Illustrated Physics
physics, drawn
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Approximations that lie — part 2 of 2

The small-angle assumption, the frictionless plane, the point mass — where each is fine, and precisely where it stops being.
A field that oscillates faster than any of its components. The oscillation of (cos x + i·2·sin x) to the power 10 over half a period, with its changing size divided out so that only the turning of its phase shows, beside the fastest wave it contains, cos 10x (dashed). Multiplied out, the function is a sum of waves e^(inx) with n running from −10 to 10 and no further. Yet in the shaded window, |x| < 0.62, it oscillates faster than cos 10x — near the origin like cos 20x, 2 times faster — completing 3.0 cycles there where its fastest component completes 2.0. Outside the window it slows down, and the next figure shows what happens to its size there. Waves

The wiggle faster than any wave in it

Light passing through a lens carries no detail finer than half a wavelength, because no travelling wave in it varies faster than that. It would seem to follow that the light itself cannot vary faster anywhere. It can. A sum of slow waves can be arranged to wiggle faster than any of them over a stretch of any length chosen — and to focus to a spot as narrow as wanted — provided the stretch sits between side lobes that grow exponentially with every extra wiggle.

 Water gets denser the deeper it is. The density of seawater at 0 °C and a salinity of 35 against depth, from the surface to the 10.99 km of the Challenger Deep, with the pressure integrated down the column from the equation of state rather than assumed. At the surface it is 1028.1 kg/m³. At the ocean's mean depth of 3.68 km it is 1045.3, 1.67 per cent more; at the bottom of the deepest trench it is 1076.5, 4.71 per cent more. An incompressible ocean would be the vertical line. The curve is almost straight; what curvature it has, too little to see here, is the water stiffening as it is squeezed: its secant bulk modulus rises by 3.24 pascals for every pascal of pressure. Fluids

The sea that stands lower because water gives

Water is the textbook incompressible fluid, and at the bottom of the ocean it is 4.7 per cent denser than at the top. That squeezing lowers the whole sea by about thirty metres, adds the weight of 266 metres of water to the pressure in the deepest trench, and speeds sound up so steadily with depth that it digs a channel in which a low note can cross an ocean.

 The area each quantum state is allowed. Phase-space orbits of the four lowest quantum levels, computed as eigenvalues of the discretised Schrödinger equation, for a harmonic well (left) and a box with hard walls (right), in units ħ = m = 1. Each orbit encloses an area that is a whole number of Planck constants plus a fixed offset: for the harmonic well 0.500, 1.500, 2.500, 3.500 h — n + ½ — and for the box 1.000, 2.000, 3.000, 4.000 h — n + 1. The harmonic orbit turns round smoothly at both ends and loses a quarter of a cycle at each; the box orbit is reflected sharply by each wall and loses half a cycle there. Mechanics

The quarter cycle a turning point costs

The old quantum rule said an orbit's action is a whole number of Planck constants. It gets the harmonic oscillator wrong by exactly half a unit, the box by exactly one, and a neutron bouncing on a mirror by three quarters. Each is the same correction: a wave loses a quarter of a cycle every time it turns round smoothly, and half a cycle every time it hits a wall.

 The thin parcel that keeps sinking. A parcel pushed one centimetre down a column in which heat stabilises and salt destabilises, with a density ratio of 2, integrated for three hours, for parcels whose pattern of sinking and rising water repeats every 2 m, 20 cm, 6 cm. The 2 m parcel keeps its heat and its salt and swings back and forth with the buoyancy period of 23.6 minutes. The 20 cm parcel crosses its starting level 8 times while its swing dies away, and is left sinking slowly, at the 0.48 e-folds an hour its width allows; the 6 cm parcel never comes back and passes five centimetres below it after 60 minutes. The thinnest loses its heat to its surroundings in 11 minutes and its salt in 18 hours, so it arrives salty and cooled, heavier than the water around it, and runs away at 2.33 e-folds an hour — the rate the linear dispersion relation gives for that width. Fluids

The salt that sinks through a stable sea

Warm salty water lying on cold fresh water is heavier at the bottom at every depth, and by every test of static stability it should stay where it is. It does not. Heat leaks sideways out of a thin tongue of sinking water a hundred times faster than salt does, so the tongue arrives cooled and still salty — heavier than its new surroundings — and keeps going. A column can be stable by density and unstable by diffusion, and most of the subtropical ocean is.

 The flow past a thread takes thousands of widths to recover. The speed of the fluid beside a body, at right angles to the stream, as a fraction of the free stream, against distance from the body's axis in body radii, in slow viscous flow. Past a sphere, from Stokes's solution, the flow is back to 90 per cent of the stream by 7.5 radii whatever the Reynolds number. Past a cylinder, from Lamb's inner solution, the speed grows only as the logarithm of distance, with a slope set by the Reynolds number: it reaches 90 per cent of the stream at 29, 232, 1,843 and 1.5 × 10⁴ radii for Re = 10⁻¹, 10⁻², 10⁻³ and 10⁻⁴ in turn. Each cylinder curve is drawn out to where the inner solution reaches the stream speed, beyond which Oseen's outer flow takes over. Each tenfold drop in the Reynolds number pushes the recovery ten times further out, because what finally stops the disturbance is inertia, at a distance of order ν/U. Fluids

The drag a distant wall decides

A sphere moving slowly through a liquid has a drag that belongs to the sphere: six pi times the viscosity, the radius and the speed, and the liquid a few radii away hardly knows it is there. An infinitely long cylinder has no such drag. The equations of slow viscous flow have no solution past it at all — Stokes found this in 1851 — because a force spread over circles instead of spheres never finishes spreading. What rescues the cylinder is whatever distant thing supplies a largest length, and its drag carries the logarithm of that length for ever after.

 The noisy orbit a true orbit stays beside. A computation of the doubling map with an error of up to 0.001 added at every one of 400 steps, and two exact orbits compared with it. The exact orbit started from the computation's own first point parts from it at once, by more than 0.1 after 8 steps, because each error is doubled at every step after it. The shadow — the exact orbit built by running the map backwards from the computation's last point, where each error is halved instead — stays within 8.0 × 10⁻⁴ of the computation at every one of the 400 steps, closer than the noise added at any single step. Every point of the noisy computation is therefore a faithful picture of a real orbit of the map — just not of the one that started where the computation did. Mechanics

The wrong orbit beside a right one

Every computer simulation of a chaotic system is wrong within a few dozen steps: rounding errors are stretched exponentially, and after fifty iterations of the simplest chaotic map two computations of the same equation disagree completely. Yet chaotic simulations are trusted everywhere, and for a reason that is a theorem rather than a hope. For a large class of chaotic systems the computed sequence, though it follows no true orbit from where it started, stays close to some other true orbit for as long as it runs.

 The area under the spacings is the bond. The gap between each vibrational level and the next, against level number, for a Morse well and a Lennard-Jones well with the same depth and the same curvature at the bottom, Λ = 100, in units of the well depth. The sum of the gaps from the lowest level up is the energy needed to break the bond from there. For the Morse well the gaps fall in a straight line to zero, and a straight line through the first 5 of them, extended to zero, encloses 0.9473 against the true 0.9473. For the Lennard-Jones well the gaps start almost the same and fall slightly faster at first — which is what the straight line sees — then trail off in a long tail of small gaps near the top; the same extrapolation from the first 5 encloses 0.7988 against the true 0.9474 — 15.7 per cent short, because the straight line cannot see the crowded levels its data never reached. Mechanics

The spectrum that weighs a bond

A vibrational spectrum measured near the bottom of a molecule's well gives its curvature, and the curvature says nothing about how deep the well is. Measured further up, it says everything. The levels of a real bond crowd together towards the top, the gap between neighbours falls towards zero, and the sum of all the gaps is the energy it takes to break the bond. A straight line drawn through the gaps that have been measured, and extended to zero, was for decades how bond energies were found — and for a bond with the tail of two neutral atoms it comes out a sixth short.

 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. Electromagnetism

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.

 Flow curves that bend back, and one that stops. Shear stress against shear rate, both on logarithmic axes, for volume fractions 0.5, 0.55, 0.565, 0.575, 0.6, from the same model: stress in units of σ, rate in units of σ over the liquid's viscosity. Below φ = 0.569 every curve rises steadily — the suspension thickens but continuously. Above it and below the frictional jamming point 0.58 the curve bends back into an S: over a range of stress, more stress gives less flow, and a suspension driven at a fixed rate must jump between branches. At φ = 0.6 the curve returns to zero rate at 2.47 σ*: pushed that hard, the suspension stops flowing altogether and holds the stress like a solid. Fluids

The suspension that seizes when pushed hard

Cornflour stirred into half its volume of water runs off a spoon like cream, and a fist driven into it bounces off as if it had hit a floor. The particles have not changed and neither has the water. What changes, above a threshold stress, is the kind of contact between neighbouring grains: lubricated, they slide past one another and the suspension can pack to 64 per cent before it locks; pressed into frictional contact, they lock at 58. A suspension sitting between those two numbers is a liquid when handled gently and a solid when handled hard.

 Restitution with nothing lossy in it. The coefficient of restitution of a perfectly elastic ball striking a large, perfectly elastic plate, against Zener's parameter λ, which grows as the square of the ball's radius over the plate's thickness and weakly with speed. At λ = 0 the ball comes back at its arrival speed. The rebound falls below half at λ ≈ 0.65, with no friction, plasticity or damping anywhere — the missing energy is travelling outward through the plate as bending waves. Marked: a steel ball 20 mm across at 1 m/s on steel plates 5 mm thick (λ = 0.94), 10 mm thick (λ = 0.23), 20 mm thick (λ = 0.06). Mechanics

The bounce an elastic plate cannot give back

Drop a steel ball on a thick steel block and it bounces almost to where it started. Drop it on a steel plate five millimetres thick and it comes back at barely half its speed, although steel on steel is about as elastic as materials get and nothing in the impact is plastic, sticky or damped. The energy has not been turned into heat. It has been turned into waves, running out through the plate faster than the ball can take them back — and a ball striking a longer rod loses energy the same way, by leaving it ringing.

 The wobble slows as the body yields. The period of the Earth's free wobble against how far the body yields to its own rotation, measured as the Love number k over its value for a fluid Earth, kₛ = 0.941, computed from the Earth's flattening. A rigid Earth, at zero, wobbles in 303.6 days, Euler's period. The observed 433 days requires k/kₛ = 0.299, an effective k of 0.281: the Earth yields about 30 per cent as much as a fluid would. The period grows without limit as the body approaches a fluid, because a fluid body's bulge re-forms round any new spin axis and leaves nothing to push the axis back. Mechanics

The wobble that is slow because the Earth gives

Euler worked out in 1765 that a spinning, flattened Earth whose axis is knocked slightly off its figure should wobble, the pole wandering in a small circle once every ten months. Astronomers looked for a ten-month period for a century and found nothing. In 1891 Seth Chandler found a wobble of fourteen months instead. The four extra months are not an error in Euler's arithmetic. They are the Earth yielding: its bulge partly re-forms round the displaced axis, weakening the torque that drives the wobble, and the lengthening measures how soft the planet is.

 The triangle of collisions that survive. Every collision of a particle ε above a filled Fermi sea with a partner from the sea, at zero temperature: the partner's energy across, one product's energy up, both from the Fermi level, the other product's fixed by energy conservation. Without exclusion every point in the plotted square would be an allowed collision. With it, the partner must come from below the Fermi level, and both products must land above it: the partner no deeper than ε below, the first product above zero, the second product above zero too. What survives is the shaded triangle, whose area is half of ε squared, and the collision rate is proportional to it. Halve the particle's excess energy and the triangle shrinks to a quarter; at the Fermi level itself it vanishes, and a particle there cannot scatter at all. Quantum

The collisions the exclusion principle forbids

A copper wire holds one free electron per atom, packed a quarter of a nanometre apart and repelling one another with the full Coulomb force. It seems impossible that an electron could travel any distance through that crowd without colliding. It travels micrometres — thousands of atoms — before another electron deflects it. The collisions are not weak. They are forbidden: to collide, both electrons need empty states to move into, and in a sea filled to the brim almost none exist. The same rule makes cold helium-3 thicken as the square of how cold it gets.

