Theme

What stays the same — part 3 of 3

Conservation laws, and the habit of solving a problem by refusing to look at the middle of it.

Two pulls that should be equal and opposite. Two bodies of equal passive mass, one of an ordinary material and one of a material whose active mass is 30 per cent larger than its passive mass (the difference exaggerated to be seen). Each pulls the other with its active mass and is pulled on its passive mass. The pull the unusual body exerts is therefore larger than the pull it receives, the two arrows are not equal and opposite, and the pair, with nothing outside it, accelerates towards the ordinary body. Newton's third law and the conservation of momentum both require the ratio of active to passive mass to be the same for every material, and that is a separate claim from the one torsion balances test, that passive and inertial mass agree.

The pull that has to equal the pull back

Every test of the equivalence principle drops two things and asks whether gravity pulls them equally. None of them asks whether they pull equally. A body's mass enters gravity twice — once as what is pulled and once as what does the pulling — and if the two ratios differed from one material to another, Newton's third law would fail and a lopsided body could push itself through space. The first test of this used a plastic cylinder floating in a liquid. The best uses the Moon, whose light crust is thicker on the side that never faces the Earth.

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

The handedness a mirror cannot keep

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

Optics

The beams that add only if they differ

Take two identical lasers and try to merge their beams into one beam twice as bright. No arrangement of lenses, mirrors and prisms will do it: the brightness of light cannot be increased by passive optics, and two copies of the same beam can only ever be put side by side. The escape is to make the beams not copies. Give them crossed polarisations and two will merge. Give each a different colour and a grating will stack dozens into a single beam. Lock their phases so that they stop being separate beams at all, and they add — but only as well as the gaps between them allow and only while their phases are held to a twentieth of a wavelength.

Three identical quarks in one state, and the label that rescues them. The Δ⁺⁺ particle, mass 1,232 MeV and charge +2, drawn as the quark model describes it: three up quarks, all in the lowest orbital state, all with their spins pointing the same way, giving spin 3/2. Left: with nothing else to tell them apart, the state is unchanged by swapping any two quarks — symmetric in where they are, in their spins and in their flavour — which the exclusion principle forbids for identical fermions. Right: give each quark a label that takes three values, and put the three in the one combination of the three values that changes sign under every swap. The whole state is then antisymmetric, as it must be, and the particle is allowed. The label is called colour, and three is the smallest number of values that can do this for three quarks.

Quantum

The label three identical quarks forced

In 1964 the quark model had one embarrassment it could not explain away. A particle called the Δ⁺⁺, well measured and entirely ordinary, had to be three identical up quarks in the same state with their spins pointing the same way — which the exclusion principle forbids outright. The way out was to give quarks a hidden property that takes three values, so that the three could be different after all. Nobody could see the property, and for years it looked like a bookkeeping trick. Then three independent measurements counted it, and every one came out at three. The exclusion principle, which keeps electrons in shells, had predicted the colour of the strong force.

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.

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

Electromagnetism

The current that squeezes what carries it

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

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.

$How bright a nebula looks through a 200 mm telescope. The surface brightness of the image of an extended object — a nebula, a galaxy, the Moon — through a lossless 200 mm telescope, as a fraction of its brightness to the naked eye, against magnification on a logarithmic axis, for eye pupils of 7 and 5 mm. It is never above one. For a 7 mm pupil it is one up to a magnification of 28.6, where the exit pupil equals the eye's; below that magnification the exit pupil is wider than the eye's and part of the collected light falls on the iris; above it the image dims as the square of the exit pupil: 0.327 at 50×, 0.082 at 100×, 0.020 at 200×. The telescope makes the nebula larger, not brighter; at the magnification where its exit pupil matches the eye's it makes it as large as it can while keeping it as bright as it is.$ Optics

The nebula no telescope can brighten

A telescope eight metres across gathers a million times more light than the eye, and a star seen through it is a million times brighter. The Orion nebula seen through it is not brighter at all. It is bigger, and spread over a bigger patch of the retina, and the two exactly cancel: no instrument made of lenses and mirrors can show an extended object brighter than the naked eye sees it. What a telescope does is decided by the small disc of light behind its eyepiece, the exit pupil, and by whether that disc is larger or smaller than the pupil of the eye looking through it.

A bottle as a mass on a spring. A 750 ml bottle with a neck 8 cm long and 19 mm across, and its equivalent: the plug of air in the neck, 32 milligrams, moving in and out as a piston on the air in the body, which acts as a spring of stiffness 15 N/m because squeezing it raises its pressure. The plug's mass and the spring's stiffness give a frequency of 109 Hz, the note heard when air is blown across the mouth. The effective neck length includes about one and a half radii of air beyond its two ends that moves with the plug. Nothing in the body's air moves much — it is compressed and released almost uniformly, because the wavelength, three metres, is far larger than the bottle. That is what makes the resonator lumped: one mass, one spring, one note.

Waves

The note a bottle sings whatever its shape

Blow across the mouth of an empty wine bottle and it sounds a low note, about 109 hertz for a 750-millilitre bottle. Its wavelength is three metres, ten times the bottle's height, so it is not a standing wave in the bottle the way a flute's note is a standing wave in the flute. It is a mass bouncing on a spring: the plug of air in the neck, thirty milligrams of it, riding on the springiness of the air in the body. The note depends on the body's volume and the neck's size and on nothing about the body's shape, which is why a sphere, a cube and a tall cylinder of the same volume all sing the same note — and why a car driven with one window open throbs at eighteen hertz.

The lift of a zero-pressure balloon as it climbs. The gross lift, in tonnes, of a zero-pressure balloon whose envelope holds at most 1.1 million cubic metres, against height, for launches with 3,500 and 5,000 cubic metres of helium; the dashed line is its 3 tonnes of envelope, payload and ballast. At launch the envelope is mostly empty. As the balloon climbs the gas expands exactly as fast as the air thins, so the mass of air it displaces stays the same and the lift stays at 3.69 or 5.28 tonnes all the way up. The envelope fills at 37.5 km with the larger fill and 39.9 km with the smaller; above that, gas vents through the open base, the volume is fixed and the lift falls with the air's density. Both reach the same float, 41.3 km, where the lift has fallen to the weight. More helium only makes the climb faster; the ceiling is set by the envelope's size and the mass it carries.

Fluids

The balloon that floats at a density

A helium balloon let go rises until something stops it, and what stops it depends entirely on how it is built. A party balloon or a weather balloon stretches until it bursts. An open-bottomed science balloon climbs at constant lift — its gas expanding exactly as fast as the air thins — until its envelope is full, then floats where the air is thin enough. A sealed, unstretchable balloon floats where its whole density equals the air's and bobs back if pushed away. None of them floats at a height; each floats at a density. The difference between the three is how compressible the balloon is compared with the air around it, the same comparison that decides whether a Cartesian diver sinks.

$Which way the atoms leave, against the bias. A toy version of the 2023 ALPHA-g measurement: the fraction of trapped antihydrogen atoms that leave through the bottom mirror as both mirrors are lowered over twenty seconds, against the magnetic bias, in units of the gravitational tilt, for three assumed accelerations. Each atom is given an energy between 0.02 and 0.5 K and a random starting phase, bounces between the mirrors, and leaves through the first one it reaches that has dropped below its energy. With no bias, 100 per cent leave downward if antihydrogen falls normally, 49 per cent if it does not fall and 0 per cent if it falls up. Each curve crosses one half exactly where the bias cancels the assumed gravity, and the crossing point is what the experiment measures. The curves are not steps — each falls from nine-tenths to one-tenth over about 0.6 g of bias — because the mirrors keep falling during the few milliseconds an atom takes to cross the trap: an atom that happens to arrive at the higher mirror just after it has dropped below its energy leaves that way. ALPHA-g's measured balance gave an acceleration of 0.75 g, with an uncertainty of about 0.2 g, ruling out falling upwards.$ Astrophysics

The antiatom that had to be weighed anyway

Every test of the equivalence principle for three centuries compared two lumps of ordinary matter. Whether antimatter falls down was settled on paper in the 1950s, by an engine that would run for ever if it fell up and by the electrostatic energy of nuclei, part of which is carried by fleeting positrons. It was not measured until 2023, because on a charged antiparticle gravity is ten million times weaker than a field of one volt per metre. The measurement needed neutral antihydrogen, held at half a kelvin in a magnetic bottle a quarter of a metre tall and let out over twenty seconds. It falls down, at 0.75 g with an uncertainty of a fifth of a g.

