Illustrated Physics
physics, drawn
Field

Electromagnetism

Charge, field, and the lines drawn between them.
The field of a dipole. Field lines traced from a positive charge toward a negative one. Every line does eventually close on the negative charge, but the outer ones loop far outside any frame, so this picture is a crop rather than the whole field.

Field lines are a choice, not a discovery

Nothing in space is arranged in lines. The lines are a drawing convention — and an unusually good one, because three separate facts about the field survive the translation.

 The same field, sampled as arrows. The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.

The field before the lines were drawn on it

A field is a vector attached to every point of space. Drawing it as arrows on a grid is honest and ugly; drawing it as lines is beautiful and throws information away.

 A closed surface with the charge inside. Every field line from the enclosed charge crosses the surface exactly once on its way out, so the net flux counts the charge.

Counting what comes out, and never looking inside

Draw any closed surface. The field crossing it depends only on the charge enclosed — not on where that charge sits, not on its shape, not on anything outside.

 Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

One number for every point, and nothing at all is lost

The electric field is three numbers at every point of space. Replacing it with one number loses nothing — and the reason it loses nothing is the same reason a hill can be drawn as a contour map.

 The field of a current loop, seen edge on. Magnetic field lines integrated from the Biot–Savart law for the current shown. Every line closes on itself: there is nowhere for one to start and nowhere for it to end, because no magnetic charge exists to end on.

The field with no ends, and the force that does no work

Magnetic field lines never start and never stop. That single absence is a law, it has survived every attempt to break it, and it makes the magnetic field a different kind of object from the electric one.

 The same law, three shapes of source. Field strength against distance on logarithmic axes, for a point, a long line and a wide plane carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law.

The shape decides the falloff, and the force law never changes

A point charge gives an inverse square, a line gives an inverse, a plane gives a constant. All three come from the same law, and the exponent belongs to the geometry of the source rather than to the physics.

 A conductor in a field, with the surface charge solved for. Field lines approaching an isolated conducting cylinder. The surface charge was found by requiring the conductor to be an equipotential, and the lines then end on that charge, meeting the surface at right angles and leaving the interior empty.

The inside of a conductor, where the field is exactly nothing

Put a metal object in any electric field and the field inside it is zero. Not small — zero, by an argument that takes one sentence, and with consequences that reach from lightning to the most precise test of Coulomb's law ever made.

 A loop leaving the field. A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion.

The field that makes the other, and only while it is changing

A magnet sitting next to a coil does nothing at all. Move it and a current flows. The law is not about the field but about its rate of change, and everything electrical since 1831 rests on that distinction.

 Two plates 0.20 plate-widths apart, with the field traced. The electric field between two oppositely charged plates separated by 0.20 of their own width, traced by following the field of 26 discrete charges on each plate rather than drawn as parallel lines. In the middle the lines are straight and evenly spaced; near the ends they bow outward. The field nine-tenths of the way to the edge is 80 per cent of the field at the centre.

How much charge a shape will hold, before anything is charged

Capacitance is decided by geometry alone. Two pieces of metal have a number attached to them, fixed by their shape and their separation, and it is settled before any charge arrives.

 Same field, same charge, three momenta. Electrons entering a 10 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 10.7 mm, 21.3 mm, 32.0 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment.

The force that does no work

A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.

 Four paths, one answer. The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

The field that wraps a current

Ampère's law says that going once round a closed path and adding up the field counts the current threaded through it, and nothing else about the path survives into the answer. Four different loops round one wire return the same number to five decimal places — which is exactly why the law is both easy and treacherous.

 The same loop, spanned two ways. A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all.

The term that made light

Ampère's law contradicts itself the moment a current stops being steady, and the contradiction is visible in one picture: two surfaces on the same loop, with a current through one of them and nothing through the other. The term that repairs it turns four equations into a wave, whose speed is two constants measured in a room with the lamps off.

 The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

Where the energy of a field actually is

A charged capacitor holds 1.27 µJ, and two entirely different accounts agree on the number: one built from charges and potentials, one built from joules per cubic metre of empty space. They part company at a resistor, where the power arrives sideways through the surface at 1.67 W.

 Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

The attraction that needs no charge

Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.

 The plane deleted, and one charge put in its place. A charge of 1 nC held 20 mm above an earthed conducting plane. The lines are traced through the field of the real charge plus an equal and opposite one at the mirror position, and then cut at the plane, because below it there is metal and no field whatever. Nothing in the tracing knows about the surface: each line follows the local field direction and stops where it arrives. That every one of them arrives perpendicular — the worst departure among the 9 drawn is 2.2° away from square — is the boundary condition showing itself rather than a rule imposed on the drawing. The image charge is drawn faint because it is not there: it is a way of writing a function that happens to satisfy the equation and the boundary values, which by the uniqueness theorem makes it the field and not a model of the field.

The charge that has to be somewhere else

Hold a charge above an earthed metal sheet and the field above it is exactly the field of two charges — the real one and an imaginary partner buried at the mirror position. The partner is not an analogy or an approximation. It is a legal guess, and a legal guess is a proof.

 A torque, and no force at all. A loop of 1 turn enclosing 20 cm² and carrying 10 A has a magnetic moment of 0.02 A m². In a uniform field of 0.05 T the torque on it is m B sin θ, drawn here against the angle between the moment and the field: zero when they are aligned, largest at 0.001 N m across, and zero again when they are opposed. The second curve is the energy, −m·B, whose minimum is the aligned position and whose maximum is the opposed one — which is why a compass needle settles one way round and not the other. The net force is zero at every angle on this axis, exactly and not approximately: the force is I dl × B summed round the loop, the sum of dl round any closed path is zero, and a constant B comes outside the sum. The inset shows the four forces on a rectangular loop; the pair across the axis is the couple, and the pair along it cancels.

The loop that behaves like a needle

Far enough away, a current going round in a circle is indistinguishable from a bar magnet, and one number describes both. That number tells a uniform field how to turn the loop and gives it no way to pull on it at all — which is why two magnets attract by the fourth power of the distance and not the second.

 A metre of tube, with and without the metal. The magnet's position against time over a 1.00 m drop, integrated from Newton's second law with the eddy-current drag included, and the same fall with the drag switched off. The braked descent takes 2.09 s against 0.45 s in free fall. After a transient of about 50 milliseconds — the mass divided by the drag coefficient, and the only timescale in the problem — the trace is a straight line, which is to say the magnet is falling at a constant 49.0 cm/s. Every joule of the 118 mJ of gravitational energy released has gone into resistive heating of the wall, and none into the magnet's kinetic energy, since that is the same at the bottom as it was a moment after the start.

The magnet that falls slowly

Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.

 A disc that generates 0.785 V with no changing flux anywhere. The force per unit charge on a carrier in a conducting disc of radius 100 mm spinning at 3000 revolutions a minute in an axial field of 0.5 tesla, against how far out the carrier is. Each carrier is moving through the field at ωr, so v × B pushes it radially with a force per unit charge of ωrB — zero at the axle and largest at the rim. The area under the line is the potential difference between the axle and the rim, which is ½BωR² = 0.7854 volts. Nothing in this calculation mentions a loop, a flux or a rate of change, and the flux through the circuit formed by the disc, the contacts and the external wire does not change at all while the machine runs.

The rule that is two laws wearing one coat

The flux rule folds two entirely different pieces of physics into one number, and they agree exactly, which is why nobody notices there are two. A disc spinning in a steady field generates a voltage with no changing flux anywhere, and the folding comes apart.

 The field a polarised sphere makes inside itself. A uniformly polarised sphere, its bound surface charge drawn at the size the cosine gives it, the uniform field that charge makes inside, and the exact dipole field it makes outside. The internal field is the same everywhere and points against the polarisation — that is what makes it a depolarising field — and its size is P/3ε₀, the third being the sphere's share of the one unit the three axes divide between them.

The field the matter takes away

Put a piece of glass in an electric field and the field inside it is smaller. How much smaller is not a property of glass. A needle of it keeps almost the whole field, a sphere keeps three-quarters, a slab across the field keeps a seventh — same material, same applied field, three answers, and the difference is arithmetic about shape.

 Where a magnet actually sits on its own curve. The second quadrant of a magnet's B–H curve, for a material with a remanence of 1.28 T, and the load lines four shapes of it impose. A magnet's own poles put it in a reverse field of N·M, so the working point is where the curve meets the line B = −μ₀(1−N)/N·H. A long thin magnet with N = 0.02 keeps 98 per cent of its remanence and a squat one with N = 0.7 keeps 30 per cent — the same material, cut differently.

The magnet that has to fight its own field

A bar magnet's own poles put it in a reverse field, so the same material cut short and fat is weak and cut long and thin is strong. And a magnetised material does not have a magnetisation — it has a magnetisation and a history, which is why the word for what it does is the Greek for coming late.

 The shell of news, and the kink inside it. A charge that was moving at 0.6c to the right, stopped over a short interval, and has been at rest ever since. Outside the sphere of radius ct nothing has heard: the field there still points at the position the charge would have reached had it carried on, marked ahead of it. Inside, the field is that of a charge at rest. In between is a shell one deceleration-time thick, and across it the field line has to bend, because a field line cannot simply stop in empty space — the two ends are joined here to 8.5e-14 pixels. That bend is transverse to the radius, and it is the radiation. It is not an extra thing the charge emitted; it is the join between two static fields that do not line up, and it travels outward at c because that is where the news front is.

The field that points where the charge is now

The field here was set by what the charge was doing a distance over c ago, so it ought to point at where the charge used to be. For a charge moving steadily it points at where the charge is — not approximately, exactly — and nothing has outrun light. What breaks the arrangement is a change of motion, and the break is the whole of radiation.

 Angular momentum that was in nothing at all. A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges.

The angular momentum that is in nothing at all

A charged ring and a solenoid, both at rest, with nothing moving anywhere. Switch the solenoid off and the ring starts to turn. Angular momentum is conserved, so it was there before the switch was thrown — and it was not in the matter, because nothing was moving. It was in the field, and it is qΦ over 2π whatever the geometry.

 The field a classical partition function cannot see. The momentum plane of one classical charge, in units of the root-mean-square thermal momentum. With no field the Boltzmann weight is a set of circles about the origin; with a field the same circles sit about p = qA, displaced and otherwise unaltered, because the energy depends on p only through p − qA. Integrating over the whole plane therefore cannot notice the displacement, and the numbers beside the drawing are that integral evaluated at 6 displacements: the largest departure from the zero-field value is 3.3e-16, which is the precision of the arithmetic and not a physical effect. The classical free energy has no B in it, so the classical magnetisation is exactly zero at every field and every temperature — no paramagnetism, no diamagnetism, no ferromagnetism. Every magnetic material is therefore evidence of something classical mechanics does not contain.

The magnetism classical physics forbids

Write down the partition function of any collection of classical charges in a magnetic field, and the field cancels out. Not approximately, not to leading order — the integral is over all of momentum space and the field only shifts where the middle of it is. So classical statistical mechanics predicts no paramagnetism, no diamagnetism and no ferromagnetism, and a compass needle is a quantum instrument.

 Water's permittivity across six decades of frequency. The real and imaginary parts of water's permittivity against frequency, on logarithmic axes, from 10 to 16 in powers of ten hertz. ε′ begins at 80.1 — the number a textbook prints — falls through the Debye relaxation near twenty gigahertz, is dragged down again by the librational, bending and stretching bands of the molecule, and settles at 1.777 in the visible, whose square root is 1.3330: water's refractive index. The identity n = √ε_r is exact and is an identity between two numbers taken at the same frequency; using the static permittivity in it predicts an index of 8.9 and is the most instructive wrong answer in the subject. The curve is a Debye term and four Lorentz oscillators, with two parameters solved so that the two plateaus are the measured ones rather than fitted by eye.