 A tiny ripple that grows, peaks, and gives the energy back. The highest point of a wave train's envelope against time, starting from an even train of amplitude 1 with a ripple of 1 per cent at the fastest-growing wavenumber, integrated from the nonlinear Schrödinger equation. For a while nothing happens that the eye could see; the ripple grows exponentially from a per cent. Then, near t = 5.6, the energy of a whole modulation period gathers into one crest 2.41 times the original amplitude — 1 + √2, the height of the Akhmediev breather this wavenumber produces. Then it spreads out again, and by t ≈ 11 the train is almost even once more, its highest point 1.01: the energy returns to where it came from, and the cycle repeats, the crest rising again near t = 17. The wave's total norm is conserved throughout to a part in a hundred million. Waves

The even wave train that cannot stay even

In the mid-1960s a team at the National Physical Laboratory tried to make a long, perfectly regular train of waves in a towing tank, and could not: however carefully they set the paddle going, a few hundred metres down the tank the waves had bunched into groups. Their paddle was not at fault. A uniform train of steep waves on deep water is unstable — small modulations of its height grow, the energy of many waves gathers into a few, and then, astonishingly, gives itself back. Out of that instability comes a wave three times the height of the sea around it, with no history before and none after.

 Chains counted, and then blobs. The osmotic pressure of a polymer solution in a good solvent against concentration, on logarithmic axes, for chains of 1,000, 10,000, 100,000 segments. Dilute, each chain is one particle and the pressure is van 't Hoff's, proportional to the number of chains: a line of slope one, lower for longer chains by the ratio of their lengths. Past the overlap concentration of each, marked, the coils interpenetrate and the curves bend up onto a single line of slope 2.31 that does not depend on chain length at all. The overlap concentration falls as the chains lengthen, by a factor of 5.8 for each tenfold increase in length, which is why the longest chains leave the counting of molecules first. Fluids

The pressure that stops counting molecules

Van 't Hoff's law counts dissolved particles, and for a dilute polymer each chain is one particle, however many thousand segments it has. That is how osmometers weigh polymers. Concentrate the solution until the coils overlap and the count changes. The pressure stops depending on how long the chains are and grows as the 2.3 power of the concentration, the same for chains a thousand segments long and a hundred thousand. The unit being counted is no longer a molecule. It is a stretch of chain as long as the mesh of its neighbours allows, and it has no chemical identity at all.

 The shear a pushed contact carries, and where it gives. Across a Hertzian contact of radius a, the normal pressure as a fraction of its peak, which is also the friction limit μp in units of μp₀ (dashed), and the shear traction for a sideways push of 0.3, 0.6, 0.9 times the load that would slide it, μP. Where the shear reaches the dashed ceiling the surfaces slip; inside, they stick. At 0.3 μP the stuck disc has a radius of 0.89a, so 21 per cent of the contact area is slipping; at 0.6 μP the stuck disc has a radius of 0.74a, so 46 per cent of the contact area is slipping; at 0.9 μP the stuck disc has a radius of 0.46a, so 78 per cent of the contact area is slipping. The shear is highest just inside the slipping ring and, in an elastic contact that simply stuck everywhere, would be infinite at the rim: slip at the edge is what removes that singularity. Mechanics

The contact that slips before it slides

Static friction is supposed to hold perfectly until the load reaches μ times the normal force and then give way. Two elastic bodies pressed together do something else. Push a ball sideways on a flat with any force at all and a ring at the edge of their contact is already slipping, while the centre stays stuck. As the push grows the ring spreads inward, and gross sliding is only the moment the stuck centre shrinks to nothing. Shake the push back and forth and the ring slips each way on every cycle. The contact then loses energy though it never slides, as the cube of how hard it is shaken.

 A bound state that lives outside its well. The radial probability density of the one bound state of a square well of radius R, against distance in units of R, for four depths just past the one that first binds a state. In units of ε = ħ²/2μR², a state bound by 0.00589 ε has 93 per cent of its probability outside the well and a mean radius of 7.0 R; a state bound by 0.0397 ε has 82 per cent of its probability outside the well and a mean radius of 3.0 R; a state bound by 0.245 ε has 62 per cent of its probability outside the well and a mean radius of 1.5 R; a state bound by 1.4 ε has 35 per cent of its probability outside the well and a mean radius of 0.9 R. The more weakly the state is bound, the further its exponential tail reaches, and the less of it is where the force is. Mechanics

The molecule that lives outside its own well

The harmonic picture describes the bottom of a well, where a molecule vibrates near the length of its bond. The last bound level lives at the top, and when it is bound by almost nothing it is not near the bond at all. Two helium atoms whose potential well bottoms out at three ångströms form a molecule whose atoms sit on average about fifty ångströms apart, most of the time where the force between them is negligible. Such a state forgets the shape of the well that made it. Its size, its binding and the way it scatters are all set by one number, and nothing inside the well can be read from them.

 A barrier the metal was supposed to choose. The barrier height at a metal contact on n-type silicon against the metal's work function, from Bardeen's model with interface-state densities of none, 10¹² per cm² per eV, 10¹³ per cm² per eV, 10¹⁴ per cm² per eV, a 0.5 nm gap and a neutrality level a third of the gap above the valence band. With no interface states the barrier is Schottky and Mott's φₘ − χ, rising one for one with the work function: from aluminium to platinum, 0.23 to 1.60 eV. With 10¹² per cm² per eV the slope is 0.92 and the same metals give 0.27 to 1.53 eV; with 10¹³ per cm² per eV the slope is 0.53 and the same metals give 0.48 to 1.20 eV; with 10¹⁴ per cm² per eV the slope is 0.10 and the same metals give 0.71 to 0.84 eV. Dense enough interface states pin every barrier near 0.76 eV, and the metal hardly matters. Quantum

The barrier the metal cannot choose

Put a metal on a semiconductor and a barrier forms at the interface. The obvious theory says its height is the difference between the metal's work function and the semiconductor's electron affinity. Platinum and aluminium differ by nearly an electronvolt and a half in work function, so they should make wildly different contacts. On silicon and gallium arsenide they do not: nearly every metal gives nearly the same barrier. Bardeen explained why in 1947. A thin layer of electronic states at the interface screens the metal, and it takes only a few such states per hundred surface atoms to pin the barrier wherever they put it.

 One mode that loses less the faster it is driven. The wall loss, in decibels per kilometre, of three modes in a copper pipe 60 mm across, against frequency on logarithmic axes: TE₁₁, the lowest mode, cut off below 2.93 GHz; TM₀₁, cut off below 3.82 GHz; and TE₀₁, cut off below 6.09 GHz. Each loss is infinite at its cutoff and falls just above it. Then TE₁₁ and TM₀₁ turn and rise with frequency, as the square root, the way every conductor's loss does. TE₀₁ keeps falling: 0.51 dB/km at 60 GHz and 0.20 dB/km at 110 GHz, against 21 dB/km for TE₁₁ at 60 GHz. Waves

The loss that falls as the frequency rises

Every metal conductor loses more at higher frequencies, because the current crowds into a thinner skin and meets more resistance. One wave in a round metal pipe breaks the rule. The TE₀₁ mode's electric field circles the axis and touches the wall nowhere. As the frequency rises it becomes more and more nearly a plane wave running down the middle of the pipe, and it drives less and less current in the metal. Its loss falls with frequency, to about half a decibel per kilometre at 60 GHz in a pipe 60 mm across. In the 1960s it was the best way anyone knew to send a hundred thousand telephone calls across a continent. The catch was that the pipe also carried seven hundred other modes.

 A sharp angle costs a spread of turning. Left: the probability of finding a rotor at each angle, for three states. Right: the probability of each value of its angular momentum, in units of ħ, for the same states. A state bunched to a circular spread of 0.025 has an angular-momentum spread of 2.21ħ; a state bunched to a circular spread of 0.302 has an angular-momentum spread of 0.59ħ; a state with no preferred angle has an angular-momentum spread of 0.00ħ. The angular momentum can only take whole-number values, so its probabilities are bars; the angle is continuous. The state with no preferred angle has exactly one value of angular momentum, and its angle is spread evenly round the circle — where the naive rule would say that its angle's spread times ħ/2 should be infinite, it is π/√3. Quantum

The angle the uncertainty principle cannot be written for

Position and momentum trade exactly: squeeze one and the other spreads, with a product never below ħ/2. Angle and angular momentum look like the same pair and are not. A rotor in a state of definite angular momentum has zero spread in angular momentum and a finite spread in angle, π/√3, and their product is zero. Nothing is wrong with the rotor. The relation was written with a commutator that does not hold, because an angle lives on a circle and a line has no ends to wrap round. The relation that does hold measures the angle's spread by how far its average position on the circle falls short of the rim.

 A pointer that ends up where no eigenvalue could put it. The distribution of a measuring pointer's position, in units of its own initial spread, after it has been coupled to a spin whose two values would move it by ±g. Strongly coupled, g = 3, the pointer splits into two separate spots at ±3: an ordinary measurement. Weakly coupled, g = 0.1, the two possible shifts overlap almost entirely and the pointer's centre moves by only 0.020, the average of the spin times g. Keeping only the 0.99 per cent of runs in which the spin is afterwards found pointing along x, the kept pointer is centred at 0.802 — 8.0 times g, far outside the range ±g that either value of the spin could produce, and approaching the weak value 10, which it would reach exactly for a still weaker coupling. Quantum

The value a gentle measurement returns

Couple a pointer to a spin so weakly that the pointer's shift is far smaller than its own blur, and a single run learns almost nothing about the spin. Now keep only the runs in which the spin is later found in a chosen, nearly opposite state. The kept pointers move as though the spin's value were ten, or a hundred, though it can only be plus or minus one. That weak value was predicted in 1988 and has been measured many times. It is not a hidden value of the spin, and it does not break the uncertainty principle: it is paid for exactly in discarded runs. What it does give is a way to watch a quantum system without disturbing it.

 The heaviest star free neutrons can hold up. Mass against radius for cold stars of free neutrons, each point a different central density, held up by the pressure of the neutrons' exclusion alone: in general relativity and, with the same gas, in Newton's gravity. Relativity's curve reaches a highest mass of 0.71 solar masses at a radius of 9.5 km and turns back; Newton's keeps rising towards 5.8 solar masses as the stars shrink. The measured masses of neutron stars reach about 2.08 (dashed), three times what free neutrons allow, so the matter inside has to be stiffer than a gas of free neutrons: the repulsion between nucleons at short range is holding them up. Astrophysics

The pressure that weighs down what it holds up

In Newton's gravity a star is held up by pressure, and stiffer matter can hold up more: a star of free neutrons could reach almost six solar masses. In general relativity the pressure itself has weight. Every layer squeezed harder to hold up the layers above adds to the pull it is resisting, and past a point pressing harder makes things worse. The same gas of neutrons then tops out at 0.71 solar masses. Neutron stars twice as heavy exist, which says what their insides must be like — and no matter of any kind can hold a static body smaller than nine-eighths of its Schwarzschild radius.