The rise a watt-year buys at each depth. The rise in sea level, in millimetres, from one watt-year per square metre of heat (3.16 × 10⁷ joules) stored in a thin layer at each depth, for two water columns: a subtropical one, 24 °C at the surface cooling to 1.5 °C in the deep, and a polar one, 1.5 °C at the surface and 0.5 °C below. In the warm column the yield is 2.36 mm at the surface, falls to 1.02 mm near 2.0 km where the water is cold but the pressure still modest, and recovers to 1.49 mm at 6 km as pressure takes over. In the polar column the same heat buys 0.63 mm at the surface and 1.13 mm at 4 km. A joule stored in cold, deep water raises the sea about twice as much as one stored in the cold water above it.

Fluids

The rise a joule of heat buys

Warm a column of seawater and it stands taller without weighing any more, so the sea rises while the pressure on the sea floor stays exactly the same. How much it rises depends on where the heat goes. Seawater's expansion coefficient grows with temperature and with pressure, so a joule stored in warm tropical water lifts the sea more than six times as far as the same joule in Arctic surface water. Cold water in the deep, squeezed by four kilometres above it, expands about twice as much per joule as the same cold water at the surface. That is why the budget of sea-level rise turns on thousandths of a degree in the least-measured part of the ocean.

A cell swells until its membrane runs out. The volume of a red cell of 90 fL against the salt concentration of the solution round it, in per cent sodium chloride by weight, with 0.9 per cent isotonic. By the Boyle–van 't Hoff law the water inside follows the outside's tonicity: the cell's volume is an inactive part, 40 per cent of the normal volume in haemoglobin and other solids, plus a part inversely proportional to the concentration. Its membrane's 135 μm² can enclose at most 147.5 fL, as a sphere (dashed). The swelling curve reaches that at 0.44 per cent salt: below it the cell has no more shape to give up and bursts. In a 1.5 per cent solution the same cell shrinks to 68 fL and crenates, with membrane to spare.

Fluids

The cell that bursts by geometry

Put a red blood cell in salt water weaker than blood and water flows in. The cell does not stretch to take it, because its membrane will not stretch: it can change shape almost for nothing and change its area by about two per cent before it tears. So the cell swells by changing shape, from a dimpled disc towards a sphere, and when it reaches the sphere its membrane can enclose, it has no more shape to spend. That happens at about 0.44 per cent salt, half the strength of blood, and the clinical test that measures it — called a test of fragility — turns out to measure a ratio of area to volume, not the strength of anything.

A parcel that leaves the air behind. A parcel of air lifted from the ground at 30 °C with a dew point of 23 °C, against height, beside the air it rises through, which cools at 6.5 K per kilometre to the tropopause at 15 km. Unsaturated, the parcel cools at the dry rate and is soon colder than its surroundings; at 890 m it reaches saturation, a cloud base, and from there cools along the moist adiabat. From 2080 m it is warmer than the air round it and accelerates upward, until at 13.4 km it is cold again. The shaded area, the buoyancy integrated over that climb, is the convective available potential energy: 2543 J/kg. If all of it went into motion the updraft would reach 71 m/s, the square root of twice the CAPE. The 99 J/kg below the level of free convection is the push it needs to get started.

Fluids

The cloud that cools more slowly than the air

Dry air carried upward cools at 9.8 degrees per kilometre, and the atmosphere, which cools at about 6.5, is stable against it: a dry parcel nudged upward ends up colder than its surroundings and sinks back. Saturated air is different. As it rises its vapour condenses, the latent heat warms it, and it cools at only four to six degrees per kilometre in the lower atmosphere. The same column of air is then unstable to it. Most of the troposphere, most of the time, is stable until something lifts air to saturation, and then it is not — and that one asymmetry runs from a fair-weather cumulus to a thunderstorm, to the warm wind that comes down the far side of a mountain.

$How high a jumper of any size can rise. Jump height — the rise of the centre of mass after take-off, in vacuum — against body mass, on logarithmic scales. The flat line is Borelli's limit: if muscle can do a fixed amount of work per kilogram and a fixed fraction of the body is jumping muscle, every animal rises the same height, here 0.61 m. The sloping line is the limit set by power: the legs straighten in a time proportional to their length, and a small animal must deliver its energy in a fraction of a millisecond, faster than muscle can. That limit rises as the 2/9 power of mass and meets Borelli's near 80 kg. Points mark take-off speeds reported for six animals, converted to the height they would give without air: flea 0.18 m, froghopper 1.13 m, locust 0.52 m, frog 0.62 m, galago 2.22 m, human 0.49 m. Every animal well left of the crossing jumps far above its power limit, and every one of them is known to store energy in a spring before it jumps.$ Mechanics

The jump that does not get higher with size

A flea, a locust and a person all lift their centres of mass by roughly half a metre when they jump, though their masses differ by a factor of a hundred million. Giovanni Borelli saw why in 1680: the work a muscle can do grows with its mass, and so does the work needed to lift a body, so the height cancels the size. What does not cancel is time. A small animal's legs straighten in a fraction of a millisecond, and no muscle can deliver its energy that fast — a flea would need more than five hundred times the power any muscle can give. Every small jumper reaches Borelli's height anyway, and every one of them does it with a spring.

$The oldest age a trace of modern carbon allows. The age a radiocarbon measurement would report against the sample's true age, when a fraction of the carbon in it is modern contamination — from groundwater, roots, handling or the chemistry of preparation — of 5, 1 and 0.1 per cent; dashed, no contamination. Young samples are barely affected. Old ones are ruled by the contamination, because their own carbon-14 has dwindled to less than it: with 1 per cent, a sample of any age beyond about 60,000 years reads 38.1 thousand years, and 5 per cent caps it at 24.8 thousand. Even 0.1 per cent, a microgram of modern carbon in a milligram sample, caps it at 57 thousand. The limit of radiocarbon dating, usually quoted as about fifty thousand years, is a limit on cleanliness, not on the half-life.$ Quantum

The clock read by counting atoms

Radiocarbon dating reads a clock that ticks slowly: a gram of living carbon holds sixty thousand million carbon-14 atoms, and only thirteen and a half of them decay each minute. Counting those decays to date a milligram of bone would take years. Counting the atoms themselves, one by one in an accelerator, takes an hour, because in any measurement shorter than the clock's mean life almost every atom is still there, undecayed and countable. And the method's famous limit of about fifty thousand years turns out not to be set by the half-life at all. It is set by how much modern carbon a sample has picked up, because a trace of contamination outweighs whatever carbon-14 a very old sample has left.

Relativity

The gas law relativity leaves exact

Heat a gas until its particles move at nearly the speed of light and almost everything Maxwell and Boltzmann said about it changes. The speeds pile up just below c instead of spreading out. The energy per particle climbs from three halves of kT to three. The heat capacity doubles, and the ratio of heat capacities falls from five-thirds to four-thirds. One law survives untouched: the pressure is still exactly nkT, at a thousand degrees or a hundred thousand million. The reason it survives is the reason the others change. What heat hands out equally to every way of moving is not energy but momentum times velocity, and for slow particles those happen to differ only by a factor of two.

An atom's line and a nucleus's, on one scale. Three energies, in electronvolts on a logarithmic scale, for an atomic transition, sodium's yellow line, and a nuclear one, the 14.4 keV gamma ray of iron-57: the amount by which emission and absorption by free atoms miss each other because of recoil, E²/Mc²; the natural width of the line, ħ over the excited state's lifetime; and the Doppler width from thermal motion at room temperature. For sodium the recoil miss is 2.1·10⁻¹⁰ eV, 197 times smaller than the natural width, so an atom absorbs its neighbour's light. For iron-57 it is 0.00392 eV, 839,015 times its natural width of 4.67·10⁻⁹ eV. The recoil grows as the square of the transition energy and the width does not, so a gamma ray emitted by a free nucleus misses an identical nucleus by nearly a million line widths.

Relativity

The gamma ray its own kind cannot absorb

An atom can absorb the light another atom of its kind emits; that is how a sodium flame casts a shadow in sodium light. A nucleus cannot. When it emits a gamma ray it loses the photon's energy as mass, and the photon's momentum kicks it backward, taking a small share of the energy with it; to absorb the same gamma ray, an identical nucleus would need that share added rather than removed. For iron-57's 14.4 keV line the shortfall is nearly a million times the line's own width. In 1958 Rudolf Mössbauer found that a nucleus held in a crystal can hand its recoil to the whole crystal and lose almost nothing — and the resulting line is so sharp that it could weigh the fall of light down a twenty-two metre tower.

