Illustrated Physics
physics, drawn
Theme

Fields, not forces

Replacing action at a distance with something that fills space, and what that buys.
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. Electromagnetism

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. Electromagnetism

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. Electromagnetism

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. Electromagnetism

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. Electromagnetism

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.

 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. Electromagnetism

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. Electromagnetism

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. Electromagnetism

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.

 Where the electron actually is, by radius. The radial probability density of the hydrogen 1s, 2s, 2p states — the chance of finding the electron in a thin shell at each radius, in units of the Bohr radius. 1s is most likely at 1.00 Bohr radii and averages 1.50; 2s is most likely at 5.24 Bohr radii and averages 6.00; 2p is most likely at 4.00 Bohr radii and averages 5.00. Each curve integrates to one, and each has n − l − 1 radial nodes where the electron is never found. Quantum

Where the electron probably is

The Bohr atom put the electron on a circle of definite radius. What replaced it keeps the radius as the most likely place to find the electron and gives up the circle, the speed and the trajectory entirely.

 Light crossing an accelerating box. A pulse crosses a box 6 m wide while the box accelerates at 9.81 m/s². The crossing takes 2·10⁻⁸ s, in which the far wall gains 1.96·10⁻⁷ m/s, so the pulse lands 1.96·10⁻¹⁵ m below the height it left at — and the path is a parabola. An observer sealed inside cannot tell that from a beam of light bending in a gravitational field, and the equivalence principle says there is nothing to tell. The sag is drawn 5.6·10¹⁴ times its true size. Astrophysics

The floor that cannot be told from gravity

Seal a laboratory, take away the windows, and no experiment inside it can distinguish standing in a gravitational field from accelerating through empty space. That is not a philosophical remark — it forces light to bend, forces clocks to disagree, and has a size at which it stops being true.

 Two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit. Astrophysics

The bend Newton got half right

A photon treated as a falling body passes a mass and comes out deflected. So does a photon treated as a straight line in curved spacetime — by exactly twice as much. The factor of two is not a refinement; it is the sharpest available statement that gravity is geometry.

 What a passing wave does to a ring. A ring of 8 free masses at 4 phases of a passing gravitational wave, in both polarisations. The upper row is the + mode: one diameter lengthens while the perpendicular one shortens, and half a cycle later they swap. The lower row is the × mode, which is the same pattern rotated by forty-five degrees rather than ninety — the signature of a spin-2 field, and the reason a detector is built as two arms at a right angle. The drawn strain is 0.42; a real one is 10⁻²¹, so the deformation is exaggerated 4.2·10²⁰ times. At that true strain a four-kilometre arm changes length by 4·10⁻¹⁸ m. Astrophysics

The wave that stretches one way and squeezes the other

A gravitational wave passing through a ring of free masses lengthens one diameter while shortening the perpendicular one, then swaps. The two conservation laws that forbid anything simpler are why the effect is a part in a thousand million million million.

 The classical atom, and how long it lasts. An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion. Astrophysics

A charge that turns must glow

An accelerating charge radiates, and a charge going round in a circle is accelerating. Apply that to an electron orbiting a nucleus and classical physics predicts that every atom collapses in sixteen picoseconds — a calculation with nothing wrong in it except its conclusion.

 What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it. Astrophysics

The field that cannot get out

Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.

 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. Electromagnetism

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. Electromagnetism

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 same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second. Relativity

Magnetism is electricity seen sideways

The force on a charge moving beside a current-carrying wire is magnetic in the laboratory and purely electrostatic in the charge's own frame. Both calculations give the same answer, and the drift speed that makes them agree corresponds to a Lorentz factor differing from one in the twenty-sixth decimal place.

 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. Electromagnetism

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. Electromagnetism

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. Electromagnetism

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. Electromagnetism

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.

