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The same equation again — part 3 of 3

A pendulum, a circuit, a molecule and a bridge, all obeying one differential equation and not knowing about each other.

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.

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.

Like menisci attract, unlike menisci repel. Cross-sections of a water surface with two small floating bodies, drawn with menisci of the shape K₀(r/ℓc) that a small floater makes, exaggerated vertically. Left: two light bodies, each pulling the surface up round itself as buoyancy pushes it up against the surface; each sits on the other's slope and is drawn towards it, as a bubble rises — the two climb together. Right: a light body that raises the surface beside a heavy one that depresses it; each sits on the other's slope the wrong way and they drift apart. The deformations reach a few capillary lengths, 2.71 mm for water, and the rule is the whole of the Cheerios effect: cereal clumps, and cereal gathers at the wall of a bowl the milk wets.

Fluids

The cereal that gathers at the edge of the bowl

The last few pieces of cereal in a bowl of milk clump together and drift to the side of the bowl. Nothing is pulling on them but the milk's surface. Each floating piece bends the surface round itself, and a second piece sitting on that bend has somewhere to slide: like menisci attract, unlike menisci repel, and the reach of the force is a few capillary lengths, 2.7 millimetres for water. The arithmetic is the arithmetic of charged lines in two dimensions, screened by gravity, and it says why leaves gather on a pond, why bubbles on coffee drift to the cup's rim, and why particles smaller than about ten micrometres do not clump this way at all.

How well a diamond-like crystal conducts heat, from 1 K to 1000 K. The thermal conductivity of a model crystal with diamond's sound speed and Debye temperature, from kinetic theory applied to its phonons, against temperature on logarithmic axes: natural carbon in a sample 1 mm across, isotopically enriched carbon, and natural carbon in a sample 0.1 mm across; copper's room-temperature 400 W/m K is marked for comparison. The model is calibrated to natural diamond's 2,200 W/m K at room temperature. At low temperature the conductivity rises as the cube of the temperature, limited by the sample's walls; at high temperature it falls as umklapp collisions shorten the phonons' paths. Between, it peaks: 3.7·10⁴ W/m K at 100 K for the natural crystal; 7.8·10⁴ W/m K at 94 K for the enriched crystal; 1.2·10⁴ W/m K at 126 K for the thin natural crystal. The model gets the shape — the rise as the cube of the temperature, a peak near 100 K, the fall — but overestimates the heights, because it leaves out the phonon collisions that conserve momentum and every defect but isotopes; measured enriched diamond peaks near 4 × 10⁴ W/m K. An insulator with no free electrons still conducts heat five times better than copper at room temperature.

Thermodynamics

The gas of sound that carries a diamond's heat

Diamond has no free electrons and conducts heat five times better than copper. What carries the heat is a gas — not of molecules but of quantised sound waves, phonons — and the kinetic theory that gives the conductivity of air gives the conductivity of a crystal with the same formula: a third of heat capacity times speed times mean free path. The gas is strange in two ways. At low temperature its particles cross the whole crystal without meeting anything, so the conductivity depends on how big the sample is. And at room temperature one atom in a hundred being a neutron heavier is enough to scatter its fastest-oscillating members.

$How a solute moves both ends of the liquid range. The molar chemical potential of water against temperature near its freezing point (left) and its boiling point (right): liquid (blue), ice (green), vapour (red), each a line whose slope is minus its entropy, drawn relative to the liquid at the transition. A solute at mole fraction 0.05 — about 2.9 mol/kg — lowers the liquid's line by RTx (dashed) and leaves the ice and the vapour unchanged. The crossings move: the freezing point down by RTx divided by the entropy of melting, 5.16 K, the boiling point up by RTx divided by the entropy of boiling, 1.42 K. The same vertical shift moves the freezing crossing 3.6 times as far, because ice's line is nearly parallel to the liquid's — melting changes the entropy by 22.0 J/mol K — while the vapour's line crosses steeply, with 109 J/mol K between them. Lowering the liquid makes it the winner over a wider range of temperature at both ends: the solution is liquid from below 0 °C to above 100 °C.$ Thermodynamics

The degree a spoonful of solute buys

Dissolve anything in water and the water freezes lower and boils higher. The two shifts come from one cause — the solute lowers the liquid water's chemical potential, so the liquid wins against ice and against vapour over a wider range — but they are not the same size. For every mole of particles per kilogram the freezing point falls 1.86 degrees and the boiling point rises only 0.51, three and a half times less, and the reason is in the entropies: ice and water are so alike that a small push moves their crossing a long way. The same arithmetic says salting pasta water does nothing a cook could notice, weighed molecules for a century of chemistry, and in 1887 showed that salts in water are made of ions.