 The same paste stopping or running, according to the push. A thixotropic paste, sheared hard and then held under a constant stress: its viscosity against time, on a logarithmic axis, for stresses of 1.5, 1.8, 2.0, 2.4 Pa. Its structure builds at rest and is broken by flow, and the model's critical stress is 1.89 Pa. Below it the paste's viscosity climbs without limit: it slows, its structure builds faster than the slowing flow can break it, and it creeps to a stop, its viscosity growing in the end as the 1.5 power of time. Above it the viscosity falls: it speeds up, breaks its structure faster, and runs at a steady rate. After a thousand seconds the viscosities stand at 514.1, 238.3, 2.0, 1.5 Pa s: stresses a few per cent apart have gone opposite ways — a bifurcation, not a smooth change — and the paste hesitates longest when the stress is closest to the critical value. Fluids

The paste that forgets it was stirred

A jar of clay slurry left on a shelf sets like a jelly; shaken, it pours. That is thixotropy, and the tempting description — its viscosity falls when it is stirred — misses what makes it strange. A thixotropic paste has no viscosity at a given rate of stirring until it has been told how long it has been stirring, and under a steady push it cannot settle on a slow flow at all. Push slightly less than a critical stress and it creeps to a halt; push slightly more and it runs. Between the two there is nothing, and a paste made to flow slowly splits into a layer that flows and a layer that does not.

 A mirror for X-rays that works only at a glance. The fraction of 8 keV X-rays reflected by a smooth surface of gold and of silicon, against the grazing angle between the ray and the surface, in milliradians, with absorption included (solid) and without it (dashed). Below the critical angle √(2δ) — 10.02 mrad (0.57°) for gold, 3.88 mrad (0.22°) for silicon — the X-rays are totally reflected from outside the material, because its index is below the vacuum's; above it almost nothing is. Absorption rounds the edge and keeps the plateau a little below one. A mirror for X-rays is therefore a surface met at less than a degree, and it reflects well only while it is met that way. Optics

The index that falls below one

Glass bends light because light is slower in it. X-rays are not slower in glass, or in anything: every material's refractive index for them is slightly less than one, by a few parts in a million. Everything about X-ray optics follows from that small number with the wrong sign. A lens that focuses them must be thinner in the middle than at the edges, and so weak that dozens are stacked in a row. A mirror reflects them only if they arrive at a glancing angle of less than a degree — totally, from the outside, the reverse of the way a diamond traps light — which is why the mirrors of an X-ray telescope are long, nested barrels rather than dishes.

 What correlation is worth, to a helium atom and to H⁻. The energy needed to remove the second electron, by three descriptions, as a fraction of the measured value, for the negative hydrogen ion and for helium. Helium: one shared orbital 23.07 eV, one electron in and one out 23.83 eV, measured 24.59 eV — the simple picture is 94 per cent right. H⁻: −0.74, 0.362 and 0.755 eV — the simple picture is not merely inaccurate but has the wrong sign. The energy the simple picture misses — the electrons avoiding each other — is about 1.5 eV in both; in helium it is a small correction to 24.6 eV, and in H⁻ it is everything. Quantum

The ion that exists because its electrons keep apart

Add a second electron to a hydrogen atom and it holds on, by three-quarters of an electronvolt. The simplest respectable calculation says it should not — that the two electrons, sharing one orbital, repel each other more than the proton can hold them, and the ion falls apart. The same calculation gets helium's energy right to two per cent. The difference is not a failure of the method but a measure of what it leaves out: the two electrons avoiding each other. In helium that is a small correction. In the negative hydrogen ion it is the whole of the binding, and that ion is what makes the Sun opaque.

 A beam of light pulls hardest on what comes towards it. The sideways pull of a long, thin beam of light on a body moving parallel to it at velocity βc, as a multiple of the pull the same body would feel from a static mass with the same energy per unit length, against β: 2(1 − β)²/(1 + β²). A body at rest beside the beam is pulled twice as hard as by a static mass — the beam's pressure along its length counts. A body moving with the beam is pulled less, and a ray of light running alongside it is not pulled at all (β = 1). A ray running against it is pulled 4 times as hard as a static mass of the same energy would pull it. Tolman, Ehrenfest and Podolsky found this in 1931. Two parallel laser beams do not attract each other; two opposed ones do. Relativity

The light that pulls twice as hard as its mass

Energy has mass, E/c², and a box of light weighs that much more than an empty one. But the gravity something produces is not set by its energy alone. Pressure counts too, three times over, and light's pressure is a third of its energy — so a beam of light pulls on things beside it twice as hard as its mass would suggest. The box weighs only E/c² because its walls, straining to hold the light in, pull with a negative amount that exactly cancels the difference. Two parallel beams of light do not attract each other at all; two opposed ones attract four times as hard. And a substance whose pressure is negative enough does not pull at all — it pushes.

 The same absorber, spread evenly and gathered into patches. The fraction of light transmitted by a layer holding a fixed amount of absorber, on a logarithmic axis, against the optical depth it would have if spread evenly, for the absorber spread evenly (solid) and gathered into patches covering 50 per cent and 10 per cent of the area, the rest left clear. At an average optical depth of 2 the even layer transmits 13.5 per cent; with the absorber in patches over half the area, 50.9 per cent; over a tenth, 90.0 per cent — almost all of it through the gaps. The average of e^(−τ) over a layer is never less than e^(−average τ): Jensen's inequality, which holds for any convex function. Beer's law describes a uniform absorber and overstates the attenuation of every other kind. Waves

The pigment that absorbs less for being packed

Dissolve a dye evenly in water and it absorbs a certain fraction of the light passing through. Gather the same dye into grains — the same molecules, the same amount, in the same volume — and it absorbs less. The light has not changed and neither have the molecules. The grains shadow their own insides, and the light that would have been caught by the shaded molecules slips through the gaps between grains instead. The effect is called packaging, and it is why the chlorophyll inside large algae is worth less per molecule than the chlorophyll in small ones, and why an ocean's colour does not say directly how much chlorophyll is in it.

 A target that grows as the traveller slows. The capture cross-section of a non-rotating black hole for particles arriving with speed v, in units of the horizon's own cross-sectional area π(2GM/c²)², against v/c, both on logarithmic axes (solid). For light it is 27/4 = 6.75 times the horizon's area. It grows as the particle slows, following 4c²/v² (dashed) — 404 at a tenth of light speed and 7.43·10⁶ at the 220 km/s of the Galaxy's dark matter. The dotted curve is a Newtonian ball of radius 8GM/c² — four horizon radii — capturing by ordinary gravitational focusing, and for slow particles the hole and the ball cannot be told apart. Astrophysics

The target that grows as the traveller slows

Light aimed at a black hole is swallowed if it passes within about five of the hole's gravitational radii, a disc six and three-quarter times the horizon's own. A particle arriving slowly is swallowed from much further out, and the disc grows as the inverse square of its speed: dark matter drifting at 220 kilometres a second sees a target seven million times the horizon. The reason is that slow matter carries almost no angular momentum for a given aim, and angular momentum is the only thing that keeps anything out. For slow traffic the hole behaves exactly like an ordinary ball four times its own size.

 The band in which radiation feeds a mode instead of draining it. The speed of a mode's pattern as seen from far away, in units of the star's own spin rate, against the spin rate as a fraction of the fastest a star can turn. The shaded band, between zero and one, holds patterns that run forward across the sky and backward through the star; gravitational radiation drives any mode inside it and damps any mode outside. The r-modes with ℓ = m = 2, 3 and 4 sit at 0.667, 0.833 and 0.900 at every spin, inside the band however slowly the star turns. The dashed curves are schematic modes whose crests run backward on the star at a fixed 0.2, 0.5 and 0.9 of the Kepler rate: each is dragged into the band only once the star turns faster than its crest runs, which is why an ordinary oscillation needs a star near break-up and an r-mode does not. Astrophysics

The wobble that radiating makes larger

Radiation normally drains a vibration: a struck bell goes quiet because it sends its energy away as sound, and a ringing black hole settles in milliseconds because it sends its energy away as gravitational waves. A spinning star can have the opposite. A pattern that runs backward through the star's material, while the star carries it forward across the sky, has less angular momentum than the star would have without it, and every wave it radiates takes away angular momentum it did not have to give. The pattern grows. One family of such patterns, the r-modes, runs backward on every rotating star, so every spinning neutron star is unstable unless friction wins — and that may be why none has been found spinning faster than 716 turns a second.

 The heat two surfaces exchange when they almost touch. The radiative heat-transfer coefficient between two parallel half-spaces at 300 K, in W/m²K on a logarithmic axis, against the width of the vacuum gap between them, also logarithmic, from fluctuational electrodynamics: silicon carbide (solid) and gold (dashed), with two perfect black bodies, 4σT³ = 6.12 W/m²K, for comparison (dotted). Far apart, silicon carbide exchanges 3.3 W/m²K — less than black — and gold, a good mirror, only 0.024. Closer than the thermal wavelength, about 17 µm, waves that cannot leave either surface begin to tunnel across the gap. At 100 nm silicon carbide exchanges 137 W/m²K, at 10 nm 9338, at 1 nm 9.3·10⁵ — rising as the inverse square of the gap, 1.5·10⁵ times the black-body value at 1 nm. Gold exceeds black only below a few hundred nanometres and then grows much more slowly: 1284 at 10 nm, 2561 at 1 nm. Quantum

The heat that crosses a gap too narrow for light

Stefan and Boltzmann's law is usually read as a ceiling: no surface can radiate more heat than a perfect black body at the same temperature, 6.1 watts per square metre for every degree of difference at room temperature. It is a ceiling only for surfaces that are far apart. Bring two of them closer than the wavelength of their thermal glow and they begin to exchange waves that never leave either surface — waves that cling to it and die away within a wavelength — and those waves tunnel across the gap. Two plates of silicon carbide ten nanometres apart exchange fifteen hundred times what two black bodies can, almost all of it in a single colour, and the colour is the one in which silicon carbide is too shiny to glow at all.

 Every path a rolling ball can take on a turntable, seen from the room. The paths, seen from the room, of four solid balls rolling without slipping on a turntable of radius 15 cm turning at 33.3 revolutions a minute (grey disc), over 6.3 seconds. A ball set down at rest relative to the room stays exactly where it is, spinning on the spot while the table turns under it. Every other ball moves on a circle, turning the same way as the table at 2/7 of its rate — once every 3.5 turns of the table — and comes back to where it started, whatever the push. A ball set down moving with the table does not ride with it: it circles on a path three and a half times the radius it started at, and the circle's centre is on the other side of the axis. Mechanics

The ball a turntable will not throw off

Set a marble down on a spinning turntable and the expectation is that it will be flung off the edge. It is not. Placed at rest relative to the room, it stays exactly where it is, spinning on the spot while the table turns beneath it. Given a push, it moves in a circle, turning the same way as the table at two-sevenths of the table's rate, and comes back to where it started. Tilt the turntable and the marble does not roll downhill: it wanders across the slope in a chain of loops. A rolling ball on a rotating surface obeys exactly the equation of an electric charge in a magnetic field, and the two-sevenths is how much of every push goes into turning the ball rather than moving it.

 The frequency of a mass on a spring that weighs something. The angular frequency of a mass M on a spring of stiffness k and mass mₛ, in units of √(k/M), against the ratio mₛ/M on a logarithmic axis: exact, from β tan β = mₛ/M (solid); Rayleigh's estimate √(k/(M + mₛ/3)) (dashed); and the textbook answer that ignores the spring's mass, 1 (dotted). For a spring a tenth of the mass the exact value is 0.9836, Rayleigh's 0.9837 and the textbook's 1 — 1.7 per cent too fast. With spring and mass equal the exact frequency is 0.8603, Rayleigh's 0.8660: the third is good to 0.66 per cent where the textbook answer is 16 per cent out. Mechanics

The third of itself a spring carries

Every textbook gives the frequency of a mass on a spring as the square root of stiffness over mass, and every real spring has mass of its own, which the formula leaves out. The repair is famous and oddly specific: add a third of the spring's mass to the load. The third is not a fudge. It is what the spring's kinetic energy comes to if each coil moves in proportion to its distance from the fixed end, and it is good to two-thirds of a per cent even when the spring weighs as much as the mass. Where it fails, it fails because the spring has stopped being a spring and become a column of waves, with notes of its own above the bounce — the notes that make the valve springs of an engine lose control.