The constant that depends on how fast it is asked

Water's relative permittivity is 80.1. Its refractive index is 1.333, and the square of 1.333 is 1.777. The identity n = √ε_r is exact, the two numbers differ by a factor of forty-five, and nothing is wrong with either — because a permittivity is a function of frequency and the two measurements were made eight orders of magnitude apart.

 Every line that goes in has to come out. Field lines of a 5:1 solenoid in the plane through its axis, each traced by stepping along the local direction of a field summed turn by turn from Biot–Savart. Inside the winding they are parallel and evenly spaced, which is the picture the textbook argument is about. Outside they are not absent: they are spread over the whole of the rest of space, which is why the field there is small — 1.60e-2 of the centre value at 2 radii off the axis — and why it cannot be zero. A line has no end, so every one of the lines through the bore returns outside, and a field with no outside would be a field whose lines stop.

The field outside the solenoid, which is not zero

Ampère's law says the field outside a solenoid vanishes, and every step of that argument is exact — for a winding of infinite length. A real one is a bar magnet seen from outside, its external field falls as the inverse square of its length rather than to nothing, and the "exactly zero" that makes the derivation so satisfying is the one part of it a laboratory cannot have.

 Three vector potentials for one magnetic field. Three different vector potentials, drawn as arrow fields over the same square, every one of which describes the same uniform magnetic field of 1 tesla out of the page. The symmetric gauge circulates about the origin; the two Landau gauges are unidirectional and point in perpendicular directions, and neither has any circulation about anything a reader can see. Underneath each is the field recovered from its own arrows, by adding up the potential round a small square and dividing by the area enclosed — which is what the curl is. The three numbers 1.000000, 1.000000, 1.000000 agree to the last digit computed. The picture is the argument: a vector potential has no physical direction, no physical magnitude and no physical circulation of its own, because a change of gauge alters all three and alters nothing that can be measured. What survives is the curl, and the curl is the field.

The potentials that are not unique

Nobody solves Maxwell's equations for the fields. They are solved for potentials instead, and the potentials are not unique — three completely different vector potentials describe the same uniform magnetic field, and one of them changes everywhere the instant a charge moves, at any distance, without anything having outrun light.

 Skin depth against frequency, over eleven decades. The skin depth of copper, stainless steel, seawater, mu-metal against frequency, both axes logarithmic. Every line has the same slope, −½, because the depth goes as the inverse square root of the frequency for all of them; what separates them is conductivity and permeability, which enter the same way. Three points are marked and each is a practical fact. Copper at mains frequency has a skin depth of 9.22 mm, so a busbar of that thickness carries current throughout and a thicker one does not. Copper at a gigahertz has 2.1 µm, so the current in a microwave component runs in a layer thinner than the plating and the surface finish becomes the conductor. Seawater at the frequency navies use to signal submerged submarines has 28.9 m, which is why that link is measured in tens of hertz and carries a few characters a minute. Mu-metal is the outlier and the reason it exists: its conductivity is forty times worse than copper's and its permeability twenty thousand times better, so at low frequency it is the one material on the chart that is thin.

How far a field gets into metal

Every static argument about conductors says the field inside one is zero, and every one assumes the electrons have had time to move. Give them less time and the field gets in — 9.2 millimetres into copper at mains frequency, 2.1 microns at a gigahertz — and the metal box that silences a radio does almost nothing about the cable running past it.

 How much of the answer a finite wire gives back. The field beside a straight segment of wire, divided by what the infinite-wire formula would give, against the length of the segment in units of the distance to the field point, on a logarithmic horizontal axis. The field is integrated element by element along the segment rather than evaluated from a closed form. A wire ten times as long as the distance already gives 92.8 per cent of the infinite answer, and one as long as the distance gives 45 — which is the practical content, and the reason the infinite-wire result is used for laboratory wires without apology. The second column is the part that matters for the law rather than for the number. The circulation of this field round a circle of radius d is not μ₀I; it falls short by exactly the fraction the segment fails to subtend. What makes up the difference is the displacement current of the charge piling up at the segment's two ends, and the two terms, both integrated here, sum to μ₀I to within 2.2e-8 per cent at every length. So Ampère's law is not approximately true for an open circuit and exactly true for a closed one — it is exactly true always, and it is a computation only when symmetry supplies the direction and the magnitude along the loop. Symmetry is doing the work; the law is doing the bookkeeping.

The law that is always true and rarely useful

Ampère's law holds for every loop and every current. Apply it to a wire of finite length and it gives an answer that is wrong by half — until the displacement current of the charge piling up at the wire's two ends is put back in, whereupon the two terms sum to μ₀I exactly at every length. The law was never approximate. It was never a computation either.

 Every stationary point has a way out. The potential along four lines through the most symmetric point of a symmetric arrangement of 4 equal charges at the corners of a square — the one place a trap might be expected. In the plane of the square the potential rises in every direction; out of the plane it falls. The point is stationary and it is a saddle, which is what Laplace's equation forces: the three second derivatives must sum to zero — computed here as 1.4e-5 against the individual values of order 2e+0 — so if two of them are positive the third must be negative. A particle released here rolls away along the direction that falls. No amount of ingenuity in placing the charges changes this, because the constraint is on the equation rather than on the arrangement, and it is why every real trap for a charged particle either uses time-varying fields, or a magnetic field with a velocity, or a material with a negative response.

Nothing can be held still by a static field

However many charges are arranged, however cleverly, a charge placed among them has somewhere to fall. The reason is one line of arithmetic — the potential in empty space satisfies Laplace's equation, and a solution of that equation has no interior maximum or minimum — and the consequence is that every real trap for a charged particle works by breaking one of the assumptions rather than by being cleverer.

 A refraction with no wave in it. Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 4, at incidences of 12°, 26°, 40°, 54°, 68°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 4.000, 4.000, 4.000, 4.000, 4.000, against 4. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

A refraction with no wave in it

A field line crossing from one dielectric into another bends, by a law that looks exactly like Snell's with the sines replaced by tangents. Nothing is oscillating, nothing is travelling, and no speed appears anywhere in the derivation — only the two conditions that say what a boundary may and may not do to a field.

 Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields.

The coupling that is the same both ways

A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.

 The screening that survives to zero frequency. How far a magnetic field gets into copper against how fast it is changing, drawn beside the London depth of a superconductor with a carrier density of 6.0e+28 per cubic metre. The metal's curve falls as one over the square root of the frequency — measured on the drawn curve as -0.5000 against −½ — and it is a straight line on these axes with no bottom: at a hundred hertz the field reaches nine millimetres in, and at zero frequency it reaches all the way through, because a normal metal screens by dissipating and a steady field dissipates nothing. The superconductor's line is flat at 21.7 nanometres. The frequency does not appear in the expression for it, so there is nothing for it to depend on, and the screening is as complete at zero frequency as at any other. The two lines cross at 9.0e+12 hertz, in the far infrared, and above that the ordinary metal is actually the better screen — which is a useful corrective, because a superconductor's advantage is not that it screens harder but that it does not need the field to be changing. What the picture cannot show is where the flat line stops: above the energy gap the pairs break, the superconductor becomes an ordinary metal, and the flat line turns into a sloping one.

The field that is pushed out

A perfect conductor keeps whatever field was inside it when its resistance vanished. A superconductor expels the field either way, and the difference between remembering and expelling is the experiment that showed superconductivity is a state of matter rather than a very good conductor.

 The size of the quantum says what is carrying the current. What the flux quantum would be for each candidate carrier charge, in units of 10⁻¹⁵ webers, against the measured value drawn as a line. A single electron would give 4.1357, a pair 2.0678, a triple 1.3786. The measurement is 2.0678, which picks the pair and excludes the others by a factor of two — not by a few per cent, so no question of experimental accuracy arises. The whole of the argument is that the condensate's wavefunction must come back to itself round the ring, which makes the enclosed flux a multiple of h over the carrier's charge; measuring the multiple therefore measures the charge, without any charge ever being measured. That is how the pairing was established in 1961, four years after it was proposed and by an experiment that looks nothing like a measurement of a charge.

The two in the flux quantum

A superconducting ring cannot hold whatever flux is applied to it. It holds a whole number of quanta and drives a current to make up the difference, and the size of that quantum is Planck's constant divided by twice the electron's charge. The factor of two was measured in 1961, four years after somebody predicted that the carriers are pairs — by an experiment in which no charge is measured at all.

 Two quantities that are never computed and never change. A two-dimensional field integrated for 110 steps using only the two equations that contain a time derivative — Faraday's and Ampère's. The two that do not, Gauss's law for the electric field and the statement that there are no magnetic charges, are never imposed and never checked during the run. Their residuals are plotted: the divergence of B stays below 2.4e-16 of the field's own size and the divergence of E below 2.4e-16, over the whole run, while the field itself moves and changes by a factor of 8.59. That is not a numerical coincidence. Taking the divergence of Faraday's law gives the divergence of a curl, which vanishes identically, so ∂(∇·B)/∂t is zero whatever the fields are doing; the same manoeuvre on Ampère's law gives ∂(∇·D)/∂t = −∇·J. So the two constraints are initial conditions, propagated for ever by the two that are laws of motion, and Maxwell's four equations are two dynamical ones and two statements about how the field was set up.

The two equations that are not laws of motion

Maxwell's equations are usually presented as four laws of equal standing. Two of them contain no time derivative at all, which means they cannot be evolution equations: they are conditions on the field at one instant. What makes them consistent with the other two is that the other two preserve them exactly — and one of the two preservations holds only because charge is conserved.

 The only part of the field that can pull. A dielectric slab part-way into a parallel-plate capacitor. Everywhere except at the slab's edge the field is perpendicular to the plates and therefore perpendicular to the direction the slab can move, so it exerts no force along that direction at all: inside the parallel-plate model, which is uniform between the plates and zero outside them, nothing pulls the slab anywhere. The force lives in the bowed lines drawn at the edge, where the field leaks past the end of the dielectric and acquires a component along the plates. Those lines are what the model throws away as a small correction near the boundary, and they are the entire mechanism. The energy method sidesteps the drawing altogether: differentiate the total energy with respect to the insertion and the answer is 1.328e-3 newtons, inward, without ever asking where on the slab the force is applied.

The force that lives where the model is not

A slab of glass held at the mouth of a charged capacitor is pulled in. Inside the parallel-plate model there is no force at all — the field is perpendicular to the slab's motion everywhere — and the same model's energy nevertheless gives the pull exactly right. The mechanism is entirely in the part of the field the model throws away.

 The outward pull on a charged surface. Electrostatic pressure against the field at a conductor's surface. The quantity is ½ε₀E², the energy density of the field itself, and it is outward whatever the sign of the charge — like charges repel, and a charged surface is trying to fly apart. The factor of a half is the interesting part and is where a first attempt goes wrong: the field is σ/ε₀ outside and zero inside, and the layer of charge feels neither of those but their mean, because no charge exerts a force on itself. 0.5 MV/m gives 1.1 Pa, 1 MV/m gives 4.4 Pa, 2 MV/m gives 17.7 Pa, 3 MV/m gives 39.8 Pa, 5 MV/m gives 110.7 Pa. Those are small pressures — three megavolts per metre is the breakdown field of air and pulls with about a hundredth of an atmosphere — which is why electrostatic forces shape soap films and dust and not much that is stiffer, and why the same pressure set against surface tension has a definite size of drop at which it wins.

The pressure a charge puts on its own metal

Charge on a conductor sits on the surface and tries to leave. The outward pull is half epsilon-nought E squared, the half is because a charge exerts no force on itself, and setting that pull against surface tension gives the largest a charged drop is allowed to be — a number Rayleigh wrote down in 1882 and an industry now depends on.

