the bend #

The rectangle’s corners look small in the chart, because the wire is thin compared with the loop. For a close look at a corner the wire has to be drawn wide. So take a strip with some width, bent through a right angle, potential fixed at both ends and no current allowed through the walls. Solve Laplace’s equation inside it.

Current does turn. But it does not turn the way water does, in a smooth sweep filling the channel. In steady conduction the current is proportional to the field at each point, so it obeys Laplace’s equation too, and its patterns are those of a potential flow with no inertia. It takes the shortest path it can, crowding towards the inner corner, 1.8 times as dense there on this grid. The outer corner is almost a dead zone, with less than a tenth of the current density of the straight part. At an ideal sharp inner corner the density would go to infinity. Real traces on a printed circuit board have this as a known defect, and for a high-current trace or an integrated-circuit interconnect it matters, because the local heating and the drift of metal atoms under current are concentrated at the inner corner. A designer who rounds or chamfers a corner is reducing the same crowding this figure shows, and the extra resistance a bend adds can be read in the same picture, by the counting of squares that the sheet resistance that forgets the shape made exact.

Then comes the question of where this field gets its turn. Take the potential found inside as a new boundary condition, and solve outside the strip.

The result is not what a common picture suggests. In the usual story of how a bend is set up, moving charge first runs into the outer wall, piles up there and pushes the following charge round. That transient may have some truth in how the pattern forms. But in the steady state the largest and sharpest feature of the charge is not on the outer wall. It lies at the inner corner, where the two inner walls are close to each other at slightly different potentials, with the strong field between them shared across the empty notch outside. So the inner wall is positive just upstream of the corner and negative just downstream, and that opposed pair, sitting across the corner, is what pulls the current diagonally through the turn. The outer wall varies mildly. It is a useful corrective to a mental picture: the location of the source of a field is not always where one imagines the thing being pushed.

How little it takes #

The charge is extraordinarily small, and that is why so many treatments get away with ignoring it.

One electron in about four thousand million million is displaced. It is hard to think of anything else in a practical device that does so much with so little. The reason it can is that a metre of wire has a very small capacitance and needs little charge to change its voltage, of the order of how much charge a shape will hold would give for a thin rod. The displaced electrons are also not the ones carrying the current in any distinguishable way. Conduction electrons drift at about 0.023 mm/s at an ampere in that wire. The rearrangement of the surface charge needs only as long as light takes to cross the circuit, plus a tiny further time in the metal, set by the relaxation time of the charge, which in copper is of order $10^{-19}$ s.

That huge disparity also explains why circuit theory works so well without mentioning any of this. Kirchhoff’s law, that the current into a junction equals the current out, is really a statement that charge cannot accumulate in the junction; yet some does, in a transient and then in a steady state. It is so small a share of what flows that the error in neglecting it is about one part in $10^{16}$. The lumped-circuit idealisation has worked because of the size of this ratio, and not because the charge is absent.

Junctions between materials show the same arrangement with a sharper edge. Where copper meets iron in a series circuit, the current density must be the same on both sides, so the field jumps by the ratio of resistivities. Iron’s resistivity is about six times copper’s. Gauss’s law then requires a sheet of charge at the interface equal to $\varepsilon_0(\rho_2 - \rho_1) J$. For an ampere per square millimetre that is about $\varepsilon_0 \times 8 \times 10^{-8} \times 10^6$ coulombs per square metre, or less than a picocoulomb per square metre: an almost negligible amount, in the middle of a piece of metal that the equilibrium rule says should contain none. The rule was only ever about a conductor with no current.

The energy does not go down the wire #

This surface charge settles another odd feature of a circuit. With both a field along the wire and a field pointing radially out of its surface, the Poynting vector just outside a wire has a component directed into the wire. A steady circuit’s energy reaches the resistor from the surrounding space, through the side of the wire, carried by the crossed electric and magnetic fields, as where the energy of a field actually is described. The radial field that the surface charge makes also gives the Poynting vector a part running along the wire, from battery to load. And without a surface charge, there is no radial field outside. A wire with no surface charge would have nothing to guide energy along it from battery to load.