How supersaturated a cloud is, to an ice crystal. The supersaturation over ice of air that is just saturated over liquid water, eᵥᵥ/eᵢ − 1, in per cent, against temperature; dashed, the same computed from the chemical potentials, exp(LΔT/RT·Tₘ) − 1, with L the latent heat of melting per mole, ΔT the supercooling and Tₘ the melting point. A supercooled cloud is in balance with its droplets, so to any ice crystal inside it the air is 5.1 per cent supersaturated at −5 °C, 10.5 at −10, 22.0 at −20 and 41.2 at −35. Over the droplets themselves it is exactly saturated, so they neither grow nor shrink by vapour. The excess is the chemical-potential difference between supercooled water and ice, 217 J per mole at −10 °C, expressed as a pressure ratio.

Thermodynamics

The ice that grows by stealing from the droplets

Most clouds colder than freezing are not made of ice. They are made of droplets of liquid water, some tens of degrees below their freezing point, waiting for something to start them freezing. When a few ice crystals do appear among them, the crystals grow and the droplets shrink, without touching, because supercooled water has a higher chemical potential than ice and so a higher vapour pressure: air balanced with the droplets is ten per cent supersaturated, to an ice crystal, at minus ten degrees. A few crystals among a hundred thousand droplets each take everything, grow heavy enough to fall, and that is how most of the rain outside the tropics begins — as snow.

The highest note the air can carry, with height. The frequency at which a sound wave in the air loses its amplitude by a factor e in a single wavelength, in hertz on a logarithmic scale, against height, with the band of human hearing shaded. Because viscosity does not depend on pressure, the collision time grows as the pressure falls, and the ceiling falls with it. It is about 212 MHz at sea level, 0.17 MHz at 50 km and 3 kHz at 80 km. It passes the top of human hearing, 20 kHz, at about 68 km, and the bottom, 20 Hz, at about 110 km; at 100 km it is 92 Hz. Above about a hundred kilometres the air cannot carry anything a person could hear, at any loudness: the molecules are too far apart to pass a compression on before it has dispersed.

Thermodynamics

The note too high for thin air

Sound in a gas is a compression handed from molecule to molecule by collisions, and it can only be handed on if the molecules collide many times in each period of the wave. At sea level they collide seven thousand million times a second, and the limit is far above anything audible. But a gas's viscosity does not depend on its pressure, so as the air thins with height the time between collisions grows in proportion, and the highest note the air can carry falls with it: two hundred megahertz at the ground, three kilohertz at eighty kilometres, below twenty hertz above a hundred and ten. The upper atmosphere is not silent for want of loudness. It is silent because its molecules are too far apart to pass a sound along.

A number of details that frequency cannot change. The number of separately resolvable depths along one line of an ultrasound image, the deepest echo divided by the axial resolution, against frequency, for dynamic ranges of 50, 60 and 80 dB in soft tissue; dashed, the same for a medium that absorbed like water, as the square of the frequency, with 60 dB. In tissue the count is D/(α₀Nc), the same at every frequency: 325 for 50 dB, 390 for 60 dB, 519 for 80 dB. It is a property of the tissue, through α₀, and of the electronics, through D, and changing the frequency only trades depth for sharpness one for one. In a medium with an f² law the count would fall as 1/f, from 88,548 at 1 MHz to 1,771 at 50 MHz. It is the near-linear absorption of tissue that makes the trade exact.

Waves

The depth an ultrasound image pays for its sharpness

An ultrasound scanner sees finer detail at higher frequencies, because shorter waves make shorter pulses, and sees less deep, because tissue absorbs higher frequencies more strongly. Both effects scale as one over the frequency, so they cancel in the one number that matters: how many separate depths fit along a line of the image. For soft tissue, which absorbs nearly in proportion to frequency, that number is about four hundred whatever the frequency — set by the tissue and by the scanner's electronics and by nothing else. A scan of the abdomen at three megahertz and a scan of the skin at fifty are the same picture at different magnifications.

A floating needle and the water it displaces. A cross-section of a steel darning needle 1.36 mm thick (density 7.85) floating on water, with a contact angle of 100°, its meniscus computed from the balance of surface tension and pressure; one capillary length is 2.73 mm. The needle's centre sits 1.43 mm below the undisturbed surface, and the water meets it 117° from the top, sloping at 37°. Two regions of water have been displaced. Below the level, under the needle, a region whose weight — 21 per cent of the support — is what pressure on the needle's underside supplies. Beside it, the two dents in the surface, whose weight — the other 79 per cent — is exactly the vertical pull of surface tension along the two contact lines. Together they weigh what the needle weighs: Archimedes, counting the dent.

Fluids

The water the dent displaces

A steel needle laid carefully on water floats, and the usual explanation is that surface tension holds it up instead of buoyancy, as though the two were rival mechanisms. They are not rivals. The needle presses a dent into the surface, and the upward pull of surface tension along its edges is exactly the weight of the water missing from that dent. Archimedes' principle holds without amendment: a floating needle displaces its own weight of water. Most of that water is simply not underneath it.

$A ring turns, and its vibration turns less. Four snapshots of a thin ring vibrating in its lowest flexural mode, squashing alternately into two ellipses, while the whole ring is turned steadily about its axis; the deformation is exaggerated. The dot marks one point of the ring; the dashed line, the axis of the vibration's greatest swing. As the ring turns through 30°, 60° and 90°, the pattern of vibration turns through 18°, 36° and 54° — 60 per cent of the ring's angle. Measured from the ring, the pattern has slipped back 36° by the quarter turn. The vibration is neither carried along with the material nor left fixed in space. It follows the turning by a fixed fraction that depends only on the mode's shape, which is the effect G. H. Bryan described in 1890 from the note of a turning wine glass.$ Mechanics

The vibration that turns part of the way

Strike a wine glass and turn it on the table while it rings, and the note beats: the pattern of the vibration does not turn with the glass. Nor does it stay still in the room. For a ring vibrating in its lowest mode it follows exactly three-fifths of the rotation and slips back the other two-fifths, a fraction that depends only on the shape of the vibration. That fraction, worked out by George Bryan in 1890, is how some of the most precise gyroscopes ever built measure an angle directly, with nothing spinning in them at all.

$The speed that stops while the energy keeps coming. The speed of an electron, as a fraction of the speed of light, against the kinetic energy given to it, from special relativity (solid) and from Newton's ½mv² (dashed); dots at the five energies Bertozzi used in 1964. Newton's speed passes the speed of light at a quarter of a megaelectronvolt and is 7.7 times it at 15 MeV. Relativity's speed is 0.863 c at 0.5 MeV, 0.967 c at 1.5 MeV, 0.9948 c at 4.5 MeV and 0.99946 c at 15 MeV. Between 1.5 and 15 MeV the energy grows tenfold and the speed by three and a half per cent. The energy does not stop arriving; the speed stops answering it.$ Relativity

The speed that stops while the energy keeps coming

In 1964 William Bertozzi gave electrons energies from half a megaelectronvolt to fifteen, timed them over eight and a half metres, and caught them in an aluminium disc to measure the heat they brought. By Newton the fastest should have crossed in under four nanoseconds, seven times faster than light. They took twenty-eight, and so did the slower ones, more or less. The heat, meanwhile, rose in proportion to the energy put in. The experiment is famous for showing that nothing outruns light, but its sharper result is the second half: the energy did arrive, all of it, carried at very nearly the same speed.

Two clocks that slow together. For a muon circling a 1.45 T magnetic field, its mean life as the laboratory measures it and the time it takes to go round once, against its Lorentz factor γ, on logarithmic scales. Time dilation stretches the life to γ × 2.197 μs: 6.6 μs at γ = 3, 64.4 μs at γ = 29.3, the muon g − 2 experiments' value. The orbit grows with the momentum, so the revolution takes γ times longer too: 15.3 ns at γ = 3, 149 ns at γ = 29.3. The two lines are parallel. Their ratio, the number of turns in one mean life, is 431 at every energy — set by the field and the muon's own properties, eBτ/2πm, with γ cancelled out.

Relativity

The turns a muon makes at any energy

A muon lives two microseconds, and in a storage ring it lives longer the faster it goes, because its clock runs slow. Its orbit runs slow too: a faster muon carries more momentum, bends less in the same field and takes longer to go round. The two slowings are the same factor of gamma, so they cancel. A muon in a 1.45 tesla ring makes 431 turns in a mean life whether it is moving at nine-tenths of the speed of light or at nine nines, and the only way to more turns is a stronger magnet. The same cancellation is why the first cyclotrons stopped working near twenty million electronvolts.