 How much of μ₀N²A/ℓ a real coil actually has. The inductance of a 140-turn coil of radius 10 mm, divided by the long-solenoid formula μ₀N²A/ℓ, against how long the coil is compared with its diameter. The inductance is computed as a sum of Maxwell's mutual inductances between every pair of turns plus each turn's own, so the long-coil formula appears nowhere in it. At a length equal to the diameter the real coil has 68% of what the formula promises, and at a quarter of the diameter about a third. The reason is that the formula assumes every turn is threaded by the full interior field, and near the ends of a short coil the field has already begun to spread. The ratio is Nagaoka's coefficient, tabulated since 1909, and the computed points agree with the table.

The circuit that fights its own change

Every circuit is threaded by the field its own current makes, so every circuit resists having that current altered. It is the reason a coil takes time to start and the reason breaking one makes a voltage a thousand times the supply's.

 The current crowding into a contact spot. A meridional section through a circular contact between two solids, with the spot at the centre. The closed curves are equipotentials and the curves running through the spot are current lines, each carrying an equal share. Both are exact: in the coordinates built on the spot's rim the equipotentials are confocal spheroids and the current lines confocal hyperboloids, and in this section they are ellipses and hyperbolas. What the picture shows is that the current has to converge from a region many spot radii across and then spread again, and that the potential falls almost entirely within a few radii of the contact. The equipotential drawn at 12% of the drop sits 5.2 radii away, and everything beyond it contributes that last 12%.

The resistance that is a length

Two metals touching do not touch over the area they appear to. Current crosses at a few small spots and has to converge into each one, and the resistance of that convergence contains no area and no path length — only the size of the spot, divided into the resistivity.

 A magnetisation curve, and the same curve magnified. On the left, the position of a domain wall against the applied field, over the whole of a crystal. It looks like a curve. On the right, a window 2.0% of its width, taken 42% of the way across, at the magnification a sensitive measurement reaches: the wall does not move at all while the field rises, then jumps, then stops again. The largest jump in the window covers 0.67 units of wall position in no field change at all. There is nothing smooth underneath this — the smooth curve on the left is a staircase with 1400 steps in it, drawn small.

The curve that is really a staircase

A magnetisation curve is drawn as a smooth line and is nothing of the sort. Measured finely enough it is a sequence of jumps of every size, audible as a crackle in a coil, with no typical jump and no smooth motion underneath.

 The rotation that mixes electricity into magnetism. The two Lorentz invariants of a field — E² − c²B² across and 2E·cB up — as the field is rotated by the duality transformation that takes E into cB and cB into −E. Every configuration moves on a circle, so the combination of the two invariants is preserved while neither is. A light wave sits at the origin and stays there, which is why a wave cannot be turned into anything else by this rotation; a static charge starts on the positive axis and is carried round to a pure magnetic field a quarter turn later. The source-free equations are unchanged by the whole family, so a universe with no charges in it has no way to say which field is which.

The symmetry one missing charge would complete

Maxwell's equations with no sources are unchanged by rotating the electric field into the magnetic one. With sources they are not, and the only thing missing is magnetic charge — which, if one existed anywhere, would force every electric charge in the universe to be a multiple of a fixed unit.

 The ring picks a whole number and pays for the difference. The kinetic energy of the screening current in a superconducting ring, against the flux applied from outside, in units of the flux quantum. Each parabola belongs to one winding number — the number of times the condensate's phase turns round the ring, which has to be an integer because the wavefunction has to come back to itself. The ring cannot hold an arbitrary flux, so it holds the nearest whole number of quanta and drives a current to make up the difference; the energy of that current goes as the square of the mismatch, which is what each parabola is. The lowest curve at each applied flux is the state the ring is in, and the winding number changes at exactly the half-integers — 0.50 and 1.50 and 2.50 here. So the measurable properties of the ring are periodic in the applied flux with a period of one quantum, which is 2.0678 × 10⁻¹⁵ webers and is about the flux the Earth's field puts through a square micrometre.

Two lengths, and which one is longer

A superconductor has a depth to which a field leaks in and a distance over which superconductivity itself can be built up. Which of the two is larger decides the sign of the energy of a boundary — and therefore whether the material keeps every field out or fills itself with a lattice of holes.

 A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way.

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.

 What the plane between them carries. The stress transmitted across the plane halfway between two charges of 10 nC held 1 cm apart, against distance from the axis in units of the half-separation. For two like charges the field on that plane lies entirely in it — checked here rather than assumed — so the plane sees only the pressure across the lines, the stress is negative everywhere, and the two halves are pushed apart. For opposite charges the field on the plane is entirely perpendicular to it, the plane sees only the tension along the lines, the stress is positive, and the halves are pulled together. Faraday's two words for a field line, tension along it and pressure across it, are exactly these two curves; the whole content of the tensor is that they are the same quantity, ε₀E²/2, wearing two signs. The force is what is left after integrating either curve over the plane, and both integrals come to the same magnitude — the Coulomb force — which is the next figure.

The force read off a surface that touches nothing

Draw any closed surface through empty space, measure the field on it, and the sum of one expression over that surface is the total force on everything inside — whatever the contents are, and without knowing anything about them. The expression is Maxwell's stress tensor, and it turns Faraday's guess about tension along a field line into an exact statement.

 Three charges, three orbits, one drift. Three particles released at rest in crossed fields — 1000 V/m across 0.1 T — with their paths integrated by a scheme that rotates the velocity rather than adding to it, so the magnetic part changes no speeds. The three loops have wildly different sizes and periods: the the electron turns at 2799.25 MHz, the proton turns at 1.52 MHz, the α particle turns at 0.76 MHz. Their guiding centres all creep along the same line at the same rate, measured here from the orbits at -1.000e+4, -1.000e+4 and -1.000e+4 m/s against −E/B = -1.000e+4 m/s, a spread of -0.00 per cent. Neither the charge nor the mass nor the sign appears in the answer. A plasma in crossed fields therefore moves bodily and carries no current from this drift at all, which is the opposite of what an intuition built on ions being heavier than electrons expects.

The drift that does not care what the charge is

A charge in a uniform magnetic field goes round in a circle and arrives nowhere. Add anything at all — an electric field, gravity, a gradient in the magnetic field itself — and the circle's centre creeps sideways, at right angles to both. One of those drifts is the same for every particle regardless of charge, sign or mass; the others are not, and the difference decides what a plasma does.

 Three terms, and one distance. The three terms of an oscillating dipole's electric field against distance, in units of a reciprocal wavenumber, both logarithmic. They fall as 1/u³, 1/u² and 1/u, so they are all equal at u = 1 — a distance of λ/2π, which is a property of the frequency and of nothing about the antenna. The heavy curve is the field the three actually add up to, and it is not their sum: the static and radiation terms are in antiphase and partly cancel, which is why the total dips below every one of them just inside the crossing before settling onto the 1/u the far field is made of.

The distance where a field changes its mind

An oscillating source has three fields around it, falling as the inverse cube, the inverse square and the inverse first power of distance. They are all equal at one radius, and that radius is the wavelength over 2π — a number containing nothing about the source at all. Inside it a source mostly stores energy; outside it, mostly loses it, and the two behaviours are different technologies rather than different strengths.

 What a row of sources does that one cannot. The pattern of 8 identical sources in a row, spaced 0.5 wavelengths apart, all driven in phase. Each source alone radiates the same in every direction drawn here; together they radiate almost entirely along one. The sum being performed is the sum over path differences across the row, which is the sum a diffraction grating performs over its slits — the same function with the same first null, at sin θ = 1/Nd, measured here off the curve. What has been exploited is retardation: the contributions arrive at different times, and the pattern is a map of where they arrive in step.

When the source is not heard all at once

The dipole approximation is not a statement that a source is small. It is a statement that every part of it is heard at the same retarded time, and dropping that assumption turns one source into a sum over a source. The sum is the same one a diffraction grating performs over its slits, with the same first null and the same extra orders — so a phased array and a grating are one piece of arithmetic met twice.

 Two solutions, and nothing in the equations to choose between them. A spherical pulse leaving a point and a spherical pulse arriving at one, each drawn at three times 0.35, 0.6, 0.85 in units where the speed is one. Both are exact solutions of the same wave equation, which is checked here by differencing the drawn samples twice in space and twice in time and requiring the residual to vanish for each. The one on the left is what is always used; the one on the right is discarded, and the equations do not do the discarding. The incoming pulse grows as it converges for the same reason the outgoing one decays as it spreads — the same energy through a smaller sphere — and it is as consistent with conservation as its mirror image is.

The solution that is thrown away

Maxwell's equations admit a field that converges on a charge exactly as readily as one that leaves it, and nothing in them prefers either. Retardation is a boundary condition rather than a law. Which boundary condition is right has been argued about for a century, one of the answers makes the arrow of time a property of there being absorbers, and the laboratory version of the question — whether an atom emits at all — has a measured answer that depends on what is listening.

 The gravity that grows on the way down. The acceleration due to gravity inside the Earth against distance from the centre, from Gauss's law applied to the Preliminary Reference Earth Model's density: g(r) = G M(r)/r², counting only the mass inside each radius. The model reproduces the Earth's mass to 0.02 per cent, a surface gravity of 9.82 m/s² and a moment-of-inertia factor of 0.3308, none of which it was fitted to. For a uniform Earth, drawn dashed, gravity would fall in a straight line to zero at the centre. The real one does the opposite for the first 2891 km: it rises through the whole mantle to 10.69 m/s² at the core–mantle boundary, 8.8 per cent above its surface value, and only then falls to zero through the core. Going down through the mantle removes very little mass and brings the dense core much closer, and the second effect wins.

The pull that grows on the way down

Inside a uniform ball, gravity falls in a straight line from the surface to nothing at the centre, and that is the answer usually given for the Earth. It is wrong for almost three thousand kilometres. Going down through the mantle, gravity rises, reaching nearly nine per cent above its surface value where the core begins — because Gauss's law counts only the mass inside, and the local form of the law says gravity grows inward wherever the rock is lighter than two thirds of the average beneath it.

 A current with no voltage, and then a voltage that is a frequency. Current against mean voltage for a Josephson junction shunted by a resistance, in units of its critical current and of the critical current times the resistance. Up to the critical current the junction carries a supercurrent at exactly zero voltage, set by the difference of the two superconductors' phases. Above it a mean voltage appears, growing as R√(I² − Ic²) and approaching the ohmic line at large current; the curve is integrated from the junction equation and checked against that result at three currents. The voltage is not steady. It is a train of pulses, each the phase slipping by one turn, at a frequency of 483.6 GHz per millivolt: for a junction with Ic = 1 mA and R = 1 Ω, one unit of the voltage axis is 1 mV and 484 GHz.

The voltage that is a frequency

Two superconductors separated by a barrier a nanometre thick carry a current with no voltage at all, set by the difference of their quantum phases. Push harder and a voltage appears — and a voltage makes that phase difference run, so the current oscillates at 483.6 gigahertz for every millivolt. Shine microwaves on the junction and the voltage locks to exact multiples of the frequency divided by a ratio of fundamental constants, with nothing about the junction in it. That is why a volt is now counted in cycles.

 The potential at a point is where its walkers end up. A square divided into a 24-step grid, with its top edge held at a potential of 1 and the other three edges at 0. From the probe point (0.29, 0.71), 4000 random walkers each step to one of their four neighbours with equal chance until they touch an edge; six of them are drawn, each ending with a dot on the edge it reached. The fraction that end on the held edge is 0.405 ± 0.008, one standard error, after an average of 122 steps. Solving Laplace's equation on the same grid by repeatedly replacing every value with the average of its four neighbours gives 0.408, and the series solution for the continuous square gives 0.408. The walkers were never told the equation: a value that is the average of its neighbours and a probability of ending somewhere are the same arithmetic.

The potential is where the wanderers stop

Start a random walker at a point between charged conductors and let it wander until it touches one of them. The average potential of the surfaces the walkers touch is the potential at the starting point — exactly, with no equation solved — and the charge a conductor keeps at each place on its surface is the chance that a walker arriving from far away touches it there first.

