The glow a thin wire wears before it sparks

The spark that needs room to start found that a gap of air between two flat electrodes breaks down when one free electron, accelerated by the field, starts an avalanche of ionisation large enough to replace itself — and that the condition depends on how far the avalanche has to grow as well as on how strong the field is. Between flat plates the field is the same everywhere in the gap, so the two conditions are met together: once the field is strong enough at one point it is strong enough across the whole gap, and the gap sparks.

Most real conductors are not flat plates. They are wires, edges, points and rounded bodies, and round them the field is concentrated at the surface and falls away with distance. That changes what breakdown means. The air next to the conductor can be well past its breakdown field while the air a few millimetres away is well below it, and the ionisation that starts at the surface then has nowhere to go. It stays in a thin glowing sheath round the conductor and does not cross the gap. This is corona discharge, and it was a central practical problem of electrical engineering long before anyone could write down the physics of an avalanche.

A field that falls away from the surface #

The simplest non-uniform field is that of a wire inside a coaxial metal tube. The shape decides the falloff found that the field of a long line of charge falls as one over the distance; for a wire of radius $r$ inside a tube of radius $R$, with a voltage $V$ between them, it is

$$E(\rho) = \frac{V}{\rho\,\ln(R/r)},$$

largest at the wire’s surface, $\rho = r$, and falling to $r/R$ of that value at the tube.

At sixty kilovolts the surface of the thinnest wire sits in a field of nearly five hundred kilovolts per centimetre, sixteen times the field that breaks down a centimetre of air between flat plates. Yet the field drops through 30 kilovolts per centimetre a little over three millimetres from the axis, and over the remaining nine and a half centimetres the air is far below breakdown. An avalanche starting at the wire’s surface runs outward into weakening field and dies. The thicker wires have smaller surface fields and wider regions of moderate field, and the thickest barely reaches the critical value at all.

The geometry is the one that the pressure a charge puts on its own metal found at every tight curve of a conductor: charge crowds onto the parts with the smallest radius of curvature, and the field crowds with it. A wire is a curve tight in one direction; a point is tight in two, and its field falls even faster, as one over the distance squared near its tip. Corona is the field’s response to curvature.

Why a thin wire tolerates more #

The flat-plate criterion would say that corona starts when the surface field reaches about 30 kilovolts per centimetre. Experiment says otherwise, and the disagreement grows as the wire gets thinner.

Frank Peek, working at General Electric on the first high-voltage transmission lines, measured the onset of corona on wires and cylinders of many sizes and published an empirical law in 1915 that is still used:

$$E_c = 30\,\delta\,m\left(1 + \frac{0.301}{\sqrt{\delta r}}\right)\ \text{kV/cm},$$

with $r$ in centimetres, $\delta$ the density of the air relative to its value at 25 °C and one atmosphere, and $m$ a factor less than one for surfaces rougher than polished metal — about 0.85 for a power-line conductor twisted from strands.

For a thick conductor the onset field approaches the uniform-field value. For a thin one it rises steeply, roughly as one over the square root of the radius. A wire a millimetre thick does not glow until its surface field is twice what breaks down a centimetre gap; a filament of ten micrometres holds ten times as much.

The second term is the avalanche’s need for room. An electron starting at the surface must make enough ionising collisions — around a few thousand electrons in an avalanche is typical of the threshold — before the field weakens below the point where ionisation outpaces the attachment of electrons to oxygen molecules. That boundary in air at atmospheric pressure is near 26 kilovolts per centimetre, where the rate of ionisation and the rate of attachment balance. Round a thick conductor the field stays above it for centimetres, and a modest excess at the surface is enough. Round a thin wire the field is above it only for a distance comparable to the radius, and the avalanche must be packed into that distance, which needs a much stronger field. Peek’s square root is an empirical summary of an integral — the number of ionisations along a path out from the surface, through a field falling as one over distance — reaching a fixed count.

The same reasoning explains the density factor $\delta$. In thinner air an electron travels further between collisions and gains more energy from a given field, so ionisation starts at a lower field, as the spark that needs room to start found for flat gaps; the onset field falls nearly in proportion to the density.

The best wire for a tube #

The onset voltage of a coaxial gap combines the two effects. For a wire of radius $r$ the surface field is $V/r\ln(R/r)$, so corona begins at

$$V_c = E_c(r)\; r\,\ln\frac{R}{r}.$$

A thin wire has a large surface field per volt, so it glows at a low voltage. A thick inner conductor nearly filling the tube leaves only a narrow gap, so it also fails at a low voltage. Between them is a best size.