Astrophysics

The field no symmetric flow can keep

The Earth's core would forget its magnetic field in about forty thousand years, and the field is at least three and a half thousand million years old, so something must be making it again all the time. The obvious candidate is the flow of liquid iron. In 1934 Thomas Cowling proved that the simplest version cannot work: no motion of a conductor, however fast or however arranged, can keep up a field that is symmetric about an axis. The reason is a single ring inside the field where the current it needs cannot be driven.

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.

Six cells of a two-dimensional foam, by number of sides. Cells of a two-dimensional soap froth with three to eight sides, each drawn with its films meeting three at a time at 120°, as surface tension requires. A hexagon can do that with straight sides. A cell with fewer than six sides cannot: to make its corners 120° its sides must bulge outward, so its gas is at higher pressure than its neighbours' and leaks out. A cell with more than six must have sides curving inward, lower pressure, and gains gas. The rate of change of area, in units of πκγ/3 where κ is the films' permeability, is n − 6: −3 for a triangle, +2 for an octagon.

Fluids

The bubble that grows by counting its sides

A froth of soap bubbles squeezed between two sheets of glass coarsens over a few days: small cells vanish and large ones swell. The obvious guess is that size decides which cells win, as it does for two bubbles joined by a tube, where the small one always empties into the large one. In a froth it is wrong. John von Neumann showed in 1952 that a cell's area changes at a rate fixed entirely by how many sides it has — shrinking if fewer than six, growing if more — and not at all by how big it is. The proof is a single sum of angles.

$Three orbits that are one orbit, rescaled. Three orbits round the same mass, each an ellipse of eccentricity 0.6, with sizes in the ratio 1 : 2 : 3, integrated from the inverse-square force. The dots mark twelfths of each orbit's period, at the same fractions of the way round. The shapes are exact enlargements of one another and so are the motions, with time stretched as size to the power 3/2: the measured periods are 1.000, 2.828, 5.196 times the smallest, against 1.000, 2.828, 5.196 from the rule. That is Kepler's third law, obtained without solving for any orbit.$ Mechanics

The third law that falls out of a change of scale

Kepler spent a decade of tables to find that the squares of the planets' periods go as the cubes of their distances. The law can be had in three lines, without solving for a single orbit. Gravity's potential energy halves when distances double; stretch every length of an orbit by some factor and every time by that factor to the power three-halves, and the action changes only by a constant, so the stretched orbit is an orbit too. The same move gives the swing of a spring that keeps time at any amplitude, the virial theorem that makes a star heat up as it loses energy, and the spacing of quantum levels.

Rays and the surfaces of equal optical path. Rays from a point source 20 mm in front of a curved glass surface (radius 4 mm, index 1.5), traced exactly, and the surfaces on which the optical path from the source — distance times index — takes the same value (curves, every 2.5 mm of path). In air the surfaces are spheres round the source. In the glass the rays from the edge of the surface cross the axis at 12.8 mm, well short of the paraxial focus at 20.0 mm, and the equal-path surfaces are no longer spheres. They are still at right angles to every ray: measured on a fan of 641 rays over 10 surfaces in the glass, the largest departure from 90° is 0.005°. Further in, the surfaces fold into cusps where neighbouring rays cross, and there a surface has no single normal.

Optics

The surface every ray is normal to

Light from a point, bent by any number of lenses and mirrors, comes out as a tangle of rays that no longer meet anywhere. Hidden in the tangle is a surface every ray crosses at a right angle: the set of points the light reaches at the same optical time. That a surface always exists is a theorem two centuries old, and it is the reason a modern wavefront sensor works — an array of tiny lenses measures only the direction each patch of light is travelling, and the measurements can be added up into a shape only because they are the slopes of one.

Three waves, and the points where the light is zero. The overlap of three equal plane waves travelling 120° apart: brightness (shading) and lines of equal phase every 60° (curves). Lines of every phase meet at isolated points where the brightness is exactly zero (dots); elsewhere only branches of a single phase cross, at the bright spots, where the phase is momentarily level. Round each such point the phase changes by one full turn, clockwise (red, charge +1) or anticlockwise (green, charge −1); in this window there are 18 of one and 18 of the other, arranged on a lattice in which every neighbour has the opposite sign. No adjustment of the phases removes them; shifting a wave only slides the lattice.

Optics

The dark lines a wave cannot avoid

Overlap three waves of light and there are places where they cancel exactly — not along fringes, as two waves cancel, but at isolated points, round which the phase of the light turns through a full circle. They cannot be removed by any small change, they can only be created or destroyed two at a time, and random light is full of them, about three to every square wavelength. A grating with a fork in it writes one into a beam on purpose, and the beam that comes out has a dark core and carries angular momentum in its twist.

Four operations on one half, four states of the pair. The sender holds one half of an entangled pair prepared in the state |Φ+⟩ = (|00⟩ + |11⟩)/√2, the receiver the other. To send two bits she applies one of four operations to her half alone — do nothing, flip it (X), shift its phase (Z), or both — and sends it. The table gives, computed from the state vectors, the probability that the receiver's joint measurement of both halves in the Bell basis returns each of the four Bell states: one on the diagonal, zero everywhere else. Four perfectly distinguishable outcomes from one transmitted qubit is two bits. Sent without the pair, a qubit can carry at most one.

Quantum

The qubit that carries two bits

A single quantum two-state system, sent from one person to another, can carry at most one bit of message — a theorem, not an engineering limit. If the two already share half each of an entangled pair, the sender can do one of four things to her half alone, send just that half, and the receiver can tell which of the four she did. Two bits arrive on one qubit. Nothing is free: the shared pair had to be delivered, and counted honestly the rate is still a bit per qubit. What entanglement buys is the freedom to send half the message before the message exists, and it also buys privacy, since the qubit in transit carries nothing at all on its own.

Latent heat of boiling against boiling point. The heat needed to boil one mole of a liquid at its normal boiling point, against that boiling point, both on logarithmic axes, for twenty-one substances from helium to sodium. The line is ΔH = 87.2 J/(mol K) × Tb, the mean entropy of boiling of the ordinary liquids boiling between 250 and 700 K, which scatter about it by 3.3 per cent; the ordinary liquids as a whole (blue), from neon at 27 K to sodium at 1,156 K, by 10.0 per cent, the coldest lying low. Water, ammonia and the alcohols (red) need more heat than the line predicts, acetic acid (green) less; the quantum liquids helium and hydrogen (violet) fall far below.

Thermodynamics

The entropy every liquid gains on boiling

Divide the heat it takes to boil a liquid by the temperature at which it boils, and for benzene, chloroform, acetone, hexane, ether and carbon tetrachloride the answer comes out within a few per cent of the same number — about ten and a half times the gas constant. Frederick Trouton noticed it in 1884. It is a statement about entropy, and once it is read that way the liquids it fails for become the interesting ones: water and the alcohols, whose hydrogen bonds hold the liquid in order; acetic acid, whose vapour is made of pairs; and helium, whose liquid is already as disordered as quantum mechanics allows.

$Two lifetimes a thousand times apart. The fraction of positronium atoms surviving in the ground state against time since formation, on a logarithmic time axis, for the spin singlet, para-positronium (red), and the spin triplet, ortho-positronium (blue), in vacuum. Para lives 125.1 ps and ortho 142.1 ns, a ratio of 1135. The lowest-order formulas give 124.5 ps, from α⁵mc²/2ħ, and 138.7 ns, from 2(π² − 9)/9π · α⁶mc²/ħ; the ratio is almost entirely one extra power of the fine-structure constant, the price of a third photon.$ Quantum

The atom with no nucleus

An electron and a positron can bind into an atom whose structure is hydrogen's with every level halved and every size doubled, because neither partner sits still at the centre. The atom then destroys itself, and how fast depends on one symmetry. With the two spins opposed it vanishes in 125 picoseconds as two photons; with them aligned it must make three, and lives 142 nanoseconds — over a thousand times longer — for no reason but that a rule about charge conjugation forbids the shorter route.

A rod in a shear flow turns in fits and starts. The angle between a rigid rod's axis and the flow direction, unwrapped, against the strain of a steady shear flow (shear rate times time), for spheroids of aspect ratio 1, 5, 20 lying in the plane of the shear, each starting along the flow. A sphere (r = 1) turns steadily at half the shear rate. A rod spends most of its time nearly aligned with the flow, turning slowly, and then flips through half a turn quickly; at r = 5 it is within 10° of the flow direction 46 per cent of the time, at r = 20, 82 per cent. Each flip is half of Jeffery's period, π(r + 1/r) units of strain.