 The two reflectances, and the angle one of them loses. Reflectance against angle of incidence for light going from n = 1 into n = 1.5. The upper curve is light polarised with its electric field along the surface, which reflects more and more strongly until at grazing incidence everything reflects. The lower curve is light polarised in the plane of incidence, and it does something the other cannot: it falls to exactly zero at 56.31°, where tan θ = 1.5000, and then rises again. At normal incidence the two are equal at 4.00% because there is no plane of incidence to tell them apart. The dashed curve is the transmittance, computed from the transmission coefficients and the two media's projected impedances rather than as one minus the reflectance; it agrees with one minus the reflectance to 4.4e-16 across the whole range, which is where the energy accounting can be seen to close. Optics

The angle at which reflection picks a side

At one angle of incidence, a water surface reflects no light at all of one polarisation. The Fresnel algebra says so, and says nothing about why. The reason is that the reflected ray would have to leave along the axis of the charges radiating it — and a shaking charge sends nothing along the direction it shakes in.

 Rays that turn round without meeting anything. Rays leaving an eye 1.5 m above a road, at 0.18°, 0.28°, 0.36°, 0.42°, 0.52° below the horizontal, in air whose refractive index is reduced by 3.0e-5 at the hot surface and recovers over 5 cm. Each is traced by integrating the ray equation, with the conserved quantity n cos θ fixed by where and how the ray set out. The shallow ones come back up without touching anything — the lowest gets to 0.6 cm above the surface and turns — and a ray that returns to eye level from below is seen as sky lying on the road. The steep ones run out of gradient first and hit it. The dividing angle is 0.444°, which is what the whole effect is made of: an unremarkable temperature difference and an angle a fiftieth the width of the Moon. Heights are exaggerated 128× against distances; at true scale every ray here would be indistinguishable from the axis. The conserved n cos θ holds to 4.3e-14 over every trace, which is what says the turns are the physics and not the integrator. Optics

The ray that bends without a surface

Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.

 How large a box may be before gravity shows in it. The largest freely falling laboratory in which nothing can be detected, against how long the experiment runs, for an instrument resolving 10⁻⁹ m. Free fall removes the field and leaves the gradient: two masses released a distance apart converge across the fall and separate along it, by (GM/r³)Lt²/2, so the box that stays undetectably flat shrinks as the square of the time. At one second the limits are 1.30 mm at the Earth's surface, 5.07 mm at the Sun's surface, 8.61 nm at a white dwarf, 1.86·10⁻¹⁷ m at a neutron star, 3.88·10⁻¹⁷ m at a stellar black hole, 388 mm at a giant black hole. Two things are worth reading off it. The equivalence principle is local in time as much as in space, on the same footing and with a worse exponent — patience is more expensive than room. And the tidal parameter GM/r³ falls with mass at a horizon, so a large enough black hole is one an experimenter could fall through with a metre-sized laboratory and a very good interferometer and detect nothing at all: the curve for the giant lies above the Earth's, not below it. Astrophysics

The term free fall cannot remove

Fall freely and gravity disappears. It disappears only to the extent that the falling laboratory is small — what survives is the gradient, which pulls two released masses together across the fall and apart along it. Given an instrument, the size of the box in which nothing is detectable is computable, and that number is the whole content of the word "locally".

 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. Electromagnetism

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.

 Two ways to weaken a surface. Surface tension is a property of a surface rather than a constant of a liquid, and this shows how far it moves. The steep curve is water with ethanol dissolved in it, against ethanol mole fraction, through nine measured points: two per cent of ethanol takes the tension from 72 to 56.4 mN/m, a fall of 22% for a change of composition small enough to taste and not to see. The initial slope is 837 mN/m per unit of mole fraction, because ethanol collects preferentially at the surface and a little of it covers a great deal of area. The gentle line is pure water against temperature, read on the same horizontal axis as degrees rather than fraction: about 0.15 mN/m per degree, so a difference of ten degrees across a surface is worth about a millinewton per metre. Neither dependence would be interesting if surfaces were uniform. What makes them matter is that a difference in tension across a surface is a force along it, and nothing in the liquid prevents such a difference from existing. Fluids

The surface that pulls toward the stronger side

Surface tension is usually treated as a constant of a liquid, and it is not — it depends on temperature and on what is dissolved, and a difference in it along a surface is a force along that surface — which drags the liquid underneath and needs no pressure difference at all.