 The stiffness a two-solar-mass star demands. The heaviest neutron star the model matter can support, in solar masses, against the constant square of the sound speed assumed above 300 MeV/fm³, as a fraction of c². At the conformal value, one-third (dashed), it is 1.92 solar masses; the heaviest star the model can make rises to 2.97 at the causal limit. To hold up the 2.08 solar masses of J0740+6620 (shaded) the square of the sound speed must exceed 0.394 of c² somewhere above that density — above the conformal one-third, by this model's reckoning. The conclusion depends on the model's low-density part, which the next figure varies. Relativity

The sound speed a neutron star needs

Heat any gas without limit and the speed of sound in it climbs towards a ceiling: not the speed of light, but light's speed divided by the square root of three, the sound speed of a gas of massless particles. For decades it was a natural guess that no matter could beat that ceiling, since the densest matter ought to behave like a gas of free quarks. Neutron stars say otherwise. A star of twice the Sun's mass, packed into a ball twenty-five kilometres across, can only be held up if its core is stiffer than the ceiling allows — unless nuclear physics fails at densities where it is believed to hold. Somewhere inside the heaviest neutron stars, sound almost certainly travels faster than it can in any free gas.

 Why the mixture of two equal waters is heavier than either. A temperature–salinity diagram for seawater near the surface, with lines of equal density (curves) 0.3 kg/m³ apart. Two waters lie on the same line: one at 4 °C and 33.93 g/kg, the other at 16 °C and 36.53 g/kg, both 1026.00 kg/m³. Mixing them in any proportion moves along the straight line between them, because temperature and salt add linearly; the lines of equal density curve, because cold water's density responds less to temperature than warm water's. The straight line lies on the dense side of the curve. An even mixture, at 10 °C and 35.23 g/kg, is 0.182 kg/m³ denser than either parent, and sinks between them. Fluids

The mixture heavier than either water

Two bodies of seawater can have exactly the same density — one cold and fresh, one warm and salty — and sit side by side without either wanting to sink. Mix them, and the mixture is denser than both, and sinks. Nothing has been added and nothing removed; the extra weight comes from the fact that water's density is a curved function of temperature while mixing averages temperature in a straight line. The same curvature, at its most extreme, is why a lake overturns twice a year and freezes from the top, and why the open ocean, too salty to have a density maximum above freezing, does neither.

 How much darker a surface looks under a film of water. The diffuse reflectance of a surface covered by a thin film of clear water (solid), against its reflectance dry, with the dry value for comparison (dashed): (1 − Rₑₓₜ) a (1 − Rᵢₙₜ)/(1 − a Rᵢₙₜ), where the film traps and returns light that falls outside its escape cone. A bright surface of reflectance 0.9 falls to 0.77; a mid-grey one of 0.5 to 0.32, 64 per cent of its dry value; a dark one of 0.2 to 0.11, 54 per cent. Dark surfaces darken proportionally most, because each return trip costs them most; and a coloured surface grows more saturated, because the colours it absorbs are absorbed again on every return while the colour it reflects survives. Optics

The sand that darkens when it is wet

Wet sand is darker than dry sand, a wet stone darker than a dry one, a splashed pavement darker in patches, and a wet white shirt turns grey and then see-through. Water is clear and absorbs almost nothing, so it cannot be adding darkness. What it does is change where the light goes. A film of water on a surface traps light that has been scattered back up, returning most of it for another chance to be absorbed. Water in the pores between grains replaces air with something closer to the grains' own refractive index, so each grain turns light aside less, and light wanders deeper before it gets out. Both lengthen the path, and on a longer path more light is absorbed by the material that was there all along.

 The boiling curve, which turns back. Heat flux from a horizontal surface into water boiling at one atmosphere, on a logarithmic axis, against how far the surface is above the boiling point, also logarithmic, from standard correlations. At a few degrees heat leaves by convection; from about 5 K bubbles nucleate and the flux climbs as the cube of the superheat, to the critical heat flux of 1.11 MW/m² at 20 K. Beyond it the flux FALLS as the surface gets hotter — the transition regime, drawn dashed as the interpolation it is — down to 19 kW/m² at 67 K, the Leidenfrost point, where a continuous film of vapour insulates the surface. Past that, film boiling and radiation let the flux rise again, slowly: 310 kW/m² at 1000 K above boiling. Between the crisis and the Leidenfrost point, a surface 3 times hotter passes 58 times less heat. Thermodynamics

The boiling curve that turns back

Heat a surface under water and more heat flows from it the hotter it gets — up to a point. Past about twenty degrees above boiling the bubbles crowd together, merge, and begin to blanket the surface with vapour, and from then on a hotter surface passes less heat, not more, until by a few hundred degrees above boiling it is wrapped in an insulating film and passes fifty times less than it did at its best. Shigeo Nukiyama found the turn in 1934 by melting wires. It is why a water drop skates for a minute on a very hot pan but vanishes in two seconds on a merely hot one, and why every boiler, steam generator and water-cooled reactor is designed around a number called the critical heat flux.

 The race to equilibrium that the hotter start wins. How far each system is from equilibrium with the bath — the relative entropy of its distribution from the bath's, in nats, on a logarithmic axis — against time since it was placed in the bath at 0.30, also logarithmic, for a start at 0.50 (warm, dashed) and one at 0.901 (hot, solid). The hot system starts further away: 1.243 against 0.254. Both close in quickly as their left wells equilibrate internally. Then the warm system must wait for probability to leak between the wells, and stalls, while the hot one, whose wells already had the right shares, carries on falling. The hot curve overtakes the warm one at a time of about 6.9, and from then on the hotter start is always nearer equilibrium. Thermodynamics

The hotter start that cools first

Put two identical systems, one warm and one hot, into the same cold bath, and common sense says the warm one reaches the bath's temperature first, since the hot one has further to go. The claim that hot water sometimes freezes before warm water — the Mpemba effect — has been argued over for sixty years and does not survive careful measurement. But a cleaner version of it is real and has been observed: in systems whose relaxation has a slow route and a fast one, a hotter start can miss the slow route entirely and overtake a cooler start, exponentially. Whether a system cools quickly depends not on how far it is from equilibrium but on how much of the slowest way back it needs.

 The pressure under a wheel rolling on a soft surface. The pressure under a rigid wheel of radius 30 cm carrying 3 kN per metre of width as it rolls over a rubber-like bed, against position along the contact (ahead of the axle to the right), scaled to the largest pressure and the widest contact, at three speeds. The bed relaxes with times of 2 and 20 ms. At 1 mm/s the pressure is symmetric and its resultant passes through the axle. At 1.0 m/s the material ahead is being squeezed faster than it can flow and pushes back harder than the recovering material behind, which lags and lets go early: the resultant sits 6.39 mm ahead of the axle. At 1000 m/s the bed has no time to relax at all and behaves as a stiffer elastic solid, symmetric again. A wheel is held back by where its support pushes, not by any sliding. Mechanics

The push a millimetre ahead of the axle

A wheel rolling freely on a flat road slows down, and nothing in it slides. What holds it back is where the road pushes on it: not directly under the axle but a little ahead, because the material in front of the contact is being squeezed and pushes back harder than the material behind, which is recovering and lags. A car tyre's push is about three millimetres ahead of its axle, a train wheel's about two-thirds of a millimetre, and that offset, divided by the radius, is the whole of rolling resistance. It peaks at the speed where the time under the wheel matches the time the material takes to relax, and the same loss, at a different frequency, is what gives a tyre its grip.

 A drop that sweeps up mist settles at a seventh of g. The acceleration of a water drop falling from rest through a still mist that it sweeps up completely, in units of g, against time, for drops starting at radii of 0.2 mm, 0.05 mm and effectively zero, with no air resistance. The mist holds 10 g of water per cubic metre, twenty times a typical cloud, so that the approach takes seconds rather than minutes. Every drop starts at g, because at first it has gathered nothing, and every one settles to g/7 = 0.1429 g: after 2 s the three are at 0.377, 0.172, 0.143 g, and after 8 s at 0.144, 0.143, 0.143 g. The drop that starts from nothing is at g/7 from the start. The final acceleration contains no property of the mist, the drop or the water: a denser mist makes the drop grow faster and reach g/7 sooner, and nothing else. Mechanics

The raindrop that falls at a seventh of g

A drop of water falling through a still mist, gathering every droplet in its path and with no air to slow it, does not fall at g. It settles to an acceleration of exactly a seventh of g, whatever the mist, whatever the drop, and its radius grows by the same amount for every metre it falls. Of the work gravity does on it, three-sevenths goes into the gathering and never comes back. The same reasoning gives a third of g for a chain paid out from a heap, and a single rule — g/(2n + 1) — for anything that sweeps up mass as the n-th power of the distance it falls.

 Four cosines between a lens and the edge of its picture. Light from a direction 30° off the axis passing through a lens and landing at the edge of the image, with the four factors that dim it relative to the centre. The aperture, seen from that direction, is foreshortened to cos θ of its area. The image point is further from the lens, f/cos θ instead of f, and the light spreads as the square of the distance: cos²θ. The light arrives at the image plane at a slant, spreading over 1/cos θ more area: one more cos θ. Together, cos⁴θ: at 30° the edge receives 0.563 of the light at the centre, 0.83 photographic stops darker. Nothing here is a fault of the glass. A perfect thin lens, or a pinhole, does it, because it is geometry. Optics

The four cosines at the edge of a photograph

Photograph a plain, evenly lit wall with a wide-angle lens and the corners of the picture come out darker than the middle — by more than three photographic stops with a 14 mm lens on a full-frame camera. It is usually blamed on the lens. A perfect lens does it too, and so does a pinhole, because the light reaching the edge of an image is reduced by four separate cosines of the angle it came from: three of them pure geometry, and one the slant at which it lands. Fisheye lenses avoid most of the loss by refusing to keep straight lines straight. Every wide lens is a choice between the two.

 A wave that falls with nothing pulling it. The intensity of Berry and Balazs's Airy packet, |Ai(x − t²/4)|², drawn at six moments t = 0 to 5 in units where ħ = m = 1, each profile raised by its time. It is an exact solution of the free Schrödinger equation — no potential, no force — and equally of the equation that governs a light beam in the paraxial approximation, with t then the distance travelled. Its shape never changes and never spreads, and its main lobe, which starts at x = −1.02, moves along the parabola x = t²/4 (dashed): −0.77 at t = 1, −0.02 at t = 2, 1.23 at t = 3, 2.98 at t = 4, 5.23 at t = 5. A uniform acceleration of a half, with nothing accelerating it. The packet carries infinite energy, which is the price of the trick: its oscillating tail stretches to minus infinity, and the tail is what pushes the main lobe along. Waves

The packet that bends with nothing pushing it

Every wave packet left to itself spreads, and every free particle moves in a straight line. In 1979 Berry and Balazs found a packet that does neither: it keeps its shape for ever, and its peak accelerates sideways along a parabola, in empty space, with no force acting. Nothing in it breaks Newton's laws — the packet's centre of mass stays exactly where it was, and only its bright head runs off, fed by a long dim tail — and the trick turns out to be the rainbow's: the bright lobe is the envelope of a family of straight rays. Thirty years later it was made with light, and then with electrons.

 Six ways a wave of gravity could move a ring of free masses. Sixteen free masses on a ring (dashed) and where a passing gravitational wave of each of the six possible polarisations moves them at its crest, drawn at a strain of 0.32, some 10²⁰ times larger than a real wave's. Top row: plus and cross, the two general relativity allows, which stretch one direction across the wave while squeezing the other and change no area; and breathing, which swells the ring evenly. Bottom row, drawn in planes containing the direction of travel (arrow): longitudinal, which stretches along the travel; and the two vector modes, which shear those planes. Any metric theory of gravity permits these six and no others; which of them a theory keeps is decided by the field that carries gravity. Astrophysics

The four shapes a wave of gravity leaves out

A passing gravitational wave squeezes a ring of free masses in one of two patterns, and that is usually presented as the whole menu. It is general relativity's choice from a longer one. The tide a wave carries is a symmetric table of six numbers, so any theory in which gravity is geometry allows six shapes; a theory keeps two only if its graviton has no mass and nothing else rides along with it. Give the graviton the smallest mass imaginable and it gains three more states, and one of them changes how much the Sun bends light — by a quarter, however small the mass.