If the onset field were a constant, the maximum would sit exactly at $r = R/e$, a ratio of 0.368, from differentiating $r\ln(R/r)$. Peek’s rising onset field for thin conductors moves it slightly inward, to 0.341. The same ratio, $e$ between the radii, is the classical design rule for a coaxial cable or a high-voltage bushing that must hold the most voltage for a given outer size.

The maximum also divides two kinds of failure. To its left, a conductor that starts to glow has an ionised sheath around it that behaves like a slightly thicker conductor, and a slightly thicker conductor in that region has a higher onset voltage, so the sheath stops growing. The corona is stable, and the gap sparks only at a voltage well above onset. To the right of the maximum the same growth lowers the onset voltage, the sheath runs away across the gap, and the first ionisation is a spark. A thin wire in a wide tube glows for a long range of voltages; a fat conductor in a narrow gap goes straight from silence to an arc.

An amplifier made of a glowing wire #

A thin wire’s ability to ionise the air next to it while leaving the rest of the gap alone has been turned into instruments. A Geiger counter, and its more refined cousin the proportional counter, is a thin anode wire along the axis of a gas-filled tube. A charged particle passing through the gas leaves a trail of free electrons, which drift towards the wire through a weak field without multiplying. Only in the last fraction of a millimetre, where the field rises steeply, does each electron start an avalanche, and the avalanche is confined to that sheath. In the proportional regime each initial electron produces the same number of descendants, tens of thousands, so the pulse on the wire is proportional to the energy the particle deposited. Georges Charpak’s multiwire chamber, which put thousands of such wires in a plane and earned a Nobel Prize in 1992, was the basis of particle tracking for decades. The amplification is corona held just below its own onset.

Corona at full strength is useful too. Frederick Cottrell’s electrostatic precipitator of 1907 runs thin wires at tens of kilovolts between grounded plates: the corona sheath sprays ions into the gas, the ions stick to particles of smoke and dust, and the field carries the charged particles to the plates. It still cleans the exhaust of most coal-fired power stations and cement works. A laser printer charges its drum with a corona wire. Ozone generators use a related discharge to split oxygen. The ions leaving a corona drag the air with them, producing an ionic wind, which in 2018 flew a five-metre aircraft with no moving parts. The thrust of such a wind has a simple limit. The ions cross a gap $d$ at a drift speed set by their mobility $\mu$ in the field, and the force they pass to the air is the current times the time each spends crossing, so the thrust per watt of electrical power is $d/\mu V$ — larger for wider gaps and lower voltages, which is the opposite of what a strong discharge wants. With the mobility of air ions, about two square centimetres per volt-second, and the tens of kilovolts needed to sustain the corona, it comes to a few newtons per kilowatt — enough for a light, slow aircraft, and well short of what a propeller turned by the same power delivers. The aircraft flew on that budget; the arithmetic of mobility is why it was a demonstration rather than a design.

The glow on the masthead #

Corona was seen long before it was understood. Sailors called the blue-violet glow that gathers on mastheads and yardarms in thunderstorms St Elmo’s fire, and took it as a sign of the patron saint’s protection; it is corona from the points of the rigging, where the storm’s field of a few kilovolts per metre near the sea is concentrated by the sharp ends to the tens of kilovolts per centimetre that ionise air. The same glow appears on aircraft wingtips, on church spires and on the horns of cattle on mountain ridges, wherever a sharp conductor stands in a strong field.

Benjamin Franklin’s argument for the pointed lightning rod rested on exactly this. A point, he reasoned, would draw off the charge of a cloud quietly, before it could gather into a stroke. That turned out to be only partly right — a thundercloud holds far more charge than any point can bleed away, and a rod works mainly by offering the stroke a path when it comes — but the quiet current Franklin imagined is real. Over land under a thunderstorm, corona from trees, grass and buildings carries a large share of the charge that flows between the ground and the clouds, and the space charge it releases near the ground limits how strong the field at the surface can become. It is one of the links in the planet-wide circuit that the field that keeps Greenwich time traced: thunderstorms charge the upper atmosphere, fair-weather air leaks the charge back, and corona from pointed objects under the storms is part of how the current reaches the ground.

When a corona does develop into a spark, it often does so through branching channels called streamers, thin filaments of ionisation that advance by photoionising the air ahead of their tips. Their branching has the same statistical character as the growth the cluster that grows where the walkers arrive drew — tips that stick out see a stronger field, grow faster and shield the regions behind them — which is why a long spark or a lightning leader forks.