 The interaction that orders a magnet is not the magnetic one. For six ordered magnets, two temperatures on a logarithmic scale: the energy of the magnetic interaction between two neighbouring moments, expressed as a temperature, and the temperature at which the material actually orders. iron orders at 1043 K against a dipolar scale of 0.201 K, a factor of 5182; cobalt orders at 1394 K against a dipolar scale of 0.117 K, a factor of 11961; nickel orders at 627 K against a dipolar scale of 0.015 K, a factor of 42312; gadolinium orders at 293 K against a dipolar scale of 0.790 K, a factor of 371; europium oxide orders at 69 K against a dipolar scale of 0.615 K, a factor of 112; lithium holmium fluoride orders at 1.53 K against a dipolar scale of 1.230 K, a factor of 1.24. The five ferromagnets order between a hundred and forty thousand times above the only interaction their moments have with each other, so whatever aligns them is not magnetism. The sixth is the control: lithium holmium fluoride is a magnet whose ordering really is dipolar, and its two temperatures agree.

What holds a magnet together is not magnetism

Two neighbouring moments in iron interact magnetically with an energy worth a fifth of a kelvin, and iron keeps its order to 1,043 kelvin. Whatever aligns them is five thousand times stronger than the only force they exert on one another — and it is electrostatic, with the exclusion principle deciding which of two spatial arrangements two electrons may use.

 Two energies with opposite slopes, and the width between them. The energy per unit area of a domain wall in iron, against the width the wall is assumed to have, split into the two terms that decide it. The exchange term falls as one over the width, because a reversal spread over more atoms turns through a smaller angle between each neighbouring pair. The anisotropy term rises in proportion to the width, because every atom inside the wall points away from an easy direction and pays for it. The sum has a least value, found here by scanning four hundred thousand widths: 92.9 nanometres and 4.46 millijoules per square metre. Neither constant is a dimension of the sample, so neither is the width: it is the first length in magnetism that belongs to the material. The assumed profile turns at a uniform rate, which is not the cheapest way to turn, so this width is 41 per cent above the conventional πδ of the exact profile, and this energy 11 per cent above the exact 4√(AK). A trial function can only ever overestimate, and eleven per cent is how much this one costs.

The first length that belongs to the substance

Every length in magnetism so far has been a length of the sample — a demagnetising factor is a shape, an avalanche cutoff is a sample's own restoring field. A domain wall's width is not. It is √(A/K), made of two material constants and nothing else, and across seven ordinary magnets it runs from two and a half nanometres to nine hundred.

 The barrier a reverse field takes away. The energy of a single-domain particle against the direction its moment points, in units of its anisotropy energy, for four strengths of reverse field along the easy axis. With no field the two directions are equally good and the barrier between them is exactly the anisotropy energy. A reverse field tilts the landscape and lowers the barrier out of the forward well as the square of the field: at 0.0 of the anisotropy field the barrier is 1.000 KV, at 0.1 of the anisotropy field the barrier is 0.810 KV, at 0.3 of the anisotropy field the barrier is 0.490 KV, at 0.6 of the anisotropy field the barrier is 0.160 KV. Each barrier is located by scanning two hundred thousand directions for the stationary points rather than by substituting the formula. The whole of the argument follows from the barrier being finite: a particle does not need the field that removes the barrier, only time enough to be shaken over what is left of it.

Nothing keeps a magnetisation for ever

A magnetised particle sits in a well with a barrier between it and the other direction, and a barrier of finite height is crossed eventually. So remanence has a lifetime, coercivity is a different number depending on how fast it is measured, and a grain below about twenty nanometres of iron forgets within a second at room temperature.

 Two fields of the same magnet, and inside they point opposite ways. A uniformly magnetised sphere, with the field lines of B on the left and of H on the right, both computed from the exact solution — uniform inside, a dipole outside. Outside the sphere the two pictures are identical up to a constant, because there B is μ₀ times H and nothing else. Inside they are opposite: B is 0.67 tesla pointing along the magnetisation and H is 267 kiloamps a metre pointing against it. The B lines close on themselves and never end; the H lines begin on the top face and end on the bottom, which is what a field with sources looks like. Nothing about the magnet changed between the two panels — only which currents the circulation is allowed to count.

The field that points against the magnet it is in

There are two magnetic fields in use and the difference between them is which currents a loop is allowed to count. The consequence nobody expects on being told the definitions: inside a permanent magnet H points the other way from B. It has to — a loop inside the magnet threads no wire, so its H circulation is zero, and the only arrangement left has H running backwards.

 A potential that is lower every time round. The magnetic scalar potential along a path circling a wire carrying 10 amps, against the angle turned through, for 2 complete circuits. Away from the wire the magnetic field has no circulation round any small loop, so it is the gradient of something — and it is, except that the something does not come back to its own value. Each circuit lowers it by exactly the current, 10 amps, and a second circuit lowers it by 10 again. The potential is perfectly good locally and has no single value globally, and the amount by which it fails to close is the current threaded. So nothing has been lost in going from a circulation to a potential: Ampère's law has been rewritten as a statement about the shape of the region the potential lives in.

A potential that does not come back to itself

Where no current flows, the magnetic field has no circulation round any small loop, so it is the gradient of something and a magnetic problem becomes an electrostatic one. The catch is not that the potential fails to exist. It is that walking once round a wire lowers it by the current, and walking round again lowers it by the current again — so Ampère's law survives the translation as a statement about what the path encircles rather than about where it went.

 Three bodies of one mass, one field outside, three fields inside. The gravitational field against distance from the centre, both in units of the body's surface values, for 3 spherical bodies of the same mass and radius: a uniform ball; a dense core under a light mantle; a hollow shell. Outside the surface the three curves are one curve — checked at 1.7 radii by adding up the pull of every mass element in each body, which agrees with the pull of a point of the same mass to better than a part in five hundred. Inside they part: the uniform ball's field falls in a straight line, the layered body's rises to 1.24 times the surface value at the top of its core, and the hollow shell's is zero throughout its cavity. Their moment-of-inertia factors are 0.400 (uniform ball), 0.330 (dense core under a light mantle), 0.551 (hollow shell) — a number that no measurement of the field outside can supply.

The field outside that cannot find the core

Gauss's law gives the field outside a body from what it encloses, and read backwards it is a limit. A uniform ball, a planet with an iron core and a hollow shell of the same mass have identical gravity everywhere outside. The field fixes a list of numbers — the mass, the flattening, higher moments — and leaves free everything else, including the moment of inertia; the Earth's core was weighed by its wobble, not by its pull.

 Circles that go nowhere, and a current across the line. Gyrating ions in a uniform magnetic field pointing out of the page, with 60 guiding centres drawn from a density that falls by a factor of e every 3 gyroradii to the right, and Maxwellian speeds. Every ion goes round clockwise and none of the circles moves. Of the circles that cross the dashed vertical line, those centred to its left cross it moving down and those centred to its right cross it moving up; in this sample 11 cross moving down and 7 moving up, a count a sample this small could turn either way. Because there are more circles on the left, the ions at the line move downwards on average over every speed and phase, at exactly the thermal speed squared over the gyrofrequency times L, 0.33 thermal speeds, computed by averaging over speeds and phases and checked against that value. It is a current, carried by circles whose centres are still.

The current no particle carries

A magnetised plasma holds its own pressure against the field only if a current flows across the pressure gradient, and the fluid equations say exactly how much. Follow the particles in a uniform field and none of them is going anywhere; every guiding centre is still. The current is real all the same. It is made of circles that are more crowded on one side of a line than the other, and when the field is not uniform, the drifts that do move the guiding centres flow the wrong way.

 A pendulum with unequal steps. The potential −EJ cos φ of the junction against its phase, at EJ/EC = 50, with the lowest 4 levels of the circuit drawn across the well between their classical turning points, in units of the charging energy. The transitions are 18.94, 17.79, 16.50, each smaller than the one below it; a harmonic well of the same curvature would space them all at the square root of 8·EJ·EC, 20.00. The spacing shrinks because the cosine is flatter than a parabola away from its bottom, and the shrinking is what lets a microwave pulse tuned to the lowest transition leave the next one alone.

The circuit that forgets its charge

A tiny superconducting island joined to its surroundings through a Josephson junction has discrete energy levels, and two of them make a quantum bit. The first such circuits were ruined by stray charges on nearby surfaces, which moved their levels and scrambled any superposition within a nanosecond. The cure was to make the junction's energy fifty times the charging energy. That makes the levels exponentially insensitive to charge while costing only a power-law loss in the unequal spacing that lets one transition be driven alone.

 A living cell and a dead one, told apart by a frequency. The real part of the dielectrophoretic factor for a living cell and a dead one, ten micrometres across, in a liquid of conductivity 0.01 S/m, against frequency. The cell is a conducting interior inside a membrane five nanometres thick. The living cell's intact membrane blocks slow fields, so at low frequency it behaves as an insulator and is pushed out of strong field; above 42 kHz the membrane is short-circuited by its own capacitance and the conducting interior is felt, so the cell is pulled in; above 140 MHz the permittivities take over and it is pushed again. The dead cell's membrane leaks, its interior has lost ions, and it changes sign only at 8.8 MHz. In 2 bands — below 42 kHz and from 8.9 MHz to 140 MHz — the same field pushes the two in opposite directions.

The force whose sign a frequency chooses

A neutral particle in a non-uniform field is pulled towards strong field if it polarises more than the liquid around it and pushed away if it polarises less. With conduction in the picture, which of the two it does is decided by how fast the field alternates — and a living cell, a conductor wrapped in an insulator five nanometres thick, changes sign twice, where a dead one changes once. The same electrodes can then send the living cells one way and the dead ones the other.

 The block has to move, and which way decides the momentum. A pulse of light crosses a free block of glass of index 1.5 and length L, with time in units of L/c. The pulse falls behind where it would have been in vacuum by 0.50 L while it is inside. The block's displacement, in units of EL/Mc², is obtained from momentum conservation alone: if the light inside carries Abraham's E/nc the block takes up the rest of E/c, moves forward, and ends 0.500 units along — Balazs's (n − 1). The sum of the pulse's lag and the block's shift, which is the centre of energy's departure from uniform motion, is then zero to within 0.0004 units at every moment. If instead the light carried Minkowski's nE/c, the block would have to recoil backwards and the centre of energy would end 1.25 units behind a straight line, which no isolated system can do. For a one-joule pulse through a kilogram of glass ten centimetres long the real shift is 5.6 × 10⁻¹⁹ m.

The momentum light carries into glass

A pulse of light in glass has two momenta, nE/c and E/nc, and for a century the argument was about which one is right. Both are measured. A free block that the light crosses must move forward, which only Abraham's value explains; a Cherenkov cone exists at all, which only Minkowski's allows. They answer different questions, and the difference between them is momentum carried by the glass.

 Where a moving charge keeps its field's energy and its momentum. Left: the density of field energy and of field momentum around a charged shell moving slowly to the right, by direction, each drawn as the mass per unit volume it contributes — energy density over c², momentum density over velocity. The energy density is the same in every direction; the momentum density goes as sin²θ, zero ahead and behind and largest broadside, because it is ε₀E × B and the magnetic field of a moving charge vanishes along its line of motion. Broadside it is twice the energy's; averaged over all directions sin²θ is 0.6667, and twice that is the 4/3. Right: the fraction of the field's energy, and of its momentum, lying within a distance r of the centre of a shell of radius a. Both are 1 − a/r, because both densities fall as the fourth power of distance: half of each lies within 2a and nine-tenths within 10a.

The mass a charge's field gets wrong by a third

Give a charged sphere's field its energy U and divide by c², and the field has a mass. Move the sphere and ask the field's momentum what the mass is, and it says four-thirds of that. The discrepancy took from 1881 to the 1960s to understand, and the answer is not a better calculation of the field. It is that a charge cannot hold itself together, and whatever does has energy too.