 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. Electromagnetism

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. Electromagnetism

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.

 Throws that grow into orbits. The longest throws possible from one point on the Earth, at launch speeds of 3, 5, 6.5, 7.5 km/s, ignoring air, each at its best angle, drawn to scale round the Earth. 3 km/s: launched at 42.8° above the horizontal, landing 1.0 thousand km away; 5 km/s: launched at 37.8° above the horizontal, landing 3.2 thousand km away; 6.5 km/s: launched at 29.7° above the horizontal, landing 6.8 thousand km away; 7.5 km/s: launched at 17.6° above the horizontal, landing 12.2 thousand km away. The faster the throw, the flatter the best launch and the further round the Earth it lands; at the circular-orbit speed, 7.91 km/s, the best launch is horizontal and the throw never comes down. Mechanics

The parabola that is the top of an ellipse

Every textbook throw is a parabola, the best angle is 45°, and the range goes as the square of the speed. All three are statements about a flat Earth with gravity pointing the same way everywhere. On the real one a throw is a piece of an orbit — an ellipse with the Earth's centre at one focus — and the parabola is what the top of that ellipse looks like when the Earth is large. Throw faster and the best angle falls, the range runs ahead of the formula, and at eight kilometres a second the best throw is horizontal and never lands.

 Where the spot is, and when, for three sweep speeds. A spacetime diagram of the spot a turning beam makes on a flat screen at distance D: position along the screen across, the time at which light arrives there up, both in units of D and D/c, for beams sweeping the screen's nearest point at ωD = 0.5c, 1.0c, 3.0c. Each curve is the arrival time t(x) = atan(x/D)/ω + √(D² + x²)/c. Every curve has a lowest point on the approaching side — at x = −1.25, −0.79, −0.32 D respectively — because far out the beam strikes the screen obliquely and its sweep along the screen outruns light whatever ω is. Light reaches that point first: the spot is born there, as a pair, one running back up the screen and one forward. The faster the sweep, the nearer the centre the birth. The faint dashed lines are light signals, slope one. Wherever a spot's track is flatter than they are, it is a sequence of events no signal could connect. Relativity

The spot that outruns light and carries nothing

Flick a laser pointer across the face of the Moon and the spot it makes crosses the Moon faster than light. Nothing is wrong with relativity: the spot is not a thing, and no part of it travels from one side to the other. Each lit point is lit by its own light, sent from the hand. What makes the spot worth studying is how strange such a non-thing is allowed to be. It can be born in the middle of the screen as a pair running apart; its speed can be made infinite, or reversed, merely by watching it from a moving frame; and the composition of velocities that caps everything real has nothing to say about it.

 The standing shapes of a lake and of a harbour. The water surface at its greatest tilt in the slowest standing waves of a basin of length L, exaggerated vertically. In a closed lake the water must be still at both ends' walls, which are antinodes of the surface: the first mode tilts the whole lake about a node in the middle, the second has two nodes. In a harbour open to the sea at one end, the sea holds the mouth's level fixed, so the mouth is a node: the first mode has a quarter of a wave in the harbour, the second three quarters. Their periods are 2L/(n√(gh)) for the lake, Merian's formula, and 4L/((2n + 1)√(gh)) for the harbour — a harbour of a given length rings at twice the period of a lake of the same length and depth. Waves

The tide a lake makes by itself

On Lake Geneva the water at Geneva rises and falls by a few centimetres every seventy-three minutes, and the water at the far end does the opposite, as though the whole lake were a bath slopping back and forth. It is a standing wave, set ringing by wind and air pressure, and its period follows from the lake's length and depth alone. A harbour rings the same way with its mouth as a node, and it does something odder: narrowing its entrance, the obvious way to keep waves out, makes its resonance stronger.

 Rays that leave the far side of a compact star. Light rays leaving the surface of a star of radius 2.9 Schwarzschild radii — a neutron star of 1.4 solar masses and about 12 km — and travelling to an observer far off to the right, traced along the exact paths of Schwarzschild's geometry, shown above and below the line of sight. The last ray to escape leaves the surface tangentially from a point 121° round from the point facing the observer, so the bright arc — everything the observer sees — covers 76 per cent of the surface rather than the half that straight rays would show. The dashed circle is the horizon a star of the same mass would have if it were compressed inside it. Astrophysics

The star whose far side can be seen

Look at a ball and half of it faces you; the other half is hidden behind it. That half is a fact about straight lines, and near a neutron star the lines are not straight. Light leaving the surface is bent round the star on its way out, so a distant observer sees three-quarters of a typical neutron star at once — and a star squeezed to 1.76 times its own Schwarzschild radius would show every point of its surface, the point directly behind it included.

 How fast each point of a fast wheel moves. The speed through the ground of each point on the rim of a rolling wheel, in units of the speed of light, against the point's angle round the rim measured from the bottom, for axles moving at 0.5c, 0.8c, 0.95c. The dashed curves are the everyday answer, 2v sin(α/2), which puts the top at twice the axle's speed and would carry it past light. Composed relativistically, the bottom is still at rest and the top moves at 2v/(1 + v²/c²): 0.800c, 0.976c, 0.999c, always below c. Every point of the rim stays slower than light, and the fast part of the wheel is flattened into a broad plateau over the top. Relativity

The wheel whose top cannot go twice as fast

The top of a rolling wheel moves at twice the speed of its axle, and the bottom is momentarily at rest — two facts every cyclist's spokes show, blurred at the top and sharp at the bottom. Roll the wheel at a large fraction of the speed of light and the second fact survives exactly while the first cannot: the top moves at 2v/(1 + v²/c²), never as fast as light. Caught at one instant of the ground's time the spokes are bent, and over one turn the wheel advances not the circumference measured from its axle but γ times it — the rim's own length, laid down on the ground by a contact point that does not move.

 A pile of glass plates, added two ways. The transmission of light at normal incidence through a pile of N glass plates (index 1.5) in air, on a logarithmic axis, against N. Dashed: Stokes's 1862 result for light whose reflections add as intensities, T = (1 − R)/(1 + (2N − 1)R) with R = 0.040 per surface, which falls only as 1/N: 0.107 at 100 plates, 0.029 at 400. Solid: the same pile with the reflections added as waves and every plate and gap of random thickness, as ⟨ln T⟩ over 200 random piles (red) and for one pile (grey). It falls in a straight line, by 0.0843 per plate, an exponential decay with a localisation length of 24 plates: 10⁻³ at 100 plates and 10⁻¹⁴ at 400. Interference with random phases does not average away; it traps. Waves

The pile of glass that stops every colour

A sheet of window glass reflects about eight per cent of the light that falls on it. A pile of a hundred sheets, George Stokes calculated in 1862, should still let through a tenth, because each extra sheet only adds a little more reflection to the total. That is right for sunlight, whose colours are too mixed to interfere. For a single colour, with every sheet and every gap a slightly different thickness, the same pile lets through a thousandth, and a pile of four hundred a hundred-millionth of a millionth. Interference with random phases does not average away into the intensity sum. In one dimension it traps every wave, at every wavelength, however weak each reflection is.

 Two ways for a swing to die. A mass on a spring released 10.6 dead-band widths from rest, against time in units of 1/ω. Red: sliding on a dry surface, whose friction is a constant force; each swing is exactly a cosine, and the turning points fall in a straight line by two dead-band widths per half-cycle until the mass stops, 2.5 cycles after release, at −0.60. Blue: the same mass with viscous damping (damping ratio 0.05), whose force is proportional to speed; the swings shrink by the same fraction each cycle, along an exponential, and never stop. The period is the same in both and the same as with no damping at all to within the viscous case's 0.13 per cent. Mechanics

The swing that dies in a straight line

A pendulum in air loses the same fraction of its swing every cycle, so its amplitude falls along an exponential and, in principle, never quite reaches zero. A mass on a spring sliding over a dry surface does something else. Its friction is a fixed force, so it loses a fixed amount of amplitude every half-swing — the swings shrink in a straight line — and it stops completely, after a countable number of swings, somewhere inside a band around its natural position whose width is the friction divided by the stiffness. Where in that band it stops depends on where it started, which is why a barometer with friction in it is tapped before it is read.

 Where the level goes when a floating ice cube melts. The change in level when a 30 g ice cube floating in a glass 7 cm across has melted completely, against the density of the liquid it floats in (the liquid must be denser than ice, 917 kg/m³, or the cube sinks). In fresh water the level does not move: the meltwater exactly fills the volume the ice displaced. In seawater it rises 0.21 mm and in saturated brine 1.3 mm, because the ice was displacing a denser liquid than it melts into; in castor oil, which is lighter than water and does not mix with it, the level falls 0.32 mm as the meltwater sinks beneath the oil. The change is the meltwater's volume minus the displaced volume, m/ρw − m/ρ, spread over the glass. Fluids

The floating ice that still raises the sea

An ice cube melting in a glass of water leaves the level exactly where it was, and the reason is exact: a floating body displaces its own weight, and the meltwater has that weight. The same argument is often used to say that melting sea ice and ice shelves cannot raise the sea. It nearly works. The ice was floating in salt water and melts into fresh, which is lighter, so the meltwater overfills the hole by almost three per cent — and the ice that really fills the sea is the part of a grounded glacier above the thickness at which it would float.

 The share of its energy a particle can hand to one electron. The largest fraction of its kinetic energy a muon, a pion, a proton or an alpha particle can give a free electron in one collision, against its Lorentz factor γ, on logarithmic axes. A slow proton can give at most 0.22 per cent, 4m/M, which is why a proton slows down through tens of thousands of gentle collisions rather than a few hard ones. The share rises as γ grows and reaches a half at γ = 916 for a proton and 101 for a muon — near M/2m, where the electron's energy in the centre-of-momentum frame has grown comparable to the projectile's, so that the two meet as equals and a head-on reversal can hand the electron most of the energy. Relativity

The kick a fast particle can give an electron

A slow proton crossing matter can give any one electron at most a fifth of a per cent of its energy, because an electron struck head-on by something two thousand times heavier rebounds at no more than twice the heavier one's speed. That rule of thumb fails at relativistic speeds, and fails upward: the electron's speed is capped below light, but its energy is not, and the largest kick grows as the square of the projectile's Lorentz factor until, above a teraelectronvolt, one collision can take more than half of everything a proton carries.

 Insulating a thin wire makes it lose more heat. The heat lost per metre from a wire 2 mm across held 40 K above the air, against the outer radius of the PVC insulation round it (k = 0.19 W/m K), with the surface losing 10 W/m²K to still air by convection and radiation, on a logarithmic radius axis; dashed, the bare wire. The first layers of insulation increase the loss, from 2.51 to a maximum of 12.11 W/m at an outer radius of 19 mm — the critical radius k/h — because they add surface faster than they add resistance. Beyond the critical radius the loss falls again, but so slowly — the resistance of the insulation grows only as the logarithm of its radius — that it does not get back down to the bare wire's until the PVC is 178 km thick. Any sheath a cable could carry makes it lose more heat, not less. Thermodynamics

The insulation that makes a wire lose more heat

Wrap a hot water pipe in glass wool and it loses less heat. Wrap a thin electrical wire in PVC and it loses more — four times more, at the best thickness — and runs cooler for it. The insulation adds a resistance to heat flowing through it, but it also adds surface for the heat to leave from, and around a thin enough cylinder the surface wins. The crossover is a radius, k/h, set by the insulation and the air; below it, more insulation means more heat lost, and a small enough ball cannot be insulated by anything wrapped round it.