Why power lines are carried in bundles #

On a power line corona is waste. It costs energy, from a few kilowatts per kilometre in fair weather to many times that in rain, when water drops on the conductor act as points; it produces the crackle heard under high-voltage lines on wet days; and it radiates radio noise. Line designers keep the surface field below the onset field with a margin, and at the voltages of long-distance transmission that cannot be done with a single wire of any sensible size.

The solution, adopted from the 1950s, is the bundle: two, three, four or more subconductors held apart by spacers on a circle of a few tens of centimetres. Seen from a distance, the bundle behaves electrically like a single conductor of an equivalent radius $(nrA^{n-1})^{1/n}$, for $n$ subconductors of radius $r$ on a circle of radius $A$ — for four 16 mm conductors 45 cm apart, about 21 centimetres. That large effective radius lowers the line’s charge per volt at each subconductor’s surface, and the field at the surface falls even though each subconductor is no thicker than before. A single conductor 21 centimetres in radius would do the same, but it would weigh tons per span and carry current only in its skin; the bundle puts the metal where the current flows and the field where the radius is large.

The figure is the reason a 400-kilovolt line is usually strung with pairs or triples, and a 765-kilovolt line with fours or more. Bundles also lower the line’s inductance and raise the power it can carry, a benefit that has made them standard even where corona alone would not demand them.

Higher ground, thinner air #

The density factor in Peek’s law matters wherever a line runs high.

At four thousand metres the air is about two-thirds as dense as at sea level, and a conductor glows at roughly three-quarters of the surface field it could hold at the coast. The thin wire suffers slightly less than the thick conductor, because the second term of Peek’s law, the one that rewards a steep field, grows as the density falls. A line designed for sea level and built across the Andes or the Tibetan plateau would glow and lose power in fair weather, and lines at those heights carry larger bundles — or larger subconductors — for the same voltage. The same factor explains why aircraft and their antennas carry static dischargers: a corona at altitude starts more easily and radiates noise into the radios.

Smooth cylinders, still air and one empirical law #

Every onset field here comes from Peek’s law, an empirical fit to measurements on smooth or stranded cylinders in still, dry air at densities near one atmosphere, with radii from about a hundredth of a millimetre to tens of centimetres. It is not derived, and it does not distinguish the polarity of the wire, although positive and negative corona look and behave differently — a negative wire produces a regular train of tiny current pulses, named after George Trichel, while a positive one glows more steadily and can launch streamers that branch outward. The coaxial field is exact for an infinitely long wire on the axis of a perfect tube, with no space charge; once corona starts, the ions drifting outward from the sheath alter the field and limit the current, which the static picture omits. The power-line gradients treat each phase as a single bundle above a flat conducting ground, ignore the other two phases and the sag of the span, and take the onset for stranded conductor in fair weather; rain, frost, insects and nicks all lower it, each by putting a point where the conductor was smooth, the charge-crowding how much charge a shape will hold worked out for every shape a conductor can take. The altitude curve uses the standard atmosphere and ignores humidity. The domain is air at roughly a third to one atmosphere, conductors from ten micrometres to ten centimetres, and steady or power-frequency voltages.

Still open: what decides the first avalanche on a wet conductor #

The onset in fair weather is understood well enough to design by, but a power line’s corona losses are dominated by bad weather, and there the onset is set by water drops hanging from the conductor, which elongate in the field, oscillate and eject smaller drops from their tips — the conical points the pressure a charge puts on its own metal found a charged liquid surface pulling itself into, and the same drops whose charging the water drops that charge themselves to a spark turned into a generator. The field at a drop’s tip depends on its shape, the shape on the field, and both on the surface tension and the drop’s size, so the onset is a property of a coupled fluid and electrical system. Measured losses in rain vary by an order of magnitude with rainfall rate and conductor surface condition, and new conductor coatings designed to shed water or to make it spread into films are tested in outdoor cages because the physics does not yet predict them. How the first avalanche starts on a drop-covered conductor, and how much a surface treatment changes it, is still measured line by line.

The dry case comes down to one comparison. Round a conductor of radius $r$ the field falls as $V/\rho\ln(R/r)$, so ionisation started at the surface stays in a sheath unless the field is strong over a distance long enough for an avalanche to grow; the onset field therefore rises as the conductor thins, from 32.9 kV/cm at 10 cm to 59 at 1 mm and 316 at 10 µm (Peek), a coaxial gap holds most at an inner radius near $R/e$, and a power line above about 470 kV needs its metal split into a bundle that looks, to the air, like a conductor 21 cm thick. Corona is breakdown that the geometry will not let finish.