 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. Electromagnetism

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. Electromagnetism

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. Electromagnetism

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.

 What a boost can and cannot do to a field. The electric and magnetic magnitudes of three fields, plotted against each other as the observer is boosted from -0.98c to 0.98c across them. Each field slides along a hyperbola, because E² − c²B² does not change: the recomputed value drifts by at most 2.6e-15 over every point drawn. The diagonal is E = cB, and which side of it a field starts on is permanent. Below it there is a speed at which the electric field vanishes; above it, one at which the magnetic field does; on it, a wave that no observer can slow, dim or unbalance. Relativity

The field nobody can transform away

A wire's magnetic field is an electric field seen from the wrong frame, and a charged plate's electric field is a magnetic one seen the same way. Neither trick works on a light wave. Two combinations of E and B are the same for every observer, and which side of one line a field sits on is a fact nothing about the observer can alter.

 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. Electromagnetism

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. Electromagnetism

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.

 A lens with no focal length. Where a ray crosses the axis, against how far off the axis it passed the deflecting body, for 1 solar mass of radius 1 solar radius. A glass lens deflects a ray by an angle proportional to its distance off axis, which is precisely the condition for every ray to arrive at one point — the flat dashed line. Gravity deflects by 4GM/c²b, which grows smaller further out, so the crossing distance goes as b² and each ray has its own focus. The grazing ray crosses at 548 astronomical units and a ray passing at 12 radii crosses at 78857; the square law is verified on the drawn curve to 1.5e-16. So there is no image plane at all, only a half-line of foci beginning at the first of those and running outward for ever. Anything placed on that line sees not an image but a ring, and moving along it does not refocus anything — it selects which rays are being seen. Astrophysics

The lens with no focal length

A glass lens bends a ray by an angle proportional to how far off the axis it passes, which is precisely the condition for every ray to arrive at one point. Gravity bends by an angle that falls with distance off the axis, so every ray has its own focus and there is no image plane anywhere — only a half-line of foci, beginning 548 astronomical units from the Sun and running outward for ever.

 The mass of two things that have none. The invariant mass of a pair of photons of equal energy, in units of E/c², against the angle between them. It is computed from the total energy and the vector sum of the two momenta, and agrees with 2E·sin(θ/2) to 1.0e-14. at 0° the pair weighs 0.000 E/c²; at 30° the pair weighs 0.518 E/c²; at 60° the pair weighs 1.000 E/c²; at 90° the pair weighs 1.414 E/c²; at 120° the pair weighs 1.732 E/c²; at 180° the pair weighs 2.000 E/c². Two photons flying in the same direction have no mass between them at all, because their momenta add to exactly the energy over c; anything else and they do. Nothing has been added: the constituents are massless at every angle, and the mass of the system is a property of the arrangement. At 180° the pair weighs 2E/c², which is every joule it contains — the case of a sealed box of light, where the two beams cancel in momentum and the whole energy shows up on the scales. Relativity

The box of light that weighs something

Two photons flying apart have a mass between them, though neither has one. Seal them in a mirrored box and the box is heavier than it was empty, by exactly the energy inside divided by c². Mass is not a property of stuff and it does not add up — it is a property of a system, and 99 per cent of the mass of everything anybody has ever weighed is of this kind.

 The field near a neutral point. Field lines near a magnetic null, traced by following the local field direction rather than plotted from the closed form. The field is B ∝ (y, k²x) with k = 1, whose lines are the hyperbolae y² − k²x² = constant and whose separatrices are the straight lines y = ±1x. At k = 1 the X is symmetric and the current density is exactly zero: the field is curl-free, and nothing is stored in it beyond the field itself. The four quadrants are four separate flux systems, and which of them a given line belongs to is the quantity a frozen-in field is not allowed to alter. Astrophysics

The knot the field cannot untie

A perfectly conducting fluid cannot change which field line joins which piece of it. So two flux systems pushed together may be squashed indefinitely and can never merge, and the energy of the squashing accumulates with nowhere to go. The release happens only where the perfect conductivity locally fails — in a sheet three metres thick inside a structure ten thousand kilometres across — and the rate that follows is a hundred thousand times too slow for the flares that are observed.