 A linear quadrupole, and the cone on which it vanishes. Equipotentials of a linear quadrupole — charges +q, −2q, +q in a line, a distance d apart — in a plane containing the line, positive in one colour and negative in the other. The total charge is zero and so is the dipole moment, so far away the potential falls as 1/r³ with the angular pattern 3cos²θ − 1: positive along the axis, negative around the waist, and zero on a cone. Found on circles of growing radius, the zero of the exact potential lies at 50.89° at 2d, 53.76° at 4d, 54.49° at 8d, 54.67° at 16d, closing on arccos(1/√3) = 54.74°, the dashed lines.

The shape that takes five numbers

A charge distribution's net charge is one number and its dipole three. The next term, the quadrupole, is where the distribution's shape first reaches the far field, and it is a table of nine numbers of which only five survive: two for the shape and three for which way it points. Its field vanishes on a cone at 54.74° — the same angle at which a spinning sample makes a nuclear magnetic resonance spectrum sharp.

 The same nudge in four kinds of space. A body on a circular orbit given 5 per cent more speed, integrated for up to 6 turns in the field a point mass would have in spaces of 2, 3, 4, 5 dimensions. In 2 dimensions it stays between 1.00 and 1.10 of the original radius; in 3 dimensions it stays between 1.00 and 1.23 of the original radius; in 4 dimensions it leaves after 1.93 turns; in 5 dimensions it leaves after 1.19 turns. In three dimensions the orbit is an ellipse that closes; in two it is a rosette that does not close but never wanders far; from four dimensions up there are no stable orbits at all, and a planet nudged by anything would spiral away or plunge.

The number of dimensions a planet can orbit in

Gauss's law says that a point source's field is its strength spread over the surface around it, and the size of that surface depends on how many dimensions space has. In three it grows as the square of the distance, and the field falls as the inverse square. Count differently and the field falls differently — and above three dimensions it falls so steeply that no planet could hold a stable orbit and no atom could hold its electron. The inverse-square law is not a fact about gravity or electricity. It is a fact about space, and it is one of the conditions for there being anything in space to orbit.

 Three ships, three local times, one clock. The fair-weather field at sea level, in the same model, as recorded by ships at longitudes 120° W, 0°, 120° E, each plotting it against its own local solar time. An ordinary daily effect — sunshine, heating, turbulence — would peak at the same local time everywhere. This one peaks at 11.4, 19.4, 3.4 hours local time in turn, because every ship is reading the same global potential, which follows the world's storms and peaks at one moment in Universal Time. Seeing the maxima line up in Greenwich time rather than local time, over clean ocean air far from any storm, was how the global circuit was demonstrated: the field in fair weather is the far end of a circuit whose generators are thunderstorms thousands of kilometres away.

The field that keeps Greenwich time

Over a calm ocean, a thousand kilometres from any storm, the air carries a downward electric field of about a hundred volts per metre, and it rises and falls by a sixth once a day. It peaks at the same instant on every ship on every ocean — near seven in the evening in Greenwich, whatever the local clock says. Nothing local can do that. The field is the far end of a circuit whose upper plate is a single conductor round the whole planet, and a conductor has one potential, so every patch of fair-weather sky is reading the sum of all the world's thunderstorms at once.

 A field strings a suspension into chains. 48 identical polarisable spheres, two radii apart at least, scattered at random through a periodic cell twenty radii square — 38 per cent of its area — and the same spheres after the field (vertical) has been on long enough for the viscous liquid to let them move. Each moves in proportion to the dipolar force from all the others and is stopped only by contact. At the start no two are touching; at the end 45 are bonded end to end along the field and 48 of the 48 spheres belong to chains, lying along the field lines and kept apart sideways because side-by-side dipoles repel. Chains that reach from one electrode to the other are what make the liquid resist shear, and they form in the milliseconds it takes a sphere to travel its own diameter.

The liquid a field turns solid

Stir fine starch or glass beads into silicone oil and the result pours like cream. Put a few kilovolts across it and within milliseconds it holds its shape and will not flow until it is pushed hard. Nothing in it is charged. Each particle becomes a dipole in the field, dipoles pull on each other end to end and push apart side by side, and so the particles string themselves into chains from one electrode to the other — a solid made of forces that exist only while the voltage is on.

 The drag peaks and then gives way to lift. The two forces on a magnet moving parallel to a thin conducting sheet, as fractions of the repulsion a perfect mirror image would exert, against the magnet's speed divided by the sheet's characteristic speed w = 2/μ₀σd, on a logarithmic axis. Drag grows in proportion to the speed at first, peaks at 1.27 w with 0.300 of the full repulsion, and then falls as w/v. Lift starts as the square of the speed, equals drag at v = w, where both are 0.293, and approaches the full repulsion at high speed: 0.900 at 10 w. The same eddy currents cause both. What changes with speed is how much of the image they form has time to sink away before the magnet has passed.

The drag that falls as the magnet speeds up

Drop a magnet down a copper pipe and it creeps, because the currents it stirs up in the metal pull back on it in proportion to its speed. Push it faster and the proportion fails: the drag rises to a peak and then falls away, while a second force, pointing straight up, grows to replace it. A magnet moving fast enough over a sheet of aluminium is not braked but held up, with lift thirty times its drag at the speed of a fast train, and the reason is that at high speed the metal has no time to forget the magnet has passed.

 London's attraction, and the distance where it gives out. The dispersion energy between two identical atoms times the sixth power of their separation, relative to London's constant, against the separation in units of c/ω₀ — the reduced wavelength of each atom's strongest transition — on logarithmic axes, from the Casimir–Polder integral with a one-resonance polarisability. Close in, the product is one: London's inverse sixth power. It has fallen to a half by 3.82 c/ω₀, and far out it falls as one over the distance, so the energy goes as the inverse seventh power, reaching 23.2 per cent of London's value at ten reduced wavelengths and 2.44 per cent at a hundred. For two helium atoms, whose strongest line is at 58 nm, c/ω₀ is about 9 nm; for two sodium atoms, 94 nm.

The attraction that weakens when light is too slow

Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.

 Four places where three charges' fields cancel. Field lines in the plane of three equal positive point charges at the corners of an equilateral triangle, with every point where the field is zero marked. There are four: one at the centre, and three more on the lines from the centre to the middle of each side, 0.164 of a side from the centre. Four is (n − 1)² for three charges, the largest number Maxwell conjectured any three charges could make. Each zero is a saddle: along some directions the field pushes a test charge back and along others away, so none of them can hold anything still.

The zeros a handful of charges can make

Between two equal charges there is a point where their fields cancel. With three there can be two such points, or one, or four. Every such point is a saddle, never a place where a charge could rest. In two dimensions the count is fixed by algebra: n equal line charges make exactly n − 1 zeros, the roots of a polynomial's derivative. In three dimensions nobody knows the answer. In 1873 Maxwell conjectured that n point charges can make at most (n − 1)² zeros. Three charges on a triangle make exactly four, and the conjecture has not been proved even for three charges.

 How much reflected light keeps its handedness. Circularly polarised light reflected from a flat surface: the fraction of the reflected power that keeps its handedness, against the angle of incidence, for a perfect conductor, for glass, n = 1.5, for a medium with ε = μ = 1.5 (the same index, and the same impedance as empty space), and for one in between. A metal mirror reverses the handedness of every photon at every angle. Glass reverses it at normal incidence, returns an even split at Brewster's angle (56.3°, where only one linear polarisation reflects), and keeps 60 per cent at 60°. The medium with ε = μ keeps it at every angle: it respects the duality symmetry, and so conserves what the symmetry conserves.

The handedness a mirror cannot keep

Maxwell's equations in empty space do not change if the electric field is turned partly into the magnetic field and the magnetic partly into the electric. A symmetry that exact must hand over a conserved quantity, and this one hands over the handedness of light — how many more photons spin one way than the other. Every mirror breaks it: circularly polarised light comes back from a mirror with its handedness reversed. Only a material that responds to the magnetic field exactly as strongly as to the electric one keeps it, and such a material, it turns out, sends nothing straight back.

 Two pieces of current that one law makes push each other unequally. Two short elements of current a unit distance apart: element 1 pointing along the line joining them, element 2 at 45° to it. Left, the forces by the law that follows from Biot and Savart (Grassmann's): element 1 pushes nothing on element 2, because 1 points straight at it, while element 2 pushes on element 1 with a force of 0.71 units at right angles to it; the pair would feel a net 0.71 units. Right, the forces by Ampère's own law of 1826: 0.71 units each, along the line joining the elements, equal and opposite. Both laws give the same force between any two complete circuits, so no experiment on closed circuits can say which is right.

The force between pieces of a current

Ampère spent six years finding the force one short piece of current exerts on another, and the law he found is not the one used today. The modern law, which follows from the Biot–Savart field, disagrees with his for almost every pair of pieces — his obeys Newton's third law and the modern one does not. Both give exactly the same force between any two complete circuits, which is the only thing that has ever been measured. The force between two pieces of current is a quantity with two correct values, because a piece of current on its own does not exist.

 Three shapes of film, one sheet resistance. Three thin films of the same material, of sheet resistance one ohm per square, with four small contacts on each edge labelled A to D in order. For each, the two readings van der Pauw's method takes — the voltage between D and C per unit current from A to B, and between A and D per unit current from B to C — computed by solving for the current flow on a fine grid. For a square, contacts at the corners: 0.221 and 0.221 Ω, giving 1.000 Ω per square; a disc, contacts unevenly spaced: 0.089 and 0.450 Ω, giving 0.999 Ω per square; an irregular lamina: 0.209 and 0.232 Ω, giving 0.999 Ω per square. The readings change with the shape and with where the contacts are; the sheet resistance recovered from them by exp(−πR₁/Rₛ) + exp(−πR₂/Rₛ) = 1 does not.

The sheet resistance that forgets the shape

To measure how well a thin film conducts, the obvious method is to cut it into a neat strip and pass a current along it. In 1958 Leo van der Pauw, an engineer at Philips, showed that the strip is unnecessary. Put four small contacts anywhere on the edge of a film of any shape, take two readings, and one equation returns the film's resistance per square — the shape and the spacing of the contacts drop out entirely. The proof is a single fact about logarithms and the maps that preserve angles, and it fails for exactly one kind of film: one with a hole in it.

 Rotation is a magnetic field, and a superconductor cancels it. Magnetic fields along the spin axis inside a spinning metal, in units of 2mω/e, as seen by its electrons in the frame turning with it. In that frame the Coriolis force on an electron is exactly the force a magnetic field of 2mω/e would exert, pointing against the spin for a negative charge (Larmor's theorem). A normal metal spun in no field has no real field inside, and its electrons feel the rotation's equivalent field in full. A superconductor will not let its carriers feel any field in their interior — that is the Meissner effect — so it generates a real field of exactly the same size pointing the other way, and the sum vanishes. The real field is the London moment.

The field a spinning superconductor makes of itself

Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched.

 Two ways to be wrong, and the apparatus that sees each. The fractional sag of the potential at the centre of a charged spherical shell below the shell's own, on a logarithmic axis, against the shell's radius in metres, also logarithmic, for two different departures from Coulomb's law. If the force fell as 1/r^(2+ε) with ε = 10⁻¹⁶, the sag is 3.07·10⁻¹⁷ whatever the shell's size (dashed): the law has no length in it, so neither does its failure. If the photon had a mass of 10⁻¹⁴ eV, the sag is (μR)²/6 (solid): 4.3·10⁻²⁰ for a centimetre shell and 4.3·10⁻¹⁶ for a metre. A mass introduces a length, and the signal grows as the square of the apparatus compared with it — so the strongest bounds on a photon mass come from the largest fields, not the most sensitive laboratories.