 A tilt that reaches zero at a definite time. The tilt of the disk against time from 10°, when the losses are set to end the motion after 100 s, for two ways of losing energy: viscous air squeezed out of the thin wedge under the rim (red), which takes energy faster the smaller the tilt, so that α³ falls in a straight line; and a steady rolling resistance at the contact (blue), which takes energy in proportion to the contact rate, so that α to the power 3/2 does. Both fall slowly at first and steeply at the end, and both reach zero tilt at a definite time rather than approaching it for ever: at 99 s, one second before the end, the tilt is 2.15° under the air model and 0.46° under rolling resistance. The disk's energy, (3/2)Mgaα, runs out in a finite time, and with it the tilt. Mechanics

The disk that spins faster as it stops

A coin spun on a table settles with a whirr that rises in pitch and then stops dead. Everything about it is losing energy, yet the rattle gets faster to the very end. The point where the rim touches the table races round faster and faster as the coin tips flatter, while the coin's face, and the head on it, turns more and more slowly. Two plausible mechanisms of loss both predict that the tilt reaches zero, and the rattle infinity, at a definite instant — a singularity a coin on a kitchen table runs into every time.

 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. Electromagnetism

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.

 Pushed forward, and left at rest. The path of an electron, initially at rest, overtaken by a 6-cycle pulse of linearly polarised light travelling to the right, for peak field strengths a₀ = 0.5, 1, 2, where a₀ = 1 is the strength at which the electron's quiver becomes relativistic; lengths in wavelengths of the light, the vertical axis along the electric field. While the pulse passes, the electron quivers across the beam and is pushed along it. When the pulse has gone it is at rest again — not moving, with none of the light's energy — but displaced forward by 0.14, 0.56, 2.25 wavelengths respectively, a distance growing as a₀². The quiver's width grows only as a₀: at a₀ = 2 the electron is pushed forward several times further than it swings sideways. Relativity

The light that moves an electron and pays it nothing

A pulse of light strong enough to throw an electron forward at a large fraction of the speed of light passes over it, and the electron is left exactly as it was found — at rest, with none of the light's energy — except that it is somewhere else. The reason is two quantities the electron cannot change while a plane wave is passing, and the same two quantities fix the figure eight it traces inside the wave and the angle at which it leaves a real, focused beam carrying the energy it managed to keep.

 Where the thread's line meets the floor. A spool resting on the floor on its rims of radius R, its thread wound round a hub of radius r = 0.5R and leaving the hub's underside tangentially, pulled up and to the right at 20°, 60° and 78° above the horizontal; the dashed lines continue each thread's line back down to the floor. The spool turns about the contact point, and the sense of the turn is set by where the thread's line meets the floor. At 20° it meets the floor behind the contact point, on the far side from the hand, and the spool rolls towards the hand. At 60°, where cos θ = r/R, it passes exactly through the contact point; the pull has no turning effect there, and the spool stays still while the hand pulls. At 78° it meets the floor in front of the contact point, between the spool and the hand, and the spool rolls away from the hand, paying the thread out. Mechanics

The spool that rolls towards the hand pulling it

Pull gently on the thread of a cotton reel lying on its side on a table, with the thread coming off the underside of the hub, and the reel rolls towards the hand, winding the thread up as it comes. Raise the thread and at one particular angle the reel will not move at all however hard it is pulled; raise it further and the reel rolls away, paying the thread out. Which way it goes is decided by a single line — where the thread's direction, extended, meets the table — and the same line decides which way a bicycle goes when its lower pedal is pulled backwards.

 The ladder that leaves the wall. A 4-metre ladder released from rest at 15° from the vertical, its foot on a frictionless floor and its top against a frictionless wall, drawn at equal intervals of time from release to the moment it lies flat, computed by integrating its motion; the dots trace its centre. While both ends touch, the centre moves exactly on a circle of radius L/2 about the corner (dashed). The top leaves the wall when it has come down to 2.58 m, two-thirds of its starting height of 3.86 m; from then on the centre's horizontal speed is fixed, it leaves the circle, and the ladder falls clear of the wall with its foot still sliding. The fall takes 1.34 seconds. Mechanics

The ladder that lets go of the wall

A ladder whose foot slips on a smooth floor slides down a smooth wall — and partway down, its top comes away from the wall and the ladder falls the rest of the way clear of it. The point where it lets go of the wall is not set by its length, its mass or how steeply it started: it is always two-thirds of the height the top started at. The reason is a wall that can push but cannot pull, and the number falls out of energy alone, without a single force being calculated in advance.

 The circle of electrons a wave can talk to. Electron velocities along and across the magnetic field, in units of c. Shaded: a horseshoe of 10 keV electrons (speed 0.195c), the shape a beam takes when it is driven down into a strengthening field, with the electrons moving up along the field missing. Curves: the electrons a wave travelling across the field can exchange energy with, for wave frequencies of 0.995, 0.980, 0.960 times the cyclotron frequency. Because a faster electron is heavier and circles more slowly, at Ω/γ, each resonance is a semicircle of radius c√(1 − ω²/Ω²): 0.100, 0.199, 0.280c. A wave at 0.981Ω resonates with the horseshoe's middle. Astrophysics

The radio that only heavier electrons can make

Above every aurora the Earth broadcasts a gigawatt of radio that no one on the ground can hear. It comes out just below the frequency at which electrons circle the field there, and it is a maser — amplified by the electrons themselves. The surprise is what makes the amplifier work. Without relativity it could not: summed over every electron, a wave at the circling frequency is always absorbed. The few per cent by which a fast electron is heavier, and circles more slowly, is not a correction to the mechanism. It is the mechanism.

 Fringes that identical nuclei add. The cross-section for two identical spinless nuclei repelling each other, divided by the classical answer — the Rutherford counts for the nucleus hit and the nucleus thrown, added — against the scattering angle in the centre-of-mass frame, for Sommerfeld parameters η = 1, 4 and 16 (η is half the closest approach in units of the wavelength over 2π). Because the detector cannot tell which nucleus it caught, the two amplitudes add before squaring, and the cross term oscillates in angle: at 90° the two paths are mirror images and the rate is exactly twice the classical one at every energy. Between 30° and 90° the curve completes about 2 oscillations for η = 4 and 7 for η = 16 — their number grows in proportion to η, which is to say in proportion to 1/ħ. Quantum

The scattering quantum mechanics returns unchanged

Rutherford worked out how alpha particles bounce off a nucleus by treating them as tiny planets on hyperbolic orbits, two years before Bohr's atom and fifteen before quantum mechanics. He had no right to be exactly correct, and he was: the full quantum calculation returns his formula without a single correction, at every energy. Coulomb's force is the one force for which that happens. Make the two nuclei identical, though, and quantum mechanics adds fringes no orbit can produce — doubling the rate at right angles for one kind of particle and halving it for the other.

 The spring falls from the top down. Snapshots of the falling spring, every fifth coil, at 0.00, 0.05, 0.09, 0.14, 0.18, 0.23, 0.26, 0.29 seconds after release, for a spring that hangs 1 m; the dashed line is the height of the bottom coil before release. The coils collapse onto one another from the top, and the collapsed block falls on the coils below; everything below the block hangs exactly as it did before the spring was let go, the bottom coil included. Only when the block reaches the bottom, 0.261 s after release, does the bottom begin to fall, and then the whole collapsed spring falls together. Mechanics

The spring whose bottom does not know it has been dropped

Hang a soft spring from its top, let go, and its bottom end stays exactly where it was — not falling slowly, not falling at all — while the top races down to meet it. A spring that hangs a metre takes about a quarter of a second to collapse, and only then does the bottom start to fall. Nothing holds it up. The centre of mass falls freely from the first instant, as it must, and the bottom simply has not been told: the news that the top has been released travels down the spring as a front of collapsing coils, and nothing below the front can feel anything until the front arrives.

 A floor that has to hit a rod nobody dropped. The normal force on the lower end of a uniform rod at 63° with μ = 1.6, sliding outward at √(gl) and not yet turning, against time on a logarithmic axis, for a floor whose contact stiffness is 10⁴, 10⁵ and 10⁶ times mg/l. The rigid equations have no solution here, and the compliant floor shows why: the end digs in, the deeper it digs the harder friction twists it further down, and the force climbs until the sliding stops. The stiffer the floor, the higher and briefer the spike — 20, 56, 163 times the rod's weight, ending after 41, 18, 7 thousandths of √(l/g) — a collision with no approach. Mechanics

The rod that friction cannot decide about

Newton's laws, a rigid rod and Coulomb's law of friction are the first things a mechanics course teaches, and in 1895 Paul Painlevé showed that together they can fail to give any answer at all. A rod sliding on a rough enough floor, at the right angle, has no motion consistent with the three — or has two. The way out is not a new law but a forgotten fact: real contacts give a little. With that, the rod jams, and the floor delivers a sudden blow to a rod that nothing struck — an impact without a collision, which is how a stick of chalk comes to hop.

 The force to pull a foot out of packed sand. The suction that resists pulling a foot (a sole of 250 cm², 10 cm long) straight up out of sand, as water is drawn through the sand into the space the foot leaves — Darcy's law with a Kozeny–Carman permeability — against the pulling speed, on logarithmic axes, for compacted sand (fraction 0.6) of three grain sizes and for loose sand (0.41) of 0.1 mm; the dotted line is a 70 kg person's weight. The force is proportional to the speed and inversely to the square of the grain size. Pulled at a centimetre a second from compacted 0.1 mm sand, a foot resists with 2.5 kN, 4 times the person's weight; the speed at which the suction equals that weight is 2.71 mm/s. In the loose sand before it compacted, the same pull would need 15 per cent of the force; with clay filling the pores of the compacted sand, as in natural quicksand, a thousand times more, and the speed a body's weight can pull at falls to microns a second. Fluids

The sand that lets a body in and will not let it out

Quicksand cannot swallow anyone: liquefied, it is nearly twice as dense as water, and a person in it floats with their chest clear. What makes it dangerous is the opposite asymmetry. Going in is easy, because a step breaks the gel holding the grains apart and the sand turns liquid. Coming out is hard, because the sand settles and packs around whatever sank, and a trapped foot can only rise as fast as water can seep through the packed grains to fill the space it leaves. Pulling harder barely helps; letting water in does.

 A hydrogen atom tilted by a laser field. The potential energy of hydrogen's electron along the direction of a laser's electric field at the field's peak, in atomic units (one unit of energy is 27.2 eV, one of length 0.053 nm), for peak intensities of 3 × 10¹³ and 10¹⁴ W/cm² and at the barrier-suppression intensity, 1.37 × 10¹⁴; the dashed line is the bound level, 13.6 eV below the ionisation limit. The field tips the Coulomb well into a barrier on one side. At 3 × 10¹³ W/cm² the barrier is 15 atomic units thick at the level — 0.8 nm — at 10¹⁴, 6; at the suppression intensity its top has fallen to the level and the electron can leave without tunnelling at all. The barrier is real only for the fraction of the laser's period that the field stays near its peak. Quantum

The atom a light wave opens like a wall

Red light cannot free an electron from hydrogen: each photon carries 1.55 electronvolts and the electron is bound by 13.6. Make the light intense enough, though, and the atom ionises anyway — by absorbing nine photons at once, or, at higher intensity, by having its potential tipped over by the light's electric field until the electron tunnels out through the side. Which of the two happens is decided by a single ratio, found by Leonid Keldysh in 1964: how long the electron needs to get through the barrier, against how long the field holds still. The tunnelling side of that ratio is where attosecond physics begins.