So the picture of a circuit that emerges is not one of electrons bringing energy along inside a tube. It is a field in the room, kept in place by a small layer of charge on a metal surface, delivering energy through the sides of the wire. The electrons are what is heated. This sounds like a redescription of something simple and it is more than that. It predicts that the field of a circuit should be measurable outside it, and it is. Jefimenko’s 1962 photographs, using charged grains on a painted conducting strip on glass, showed the fine lines of force tilted relative to the wire, just as the equipotentials above imply. A coaxial cable shows the same thing in a tidier form: a steady current through its resistive inner conductor gives a radial field between conductors that varies along the length, linearly, and passes through zero at a point fixed by the two resistances, the exact analogue of the zero on the far side of the loop.

The half a wire always takes found that a capacitor charged through any resistance loses half the energy to the wire. Its surprise was that the resistance does not appear. A similar absence is present here: the amount of steering charge does not depend on the resistivity of the wire. It depends on the voltage along it, on the wire’s shape and on the shape of everything around it, and the current follows.

Where this calculation stops #

The solution is two-dimensional. A real wire is a thin cylinder in three dimensions and the solution here treats it as a set of infinitely long lines coming out of the page, whose charge per length is what a section through such a set would have. The result that matters, the pattern, carries over: positive on one half, negative on the other, a zero on the far side and concentrations near the battery and the corners. The absolute numbers do not. The value of 10⁻¹¹ C per metre is an estimate from a rough capacitance, good to a factor of a few for a loop about a metre across, and it depends logarithmically on the wire’s radius.

The outer boundary is a box at the mid-potential, standing in for a distant room. Moving it changes the strengths slightly, and in reality the room contains other conductors, a bench and the experimenter, each of which takes some share of the field lines. So the zero at the far side of the circuit is true of this symmetric layout and may move in another one. The surface charge of a real circuit depends on where it has been put.

The figures are a steady state. They say what the charge is after everything has settled, in a few nanoseconds for a bench-sized circuit, and not how it gets there. That transient is a travelling wave along the wire, which the distance where a field changes its mind and ordinary transmission-line theory treat better than a Laplace solution can. The grid also limits the corner: a cell-sized corner has a cell-sized answer, and the true current density at a perfect corner diverges. And nothing here includes the magnetic field of the current, the Hall effect it causes, or the small self-pinch, which themselves produce a very small extra transverse field. In a wire those are smaller by a further factor of order $v/c$, for a drift speed $v$ of millimetres per second.

Still open: how much steering a very small wire can afford #

The arguments above treat the metal as a continuum and the charge as a smooth surface density. Modern interconnects on a chip are around ten nanometres wide, with a few atomic layers of surface at each side, and the current turns corners within a space in which grain boundaries and rough walls change the local resistivity. In those wires the current density is so high that the steady crowding at an inner corner slowly moves metal atoms, in the failure called electromigration. Whether a simple potential-flow model, with a local resistivity varying from point to point, gets the crowding right at that scale is something that chip manufacturers test empirically, and the surface scattering that dominates at those widths does not obey a local Ohm’s law at all. In very pure metal at low temperature, the same concerns lead to another regime, in which electrons flow more like a viscous fluid and whose patterns at a bend differ from those here; that regime has been seen in graphene and in a few metals, and how far it extends is still being worked out.

The current follows a wire because a falling potential along the wire requires a surface charge, positive on one half and negative on the other, that at about 3 × 10⁻¹⁶ of the conduction charge sets up the field inside; at a bend the field turns the current, crowded 1.8 times at the inner corner, by way of a sharp pair of opposite charges straddling that corner. A rule is true of what it was derived for, and a conductor with no field in it is a conductor with no current. The habit worth carrying is to ask, when something happens without obvious cause, what could carry the instruction, and to follow the answer however small it turns out to be.