Fluids

The rod that tumbles for ever in a stream

Drop a short rod into honey being sheared between two plates and it does not settle into line with the flow. It lies nearly along the stream for a long time, then flips end over end in a moment, then lies along it again, and it keeps doing that for as long as the shear lasts. George Jeffery worked out the motion in 1922: every orientation belongs to a closed orbit, every orbit takes the same time, and the flow has no way to choose between them — so a suspension of rods never reaches a steady state, and its thickness rises and falls in time with the tumbling.

$Rays inside a drop never reach the critical angle. Rays of light entering a drop of water (index 1.333) from the left at four heights — 0.3, 0.6, 0.85, 0.97 of the radius — followed through six internal reflections, each chord drawn with a width proportional to the light it carries. Every chord inside a sphere meets the surface at the same angle at both ends, so each ray strikes the inside of the drop again and again at its refraction angle: 13.0°, 26.8°, 39.6°, 46.7°, all below the critical angle of 48.6°. None is ever totally reflected; at each contact most of the light leaves, and what is left inside dwindles — the light that, leaving after one and two reflections, makes the primary and secondary rainbows.$ Optics

The drop that cannot hold the light it lets in

Light inside water meeting the surface at more than 48.6° is totally reflected, and every raindrop is made of water. Yet no ray of sunlight that enters a raindrop is ever totally reflected inside it. A sphere's geometry forbids it: a ray refracted on the way in meets the inside of the surface at its refraction angle, and keeps meeting it at that same angle every time it bounces, and that angle is always smaller than the critical one. The light leaks out a little at every bounce, which is why there are rainbows at all. Light born inside the drop is another matter, and a sphere can keep it for ever.

The phase a beam gains through its focus. How far the phase of a focused beam, measured on its axis, runs ahead of a plane wave of the same frequency, in cycles, against distance from the focus in units of the Rayleigh range π w₀²/λ, for a beam focused to a point (solid) and one focused only in one direction, to a line (dashed). The point-focused beam lags a quarter of a cycle far before the focus and leads by a quarter far after, gaining half a cycle in all, most of it within a Rayleigh range of the focus: 0.25 of a cycle by z = zR, 0.40 by 3 zR. The line focus gains half as much, a quarter cycle. This is the Gouy phase, measured by Louis Georges Gouy in 1890.

Optics

The half cycle a focus adds

A beam of light brought to a focus and allowed to spread again comes out half a wavelength ahead of where a plain wave of the same colour would be, although no part of it travelled faster than light. Louis Georges Gouy found the shift in 1890 by interfering a focused beam with an unfocused one. It is the price of confinement: a beam narrow enough to focus is a spread of directions, the parts travelling at an angle make slower progress along the axis, and the spread is greatest at the focus. The same half cycle decides the frequencies at which every laser cavity can resonate.

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.

$How much of supercooled water freezes in an instant. The fraction of a body of supercooled water that freezes at once when it is nucleated, against how far below 0 °C it had been cooled: the ice that forms releases latent heat until the whole of it is back at 0 °C, and freezing stops there until heat is taken away. Solid: with water's measured heat capacity, which rises steeply below about −20 °C; dashed: with the heat capacity it has at 0 °C. At −15 °C, 19.1 per cent freezes at once; at −20 °C, 25.7 per cent; at −38 °C, the coldest liquid water reaches before it freezes by itself, 51 per cent. The rest stays liquid at 0 °C. Water would have to be cooled about 80 degrees for all of it to freeze at once, and no liquid water survives that long.$ Thermodynamics

The supercooled water that only partly freezes

Water can be cooled well below 0 °C and stay liquid, and when a crystal finally starts, the whole bottle seems to turn to ice at once. It does not. Freezing releases heat, and with nowhere for that heat to go in the instant of freezing, it warms everything back to 0 °C — at which point freezing must stop. Water cooled to −15 °C freezes about a fifth at once; to −38 °C, the coldest liquid water gets, about half. The rest has to wait for the surroundings to take the heat away, and a hand warmer is the same arithmetic run above body temperature.

The number of collisions for heavier and heavier blocks. The number of collisions, simulated one by one, for mass ratios of 1, 10², 10⁴, 10⁶, 10⁸, on a logarithmic axis: 3, 31, 314, 3141, 31415. Each factor of a hundred in the mass ratio adds one digit of π. Nothing in the setup mentions a circle — two blocks, a wall and the conservation of energy and momentum — but energy conservation makes the velocities live on a circle, and the count is how many equal steps of a small angle fit into half of it.

Mechanics

The collisions that count the digits of π

Slide a heavy block towards a light one resting near a wall. The light block is batted between the heavy one and the wall until the heavy block turns round and leaves. Count the collisions. With equal masses there are 3; with one block a hundred times heavier, 31; ten thousand times, 314; a million, 3141. Nothing in two blocks and a wall mentions a circle, but conservation of energy puts their velocities on one, and each collision steps round it by the same small angle. The count is how many such steps fit in half a turn.

Astrophysics

The velocity that always lies on a circle

A planet on an elliptical orbit speeds up and slows down, its velocity swinging round as it goes. Draw all its velocities from a single point and their tips fall exactly on a circle — not an ellipse, a circle — offset from the point by the orbit's eccentricity. Hamilton noticed this in 1846; it holds for every orbit under an inverse-square force and for no other force; and it turns the ellipse into something that can be proved with a ruler and compass, as Feynman did in a lecture that was lost for thirty years.

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.

$One lever, two frames, two different torques. A right-angled lever pivoted at its corner, with arms of length L, held still by a force F at the end of each arm, perpendicular to it. Left, in the lever's own frame: torque FL anticlockwise from the horizontal arm's force, FL clockwise from the vertical arm's, and they cancel. Right, the same lever seen moving to the right at 0.8 of the speed of light: the horizontal arm, along the motion, is contracted to L/γ = 0.60L, and the force on it, which is across the motion, is reduced to F/γ = 0.60F; the vertical arm and its force, along the motion, are unchanged. The torques are now 0.36FL and FL, and the lever is twisted clockwise by 0.64FL — the square of its speed as a fraction of light's. It does not turn.$ Relativity

The lever that is twisted and does not turn

A right-angled lever held still by two equal forces has no net torque on it. Watch the same lever go by at high speed and one arm is shorter, one force is weaker, and the torques no longer cancel — yet the lever goes by without turning. Nothing is wrong with either frame's arithmetic. The forces are doing work on the moving lever, the energy they put in is flowing through its arms, and a flow of energy carries momentum that points across the lever's motion: a body that moves straight with a momentum that does not.

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

Electromagnetism

The field at which no electron reaches the anode

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

The 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.

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.

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.

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

The half a wire always takes

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

One height inside the bottle where the pressure is the air's. Gauge pressure against height inside a sealed bottle with an open tube reaching down to 10 cm above its outlet, while it is bubbling, at three water levels: 30.0, 22.0, 13.0 cm. Each line rises at the same hydrostatic slope, ρg, from the air pocket at the top, which is below atmospheric, to the outlet at the foot. All three cross atmospheric pressure at the tube's tip — 7.4 mm below it, the extra a bubble needs to leave a 2 mm tube — and so all three reach the outlet at the same 906 Pa above atmospheric, whatever the level. Dashed: an open bottle at the same levels, whose pressure at the outlet is 2937, 2154, 1273 Pa.

Fluids

The bottle that pours at one rate however full it is

Water running from a hole in the foot of an open bottle slows as the bottle empties, because the pressure at the hole is the depth of water above it. Seal the bottle and push an open tube through the stopper, down into the water, and it stops slowing: the flow holds steady from full until the level reaches the bottom of the tube. Nothing regulates it. The tube's tip is the one point inside where the pressure is the atmosphere's, the water above it hangs from an air pocket below atmospheric, and the outlet sees only the height of the tip.

Mechanics

The kick that does not change the beat

A pendulum clock has to push its pendulum, or friction would stop it within minutes, and every push risks disturbing the very period the clock exists to keep. In 1826 George Airy worked out exactly when a push does no harm: at the bottom of the swing, where the pendulum is moving fastest, an impulse changes how far it swings and leaves the timing untouched. A push before the bottom makes the clock gain; after it, lose. Three centuries of escapement design are a search for a mechanism that pushes where Airy said — or that pushes in the wrong place in a way that cancels some other error.

Salt that takes in water from the air at three-quarters humidity. The water held by a crystal of common salt, in grams per gram of salt, against the relative humidity of the air around it at 25 °C, with the humidity rising (solid) and falling (dashed). Rising, the crystal stays dry until 75.2 per cent and then dissolves at once into a drop holding 2.79 g of water per gram — the saturated solution — which goes on drinking as the air gets damper: 4.24 g at 85 per cent, 11.5 g at 95. Falling, the drop does not crystallise at 75 per cent. It sheds water along the same curve, becomes supersaturated, and stays liquid down to about 45 per cent before a crystal finally nucleates and it dries out at once.