 The one frequency a broken repeat lets through. Transmission through a quarter-wave stack whose repeat is broken once, on a logarithmic scale, against frequency in units of the quarter-wave design frequency. The perfect stack forbids this whole band; adding a single half-wave layer at the centre opens one line inside it, at exactly the middle of the gap, through which the stack transmits everything. With 4 pairs either side the line is 2.50e-3 wide, a quality factor of 400, on a background of 4.5e-4; With 6 pairs either side the line is 1.56e-4 wide, a quality factor of 6410, on a background of 6.3e-6; With 8 pairs either side the line is 1.00e-5 wide, a quality factor of 100000, on a background of 9.2e-8. The line narrows geometrically as the mirrors are made thicker, because the field inside the defect leaks out through a barrier whose transmission falls exponentially with its thickness — so the useful quantity of a band gap turns out to be not what it excludes but how well it can trap what a single flaw is allowed to hold. Waves

The mode that lives in the mistake

A perfect stack of alternating layers refuses a whole band of frequencies — not weakly, not with loss, but not at all. Break the repeat once, by inserting a single layer of the wrong thickness, and exactly one frequency inside that band passes through the whole stack with a transmittance of one. The useful thing about a forbidden band turns out to be not what it excludes but what a single flaw is thereby allowed to hold.

 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. Electromagnetism

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.

 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. Electromagnetism

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. Electromagnetism

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.

 A click that arrives sorted by pitch. Arrival time against frequency for a broadband pulse travelling 40 megametres along a magnetic field line through a plasma of 1e+8 electrons per cubic metre in a field of 300 nanotesla — the magnetosphere, roughly. The gyrofrequency is 8.4 kilohertz and the plasma frequency 90 kilohertz, so this is the regime where a right-hand circularly polarised wave travels happily below both of them. It does not travel at one speed. The group velocity is 2c√ω(ω_c − ω)^(3/2)/(ω_p ω_c), which is zero at zero frequency and zero again at the gyrofrequency and peaks in between — at 2.10 kilohertz here, which is a quarter of the gyrofrequency, found by searching the drawn function rather than quoted. Everything either side of that peak is slower, so the curve has a nose: 3.21 s at 500 Hz, 2.20 s at 2000 Hz, 3.59 s at 5000 Hz. The branch below the nose is the classic whistler — a lightning stroke in one hemisphere reaching a receiver in the other as a note gliding downward over about a second, which is what gave the phenomenon its name — and the branch above it arrives as a rising tone at the same time. Both are observed, and a recording that shows the two joined at the nose is how the gyrofrequency along the path is read off directly. Read the other way it is an instrument: the product of the delay and the square root of the frequency is 79 s·Hz^½ here, nearly constant across the low end of the band, and it measures the electron content of a path through the magnetosphere that nothing else could reach. Astrophysics

The whistle that arrives sorted

A lightning stroke in one hemisphere reaches a receiver in the other as a note gliding downward over about a second. Nothing dispersed the sound; there was no sound. A radio pulse travelled forty megametres along a magnetic field line through a plasma whose group velocity depends on frequency, and arrived with its frequencies separated by up to three seconds.