 Light that turns with the molecules it passes. Through a layer of liquid crystal whose molecules turn steadily through 90° from the entrance face to the exit (dashed line), the direction of the light's polarisation against depth, for linearly polarised light entering along the molecules, in a thick cell and a thin one; u = 2dΔn/λ measures the layer's birefringence against the wavelength. In the thick cell the polarisation turns with the molecules, never more than a few degrees behind, and leaves rotated by 90°: 100.0 per cent of the light passes an analyser set along the exit molecules. In the thin cell the light barely notices the twist; its polarisation lags further and further behind and leaves turned only part of the way, with 31 per cent passing. Mauguin saw the first case in 1911: a birefringent medium that twists slowly enough drags the light's polarisation round with it. Optics

The twist that carries light round a corner

Between two crossed polarisers, a sandwich of liquid crystal a few micrometres thick turns black to white. Its long molecules lie flat, but turned a quarter-turn from one face of the layer to the other, and polarised light entering along the molecules at one face comes out along them at the other — rotated by ninety degrees and able to pass the second polariser. Charles Mauguin found in 1911 that a slowly twisting birefringent layer drags light's polarisation round with it. Sixty years later the effect became the liquid-crystal display. The layers in those displays, though, are too thin to drag anything: the light inside them wanders off and comes back, at exactly the right thickness.

 The bending a falling column must carry, along its height. The bending moment at each height of a uniform column toppling about its base, in units of its weight times its length, at tilts of 10°, 30°, 60° from upright. Every curve peaks at a third of the height, at mgL·sinθ/27: 0.0064, 0.0185, 0.0321. The base and the top carry none. The same column held still at the same tilt would carry its greatest moment at the base, mgL·sinθ/2 — 0.250 at 30°, 13.5 times as much. Falling relieves the bending by a factor of 13.5 and moves it a third of the way up. Mechanics

The chimney that breaks a third of the way up

A tall brick chimney felled by demolition rarely reaches the ground in one piece. It snaps partway up, early in its fall, and the top falls behind the rest. Nothing has hit it. The cause is a free-body diagram: as the column rotates about its base, the material near the top must be dragged round faster than its own weight would take it, and the material lower down holds back. The bending this needs is largest a third of the way up, at a thirteenth and a half of what the same column would carry standing still at the same tilt — and brick, which carries no tension, cracks there after only a few degrees of lean.

 The strong coupling, run from the proton to the Planck mass. The inverse of the strong interaction's coupling, 1/αₛ, against energy on a logarithmic axis, run at one loop from its measured value at the Z mass (1/8.47, dot), with its slope changing at the charm, bottom and top masses as each quark joins in. On this plot the running is a set of straight lines: the coupling weakens by a fixed amount for each factor of ten in energy, about 2.57 per decade above the top mass. At the Planck mass, 1.22 × 10¹⁹ GeV, 1/αₛ is 52.5 — a coupling of 0.0191. Run down instead, it reaches zero — the coupling runs off to infinity — at 143 MeV, the three-flavour scale of the strong interaction at this order. Nineteen decades of energy are crossed by a line that falls from 52 to 0. Astrophysics

The nineteen decades a slow coupling buys

A proton is ten million million million times lighter than the Planck mass, and that ratio, squared, is why gravity between two protons is 10³⁶ times weaker than the electric force between them. It looks like a number demanding an explanation, and it has one. Almost none of the proton's mass comes from the masses of its quarks; it comes from the strong force, whose strength changes with energy by a logarithm. A coupling of about 1/50 at the Planck mass, weakening by a fixed step for every factor of ten in energy, takes nineteen factors of ten to become strong — and the proton's mass is where it does.

 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.

 Thrown across a uniform field, a fast charge does not draw a parabola. The paths of a charge launched across a uniform electric field (which points up the page) at 0.30, 0.60, 0.90, 0.99 of the speed of light, from the exact relativistic motion (solid), beside the parabolas Newton's law gives for the same launches (dashed); lengths are in units of mc²/qE, the distance over which the field does work equal to the charge's rest energy. Near its launch each exact path is a parabola, but one with the inertia γ₀m in place of m, so it is deflected less than Newton says — 1.05, 1.25, 2.29, 7.09 times less for the four. Further out each turns steeply along the field: one unit across, the 0.3c charge is 11.6 units along where Newton says 5.6, because its sideways speed has fallen while the field kept pushing. Each exact path is a hyperbolic cosine. The field adds to the momentum along it while the sideways momentum stays fixed, and the charge's growing energy is a growing inertia that slows its sideways progress. Relativity

The throw a uniform field bends into a chain

Throw a stone sideways across the Earth's gravity and it draws a parabola. Throw a charge sideways across a uniform electric field and, at low speed, it does the same. At high speed it draws a different curve, a hyperbolic cosine — the shape of a chain hanging between two posts, stretched — and the reason is that nothing pushes it sideways yet its sideways speed falls. Its sideways momentum is fixed, the field keeps adding energy, and energy is inertia. The charge gets only logarithmically far across the field however long it flies, and its sideways distance turns out to be a reading of its own clock.

 Four regimes along one line of mixed states. The weight p of the singlet in Werner's mixture of a singlet and complete noise, from 0 (pure noise) to 1 (a perfect singlet), divided by what is known about it. Below p = 1/3 the state is separable: it can be made by handing out unentangled pairs. Above 1/3 it is entangled, by the partial-transpose test. Yet up to at least p = 1/2 — Werner's explicit model of 1989 — and, by a later argument through Grothendieck's constant, up to about 0.66, a local hidden-variable model reproduces the outcomes of every pair of projective measurements on it, so no Bell experiment of that kind can reveal the entanglement. Above 1/√2 ≈ 0.707 the CHSH inequality is violated. Between about 0.66 and 0.71 (and, by more recent searches, between about 0.68 and 0.70) it is not known whether any Bell inequality is broken. Quantum

The entangled pair a list of answers can imitate

Entanglement and the violation of a Bell inequality are usually spoken of as the same thing, and they are not. Mix a perfect singlet with random noise and the pair stays entangled until the noise is two-thirds of the mixture. It stops violating the CHSH inequality much sooner, when the noise passes three-tenths. In between lies a family of genuinely entangled states whose every measurement can be reproduced by particles carrying instructions fixed in advance — Reinhard Werner wrote the instructions down in 1989. The entanglement in those states is real and can be spent, on teleportation for one thing. It simply cannot be seen by asking two distant analysers what they find.

 The extra work in letting the gas expand further. Pressure against volume, the volume in units of the combustion chamber and the pressure on a logarithmic axis, for two ideal engines that compress the charge by the same ratio of 10 and add the same heat: an Otto cycle that expands by 10 too and throws away gas still at 3.8 bar (dashed), and an over-expanded cycle that lets the burned gas expand 26.8 times, until it is back at the 1 bar it came in at (solid). The shaded region between the expansion line and the intake pressure, beyond the Otto engine's stroke, is where the extra work is done (on this logarithmic axis its area is not to scale): 136 kJ per kilogram of charge, raising the efficiency from 55.3 to 63.6 per cent. It is taken from gas that an Otto engine releases through its exhaust valve still pushing. Thermodynamics

The engine that expands further than it compresses

A petrol engine's efficiency is usually quoted as a function of its compression ratio, and its compression ratio is held near ten because a higher one makes the fuel detonate. But the efficiency does not come from the compression. It comes from the expansion — from how far the burned gas is allowed to push the piston before it is thrown away — and an ordinary engine expands its gas exactly as far as it compressed it, simply because the same cylinder does both. Let the gas expand further than it was compressed and most of what a higher compression ratio would have given comes back with no knock at all. The price is paid in power, and it is why the most efficient petrol engines on the road are built this way.

 Two measures of how uncertain a split state is. For the two-peaked state, against half the separation of the peaks in units of their width: Heisenberg's product σₓσₚ divided by its minimum ħ/2 (solid), and the entropic sum Hₓ + Hₚ minus its minimum ln(πeħ) (dashed), both computed from the state's distributions. As the peaks separate the product grows without limit, 8.1 times the minimum at the right of the figure, because the variance counts the empty space between the peaks as uncertainty. The entropic excess levels off at 2 ln 2 − 1 = 0.386 (0.386 at the right): the position's entropy gains ln 2, one bit, for not knowing which peak, and the momentum's gives back 1 − ln 2 because its fringes pack it into narrow bands. The entropic relation says what is true of the state: it is a fraction of a coin toss away from a minimum-uncertainty packet, however far apart the two places are. Quantum

The sharpness a variance cannot see

Heisenberg's inequality is a statement about standard deviations, and a standard deviation is a poor measure of how sharply something is known. A particle that is certainly in one of two narrow places a metre apart has a position spread of half a metre, and satisfies the inequality with room to spare while being sharply localised. Measure uncertainty by information instead — by the Shannon entropy of each distribution — and the relation tightens: the sum of the two entropies has a floor that the split state cannot get under either, and the floor survives in places where the variance form says nothing at all.

 A chaotic rotor heats without limit, and its quantum version stops. The mean kinetic energy of a rotor kicked once a period by a potential of strength K = 5, against the number of kicks, starting at rest. Classically (an ensemble of 4,000 rotors with random angles) the motion is chaotic and the momentum random-walks, so the energy grows in a straight line, 6.3 per kick, near the K²/4 = 6.25 a random walk of kicks K sin θ gives. Quantum mechanically, with an effective Planck constant ƛ = 1 and 0.5, the energy follows the classical line for a while and then stops, fluctuating about 58 and 123: the smaller ƛ lets it follow the classical growth for longer and climb further before it stops. Nothing in the quantum rotor's equations damps it; the growth stops by interference. Quantum

The heating a quantum rotor refuses

Kick a freely turning rotor hard, at regular intervals, and classical mechanics says its motion becomes chaotic and its energy grows without limit, the momentum random-walking upward with every kick. Do the same to a quantum rotor and it follows the classical heating for a while, then stops — not because anything damps it, but because the waves of its momentum states interfere so as to cancel further spreading. The correspondence principle, which promises classical behaviour when quantum numbers are large, is broken here in the most direct way: by a quantum system that refuses to get hot.

 The same damped motion in two phase planes. A cloud of damped oscillators that start on a small circle of positions and velocities, drawn at times 0, 2, 4, 6 (outlines) with the centre's path. Left: position against the physical momentum mẋ. The cloud spirals inward and shrinks: its area falls as e^(−γt), to 0.30 of its start by t = 6, as friction squeezes every motion towards rest. Right: position against the canonical momentum of the Caldirola–Kanai Lagrangian, p = e^(γt)mẋ. The same cloud keeps exactly its starting area while it is stretched out of shape, as Liouville's theorem requires of any Hamiltonian flow. The friction has been hidden in the definition of momentum: the canonical momentum grows exactly as fast as the real one decays. Mechanics

The friction an action can hold only by changing its clock

The principle of least action generates every conservative law of motion from a single integral, and it has no place for friction: there is no ordinary Lagrangian whose equation of motion is a damped oscillator's. Multiply the Lagrangian by an exponential of time and friction appears — but only by redefining momentum so that it grows as fast as the real momentum decays. The phase-space area the principle conserves is then no longer the one that counts motions, and quantised, the damped oscillator becomes more sharply known than the uncertainty principle allows. Friction belongs to a system and the surroundings it loses energy to, and an action for the system alone can only fake it.

 How fine a grid has to be to polarise. The fraction of the power transmitted by a grid of thin parallel wires, against the ratio of the wire spacing d to the wavelength, on logarithmic axes — for the field along the wires with three wire radii (solid), and for the field across the wires with the thickest (dashed), from the quasi-static theory of strip gratings, valid while the spacing is under about half a wavelength. Across the wires nearly everything passes at every spacing. Along them the leak is about four times the square of (d/λ)·ln(d/2πa): 0.1% of the power at a spacing of a hundredth of a wavelength, 10% at a tenth, 64% at four-tenths, for wires of radius 3% of their spacing. Thicker wires block better because the inductance of the grid falls. The diamond marks Hertz's grid of 1888, 4% leaking along the wires; a visible-light polariser needs the same ratio at a wavelength of half a micrometre, which means wires about 140 nanometres apart. Optics

The grid that stops the light along its wires

A row of parallel wires looks like a fence, and a fence ought to let through a wave that shakes along its slats. A grid of wires does the opposite. A field along the wires drives electrons up and down them, the currents re-radiate, and the wave is reflected; a field across the wires can move charge only through a wire's width, and passes. Heinrich Hertz used such a grid in 1888 to show that his new waves were polarised. The same grid, shrunk two hundred thousand times to wires 140 nanometres apart, polarises visible light in projectors — and every sheet of polarising plastic is the same idea made from chains of iodine.