The mass that would make Gauss's law leak

Gauss's law says the flux out of any closed surface counts the charge inside, whatever the surface's size. If the photon had even a tiny mass, that would stop being true: the field of a charge would die off exponentially beyond a length ħ/mc, the flux through a large sphere would fall short of the charge, and the inside of a charged shell would no longer be field-free. There are two different ways Coulomb's law could fail, and they are caught by opposite strategies. A wrong exponent has no length in it and shows equally in any apparatus; a mass has a length, and shows as the square of the apparatus's size. That is why the tightest limit on the photon's mass comes not from a laboratory but from the magnetic field the solar wind carries past Pluto.

 The Casimir pressure between warm plates, from vacuum to heat. The attractive pressure between two parallel, perfectly reflecting plates at 300 K against their separation, on logarithmic axes, from the Lifshitz sum over Matsubara frequencies. Dashed: the zero-temperature law π²ħc/240a⁴. Dotted: the classical law ζ(3)kT/4πa³, which contains no Planck constant. At 1 μm the full pressure is 0.0013 Pa, within 0.2 per cent of the vacuum value; at 10 μm it is 3.96·10⁻⁷ Pa, 3.05 times the vacuum value and within 0.002 per cent of the classical law. The two laws cross at 3.28 μm; the thermal length ħc/kT is 7.63 μm. The second solid curve removes one term — the static magnetic fluctuation, which a metal of finite conductivity does not reflect — and sits 15 per cent lower at 1 μm and a factor of two lower far out. The plates are otherwise ideal mirrors: a model, chosen to isolate that one term.

The force that forgets Planck's constant

Two mirrors in a vacuum attract, with a pressure that contains Planck's constant and nothing else about the world but the speed of light. Warm the mirrors and pull them apart, and past a few micrometres at room temperature the Planck constant drops out of the answer. The force is still there and is now larger than the vacuum alone would give, but it has become classical: a count of thermal fluctuations of the static field, with kT of energy each. Half of that classical force comes from one fluctuation — a static magnetic field — and whether a real metal mirror reflects it is a question that has divided the people who measure the force for twenty years.

 Four plasma columns that need exactly the same current. Left: four pressure profiles across a 1 cm column — flat-topped, parabolic, peaked and hollow — each holding the same 10¹⁹ ions per metre at the same 10 keV, so the same total ∫2πr p dr. Right: the current each needs to hold itself together, from solving the force balance J × B = ∇p with Ampère's law, radius by radius. All four come out at 0.800, 0.800, 0.800, 0.800 MA: Bennett's 0.800 MA. The shape of the column decides where the current flows and how strong the field is inside it, and has no effect at all on how much current there must be.

The current that squeezes what carries it

A current flowing along a column of hot gas makes a magnetic field that wraps round the column, and the field pushes the current inward. If the column is a plasma, free to move, it is squeezed. Ampère's law and a balance of forces then fix how much current a column of given temperature needs to hold itself together — and the answer does not depend on how wide the column is or how its plasma is arranged inside. Ten thousand electronvolts in ten million million million ions per metre needs 0.8 million amperes, whatever the shape. Above about 1.4 million amperes the same arithmetic says a hydrogen column radiates faster than its current can heat it, and collapses.

 Three cylinders that focus electrons. An einzel lens: three coaxial tubes of radius R, the outer two at the potential the electrons arrived with and the centre one, 2R long, at 4 times it. Electrons enter from the left parallel to the axis at five heights, from 0.1R to 0.7R, and are traced through the field without approximation. In each gap the equipotentials bulge between the tubes, and an electron crossing them is refracted as at a curved glass surface: pushed towards the axis where the potential is low and away from it where the potential is high. It is slower where it is pushed inward, so it spends longer there, and each gap converges on balance; the electron leaves with the speed it came in with. The rays cross the axis at 4.00R, 3.92R, 3.78R, 3.56R, 3.25R beyond the lens centre: the outer ones are bent too much and cross first, which is spherical aberration.

The voltage that bends an electron like glass

An electron that has fallen through a potential difference V moves at a speed proportional to √V. Crossing a surface of constant potential, it keeps the part of its momentum that runs along the surface and changes the part across it, and the two rules together are Snell's law with √V in place of the refractive index. Every arrangement of charged electrodes is therefore a piece of glass for electrons. Three metal tubes make a lens that focuses whichever way its middle tube is charged — and, by a theorem proved in 1936, no lens built from round electrodes can be free of spherical aberration. That theorem kept electron microscopes some fifty times coarser than their wavelength for sixty years.

 One number decides the efficiency. The best efficiency a link of two resonant circuits can reach, with its load chosen to maximise it, against the figure of merit U = k√(Q₁Q₂) on a logarithmic scale: the coupling coefficient times the geometric mean of the two coils' quality factors. The curve is η = U²/(1 + √(1 + U²))², and nothing else about the coils, the frequency or the distance enters: 0.25 per cent at U = 0.1, 17 per cent at U = 1, 52 per cent at U = 3, 82 per cent at U = 10, 94 per cent at U = 30. A weak coupling is rescued by a high Q: k = 0.003 with Q = 1,000 on each side gives U = 3 and 54 per cent, where the same coils used without tuning capacitors, as a plain transformer, deliver under one per cent, with the source driving mostly reactive current that does nothing.

The coupling a resonance rescues

Two coils a hand's breadth apart share a few thousandths of their magnetic flux, and as a transformer they deliver almost nothing. Tune each with a capacitor to the same frequency and the same pair can carry most of the power across. Resonance does not strengthen the coupling, which is fixed by the geometry and falls as the cube of the distance. It changes how many times the energy goes round before it is lost. The efficiency then depends on one number, the coupling times the quality factor, and that one number explains a toothbrush on its stand, a phone on its pad, a car parked over a plate in the road and a light bulb lit two metres away.

 The voltage a gas gap can hold, on one curve. The breakdown voltage of a gap in air between flat electrodes against the product of pressure and gap width, on logarithmic scales — Paschen's law, in which pressure and gap enter only as their product because both count how many collisions an electron makes on the way across. Townsend's model, with γ = 0.01 electrons freed per ion at the cathode, has its minimum of 305 V at 1.12 Pa·m — a gap of 11 μm at atmospheric pressure, or 112 Pa across a centimetre. Measured minima in air are about 330 V. To the left the gap is too short, or the gas too thin, for an electron to make enough collisions, and the voltage climbs steeply; below about 0.41 Pa·m there is no gas breakdown at any voltage. To the right, collisions are so frequent that an electron gains too little energy between them, and the voltage rises nearly in proportion to pd. Dashed, the empirical law for air at larger pd: 31.4 kV for a centimetre at one atmosphere, against the model's 35.5.

The spark that needs room to start

Air is an excellent insulator until it is not, and the usual statement of when — three megavolts a metre — is true only for gaps of about a centimetre. A gap breaks down when one electron's avalanche of ionisation sends back enough ions to the cathode to free one more electron, and that condition depends on the pressure and the gap only through their product. The result is a curve with a floor: about 330 volts in air, below which no gap of any width will spark, and above which both a thinner gas and a denser one insulate better.

 Two stiffnesses, positive together only in a band. The stiffness of the trap for a field-following magnet above a ring base, against height in units of the ring's radius: vertically, proportional to the curvature of the field strength along the axis; sideways, proportional to the curvature of |B| across it, B′²/4B − B″/2. Below half a radius the field's fall is still steepening, the vertical stiffness is negative, and a magnet nudged downward keeps going. Above √(2/5) = 0.632 radii the field strength no longer rises away from the axis, the sideways stiffness is negative, and a magnet drifts off to the side. Only between 0.500 and 0.632 radii are both positive — 6.6 mm of height for a 10 cm ring. For the nominal top, floating at 0.565, the trap's natural frequencies are 1.36 Hz vertically and 0.82 Hz sideways.

The window a spinning magnet floats in

No arrangement of fixed magnets can hold another magnet still in mid-air, and a spinning top made of a magnet floats above a magnetised base for minutes. The spin does not cancel the theorem; it changes what the top's energy depends on, from one component of the field to the field's strength, which can have a minimum. But the minimum exists only in a band of height a few millimetres thick, for a weight right to a couple of per cent, at a spin neither too slow nor too fast, and the well holding the top is about as deep as the energy of a twenty-gram weight dropped fifty micrometres.

 The armature sits where the field has halved. The magnetic field halfway between two parallel rails carrying a current out along one and back along the other, against the distance along the barrel in units of the rails' spacing, as a fraction of the field between infinitely long rails. The rails end at zero, where the armature closes the circuit. A few spacings behind the armature the field is the full value: 98.5 per cent at two spacings back. At the armature it is exactly half, because each rail there is a wire that stops, and a wire that stops gives half the field of one that does not. Ahead of the armature it falls away: 15 per cent half a spacing in front. The current in the armature crosses this field and is pushed forward; the push is the current times the field it sits in, which is half the field behind it, and that is where the half in ½L′I² comes from.

The push at the end of a pair of rails

A railgun is two metal rails, a sliding conductor across them and a very large current, and the push on the slider is half the inductance per metre times the current squared. The half is not a fudge. The slider sits at the end of two rails that stop, and a wire that stops gives exactly half the field of one that does not. Followed through, the same circuit pushes its own rails apart six times harder per metre than it pushes the projectile, stores as much energy in its field as it gives the projectile, and wastes less the faster it shoots.

 A cage of wires in a uniform field. Equipotential lines round a cage of 12 parallel wires, each 2 per cent of the spacing in radius, joined so that they share one potential and carry no net charge, in a uniform field running left to right. Outside, the lines bunch and bend round the cage as they would round a solid cylinder. Inside, they do not vanish: they run straight across, carrying a uniform field 25.8 per cent as strong as the one outside (dashed, drawn at a fifth of the outside spacing because the potential differences inside are so much smaller). The wires' charges are computed so that every wire is at the same potential.

The field a ring of wires lets through

A cage made of wires is supposed to work almost as well as a solid box, because the field that leaks through the gaps dies away exponentially within a gap's width. The exponential is real and it is the wrong part of the story. It describes the ripple the separate wires make; underneath it the average field goes straight through, at a level that falls only in proportion to the spacing and depends on the wires' thickness through a logarithm. A cage of twelve thin wires lets a quarter of the field in.

 How a field enters a superconductor that pins it. The magnetic field inside a slab of type-II superconductor, against position across its thickness, as an applied field is raised from zero, in units of the full-penetration field H = Jc a. Each line is the profile at an applied field of 0.5 H, 1.0 H, 1.5 H, 2.0 H. The field enters from both faces as a wedge whose slope is the most current the pinned vortices can carry, Jc; at 0.5 H it reaches halfway to the middle, at H* it just reaches the middle, and beyond that the whole slab carries Jc and the profile rides up with the applied field. A type-I superconductor would show no field inside at all, and a normal metal the applied field everywhere.

The field a superconductor keeps after all

A superconductor is famous for throwing magnetic field out, and the experiment that showed it is the one that made superconductivity a state of matter. Most superconductors that are actually used do something closer to the opposite: they let the field in as threads, grip the threads where they are, and keep them when the field is taken away. The result is a slope of field inside the material that is as steep as the grip allows and no steeper, which is all that is needed to explain why such a superconductor has a hysteresis loop, heats up in an alternating field, can hang a magnet underneath itself, and can be made into a magnet stronger than any of iron.

 Jefimenko's two sources of a magnetic field. The magnetic field one metre from a long wire whose current is switched on, against time in light-metres, split into the two terms of Jefimenko's equation: the current at the retarded time divided by the square of the distance (solid blue), which is Biot and Savart's law with each piece of the wire seen as it was when its light left; and the rate of change of the current at the retarded time divided by c times the distance (solid green). Their sum (red) is the field. Nothing arrives before one metre of light-time. At 1.5 light-metres the two terms are 0.68 and 0.78; at 4, 0.97 and 0.07; the second term dies away as the switch-on recedes along the wire, and the first grows to the steady field. Neither term contains a changing electric field.