 The charge a plasma hides. The potential around a charge in a plasma, as a fraction of the bare Coulomb potential at the same distance, against distance measured in screening lengths. The electrons crowd toward the charge and the ions move away until the rearrangement cancels the field, and what is left falls as exp(−r/λ_D) on top of the ordinary 1/r. At one screening length the potential is already down to 37 per cent of the bare value, at three to 5 per cent, and at ten to 4 × 10⁻⁵ — so a charge in a plasma is invisible beyond a few λ_D, and the long range of the Coulomb force, which is what makes electrostatics awkward everywhere else, is simply gone. The screening length at 10¹¹ m⁻³ is 3.8 mm at 300 K, 6.9 mm at 1000 K, 21.8 mm at 10000 K, rising as the square root of the temperature because a hotter electron is harder to hold in place. Drawn this way the three curves coincide exactly: the shape is universal and the only thing a plasma's density and temperature decide is the length written on the axis. Astrophysics

The long-range force that does not reach

The Coulomb force falls off as slowly as gravity does, which is what makes electrostatics awkward — every charge is in principle in contact with every other. Put the same charge into a plasma and it becomes invisible beyond a few millimetres, because the mobile charges around it rearrange until its field is cancelled. What is left is a screened potential with a definite range, and that range is what makes a plasma a plasma.

 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. Electromagnetism

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.

 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. Electromagnetism

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. Electromagnetism

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.

 A step, a wave from the rim, and what they make together. The pattern behind a straight edge, split into the two things Young said it was: the incident wave where the edge does not block it, which is a step, and a wave that appears to come from the rim itself. The step is discontinuous at the shadow boundary by the whole incident amplitude. The edge wave's magnitude is continuous there, to 0.0e+0 — it changes phase by π instead of changing size — and it is exactly half the incident amplitude on both sides, which is why the total intensity on the geometrical shadow boundary is 0.250000 of the unobstructed value rather than a half. Everywhere else the two add, and their sum reproduces the directly computed field to 0.0e+0. The fringes in the lit region are the interference of the two, which is why they are fringes at all: a monotonic decay has nothing to beat against. In the shadow there is only the edge wave, so there is nothing to interfere with and the intensity falls smoothly. Waves

The wave that comes from the rim

A shadow's edge is not a boundary between light and no light, and the fringes on either side of it are not a smudge. The whole pattern is the sum of two things — the light nothing blocked, and a single wave that behaves in every respect as though the rim of the obstacle were radiating it — and splitting it that way is exact rather than a picture.

 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. Electromagnetism

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. Electromagnetism

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.

 A force that points across the beam, not along it. The intensity of a beam focused to a waist of 0.5 µm, and the force it exerts on a 60 nm polystyrene sphere in water, across the beam. The force is not a pressure and does not point along the light: it is the pull on an induced dipole sitting in a non-uniform field, proportional to the gradient of the intensity rather than to the intensity, and it points up the gradient — towards the bright axis from either side. It vanishes exactly on the axis, which is what makes the axis a trap rather than a place. The well is 47 times kT deep at 100 milliwatts, which is why the particle stays: thermal motion explores it and does not escape it. The stiffness near the bottom is 3.070 piconewtons per micrometre, and a particle in a spring that stiff wanders 36 nanometres from centre. Astrophysics

The light that pulls rather than pushes

Radiation pressure is momentum arriving, and it points along the beam. A gradient of intensity does something else entirely — it polarises a particle and then pulls the induced dipole up the gradient — so a focused beam holds a particle at its waist against the push, and the force that does it is not a pressure at all.

 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. Electromagnetism

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 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. Electromagnetism

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.

 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. Electromagnetism

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.

 Outside the winding, where nothing is supposed to be. The mid-plane of a solenoid 20 radii long, wound at 8 turns per radius, with the field summed turn by turn rather than assumed. The upper curve is the axial field against distance from the axis: it holds up across the winding and collapses outside, reaching 0.47 per cent of its central value at 2 radii. The lower curve is the flux enclosed by a circle of that radius, which is what the vector potential integrates to. The two behave completely differently, and that difference is the whole subject: the field an electron outside can feel has effectively gone, and the flux it encircles has not. The enclosed flux does fall, by 3.29 per cent out to four radii, because the lines that leave the ends come back through the plane outside the coil — that return flux is the leak a real experiment has to defeat, and the reason the definitive versions used a closed toroidal magnet with no ends at all. Quantum

The phase a magnet leaves on a path it never touched

An electron beam split around a solenoid comes back with its fringes displaced, although neither path ever entered a magnetic field. What the electron responds to is the flux it went round, and the only local quantity that knows about that flux is the potential.