Thermodynamics

The salt that dissolves in the air's own water

Leave a dish of salt in a bathroom and it does not just clump; on a humid enough day it turns into a puddle, though no water was poured on it. Above 75 per cent humidity, common salt pulls water out of the air until it has dissolved itself. The number is not a property of salt's attraction for water in any vague sense. It is the vapour pressure of a saturated salt solution, read as a fraction of pure water's: wherever the air holds more water than that, the solution is the state with the lower chemical potential. Bring the humidity back down, and the puddle does not turn back into salt until the air is far drier — about 45 per cent.

The same energy in less and less whip. The speed of the moving part of a whip 2 m long and weighing 250 g, against how far the loop the arm threw into it still has to go to the tip, on a logarithmic scale, if the 15 J the arm gave it stays in the part still moving. Three whips: uniform, tapering linearly, and tapering faster, each ending in a 0.2 g cracker. At the handle every part moves at 11 m/s. Half a metre from the tip the uniform whip is at 22 m/s and the faster taper at 72; the uniform whip passes the speed of sound (dashed) only within its last millimetre, the tapers 1.6 cm and 1.5 cm from it. All three end at the same ceiling, √(2E/m) = 387 m/s, set by the cracker alone: the taper decides how gradually the speed builds, not how fast the end goes.

Mechanics

The crack at the end of a whip

An arm moves a whip's handle at ten metres a second, and a fifth of a second later its tip passes the speed of sound. Nothing in between is a lever or a spring. The arm throws a loop into the whip, the part behind the loop comes to rest, and the energy stays with the part still moving, which grows lighter as the loop runs out towards the tip. Energy shared by less and less mass means speed, and what caps it is the tip's own mass, the tuft of string at the end. The crack is a sonic boom, made by something the size of a shoelace's end.

The swing's plane, turning a step at a time. The direction in which a swinging spring swings, seen from above, against time, for a spring whose bounce is twice as fast as its swing, started bouncing almost vertically with a trace of sideways motion. Points are coloured by how much of the energy is in the swing (darker: more). While the swing holds the energy the direction is fixed; while the bounce holds it, the direction changes, and the swing returns in a new direction each time — 9 steps in 1500 units of time, each 43.6° on average (43.6, 43.6, 43.6, 43.6…), always in the same sense. In a plane the swing could only come back along its old line. In space it comes back turned, and the turn is the same every time.

Mechanics

The swing that turns a step at a time

A weight on a spring whose bounce is twice as fast as its swing trades its energy back and forth: the bounce becomes a swing, and the swing a bounce again. Let it move in space rather than a plane, with the slightest sideways twist, and each swing comes back in a new direction, turned by the same angle every time. The turn happens all at once, while the weight is only bouncing and the swing is too small to see, and its size is set by a single ratio, of the twist to the energy carrying it. The steps are a geometric phase made mechanical, and they blur into chaos when the spring is bounced too hard.

$What one scattering does to a photon's energy. The average fractional change in a photon's energy when it scatters off an electron in a gas at temperature T, against the photon's energy: (4kT − ε)/mc², for gases at 10⁶, 10⁷ and 10⁸ K. A photon below 4kT gains energy, taking it from the electrons' motion; one above loses it, as recoil, to the electrons. The crossing sits at ε = 4kT: 0.34 keV at 10⁶ K, 3.45 keV at 10⁷ K, 34.47 keV at 10⁸ K. The changes per scattering are small, a few per cent at most here, but a gas whose electrons scatter each photon many times, or that is bathed in light for long enough, is heated or cooled until the gains and losses balance over the whole spectrum.$ Relativity

The temperature a gas takes from the light around it

A thin, ionised gas bathed in light settles at a temperature that has nothing to do with how much light there is and everything to do with its colour. Every time a photon scatters off an electron, the two exchange a little energy: a photon softer than four times the electrons' thermal energy gains, a harder one loses by recoil. Balanced over a spectrum, the rule sets the gas at a quarter of the light's mean photon energy. That puts gas near a quasar at tens of millions of kelvin, and it caps the heating once photons approach the electron's own rest energy. For blackbody light the rule falls four per cent short, and the shortfall is the photon being a boson.

$A peak at half the mass, made by a sine. The momentum across the beam of the electron from 200,000 simulated W decays, as a probability density per GeV, for a W of mass 80.37 GeV. For a W exactly at rest across the beam, of exactly its mass (dashed), the electron carries M/2 = 40.19 GeV in the W's frame and shows a fraction sin θ of it across the beam, so the distribution climbs to a sharp peak at M/2 and stops: the Jacobian peak, piled up because sin θ hardly changes near θ = 90°. The W's natural width of 2.1 GeV rounds the peak (solid, peak at 39.9 GeV); its motion across the beam, a few GeV on average, smears it further and pushes a tail past M/2 (lighter). The position of the peak measures the mass, and is shifted by anything that moves the W.$ Relativity

The mass weighed without one of its pieces

A W boson decays into an electron and a neutrino, and the neutrino leaves every detector without a trace. Its mass cannot be found the usual way, by adding up the energy and momentum of what it decayed into, because half of what it decayed into is missing. What survives is the part of the motion across the beam, where the neutrino's momentum can be inferred from the imbalance it leaves. There the electron's momentum piles up at half the W's mass, and a quantity built from both particles ends at a sharp edge at the full mass. The world's best measurements of that edge disagree by seventy-five parts in eighty thousand.

How a sphere of gas rings. The acoustic pressure against distance from the centre of a spherical cavity, for its three lowest radial modes, in which the gas breathes in and out in spherical shells: sin(zr/a)/(zr/a), with z = 4.4934, 7.7253, 10.9041 the roots of tan z = z that make the gas's motion vanish at the wall. Filled with argon at the triple point of water, 273.16 K, a sphere 62 mm in radius rings in these modes at 3551, 6104, 8616 Hz. Radial modes are the ones precision work uses: they have no motion along the wall, so the wall's roughness and the gas's viscosity there barely touch them, and their frequencies are set by the volume rather than the shape — a slightly imperfect sphere shifts them only at second order.

Thermodynamics

The note that fixed the kelvin

A sphere of argon, struck, rings at a pitch set by the speed of sound in it, and the speed of sound in argon contains only one quantity nobody could compute: the constant that converts a temperature into an energy. Argon's atoms can hold energy in exactly three ways, so its heat capacity is a count rather than a measurement, and its speed of sound is (5/3)kT/m with nothing left over. Ring the sphere at the temperature of melting ice, measure its size with radio waves in the same cavity, and Boltzmann's constant comes out to a part in a million. In 2019 that number stopped being measured and became the definition.

Loops that swell as the electric field approaches cB. A charge starting from rest in crossed uniform fields — the electric field up the page, the magnetic field out of it — for E/cB = 0.3, 0.6, 0.85, integrated exactly; lengths in units of mc/qB. Each traces a chain of loops drifting to the right at E/B. Newton's law gives loops of height 2mE/qB² — 0.6, 1.2, 1.7 — whatever the speed. The exact loops are 0.66, 1.88, 6.13 high, larger by γ² of the drift speed, 1.10, 1.56, 3.60: in the frame moving at the drift the electric field vanishes and the magnetic field is weakened to B/γ, and the charge circles at the drift speed on a larger orbit than a weaker field would explain alone. As E approaches cB the drift approaches c and the loop grows without limit.

Relativity

The orbit that opens when E reaches cB

A charge in crossed electric and magnetic fields drifts sideways at E/B, looping as it goes, and the textbook leaves it there. The textbook result has a speed in it that can be made larger than light, and something has to give. What gives is the loop. As the electric field approaches c times the magnetic one, the loops swell without limit, and at E = cB the charge stops looping and is carried off for ever. Above that ratio no frame can make the electric field vanish; one can make the magnetic field vanish instead, and in that frame the charge simply runs away. The transition from a bound orbit to a run-away is decided by a single quantity every observer agrees on, E² − c²B².

A gridiron of steel and brass drawn to the length that holds. The rods of a gridiron pendulum, to scale, for a seconds pendulum of 0.994 m: steel rods (dark) hang from the suspension and carry cross-bars from which brass rods (light) push upward, and from their tops the next steel rods hang down again, until the last carries the bob. Steel expands downward, lowering the bob; brass expands upward, raising it. They cancel when the steel's total length times its expansion equals the brass's: 2.52 m of steel against 1.52 m of brass, a ratio of 0.605. That is why gridirons are built of so many rods — the steel alone is 2.53 times the pendulum's length, folded into 3 stages. With zinc instead of brass, which expands 1.58 times as much, 1.61 m of steel would do. Real gridirons double every rod, side by side, so that the bob hangs symmetrically.