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

The voltage that bends an electron like glass

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

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

Electromagnetism

The coupling a resonance rescues

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

The 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.

$A gas pressed against the wall of a spinning tube. The density of uranium hexafluoride gas in a rotor spinning at a steady rate, relative to its density on the axis, on a logarithmic scale, against the distance from the axis as a fraction of the radius, for wall speeds of 300, 450 and 600 m/s at 300 K. In the turning frame the gas sits in the centrifugal potential −½mω²r² and settles into the same exponential that thins the atmosphere with height: n ∝ exp(mω²r²/2kT). At 600 m/s the density at the wall is e^25.4 — about 10¹¹ — times that on the axis; at 300 m/s, e^6.4. Almost all the gas lies in a thin layer against the wall, and the middle of the rotor is close to vacuum.$ Mechanics

The spin that sorts by a difference of mass

In a spinning tube a gas settles against the wall the way the atmosphere settles towards the ground, with an exponential whose scale is set by the rotor's speed rather than by gravity. Heavier molecules settle more steeply, and the ratio of two isotopes changes from axis to wall by a factor that depends on the difference of their masses, not on the ratio. That one change from a ratio to a difference is why a spinning rotor separates uranium-235 from uranium-238 more than fifty times as strongly per stage as the porous barriers that first did it — and why the history of the machine is a history of how fast a material can turn before it flies apart.

Optics

The surface that cancels whatever skims it

Glass reflects four per cent of light falling straight on it, and water at radio frequencies two-thirds of the wave's amplitude. Tilt either towards grazing and the difference disappears: every surface, whatever it is made of, reflects a wave that only skims it completely, and turns it upside down. A wave travelling along a surface therefore meets its own reflection reversed and cancels on the surface itself. That one limit explains a dark fringe Humphrey Lloyd saw with a single mirror in 1834, why a mobile signal over flat ground fades with the fourth power of distance rather than the second, why a radar above the sea is blind at the horizon, and how a cliff in Sydney became the first radio interferometer.

Three planes that meet in one line. A thin lens of focal length 50, drawn as a vertical line, and an object plane tilted at 45° to the axis, crossing it 150 units in front of the lens. Four points on the object plane are traced to their images through the lens's centre and along the axis-parallel ray. The images lie on a straight line too — a thin lens maps planes to planes — but one tilted far less steeply, and when the object plane, the lens plane and the image plane are extended they meet in a single line, here 150 units below the axis. That is the Scheimpflug rule. A sensor laid along the tilted image plane records the whole tilted object plane sharp, which no sensor square to the axis can do.

Optics

The plane of focus a tilted lens tilts

An ordinary camera focuses on a plane square to its axis, so a road receding from it, a table top or a field of flowers can be sharp only over the depth an aperture allows. Tilt the lens, or the sensor, and the plane of sharp focus tilts too, by a rule a surveying officer patented in 1904: the plane of the subject, the plane of the lens and the plane of the image always meet in a single line. Turn the focus and the plane of focus does not slide back and forth; it swings about a fixed hinge. The same geometry lets an eye camera see the whole front of the eye in one exposure and a lidar keep a laser beam kilometres long in focus.

Position written as frequency. Four tubes of water side by side in a 1.5 T magnet, their protons all precessing at 63.87 MHz, until a gradient of 10 mT/m is switched on along the row. Then the field, and so the precession frequency, rises steadily across the row, by 425.8 Hz for every millimetre. The signal the receiving coil picks up is a sum of waves at different frequencies, and its spectrum is a picture: the amount of signal at each frequency is the number of protons at the corresponding position. The axis below is position and the axis above it frequency, and they are the same axis. Nothing has been focused. The radio wave at 64 MHz has a wavelength of 4.7 m in air, and the tubes are centimetres apart.