 The bend that depends on when the planet was there. The difference, in microarcseconds, between the quasar's deflection computed from Jupiter's position when the light passed it — its position as seen from the Earth, 53 minutes earlier — and from its position at the moment of observation, over the day of the 2002 pass. In those 53 minutes Jupiter moves 40998 km, 0.57 of its own radius, and the deflection, a vector of 1.14 mas pointing away from the planet, turns and shrinks by up to 46 µas — about one twenty-fifth of itself. That small term was what the experiment was designed to see. The VLBA measured 51 ± 9 µas, the value general relativity predicts for a deflection pointing away from the position light reveals. Astrophysics

The bend a moving planet gives

Jupiter bends starlight by sixteen thousandths of an arcsecond at its edge, a hundredth of what the Sun manages, and in 2002 radio telescopes spread across a continent measured the bend on a quasar three and a half arcminutes from the planet. The experiment was designed to see something smaller still: the fact that Jupiter moves, by half its own width in the time light takes to reach the Earth, so that the bend depends on where the planet was when the light went past. The measured term was announced as the speed of gravity, and the argument over whether it was anything of the kind is the clearest lesson there is in what an experiment's number is a number of.

 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. Electromagnetism

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.

 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. Electromagnetism

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.

 The push that makes an electron swing. An electron in a band of width 4J, pushed by a constant force F = 0.25 J per lattice spacing, over two Bloch periods h/Fa. Above: its crystal momentum, which grows steadily as Newton's law says it should, and is folded back each time it reaches the edge of the Brillouin zone, ±π/a. Below: its position, in lattice spacings, which does not grow steadily at all. The electron's velocity is the slope of the band, and the slope reverses at the zone edge, so it moves forward, slows, stops 16 lattice spacings along — 4J/F — and comes back, oscillating instead of accelerating. A free particle with the band-bottom effective mass, pushed by the same force, is dashed: the two agree only for the first fraction of a period. The electron is not stopped by anything; it is turned round by the lattice, which takes the momentum the force gives it. Quantum

The push that makes an electron swing

Push a free electron with a steady force and it accelerates for as long as the force lasts. Push an electron in a crystal with the same force and, according to the band theory every transistor depends on, it should do something else entirely: move forward, slow, stop, come back and do it again, oscillating on the spot at a frequency set by the force and the lattice spacing. Felix Bloch's oscillation was predicted in 1928 and not seen for sixty-four years, because in a real metal the electron is knocked off course ten thousand times before it completes one swing. It was found in crystals built with a thirty-times-larger spacing, and in lattices made of light where nothing scatters at all.

 A longer road against a faster car. For a proton with 0.1% more momentum than the design proton in a ring 628 m round, the change in its time per turn, in picoseconds, against the Lorentz factor: from its longer orbit, which adds about 56 ps a turn at every energy; from its higher speed, which subtracts 605 ps at γ = 2 but only 5.2 ps at γ = 20, because near the speed of light more momentum buys almost no more speed; and the net, computed exactly. The speed wins below γₜ = 6.1 and the path wins above it. In special relativity a body can always be given more momentum, but its speed is capped, and that is the whole cause. Relativity

The energy above which faster arrives later

In a racing circuit a faster car laps sooner. In a proton synchrotron it does too, at first — and then, past a definite energy, the faster proton laps later. Extra momentum sends a proton round a slightly longer orbit and makes it slightly faster, and relativity decides which wins: at low energy the speed does, but near the speed of light more momentum buys almost no more speed, and the longer road takes over. At the crossing, called transition, a spread of momenta stops producing a spread of arrival times, the bunch stops oscillating in time, and the accelerating wave has to jump from one slope to the other in a few turns. The first machine to try it did so in 1959.

 Every toss already decided. The outcome of a coin toss as a map of how it was launched: the upward speed in metres per second across, the spin in radians per second up, with the region where the coin shows the face it started with shaded and the region where it shows the other face left clear. A coin spinning about a diameter and caught at the height it left is in the air for 2v/g and turns 2vω/g radians, so the boundaries are the hyperbolae vω = nπg/2, one for every half-turn. Near the origin the bands are wide: a slow toss with little spin is easy to control, and a conjuror's "honest" toss lives there. Nothing in the map is random. Every point is a definite outcome, and the bands are smooth curves — there is no fractal boundary, no region where a tiny change has an unpredictable effect. What makes a toss a fair bet is only how narrow the bands become further out. Mechanics

The toss that is random without chaos

A coin toss is the textbook random event and the textbook deterministic one at the same time. Its flight is a few lines of Newtonian mechanics with no chaos anywhere: two tosses that start almost alike stay almost alike, drifting apart only in proportion to time. Its outcome is fixed the moment it leaves the thumb, by a map of launch speed and spin carved into bands of heads and tails. It is a fair bet because the bands, at a real toss, are only a few per cent wide and no hand repeats itself that well — and it is not quite fair, because a coin wobbles as it flips, and the wobble keeps its starting face up a little more than half the time: 50.8 per cent, predicted in 2007 and counted in 2023 over 350,757 tosses.

 Three throws in a spinning room. Balls thrown at 5 m/s from the floor of a station 10 m in radius, spun at 0.99 rad/s (9.5 turns a minute) so that its floor presses with one g, as seen by someone standing in it: at 45° spinward, at 45° antispinward and straight up; with the Earth's parabolas for the two 45° throws dashed. Distances along the floor and heights in metres. On the Earth both 45° throws land 2.55 m away. In the station the spinward throw lands 1.52 m away and the antispinward one 3.06 m away, and the ball thrown straight up comes down 1.30 m ahead of the thrower, spinward. Nothing acts on any of the balls in flight. Each flies straight across the station while the floor turns beneath it, and what the thrower sees as gravity bending the path is the room rotating. Mechanics

The throw a spinning station bends

A space station spun fast enough presses its floor up against its crew with the force of the Earth's gravity, and from the inside it can feel like home. Throw a ball, and the illusion fails. Nothing acts on the ball once it leaves the hand: it flies in a perfectly straight line across the station while the floor turns beneath it, and what the thrower sees as a parabola is a chord of a circle seen from a turning room. In a station ten metres across, a ball thrown up a metre comes down a metre ahead of the hand, a ball thrown with the spin falls short and one thrown against it carries long, and a dropped spanner lands a metre behind the point beneath it. How big a station must be to feel like the Earth is a question projectiles answer.

 The focus that moves out of the eye. Parallel rays from a distant object entering a simplified eye 24 mm long, in millimetres along its axis: in air the eye's 61 dioptres bring them to a focus 21.9 mm behind its principal plane, on the retina; under water its 18 dioptres would bring them to a focus 73 mm behind, about three times the length of the eye. On the retina itself the underwater rays have converged only a third of the way, and every point of the scene is spread into a disc the size of the pupil's shadow. The eye is not out of focus by a little, as a short-sighted eye is; it is missing most of a lens. Optics

The eye that loses its lens in water

Eyes opened under water see only a blur, and the reason is not water in the eye or salt or murk. The human eye does two-thirds of its focusing not with its lens but with the curved front of its cornea, whose power comes from the jump in refractive index between air and tissue. Water is almost as dense optically as the cornea, and the jump all but disappears: about forty-three of the eye's sixty dioptres vanish, and the focus moves to three times the eye's own length behind the retina. A mask gives the cornea back its air; a small pupil and a hard squint buy back a little; animals that hunt in both worlds have flattened their corneas and given the work to the lens.

 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. Electromagnetism

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 band of speeds in which a bicycle rides itself. The growth rates of the small motions of a riderless bicycle against forward speed, from the four roots of the linearised equations for a benchmark machine, with the published eigenvalues reproduced before anything else was drawn. A positive rate means the motion grows. Below 4.29 m/s a pair of roots has positive real part: the weave, a swaying of the steering and lean together, grows. Above 6.02 m/s one real root turns barely positive: the bicycle slowly falls over into a spiral, the capsize. Between these speeds, in the shaded band, all four roots have negative real parts, and a shove dies away on its own with no hand on the bar. Nothing is stabilising the machine except the coupling of lean to steer. Mechanics

The bicycle that steers itself upright

Give a riderless bicycle a shove and, at the right speed, it wobbles and straightens up, and rides on with nobody on it. People have long credited the gyroscopic wheels or the trail of the fork. A machine built in 2011 had neither, and it stayed up. Four numbers, the roots of a quartic, tell the whole story of when a bicycle rides itself, and they do not contain a simple reason.

 A fountain in proportion to the drop. The height of the arch above the jar against the drop below it, for three values of α, the share of each link's momentum that the pile supplies. A steady chain has one tension at the top of the arch, equal to its momentum flux λv². Going down either leg the tension falls by the weight of chain below. At the jar the tension must be (1 − α)λv²; at the floor, with the floor taking all of the arriving momentum, it is zero. That gives v²α = gh₁ and h₁ = h₂ α/(1 − α), a straight line through the origin, and no arch at all if α = 0. A measured α near 0.12 gives 27 cm of fountain at a 2 m drop, and this proportion, about a seventh of the drop, is what the experiments found. Mechanics

The chain that climbs out of the jar

Pull the end of a long bead chain over the rim of a jar and let it fall to the floor, and the rest follows, as a siphon does. That much is unsurprising. What a video showed in 2013 was stranger: the chain does not run over the rim. It leaps up from the jar in a standing arch, tens of centimetres above it, with nothing supporting it. The explanation took a year and turned on something a string does not have: a link is stiff, so the pile has to push back.

 The rings the full solution finds. The light scattered back by water drops of about 10 µm radius, at 550 nm, from the exact Mie series with around 134 terms for each drop, averaged over a ten per cent spread of sizes so that the fine ripple each single drop has is washed out. Instead of the smooth, weak background of the ray picture, the intensity rises sharply toward 180° through a series of rings about one and a half degrees apart, each brighter than the last, to a first ring 1.5° from straight back. These are what an observer sees round the shadow of an aircraft on a cloud, or round a shadow on a mist at a mountain top: the glory. It is present in the same mathematics that gives the rainbow, and it takes the whole series, with its surface-wave terms, to find. Optics

The rings no ray can reach

Look down from an aircraft at a cloud, and round the aircraft's shadow there may be coloured rings, centred exactly on the point opposite the Sun. Mountain climbers have seen them round their own shadows for centuries, and every drop in that cloud is a sphere of water. Yet ray optics says that a drop of water cannot send light straight back. Its best ray falls fourteen degrees short. The glory lives entirely in those fourteen degrees.

 A cooling planet, read from the top. Temperature against depth in a body cooling by conduction from a uniform start at 3870 °C, with a surface held at nought, at four ages: the exact error-function solution for a half-space. Cooling reaches downward as the square root of time, so after a hundred million years the upper few hundred kilometres have cooled and the interior knows nothing about it. The gradient at the surface is what a mine measures, and it falls as the inverse root of time. A thermal gradient therefore gives an age, and this is what Kelvin saw in 1862. He assumed the Earth was a solid conductor from the start, on a good basis, as everyone believed then, and these curves are the mathematical consequence of that belief. Thermodynamics

The age a cooling Earth gives

In 1862 William Thomson, later Lord Kelvin, took a thermometer reading from the bottom of a mine and a piece of mathematics from Fourier and computed the age of the Earth: about a hundred million years. Geologists, needing far longer, had no physics with which to reply. The arithmetic was right. The usual story blames radioactivity for overturning it, but half the heat is not radioactive, and the error lay in assuming the inside of the planet stays still.

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