 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. Electromagnetism

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. Electromagnetism

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.

 One magnitude, and a direction that turns the wrong way. The force per unit area a magnetic field of 0.01 tesla exerts on a surface, drawn for surface normals at 0°, 30°, 45°, 60°, 90° to the field. The short grey arrows are the normals; the long coloured ones are the tractions. Every traction has the same length, 40 pascals, because the magnitude of T·n does not depend on the orientation of n at all — and the direction is the normal reflected in the field rather than carried round with it, so as the surface turns one way the force turns the other. A surface cutting across the field is pushed (the magnetic pressure); a surface cutting along it is pulled (the magnetic tension); and at 45° the traction lies in the surface, which is a shear and is neither. Astrophysics

The same force whichever way the surface faces

A magnetic field's stress is usually described as a pressure across the lines with a tension along them, as though it were two effects. It is one. The force per unit area is B²/2μ₀ on every surface however it is oriented, and only the direction changes — the surface normal reflected in the field, so that turning the surface one way turns the force the other. At 45° neither name applies.

 Three speeds, drawn against direction. The phase speed of the three magnetohydrodynamic waves against the angle between the wavevector and the field, drawn as a polar diagram with the field horizontal and the sound speed 0.6 times the Alfvén speed. Along the field the two magnetosonic branches take the values of the sound speed and the Alfvén speed themselves and swap which is which as the ratio crosses one; across the field the slow branch vanishes entirely and the fast branch runs at the quadrature sum. The shear branch is the figure-of-eight, v_A cos θ, and it is the only one of the three whose speed contains no thermodynamic quantity — no pressure, no temperature, no sound speed. It does not know what the gas is made of. Astrophysics

The wave that does not know what the gas is made of

Give the magnetic tension of a bent field line an inertia and it becomes a string. The wave that runs along it goes at the same speed at every wavelength, compresses nothing anywhere, and has a speed containing no temperature, no pressure and no sound speed — it is the only wave in classical physics that is indifferent to what the medium is. Its energy travels along the field whatever direction the wave was sent in.

 An angle set by a voltage. The contact angle of water on a fluoropolymer coating with a rest angle of 115°, against the voltage across the coating, for coatings 1 and 3 µm thick with a relative permittivity of 1.93. The Lippmann–Young law cos θ = cos θ₀ + ε₀εᵣV²/2dγ makes the cosine rise as the square of the voltage, so the curve is symmetric about zero volts and steepest where the angle is near 90°. For 1 µm the law reaches complete wetting at 110 V; for 3 µm the law reaches complete wetting at 191 V. It does not get there. In most reported experiments the angle stops falling somewhere in the shaded band and then stays put however high the voltage is raised — contact-angle saturation — and why is still argued: charge trapped in the insulator, ionisation of the air at the sharp edge of the drop, and the thermodynamic stability of the contact line have all been proposed and none accounts for every case. Fluids

The angle a voltage can set

A contact angle is treated as a fact about three materials — a solid, a liquid and the air — fixed the moment they are chosen. Put a voltage across a micrometre of insulator under a drop and the angle falls as the square of the voltage, with nothing about the materials changed. Drops can be steered across a chip with no pump, and a lens can focus with no moving part. The law that describes it is exact in its model, and real surfaces stop obeying it at an angle nobody has fully explained.

 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. Electromagnetism

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.

 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. Electromagnetism

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 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. Electromagnetism

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.