Quantum

The image made of frequencies

A hydrogen nucleus in a magnetic field precesses at a frequency proportional to the field — 63.87 megahertz in a 1.5 tesla scanner. Make the field rise steadily across the body and every position precesses at its own frequency, so the radio signal the body gives off is a chord whose spectrum is a map of where the protons are. A magnetic resonance image is never focused. It is computed, as a Fourier transform of a signal recorded one number at a time, and its resolution, a millimetre, is set by how strong the gradient is and how long the signal is listened to — not by the wavelength of the radio waves, which is several metres.

Relativity

The neutrino energy an angle fixes

A pion breaks into a muon and a neutrino, and the neutrino carries off a fixed 29.8 MeV in the pion's own frame. In the laboratory, straight along the pion's path, it carries a fixed share of whatever energy the pion had, so a beam of pions of every energy makes neutrinos of every energy. Two and a half degrees to one side, something else happens. The neutrino's energy, as a function of the pion's, rises to a maximum and falls again, and around the maximum pions from two to ten GeV all give neutrinos within a tenth of a GeV of each other. An experiment in Japan builds its detector deliberately off the axis of its own beam for that reason — trading most of its neutrinos for the ones at the energy it needs.

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.

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.

Waves

The wave that arrives first by going the long way

A shot fired at the ground sends a wave straight along the surface layer to a distant listener, and another down to a faster rock beneath, along the boundary and back up. The second path is longer, and beyond a certain distance it arrives first, because most of it is run at the faster speed. Huygens's construction shows what the second wave is: a front drawn by wavelets that the disturbance racing along the faster floor sends up into the slower layer, the Mach cone of a source moving faster than the layer can carry sound. Reading where it overtakes the direct wave gave Andrija Mohorovičić the thickness of the Earth's crust in 1909, and gives every refraction survey since the depth to whatever lies beneath.

$Where a speck lands decides how much it weighs. The fractional drop in frequency of a cantilever's first and second bending modes when a small mass lands on it, per unit of that mass as a fraction of the beam's, against where along the beam it lands, from the clamp to the free tip. The response is the square of the mode's displacement at the landing point divided by twice the mode's effective mass: a quarter of the beam for both modes (0.250 and 0.250), since each is measured against its own tip. At the tip both modes respond fully — the first by 2.0 times the landed fraction, the second by 2.0. Near the clamp neither responds, so a speck landing there is invisible. Halfway along, the first mode drops by 0.23 and the second by 1.02; at 0.783 of the length the second mode has a node and does not notice the speck at all. A single frequency cannot tell a light speck at the tip from a heavier one further in.$ Mechanics

The weight a frequency can find

A spring's frequency is the square root of its stiffness over the mass it carries, so adding a little mass lowers the note by half the fraction added. Nothing in that rule cares about gravity, and nothing in it cares about size. It weighs astronauts in orbit by the swing of a chair, films a fraction of an atom thick on a quartz plate, and single molecules on a beam a few micrometres long — and it improves without limit as the thing that vibrates gets lighter. What it cannot do on its own is say where the added mass sat, and that turns out to matter as much as how much there was.

$A filter made of glass powder and oil. The fraction of light transmitted straight through a 5 mm layer of N-BK7 grains packed to half its volume in the matching liquid, against wavelength, for grains of radius 5, 10 and 20 μm, from van de Hulst's extinction for large, weakly refracting particles. At the crossing the grains vanish and the layer is clear. A few tens of nanometres away each grain shifts the light passing through it in phase by enough to scatter a little of it out of the beam, and a thousand grains in a row scatter all of it: the passband is 51.7 nm wide at half height for 5 μm grains, 36.3 nm for 10 μm, 25.5 nm for 20 μm. Larger grains make a narrower filter, because the phase a grain imposes grows with its size; the same mismatch of index that a fine powder lets through, a coarse one scatters.$ Optics

The colour at which glass disappears

Lower a glass rod into an oil of the same refractive index and it vanishes, except for a faint coloured outline. The outline is the clue. Glass and oil disperse light differently, so their indices can agree at only one wavelength; at every other the glass is still there, bending and scattering a little. Grind the glass to powder and pack it in the oil, and the mixture is clear at that one wavelength and opaque at the rest — a colour filter made of nothing but two transparent materials, whose colour moves by a whole hue for a few degrees of warming.