The field a changing field does not make

Faraday's law says a changing magnetic field comes with a curling electric field, and Maxwell's correction to Ampère's law says the reverse, and the two are almost always read as a chain of causes: each field makes the other, and that is how light carries itself across empty space. Solved for what the fields actually are, Maxwell's equations say something different. Every field at every place is an integral over the charges and currents on the observer's past light cone, and over nothing else. The changing fields are not sources; they are two descriptions of the same retarded currents, and the laws that relate them hold because both are computed from one history.

 The field along the axis of two coils. The magnetic field along the common axis of two identical coaxial loops carrying the same current, against distance from the midpoint in units of the loops' radius, each normalised to its value at the midpoint, for separations of 0.6, 1, 1.4 radii. Closer than one radius the field peaks at the centre; further apart it dips there between two peaks; at exactly one radius — Helmholtz's spacing — the curvature at the centre vanishes and the field is flat, staying within a part in a thousand of its central value out to 0.17 radii either side.

The two coils one radius apart

A single loop of wire makes a field that is strongest at its centre and falls away in every direction. Two loops on the same axis, spaced exactly one radius apart, make a field that is flat at the midpoint to a part in a thousand over a region a third of the coils' size — and flat sideways as well as along the axis, although only the axis was designed. The sideways uniformity is free because a magnetic field in empty space obeys Laplace's equation, which ties its variation in one direction to its variation in the others, term by term.

 A ring pushed and pulled, and pushed a little more. Over two cycles of the alternating supply, for a ring whose reactance equals its resistance (ωL/R = 1): the flux through the ring (grey), which also sets the radial field that pushes the ring along the core; the voltage induced round the ring (dashed); the current, which lags that voltage by 45° because of the ring's own inductance (blue); and the force along the axis, current times radial field (red). The force pushes the ring away from the coil for part of each half-cycle and pulls it back for the rest; with the lag, the pushes outweigh the pulls, and the average (dotted) is 0.250 of the peak flux–current product. With no lag the pushes and pulls would cancel exactly. The ring jumps because its current is late.

The ring that jumps because its current is late

Slip an aluminium ring over the iron core of a coil, switch on the alternating current, and the ring leaps off the top. The usual explanation is Lenz's law: the ring's induced current opposes the change, so the ring is repelled. But an alternating field changes both ways every cycle, and an induced current that simply opposed the change would be pushed and pulled equally and stay put. The ring jumps only because its current runs late, held back by its own inductance — which is why a cooled ring jumps far higher and a ring with a saw cut through it does not move.

 Water drops that charge two cans to a spark. The voltage between the two cans of a Kelvin water dropper against time, on a logarithmic axis, starting from a stray difference of 1 V, for 10, 20, 40 drops a second in each stream; drops 3 mm across, each with a capacitance of 0.17 pF to its ring, cans of 20 pF. Each drop carries a charge induced by the opposite can's potential into its own can, which raises the opposite can's induced charge on the next drop: the difference grows exponentially, multiplying by e every 6.0 s at 20 drops a second. It passes the 10 kV that will jump a few millimetres of air after 55 s — from one volt to a spark, with nothing but falling water.

The water drops that charge themselves to a spark

Two streams of water drip from one earthed pipe through two metal rings into two metal cans, and each ring is wired to the opposite can. Nothing is charged to begin with, nothing is rubbed, nothing is plugged in. Within a minute a spark cracks between the cans at ten thousand volts. A stray volt between them makes the drops forming in one ring pick up a slight charge, which they carry to the can that makes the other ring's drops pick up the opposite charge — a loop in which every drop strengthens the imbalance that charged it. Kelvin built it in 1867; it is a dynamo made of water and gravity.

 A current limited by the wire's circumference. The largest supercurrent a wire can carry before the field of its own current destroys superconductivity at its surface, Ic = 2πaHc, against the wire's diameter, for lead, mercury and tin at 1.5 K, on logarithmic axes; dashed, a niobium–titanium wire carrying 3,000 A/mm² in a field of 5 tesla, which grows with the cross-section rather than the circumference. A lead wire 1 mm across carries 192 A — 244 A/mm² — and one 1 cm across only ten times as much, at a tenth of the density. Below 81 micrometres lead carries more than niobium–titanium; above it niobium–titanium wins by a factor that grows with the wire, and it does so in a field of 5 tesla, where lead cannot superconduct at all.

The current a superconductor stops with its own field

A superconducting wire has no resistance, so it seems it should carry any current at all. It carries a definite amount and no more, and the amount is set not by the metal's ability to conduct but by the field the current itself makes: when that field at the wire's surface reaches the critical field, superconductivity fails there. The rule makes the largest current proportional to the wire's radius rather than its area, makes a thick wire worse per square millimetre than a thin one, and explains why the first superconducting magnet, wound in 1913, did not work.

 Electrons that reach the anode, and electrons that turn back. Paths of electrons leaving a cathode 4 mm across at rest, in a cylindrical diode with an anode 8 mm across held at 4000 V, with no space charge, for axial magnetic fields of 0.80, 0.98, 1.02 and 1.30 times Hull's cut-off field of 142.2 millitesla; computed by integrating the Lorentz force. Below the cut-off the field only bends the path, and the electron still reaches the anode — at 0.98 of the cut-off it arrives almost tangentially. Just above it the electron turns back just short of the anode and returns to the cathode, then repeats the arc further round; at 1.30 times the cut-off it turns back well inside. Nothing reaches the anode, and the current stops completely within a few per cent of the field.

The field at which no electron reaches the anode

Put a hot wire inside a metal tube, make the tube positive, and electrons stream across from wire to tube. Add a magnetic field along the axis and they curve on the way, and still arrive — until, at one definite field, every one of them turns back a hair's breadth short of the tube and the current stops dead. The field at which it happens can be found without knowing anything about how the voltage is distributed between the wire and the tube, because one quantity the electron carries is fixed by the magnetic field alone. The microwave oven runs just inside that line.

 The diamonds where the island's charge is fixed. The stability diagram of a single-electron island with identical junctions: bias voltage (units of e/CΣ) against the charge the gate induces (units of e). Inside each shaded diamond, labelled with the number of extra electrons the island holds, every possible tunnelling step costs energy and no current flows; outside, at least one step pays for itself and electrons pass one at a time. The diamonds repeat every time the gate charge rises by one electron, and touch at gate charges of e/2, 3e/2, … where two charge states cost the same and the island conducts at any bias. Their height is e/CΣ in each direction. Probing the model at three points outside the diamonds gives currents of 0.60, 0.25, 0.53; inside them, nothing.

The island that takes electrons one at a time

A capacitor holds charge in proportion to its voltage, and the proportion is a property of its shape — on any piece of metal large enough to see. Make the metal small enough and cold enough, and joined to the world only by tunnel junctions, and the proportion breaks into steps: one electron's charge costs more energy than the heat or the battery can supply, so no current flows at all until it can, and then the electrons cross one by one. A charge of a single electron becomes something a voltmeter can see.

 The tilt that starts the reversal, in a well of soft crystal. A planar defect three wall parameters wide (three times √(A/K)) in which the crystal's anisotropy falls to zero, drawn as the anisotropy K(x) across it; the curve is the shape of the first tilt of the magnetisation to become unstable as a reverse field is raised, the lowest mode of −A d²/dx² + K(x), and the dashed line is its eigenvalue. The equation is Schrödinger's, with the exchange stiffness A in place of ħ²/2m and the anisotropy as the potential, and the reversal begins in its ground state. Here the ground state lies at 0.373 K, so the magnet reverses at 37 per cent of the anisotropy field. The mode reaches out of the well a distance of about √(A/K) on each side, because exchange will not let the tilt change abruptly.

The magnet that reverses where its crystal is weakest

A perfect crystal of neodymium-iron-boron should hold its magnetisation against a reverse field of 7.7 tesla. The best magnets made of it give way at one to three. The shortfall, known for eighty years as Brown's paradox, is not a failure of the crystal but of its edges: reversal begins wherever the crystal is weakest, in a layer a few nanometres thick. Working out how weak that layer can afford to be turns out to be a problem already solved in quantum mechanics — the nucleation field is the ground-state energy of a particle in a well shaped like the defect.

 A row of magnets with its field on one side. Two wavelengths of a Halbach row: blocks of permanent magnet, 4 to a wavelength, each magnetised a 90° turn from the last (block arrows), with the magnetic field drawn as arrows above and below, their lengths proportional to its strength. Just above the row the field is 5 times stronger than the same distance below it. The magnetisation turning one way along the row puts the field on that side; turned the other way, it would put it underneath. With finer blocks the weak side empties further, and a continuously turning magnetisation leaves it with no field at all.

The magnets that put their whole field on one side

Lay bar magnets in a row, each turned a quarter turn from the last, and the row's field appears almost entirely on one side, with almost nothing on the other — although every magnet in it, alone, is perfectly symmetric. Turn the magnetisation smoothly instead of in steps and the weak side has no field at all, exactly. Nothing is shielding it. The two halves of the pattern each make a symmetric field, and their fields line up on one side and cancel on the other, because a magnetisation that turns as it goes can only make the field that decays in one direction.

 Where the energy of a charged capacitor goes. The energy stored in a 1 nF capacitor charged to 10 V, in an identical empty one connected to it through a 100 Ω resistance, and dissipated as heat, as fractions of the starting energy, against time in units of the time constant RC/2 (50 ns here). Charge flows until both capacitors sit at 5 V. Each then holds a quarter of the original energy, and the other half has gone into the resistance, 86 per cent of it within the first time constant. The stored total (dashed) falls to one half and stays there.

The half a wire always takes

Charge a capacitor, connect it to an identical empty one, and wait. Charge flows until both sit at half the voltage, and when it stops, the two together hold exactly half the energy the first one started with. The other half has gone, and it has gone whatever the wire was made of: a thick copper bar and a megohm resistor take the same half, one in picoseconds and the other in milliseconds. Remove the resistance altogether and the energy still goes, as radio waves. The loss is not a property of the wire. It is what sharing a conserved quantity between two quadratic stores always costs.

 The field round a thin wire. The electric field against distance from the axis, on a logarithmic axis, for a wire inside a coaxial cylinder of radius 10 cm at 60 kV, for wire radii of 0.2, 1 and 5 mm; the dashed line is 30 kV/cm, the field at which air breaks down across a gap of a few centimetres, and each curve starts at its wire's surface. The field falls as 1/ρ, so it is concentrated at a thin wire: 483 kV/cm at the surface of the 0.2 mm wire, 130 at the 1 mm wire and 40.1 at the 5 mm one. The field exceeds 30 kV/cm only within 3.2 mm of the axis for the thinnest wire, 4.3 mm for the middle one and 6.7 mm for the thickest; across the remaining nine centimetres the air is far below breakdown, so ionisation can start at the wire and has nowhere to go.

The glow a thin wire wears before it sparks

Raise the voltage on a thin wire inside a metal tube and, long before anything sparks across, the wire starts to glow faintly violet and to hiss. The air right next to it is ionising, and the air a millimetre further out is not. The field round a wire falls away so fast that an avalanche of ionisation can start at the surface and find nowhere to go. That partial breakdown, corona, costs power lines energy in the rain, sets the size of the wires they are made from, cleans the smoke from chimneys and counts particles in detectors — and the field it needs is not a property of air but of the wire's radius.

 An electron held above helium by its own reflection. The potential energy of an electron at height z above liquid helium (ε = 1.0572), attracted by its image, −Λe²/4πε₀z with Λ = 0.00695; the surface itself is a wall the electron cannot cross. The lowest three levels (horizontal lines) are −0.657, −0.164, −0.073 meV — hydrogen's 1/n² series at 0.657 meV instead of 13.6 eV — and each is drawn with its probability density above it. The ground state's electron hovers 11.4 nm above the surface on average, about thirty helium atoms' diameters; the first excited state 45.7 nm.