 Why the meander has to move west. A line of fluid columns at 45° pushed into a meander 3,000 km long and 300 km in amplitude, seen from above with north up and east to the right. Columns pushed north find the planet spinning faster beneath them and turn clockwise; columns pushed south turn anticlockwise. Each turning column drives the fluid north on one side of it and south on the other, and the arrows are that flow, from integrating the columns' spin along the line: it peaks at 2.32 m/s, a quarter of a wavelength west of each crest, so the columns just west of a crest are carried north and the crest itself is rebuilt further west. The dashed meander is the same pattern 2 days later, moved 638 km west at β/k² = 3.69 m/s. Nothing in the argument depends on which way the meander was first pushed. Mechanics

The wave that can only travel west

A column of air or water on a spinning planet keeps one quantity as it moves: its total spin divided by its depth. Carry it north, where the planet's own spin about the local vertical is larger, and it must turn the other way to compensate. That single bookkeeping rule makes a wave that exists nowhere else — one whose crests can only ever move west — and the same rule holds the jet stream's meanders still over the mountains, makes the ocean take decades to notice a change in the wind, and pins the Gulf Stream to the western side of the Atlantic.

 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. Electromagnetism

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. Electromagnetism

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.

 Two pushes that nearly cancel. The forces between two equal charges moving side by side at the same speed, measured in the laboratory, in units of the Coulomb force the same two charges exert at rest, against their speed. Electric repulsion grows as γ, because a moving charge's field is stronger across its motion: 2.29 at β = 0.9, 7.1 at 0.99. The magnetic attraction of two parallel currents grows as γβ² and eats it almost entirely: the net push is γ(1 − β²) = 1/γ — 0.436 at 0.9 and 0.141 at 0.99. Measured in the pair's own frame the push is exactly the Coulomb force at every speed; the difference is not a force that exists in one frame and not another but a difference in how fast the two frames' clocks run while the charges drift apart. Relativity

The beam that stops pushing itself apart

A beam of electrons is a crowd of like charges, and a crowd of like charges flies apart. A slow beam does, within centimetres. A fast one hardly does at all: at a gigaelectronvolt the same current travels kilometres before its width doubles. The electric repulsion has not weakened — it has grown, by the factor γ. What has grown faster is a magnetic attraction between parallel currents, which cancels all but 1/γ² of the push. Seen from the beam itself there is no magnetism at all, only Coulomb repulsion acting while the beam's clocks run slow.

 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. Electromagnetism

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.

 A muon's spin lapping its own momentum round a storage ring. Snapshots of a muon circling a storage ring at γ = 29.3, taken every four turns (not to scale in time: the muon is drawn at a different place on the ring for each snapshot so they do not overlap). The dashed arrow is the direction of motion, tangent to the ring; the solid arrow is the spin. On each turn the spin gains 12.3° on the momentum — 2πaγ — so after 29.3 turns it has gained a full lap. That lap takes 4.37 microseconds, and the muon, whose lifetime is stretched from 2.2 to 64.4 microseconds by its speed, lives for about fifteen of them. Relativity

The spin that runs ahead of its own motion

A charged particle circling in a magnetic field has two directions that turn: the direction it is moving, and the direction its spin points. If the particle's magnetism were exactly what Dirac's equation first gave — a g-factor of exactly two — the two would turn together at every speed, the spin locked to the motion as a compass needle is locked to a car. The muon's g is not quite two, and its spin creeps ahead by a fixed rate that is the same at a crawl and at 99.94 per cent of the speed of light. That constancy is a delicate cancellation between a magnetic field made stronger by motion and a rotation that exists only because the particle is being turned, and it is what made the most precise measurements of the muon possible — at one magic speed, where the electric fields that steer the particles stop affecting the spin at all.

 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. Electromagnetism

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.

 The cone a charge leaves behind in water. A charged particle crossing water at 0.99 of the speed of light in vacuum, which is 1.317 times the speed of light in water, drawn at one instant. From each point it passed, its field set off a spherical wavelet travelling at c/n; the wavelets from earlier points have grown larger. Because the particle outruns them, they all touch one cone with its tip at the particle, at an angle θ = arccos(1/nβ) = 40.6° to the path — the light of Cherenkov radiation, which travels out perpendicular to the cone. No point on the path is special and the particle never accelerates: the radiation is a property of the field moving faster than the medium can carry it, the electromagnetic counterpart of a sonic boom or a boat's wake. Astrophysics

Why the glow of a fast charge is blue

A charge moving in a straight line at constant speed does not radiate — in a vacuum. In water, if it moves faster than light does in water, it glows, and the glow is blue. The cone of the light is the familiar half of the story. The other half is a count: Frank and Tamm's formula says each centimetre of water yields about two hundred visible photons, in numbers that rise as the inverse square of the wavelength, which is why a reactor pool shines blue and why the light keeps getting stronger into the ultraviolet until water stops being transparent. The charge never accelerates. What radiates is the water.