Thermodynamics

The frost that lifts by drinking

Frozen ground rises in winter, sometimes by tens of centimetres, and the usual explanation — water expands by nine per cent when it freezes — is wrong: soil soaked in benzene, which shrinks as it freezes, heaves too. The lift comes from layers of clear ice that grow by pulling liquid water up through the soil from below. Ice and water touching each other need not be at the same pressure, and when the water is free to flow, ice a degree below its melting point can push with 1.12 megapascals while it drinks — twelve times less than freezing water in a sealed pipe, and enough to lift a house at a tenth of a degree.

$Light added to its own reflection. Green light of 550 nm in an emulsion of refractive index 1.5, falling on a metal mirror at the left and adding to its reflection, which comes back inverted. Thin lines: the total electric field across the first 750 nm at three moments; heavy line: its envelope, 2|sin kz|; dashed: the envelope of the magnetic field, 2|cos kz|. The sum does not travel. Its electric field is zero at the mirror at every instant and again every 183 nm, half a wavelength in the emulsion, with maxima between; the magnetic field is largest exactly where the electric field vanishes. A light-sensitive layer anywhere in that space is exposed in sheets parallel to the mirror, and which field it responds to decides whether the first sheet lies at the mirror or a quarter-wavelength above it.$ Waves

The photograph that keeps the spectrum

Light falling on a mirror adds to its own reflection, and the sum does not travel: it stands, with sheets of darkness every half-wavelength. Otto Wiener photographed those sheets in 1890 and found the film clear where it touched the mirror, which settled which of light's two fields acts on matter. A year later Gabriel Lippmann recorded the sheets through the depth of an emulsion and developed them into stacked layers that, lit with white light, send back exactly the colours that wrote them. His photographs store the spectrum itself, so they can tell apart two yellows that every colour film and every eye records as one.

Water's surface tension, all the way to the critical point. The surface tension of water against its liquid vapour, from the international standard correlation, against temperature from 0 °C to the critical point at 373.946 °C. 20 °C: 72.7 mN/m; 100 °C: 58.9 mN/m; 200 °C: 37.7 mN/m; 300 °C: 14.4 mN/m; 370 °C: 0.4 mN/m. It falls almost in a straight line for most of the range and then bends down to reach zero exactly at the critical point, where liquid and vapour stop being two things. The bend is a power law, the gap to the critical temperature raised to 1.256.

Fluids

The tension that vanishes where the phases meet

Surface tension is usually treated as a property of a liquid, like its density. It is a property of a boundary between two things, and it can only exist while there are two things. Heat water in a sealed tube and its surface tension falls steadily, then faster, and reaches exactly zero at 374 °C, where liquid and vapour stop being different. On the way the surface stops being a surface: the step in density spreads from a molecule's width to thousands, and the rate at which the tension dies turns out to be a number the whole family of critical points shares and the simplest theory gets wrong.

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

Light inside an accelerating rocket travels on circles. Rays of light leaving one point inside a rocket with constant acceleration, drawn in the rocket's own coordinates: height above the horizon up the page, in units of c²/a (one light-year for one gravity), and distance across. Each ray is a straight line in an inertial frame, transformed event by event. Launched across the rocket (0°) or at +30°, +60°, −30°, −60°, every ray is an exact semicircle centred on the horizon, with radius 1.000, 1.155, 2.000, 1.155, 2.000: the rays bend down towards the horizon and meet it at right angles. A ray sent straight up stays straight. These are the straight lines of the hyperbolic plane, although the spacetime inside the rocket is flat.

Relativity

The rocket in which light travels on circles

Inside a rocket that accelerates for ever at a constant rate, a beam of light sent across the cabin sags towards the floor, as Einstein argued it must if acceleration and gravity are indistinguishable. Followed further than any cabin, the sag turns out to be exact geometry: every ray inside the rocket is a perfect semicircle centred on the horizon below, and the rays obey the rules of a non-Euclidean plane in which the angles of a triangle add to less than two right angles. The spacetime in the rocket is flat; the curvature belongs to the clocks.