Mechanics

The pendulum that must not grow when it warms

A pendulum clock keeps time by the length of its pendulum, and every material grows when it is warmed. A steel rod a metre long lengthens by a hundredth of a millimetre for each degree, and the clock loses half a second a day — a minute a fortnight between a cold winter and a warm summer, far worse than anything the clockmaker's escapement could cause. The cure, found in the 1720s, was to let the pendulum grow in two directions at once: one part downward, another upward, in exactly the proportion that leaves the bob where it was. The same trick, with different metals, kept the best clocks to a hundredth of a second a day for two centuries, and its limits are set by heat that does not spread evenly or quickly.

$Three shapes of one cell, delaying light by the same total. The phase delay of 633 nm light across the middle of a model cell holding 300 pg of dry matter, in three shapes — rounded up, spreading and flattened — each with the same contents dissolved in volumes of 942, 965, 982 μm³. The rounded cell delays the light by 3.41 rad at its centre, the flat one by only 0.55. But the delay summed over the whole image is the same for all three, 536 rad·μm², and it gives back 300.0 pg through the refractive increment α = 0.18 mL/g: a cell is weighed by the light it delays, whatever shape it takes.$ Optics

The cell weighed by the light it delays

A living cell is nearly transparent, and the light that passes through it comes out almost as bright as it went in. What changes is its timing: the cell's proteins and nucleic acids raise its refractive index a little above the water around it, and the light is delayed by a fraction of a wave. Measure that delay at every point of the image and add it up, and the sum is the cell's dry mass — the weight of everything in it except water — whatever shape the cell takes and however much water it holds. The water drops out because the medium is water too; the shape drops out because the delay is an integral of concentration along the path. Light becomes a balance that weighs a living cell to a part in a thousand without touching it.

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

The imbalance of charge that would outweigh gravity

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

Reversing the magnetism in step with the swing. The rod hung from a fibre as a torsion pendulum with a period of 10 s and a quality factor of 200, its magnetisation reversed 40 times, once every half swing, in step with its motion, then left alone; the angle it turns through, integrated, in degrees. Each reversal adds the same small kick of angular momentum and the kicks add up, as pushes on a swing do: the swing grows to 15.46° if the magnetism is spin, 30.92° if it were orbital currents. A single reversal would turn the rod by about 0.45° — measurable in principle, buried in practice by the much larger twist that the coil's own field puts on any slight misalignment, which is why Einstein and de Haas drove the rod at its resonance and why their first result, g′ = 1, came out wrong.

Quantum

The rod that turns when its magnetism flips

Hang an iron rod on a thin fibre inside a coil, reverse the current, and the rod turns a little. Nothing has pushed it sideways: reversing the magnetisation reversed the angular momentum the electrons inside it carry, and since nothing outside can twist the rod about its axis, the iron lattice has to take up the difference. Einstein and de Haas did it in 1915 to prove that magnetism is electric charge going round. They got the right effect and the wrong number, and the right number — twice what they expected — was evidence, years before anyone proposed it, that the electron's magnetism comes from something that is not a rotation at all.

An attraction found by counting. How often two large discs are found at each gap between their surfaces, relative to how often they are found far apart, in a direct simulation of two discs among 270 small particles that pass through each other freely and do not enter the discs (dots), and the prediction from counting the small particles' available area (line); small particles a quarter of the large radius. Nothing in the simulation attracts anything: every move that does not put a small particle inside a large one is accepted. Yet the discs are found touching 2.7 times as often as apart (2.6 predicted), because touching returns the overlap of their exclusion zones to the small particles, and the probability of a separation is proportional to the number of arrangements the small particles have at it. Beyond a gap of twice the small radius the zones no longer overlap and the effect vanishes. Written as a potential, −kT times the logarithm of that ratio, it is an attraction about 1.0 kT deep.

Thermodynamics

The attraction that is the small ones' room

Put large particles into a crowd of small ones that cannot get inside them, and the large ones begin to stick together — though nothing in the mixture attracts anything. Each large particle keeps the small ones' centres out of a thin shell around it. When two large particles touch, their shells overlap, and the overlap is handed back to the small particles as room to move in. More room is more arrangements, more arrangements is more entropy, and the large particles are pushed together by the pressure of the small ones on their outer sides. Fumio Oosawa and Sho Asakura worked it out in 1954. The force is now used to purify DNA, sort colloids onto patterned surfaces and explain why the inside of a cell behaves as though its proteins were stickier than they are.

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

The hum at twice the mains

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

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.

A mass that goes missing as light. For six captures of a slow neutron — ¹H to ²H, ¹²C to ¹³C, ⁵⁶Fe to ⁵⁷Fe, ²⁸Si to ²⁹Si, ³⁵Cl to ³⁶Cl and ³²S to ³³S — the mass that disappears, the atom plus the neutron less the atom formed, in thousandths of an atomic mass unit, against the energy released as gamma rays, in MeV, both computed from the tabulated atomic masses. All six lie on the line Δm = E/c², whose slope is one atomic mass unit per 931.494 MeV: ¹H → ²H loses 2.388 mu and releases 2.225 MeV; ¹²C → ¹³C loses 5.310 mu and releases 4.946 MeV; ⁵⁶Fe → ⁵⁷Fe loses 8.208 mu and releases 7.646 MeV; ²⁸Si → ²⁹Si loses 9.097 mu and releases 8.474 MeV; ³⁵Cl → ³⁶Cl loses 9.211 mu and releases 8.580 MeV; ³²S → ³³S loses 9.277 mu and releases 8.642 MeV. Drawn this way the line is a definition — the energies were computed from the masses. The experiment of 2005 measured the two quantities for silicon and sulphur independently, the masses as frequencies and the energies as angles, and asked whether they still fell on it.

Relativity

The test of mass and energy that needs a ruler

The most direct test of E = mc² weighed two silicon atoms and measured a gamma ray. A silicon-28 nucleus that captures a slow neutron becomes silicon-29 and loses three ten-thousandths of its mass as gamma rays; weigh both atoms in a trap where they circle a magnetic field, measure the gamma rays by the angle at which a silicon crystal diffracts them, and the two sides agree to four parts in ten million. What the comparison needed is the interesting part: the recoil of the atom, a correction four hundred times larger than the precision, computed from the momentum relativity gives to light; angles of a few hundred picoradians; and a chain of constants tying a frequency and a ruler to the same number.

The tube that is stiff because it is full. The bending moment a thin-walled tube 1 mm in radius can carry when inflated, in newton-millimetres, against its internal pressure in megapascals: the moment at which the side in compression goes slack and wrinkles, πPr³/2, and the moment at which the whole section folds, πPr³. Both are proportional to the pressure and contain nothing about the wall's own stiffness: a membrane that cannot resist compression at all carries bending only by being pre-stretched, and the pressure is the pre-stretch. At 0.74 MPa the tube wrinkles at 1.17 N·mm; with no pressure, at nothing. That is the whole of how a lettuce leaf, a petal or a young shoot keeps its shape. It is also how an inflatable boat, a bouncy castle or the boom of an inflatable antenna holds up a load, and how they fail: by wrinkling on the compressed side first.

Fluids

The stem that stands by pressure

A lettuce leaf has no bones, and a young sunflower stem has hardly any wood, yet both hold their shape against gravity and wind. They do it with pressure. Each cell holds more dissolved solute than the water around it, water flows in until the cell wall pushes back, and the wall ends up stretched taut by a pressure of seven or eight atmospheres — three times a car tyre's. A tube of membrane under tension resists bending in exact proportion to its pressure, whatever the membrane is made of, so a soft plant is stiff exactly as long as its cells are full. Lose a few per cent of their water and the pressure is gone, and the plant wilts.

The parabola every quartz watch follows. The frequency error of a watch's quartz fork, in parts per million, against its temperature: a downward parabola, −0.034 ppm per degree squared, about a turnover at 25 °C, where the fork runs exactly at its nominal rate; with the right-hand scale in seconds a day, since one part per million is 0.086 s a day. On a wrist, at about 31 °C, the watch runs 1.2 ppm slow, 0.11 s a day; in a drawer at 10 °C, 7.7 ppm, 0.66 s a day; at 0 °C 1.8 s a day; at 60 °C 3.6. Every departure from the turnover, hotter or colder, makes it lose. The turnover is set by the angle at which the fork is cut from the crystal, and is chosen near the temperature of the place watches are worn — the parabola is a fact about quartz, and its vertex is a fact about people.