 Light from a charge that crosses into glass at a steady speed. The transition radiation emitted as an ultrarelativistic charge crosses from vacuum into a dense medium, per unit solid angle, against the angle to the path in units of 1/γ, for photons at three frequencies: one tenth, three tenths and one times γωₚ, where ωₚ is the medium's plasma frequency. The scale is αħγ²/π² per unit frequency. Nothing is emitted straight ahead; the light comes out in a narrow hollow cone with its brightest ring at γθ = 0.98, 0.88, 0.68 for the three. At a tenth of γωₚ the peak is about ten times higher than at γωₚ. The charge never accelerates: it is the field's sudden change of shape at the surface, from the flattened disc it carries in vacuum to the one the medium's electrons allow, that is radiated. Astrophysics

The light a charge makes by changing medium

A charge moving at constant speed in a straight line radiates nothing in a vacuum, and nothing in glass either if it is slower than light there. Let it cross from one into the other and it radiates at the boundary, though it neither turns nor slows. The field it carries has one shape in each medium and has to be rebuilt as it crosses, and what is shed in the rebuilding is light. The energy shed grows in proportion to the particle's Lorentz factor, the one property of a fast particle that almost nothing else measures at high energy, and this is why detectors stack hundreds of plastic foils in a particle's path to collect one or two X-rays.

 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. Electromagnetism

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.

 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. Electromagnetism

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 pattern in which a spinning Earth turns gyroscopes. The direction in which the Earth's rotation turns a gyroscope near it, in a plane through the Earth's axis (spin upward): the lines are those of the precession rate Ω = (G/c²r³)[3(J·r̂)r̂ − J], which has exactly the shape of the magnetic field of a spinning charged ball. On the circle of a polar orbit 650 km up, arrows show the direction and relative size of the turning at twelve points. Over the poles a gyroscope is turned the same way the Earth spins, twice as fast as anywhere else at that distance; over the equator it is turned the opposite way. Averaged round a polar orbit, what is left is half the size of the over-the-equator rate and opposite to it — in the sense of the Earth's spin. Relativity

The gyroscope a spinning Earth turns

A moving charge makes a magnetic field, and a spinning charged ball makes a dipole field around it. Gravity has the same structure one level down: a spinning mass drags the local standards of rotation around with it, and a perfect gyroscope near the Earth turns, very slightly, in the pattern of a dipole field — with the Earth's spin over the poles and against it over the equator. Averaged round a polar orbit the turning is 41 milliarcseconds a year. It took forty-four years and a satellite carrying the roundest objects ever made to measure it.

 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. Electromagnetism

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.

 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. Electromagnetism

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.

 A drift that twist turns into a circle. An ion's guiding centre at 10 keV in a torus the size of ITER, seen in the plane across the bore, with a vertical drift of 609 m/s from the field's curvature and gradient. With no twist it drifts straight up and reaches the wall in about 2.8 ms, ions one way and electrons the other. With the field lines twisted so that they circle the bore once for every three trips round the torus, the same drift is carried round by the motion along the line, up on the outside and back down on the inside, and the integrated path closes into a circle offset by only 1.3 cm. The offset is the drift speed over the rate of circulation. The twist does not stop the drift; it makes the drift average to nothing, which is all a bottle needs. Astrophysics

The twist a bottle of plasma has to have

A straight magnetic bottle leaks out of its ends, so the obvious repair is to bend it round into a ring. A ring of field with no twist in it loses its plasma in about three millionths of a second. Adding a twist repairs that by making a drift average to nothing, and adding too much makes the whole column kink like an overwound rubber band. A tokamak is built in the narrow band between those two failures.

 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. Electromagnetism

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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