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.

The mass law: how much sound a limp wall stops. The transmission loss of a wall that only has mass — no stiffness — for sound arriving square-on, against frequency on a logarithmic axis, for surface masses of 10 kg/m², 20 kg/m², 40 kg/m². It is 10 log(1 + (ωm/2ρc)²) decibels. At 500 Hz, 10 kg/m² stops 31.6 dB, 20 kg/m² stops 37.6 dB, 40 kg/m² stops 43.6 dB. Doubling the mass or the frequency adds six decibels, and nothing else about the wall matters: it is the wall's inertia, set against air's characteristic impedance ρc = 413 rayl, that turns the sound back.

Waves

The wall that stops sound by its weight

A wall between two rooms keeps out sound not by absorbing it but by being hard to shake. The sound's pressure pushes the wall back and forth, the moving wall becomes a loudspeaker on the far side, and how much gets through depends almost entirely on the wall's mass per square metre against the frequency: six decibels more for every doubling of either. Read as impedance, a wall is a mass placed in series with the air, and the same reading explains the frequency at which a stiff pane of glass nearly vanishes and why two thin walls with a gap beat one thick one.

Absorption below and above the gap. The light absorbed by a direct-gap crystal against photon energy measured from the gap, in units of the exciton's Rydberg, from Elliott's formula with each line broadened to a half-width of 0.06 Rydberg (solid), beside the absorption the same crystal would have if electron and hole did not attract (dashed), which rises from zero as the square root of the energy above the gap. Below the gap the attraction adds a series of lines at −1, −1/4, −1/9 … Rydberg, the hydrogen series scaled down, with strengths falling as 1/n³, crowding towards the gap. Above it the continuum is lifted: the attraction pulls each free pair together, and at the gap itself the absorption is a finite step rather than zero.

Quantum

The hydrogen atom inside a crystal

Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature.

$Where skin and the thing it touches meet. Temperature across the contact between skin at 33 °C (left) and a block at 20 °C (right), against distance from the contact in millimetres, at 0.1, 1 and 10 s after touching, for steel (blue) and pine wood (red). Each pair of curves is two error functions meeting at the contact, and the meeting temperature does not change with time: 21.0 °C against steel and 30.3 °C against wood. The nerve endings a fraction of a millimetre under the skin report something close to that interface temperature, so the steel feels like an object at 21 °C and the wood like one at 30 — although both are at 20. The profiles spread as the square root of time, much further into the steel than into the wood.$ Thermodynamics

The metal that feels colder than the wood

A steel railing and a wooden bench in the same shade are at the same temperature, and the railing feels much colder. The usual explanation, that metal conducts heat better, is half the answer, and it gets water wrong. When two bodies touch, the surface between them jumps at once to a temperature that then stays fixed, a weighted average of the two in which each body's weight is the square root of its conductivity times its heat capacity per volume. Skin at 33 °C touching steel at 20 °C meets it at 21; touching wood, at 30. The nerves report the meeting temperature, not the object's.

Waves

The speed below which nothing on water leaves a wake

Every wave on clean water travels at 23 centimetres a second or faster: long waves are fast because gravity drives them, short ones because surface tension does, and in between there is a slowest wave. A boat, a duck or a fishing line can only make waves that keep pace with it, so anything crossing the water more slowly than 23 centimetres a second makes no waves at all. Faster, it makes two kinds at once — short ripples that run ahead of it and long waves that trail behind — and the threshold between nothing and both is set by gravity, surface tension and density alone.

Mechanics

The spring that swings when it was only bounced

Hang a weight on a spring and pull it straight down: it bounces straight up and down. Unless its bounce happens to be twice as fast as its swing would be as a pendulum. Then the bounce dies away and the weight starts to swing, wider and wider, until the bounce has gone completely — and then the swing dies and the bounce comes back. Two motions that look independent are trading their energy through a resonance, regularly and for ever at small amplitude. Give the spring more energy and the trading becomes chaotic: the same simple rule, with nothing random in it, produces motion that never repeats and that a hundred-millionth of a radian can redirect.