The atom whose nucleus is a reflection

An electron set down on liquid helium cannot go in, because the helium atoms' closed shells repel it, and it cannot go far, because the liquid polarises and pulls it back with the field of an image charge. The image is weak, three-hundredths of the electron's charge. So the electron sits in a one-dimensional hydrogen atom whose nucleus is its own reflection: its levels follow hydrogen's 1/n² series at 0.66 milli-electronvolts instead of 13.6 electronvolts, and it hovers eleven nanometres above the surface. Microwaves at 126 gigahertz lift it to the next level, and a field pressing down on it tunes the line.

 Two descriptions of gold that agree everywhere but at zero. How strongly gold answers a slowly varying field, (ε(iξ) − 1)ξ²/ωₚ² in the two forms of its permittivity at imaginary frequency ξ, against ξ on a logarithmic axis: the plasma model (dashed), which ignores the electrons' collisions and gives exactly 1, and the Drude model (solid), which includes them at the measured rate ħγ = 35 meV and falls to zero below about γ = 5.3·10¹³ per second. The Lifshitz force samples this curve only at the Matsubara frequencies 2πlkT/ħ (dots, 300 K): the first, 2.47·10¹⁴ per second, lies above γ, and there the two descriptions differ by 18 per cent, at the next by 10, and less beyond. At l = 0, ξ = 0, they part completely: 1 against 0. Everything the two models disagree about is in that one point.

The static field a metal may or may not reflect

The Casimir force between two gold plates at room temperature has been computed two ways for twenty-five years, and the two answers differ by forty per cent at three micrometres. Both use the same measured optical properties of gold. They part company over a single number: whether gold reflects a magnetic field that does not change in time. Common sense and every electrical engineer say it does not, since a static field passes straight through a copper sheet. The most precise force measurements say it does. The disagreement is not about the Casimir effect at all; it is about what a metal's electrons do with a fluctuation slower than their collisions.

 Charges released on a field line do not stay on it. A positive and a negative charge (left and right) with their field lines (thin), and the paths of a light positive test charge released at rest from four points 0.32 charge-separations from the positive charge, at 20°, 40°, 55°, 70° from the axis (thick), each beside the field line through its own starting point (dashed). Each path sets off exactly along its line — at rest, the acceleration is the only thing with a direction — and then swings wide of it, because the charge carries its velocity round the bend while the line turns. All four miss the negative charge and leave the frame, never to return, although the field line from every starting point ends on it.

The path a charge takes is not its field line

A field line is drawn by stepping along the direction of the force, and it is natural to read it as the route a charge would take. It is not. A field line is the curve whose direction is the force; a charge's path is the curve whose bending is the force, and the two agree only where the charge starts from rest. A charge released beside a dipole sets off along its line and then swings wide of it, carried by the speed it has gained; a charge launched round a point charge circles it on an ellipse that crosses every line. Only two things make a charge keep to its line: enough friction that it never has a speed of its own, or a magnetic field.

 Flux leaving a plate in stripes and spreading into a uniform field. A cross-section through a superconducting plate in the intermediate state near its top surface (below the line), with a normal fraction of 0.40: normal laminae (shaded) each carry the critical field straight up, the superconducting laminae between them carry none. Lengths are in units of the period of the pattern. Above the surface the field lines (solid) are computed from the stray field of the stripes: they fan out from each normal lamina and close up over the superconducting ones, and within about a third of a period the field is uniform at 0.40 of the critical field — the applied field, which is all a distant observer measures. The fanning field stores energy, and it stores more the wider the stripes are, because a wider pattern's non-uniformity reaches further out. That is half of what sets the period.

The stripes a superconductor makes of a field

A superconductor expels a magnetic field, but only if the field it pushes aside stays below the critical field everywhere at its surface — and for anything but a long rod held along the field, the field it pushes aside piles up at its edges. A flat plate laid across a field cannot expel any of it. It splits instead into interleaved bands of normal metal carrying exactly the critical field and superconducting metal carrying none, in the proportion flux demands, and the bands have a definite width. Walls between the two cost energy and favour few, broad bands; the field fanning out of the surface favours many narrow ones. The compromise is the geometric mean of the plate's thickness and a microscopic length.

 Where the two theories part: the magnetic permeability. The voltage across the spinning wall as a fraction of a metal's, against the material's relative permeability μ, for a permittivity ε = 6. Lorentz's electron theory (flat) polarises the material only through the force on its charges, so μ does not enter: (ε − 1)/ε = 0.833 at every μ. Relativity gives (εμ − 1)/εμ, which agrees exactly at μ = 1 — so no ordinary insulator, such as the ebonite H. A. Wilson spun in 1904, can tell them apart — and climbs towards a metal's value as μ grows. At μ = 3 (dot) the relativistic voltage is 13.3 per cent larger. That is why Wilson and Wilson needed an insulator that was also magnetic, and made one from steel spheres set in wax.

The voltage a moving magnet adds to an insulator

Spin a hollow cylinder of insulator in a magnetic field along its axis and a voltage appears between its inner and outer surfaces, because every piece of it is moving through the field. How large that voltage is turns out to depend on whether the insulator is magnetic, and the dependence was predicted by one theory and missed by the other. Maxwell's equations hold in a moving material only if the material's response is written in its own frame and carried into the laboratory by relativity — and then a moving magnet is also an electric dipole.

 How much a tiny imbalance of charge would take off a body's weight. The electric repulsion between the parts of any body, as a fraction of their mutual gravity, if every nucleon carried an unbalanced charge δ (in units of the electron's), on logarithmic axes. It is (δ/δc)², with δc = 8.93 × 10⁻¹⁹, for a grain of sand, a planet or a galaxy alike. At the imbalance Lyttleton and Bondi proposed, 2 × 10⁻¹⁸, repulsion would beat gravity 5.0 times over, everywhere — their mechanism for an expanding universe. Experiments with gas let out of tanks had bounded the imbalance near 10⁻²⁰ within a year, where the ratio is 1.3·10⁻⁴, and modern ones near 10⁻²¹, 1.3·10⁻⁶.

The imbalance of charge that would outweigh gravity

Gravity and electricity both fall as the inverse square, so Gauss's law treats them identically: the field of any round body counts what is inside it. If every nucleon in ordinary matter carried a tiny unbalanced charge, the body's charge would follow its mass everywhere, and its electric repulsion would be a fixed fraction of its gravity at every size from a grain of sand to the universe. Below about one part in a billion billion of an electron's charge, gravity always wins; above it, repulsion always does. In 1959 two astronomers proposed that the universe expands because the true imbalance is just above that line.

 Where the charge goes on a conducting egg. The surface charge on three conducting spheroids of the same length, a sphere and spheroids 3 and 10 times as long as they are wide, drawn as a band whose thickness is proportional to the charge per unit area at each point. On the sphere it is uniform. On an ellipsoid the density is exactly proportional to the fourth root of the surface's Gaussian curvature, and so it is 3 and 10 times as dense at the tips as at the waist — not the 9 and 100 times by which the tips' curvature exceeds the waist's, and not the ratio of their mean curvatures either. The charge does crowd to the sharp ends, but by how much is set by the whole shape, not by any property of the surface at the point.

The charge that follows no local rule

Charge on a conductor crowds where the surface curves sharply, and every high-voltage engineer rounds the edges for that reason. But curves sharply by what measure? On a conducting egg the density is exactly the fourth root of the surface's Gaussian curvature, so a tip a hundred times more curved than the waist carries only ten times the charge. On two spheres joined by a wire it is the square root, and on a flat disc the density goes without limit at the rim, where no curvature is defined. No local rule can be right, because where charge sits is decided by the whole shape at once.

 A loop, and the butterfly it makes. Left: magnetisation against applied field for a model ferromagnet, as a fraction of saturation, with the field in units of the coercive field — the rising branch and the falling one, crossing zero magnetisation at plus and minus one. Right: the strain the same material shows along the field, as a fraction of its saturation magnetostriction, on the same two branches. Because the strain goes as the square of the magnetisation it does not care which way the magnetisation points, so it is the same at full positive and full negative saturation, and it falls to nothing at the two points where the magnetisation passes through zero — the coercive fields. A loop traced once produces a butterfly traced once: the strain goes through two full stretches for every one cycle of the field.

The hum at twice the mains

A transformer hums, and the note is not the frequency of the mains but twice it: a hundred hertz in Europe, a hundred and twenty in America. The reason is that iron changes shape when it is magnetised, by a few parts in a million, and the change does not care which way the magnetisation points. A core driven at fifty cycles a second therefore stretches a hundred times a second. The same evenness makes the strain trace a butterfly against the field, puts harmonics at every even multiple of the mains and none at the odd ones, and makes the hum climb steeply as the steel nears saturation.

 The resistance a pan offers the coil. The surface resistance — the square root of πfμρ — of five pan materials against frequency, on logarithmic axes, with the band where induction hobs work, 20 to 100 kHz, shaded. All five rise as the square root of the frequency. At 25 kHz carbon steel offers 9.9·10⁻⁴ Ω and ferritic stainless 0.0024 Ω, while austenitic stainless, which is not magnetic, offers 2.7·10⁻⁴ Ω, aluminium 5.1·10⁻⁵ Ω and copper 4.1·10⁻⁵ Ω. For the same coil current the heating goes as this resistance, so steel heats 19 times as hard as aluminium. Matching steel with aluminium by frequency alone would need 377 times the frequency — 9 MHz, where the hob's own electronics and coil would be the problem.

The pan a better conductor will not heat

An induction hob heats a pan with currents the pan carries itself, driven by a field oscillating twenty-five thousand times a second, and it will heat a cast-iron skillet furiously while an aluminium saucepan, which conducts electricity six times better, barely warms. The better conductor is the worse load. What a pan presents to the coil is a surface resistance that goes as the square root of permeability times resistivity, and on that measure iron beats aluminium twenty times over. The same arithmetic explains why kitchen foil on the same hob can catch fire, and why a steel tip heated this way switches itself off at a fixed temperature.

 Two fields that part company in the middle. The direction of the electric field of two opposite charges, on the left, and of the magnetic field of a loop of current cut across, on the right, each with a dipole moment pointing up the page. Outside, the arrows of the two pictures are nearly the same and become identical with distance. In the middle, drawn darker, they point opposite ways: the field between the two charges runs from the positive to the negative, downward, against the moment, and the field through the loop runs upward, with it. Field lines of the magnetic dipole close through the middle; those of the electric dipole begin and end there. No measurement made from a long way off can tell them apart.

Two dipoles that agree everywhere but inside

A pair of opposite charges and a small loop of current make the same field at any distance worth measuring. Between the charges that field points one way, and through the loop it points the other. Shrink both to a point and the disagreement does not vanish: it becomes a term concentrated at the origin, and hydrogen's 21-centimetre line depends on it. The level order of the hydrogen atom says which kind of dipole a proton is.

 The potential a stream of charge bends. The potential and electric field across a vacuum gap of width d with a voltage V across it, drawn in units of V and V/d. With no charge in the gap the potential is a straight line and the field uniform. With a stream of charge crossing, the charge's own field pushes the potential curve down, and the current can rise only until the field at the emitter falls to zero: past that the next particle would be pushed back. At that limit the potential goes as the four-thirds power of the distance and the field as the cube root, so the emitter sees no field at all and the far side sees four-thirds of what it saw before. Child and Langmuir found the same curve independently, in 1911 and 1913, and it fixes the current completely.

The current a vacuum will not carry

A vacuum has no resistance, and yet a gap of empty space will carry only so much current for a given voltage, however freely its cathode offers electrons. The charge already crossing cancels the field that would pull the next one off. Gauss's law gives the result in a few lines, and it says something odd about ion engines: their thrust per square metre does not depend on what the ion is, and it is exactly the electric stress on their grid.

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