Waves

The fork inside a watch

Every quartz watch keeps time with a tuning fork three millimetres long, cut from a crystal of quartz and ringing thirty-two thousand seven hundred and sixty-eight times a second. Each number in that sentence is a decision. The frequency is two to the fifteenth, so that fifteen halvings make one tick a second, and it sits just above the range of human hearing. The shape is a fork because two tines swinging against each other leave their base still and lose almost no energy into it. And the fork's rate drifts with temperature along a parabola whose vertex is set near twenty-five degrees, which is why a watch worn on a wrist keeps better time than one left in a cold drawer.

Astrophysics

The product a fusion plasma has to beat

A star makes less power per litre than a compost heap, and it has burned for four and a half thousand million years. A fusion plasma makes thousands of times more and has to be held for seconds, and it has taken seventy years to get close. Both are obeying one inequality, written down in a classified report in 1955, in which density and time appear only as their product.

$The one number a burn depends on. The fraction of a deuterium–tritium fuel ball that burns before it flies apart, against its density times its radius. The fuel is held by nothing: a sound wave crosses it in a time R/c, the burn rate goes as the density, and so the fraction burned goes as the product ρR, flattening as the fuel runs out, f = ρR/(ρR + 7 g/cm²). A disassembly-time estimate at 30 keV gives 7.6 g/cm² for the same constant, close to the fitted value. The radius and the density do not appear separately. A third of the fuel burns at ρR = 3 g/cm², and that value does not depend on how large the ball is; what sets the size of an inertial-fusion capsule is what has to be done to reach it.$ Astrophysics

The fuel held together only by its inertia

On 5 December 2022 a laser put 2.05 megajoules into a gold can the size of a pencil rubber, and a speck of fuel inside it released 3.15. Nothing held that fuel together for the tenth of a nanosecond it burned. It needed to be crushed to a thousand times the density of solid, and it was pushed there by something light cannot do by itself: light's own pressure is hundreds of times too feeble, so a capsule is a rocket that fires inward.

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

Electromagnetism

The current a vacuum will not carry

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

The spectrum two rates would make. Balance absorption against spontaneous emission alone, with level populations in the Boltzmann ratio, and the radiation comes out as x³ times e to the minus x, Wien's law; add stimulated emission, and the same balance gives x³ over e to the x minus one, which is Planck's. The two agree on the right, where a photon is much more energetic than the thermal energy and stimulated emission is negligible, and disagree on the left, where only Planck's approaches the classical count of waves. At x = 1 Planck's is 1.58 times Wien's. Measurements around 1900 at long wavelengths had already found that Wien's law failed, so experiment required the third rate seventeen years before anyone saw it in a laser or a maser.

Quantum

The third process equilibrium demands

In 1916, with general relativity just finished, Einstein asked how atoms and light could reach equilibrium at all. Absorption and emission, balanced against each other, gave the wrong spectrum. A third process was needed, in which light makes an excited atom emit a copy of itself. Nobody had seen it, and it took forty years to be put to use. It also fixes a ratio that goes as the cube of the frequency, which is why X-ray lasers are so hard.

The peak no pion spectrum can move. The photon spectrum from a population of neutral pions, summed from their boxes, for pion spectra falling as three different powers of the energy. Whatever the pion spectrum, each curve peaks at the same 67.5 MeV and is symmetric on the logarithmic axis, since every box contains that point and a box's height falls as it widens; checked here at the point and its mirror, the ratio is 1.000. This is the shape, known as the pion bump, that Floyd Stecker worked out in 1971. Gamma-ray telescopes found it in 2013 in two supernova remnants, which was taken as direct evidence that such remnants accelerate protons, because the electrons that produce most other gamma rays have no reason to make a peak here.

Relativity

The flat spectrum a fast pion must paint

A neutral pion falls apart into two photons, each with half its rest energy, in a hundred-million-billionth of a second. Seen in flight, those photons could have any energy between two edges, and every energy in that range is equally likely, whatever the pion's speed. Add up a whole population of pions and something unexpected appears: a peak at exactly the 67.5 MeV that no individual decay shows. Astronomers took it as evidence of where cosmic rays are made.

The price of a pair depends on who pays it. The lowest energy of a photon that can make an electron–positron pair when struck on a stationary partner, against the partner's mass: 2mc²(1 + m/M), from requiring that the total four-momentum have the invariant mass of what is left at rest in the centre of momentum. For a lead nucleus it is 2mc² to better than one part in a hundred thousand, the figure every textbook quotes. For a proton it is 0.054 per cent above. For a free electron it is exactly twice, 4mc², or 2.04 MeV, even though the partner is the very same mass as the pair's members. The rest energy created is unchanged; what rises is what has to be carried off by the partner and the pair together.

Relativity

The partner that sets the price of a pair

Every textbook says it takes 1.02 MeV to make an electron and a positron from light, twice the electron's rest energy. Done in empty space, it cannot be done at any energy at all. Next to a lead nucleus it costs 1.02 MeV. Next to a lone electron, a particle exactly as heavy as the ones being made, it costs 2.04. Nothing about the pair has changed. What changes the price is who has to carry the momentum away.

Fluids

The sandcastle a spoonful of water holds

A dry heap of sand stands at thirty degrees and no steeper. Add a spoonful of water to a bucketful, and the same grains will stand in a vertical wall and carry a person's weight. Add ten times as much water and the wall is no stronger, because a bridge of water between two grains pulls with almost the same force whatever its size. Its strength belongs to the grain and the surface tension, and the amount of water hardly enters.

A spin set by the length of a cord. The body's spin against the cord let out, from conservation of energy and of angular momentum alone, for three values of c, one plus the body's moment of inertia over the weights' mR². Every curve passes through zero at a cord length of exactly √c radii, and a direct integration of the equations of motion, which assumes neither conservation law, lands on the same curve to a part in a thousand. Past that length the spin reverses. The starting spin does not appear in the zero's position: only the shape of the machine does. A weight on a cord cut to length therefore removes all the spin whether it was spinning at 20 or 100 revolutions a minute, provided the cord is released at that length.

Mechanics

The weights that stop a spinning satellite

A rocket stage is often spun up to keep it pointing straight during its burn, and the spin then has to be removed. A small cord and a weight wrapped round its rim do the whole job, with no fuel, no sensors and no moving parts besides the cord. The cord's length has no dependence on how fast the stage is spinning, so the same two weights stop it at any speed. And the cord unwinds at exactly the rate the stage was first spinning, while its spin falls away to nothing.

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.

$A facet too wide and a facet too narrow. The blur of an insect's compound eye of radius 1 mm against the width of one facet, in green light. A wide facet looks at a wide patch of sky compared with its neighbours, so the eye's sampling is coarse, and that blur rises with the width. A narrow facet is a small aperture, so diffraction spreads what it collects, and that blur falls with the width. Added in quadrature they make a minimum at D = √(λR), here 22 µm, and that is close to the facets of real insects with an eye of that size. No choice of facet width can get below about 1.8°: that figure is forced by the size of the head. Horace Barlow worked this out in 1952 and compared it with bee eyes.$ Optics

The eye a fly cannot afford

A compound eye is built from thousands of tiny straws, each looking in one direction. Wide straws look coarsely at the world, and narrow ones blur what they collect by diffraction, so there is a best width, and it depends only on the size of the head. To see as sharply as a person, an eye of this kind would need to be about twelve metres in radius. A single-lens eye gets the same result with a pupil of under two millimetres.

Lamp noise that a decay does not notice. 400 simulated measurements of an absorption of 1 × 10⁻⁷ per metre, with a source whose brightness jumps by 5 per cent between shots and a detector with 0.1 per cent noise. Read directly, as transmission through a metre of gas, each measurement scatters by 7 × 10⁻² per metre and the true value is five orders of magnitude below that. Read as a fitted decay from a cavity with 99.99 per cent mirrors, it scatters by 5 × 10⁻⁸, about a million times less. The slope of a logarithm has no information about how bright the pulse started, so a source's noise simply is not in the answer. Most of the method's sensitivity comes from this and only some from the long path.

Optics

The absorption measured with a stopwatch

The usual way to measure how much light a gas absorbs is to compare a beam before and after, and a very weak absorption vanishes in the flicker of the lamp. Trap a pulse between two excellent mirrors, and it bounces for ten kilometres inside a tube a metre long, leaking out a little at each bounce. How fast that leak dies measures the gas, and how bright the pulse began does not matter at all. A brightness measurement has become a measurement of time.

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