The waterfall hidden in a river mouth. The height of waterfall that would release as much energy per cubic metre of fresh water as mixing it into a large body of salt solution, against the solution's salt content by mass, for sodium chloride at 25 °C: the osmotic pressure divided by the weight of a cubic metre of water. Seawater is worth a 290 m fall; the brine rejected by a desalination plant, about twice as salty, 617 m; a saturated salt solution 3864 m. The curve bends upward because concentrated brine is less ideal than dilute and its osmotic coefficient rises above one. Every river mouth on Earth is a waterfall of this height that nobody sees, since all of its energy goes into the stirring of the estuary.

Fluids

The waterfall at every river mouth

Where a river meets the sea, fresh water mixes into salt and the free energy of mixing is released as nothing more useful than a little warmth and stirring. It is not a little energy: a cubic metre of river water entering the sea gives up as much as if it fell 290 metres. A desalination membrane run backwards can catch some of it, and the arithmetic of doing so is the arithmetic of every source with an internal resistance — the most power comes at half the pressure, and half the work is the price of getting it quickly.

The cavities that hold a beam. Every two-mirror cavity as a point (g1, g2), with g = 1 − L/R for each mirror. The shaded region, between the axes and the hyperbolas g1 g2 = 1, holds a ray for ever; everywhere else a ray is walked out. The commonest designs sit on the boundary: two flat mirrors (1, 1), the confocal cavity with the mirrors' foci together (0, 0), the concentric cavity with their centres together (−1, −1), and a flat mirror facing a curved one at its own radius (1, 0). A boundary cavity is stable in principle and marginal in practice, which is why the confocal one, sitting at the corner where the two shaded regions touch, is usually built a little inside it.

Optics

The two mirrors that hold a beam

A laser is two mirrors facing each other, and the light in it makes thousands of round trips before it leaves. Whether a ray stays between the mirrors for ever or is walked out after a few bounces depends on a single number, built from their spacing and their curvatures, and the answer has a sharp edge: two flat mirrors sit exactly on it, which is why the first lasers were so hard to align. The criterion is the one that decides whether a pumped swing grows, whether a wave crosses a crystal, and whether a particle beam stays in an accelerator.

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

Relativity

The gyroscope a spinning Earth turns

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

Too little noise, the right amount, too much. The particle's position over 8 cycles of the push (thin grey line, scaled to the wells), with noise of intensity 0.03, 0.085, 0.3 (top to bottom), the barrier being 0.25. With little noise it stays almost always in its starting well and only trembles: 0 hops in eight cycles. With noise 0.085 it hops 30 times, almost always once per half-cycle and almost always while the push is helping — a clean square wave following a push too weak to cause a single hop. With noise 0.3 it hops 285 times, at random, and the push is lost again. There is a best amount of noise, and it is not zero.

Waves

The noise that helps a weak signal through

A system that can sit in either of two states, nudged back and forth by a periodic push too weak to switch it, does nothing at all — until noise is added. Then it begins to switch, and with the right amount of noise it switches in step with the push, once each half-cycle, as though the push had been strong enough all along. Too little noise and it never switches; too much and it switches at random. The response peaks at a nonzero noise, which is why the effect is called a resonance, and why crayfish, paddlefish and human fingertips appear to use it.

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.

$A sound that survives only in a hot-electron plasma. The damping of an ion-acoustic wave at kλD = 0.3, as the fraction of its amplitude lost per radian of oscillation, −γ/ω, against the ratio of electron to ion temperature, for hydrogen and argon plasmas, on logarithmic axes: from the kinetic dispersion relation (solid) and from the weak-damping formula (dashed), which is valid only where the damping is small. With equal temperatures the wave loses 0.45 of its amplitude per radian in hydrogen — it is gone within a cycle. At Tₑ/Tᵢ = 10 the loss is 0.031; at 100 it is 0.0126, set by the electrons alone. Argon's floor is lower, 0.0020, because its ions are heavier and its electrons a smaller fraction of a sound wave's speed.$ Astrophysics

The sound a plasma carries on its electrons' heat

Sound in a gas needs two things: something heavy to be pushed, and a pressure to push it. In an ordinary gas the same molecules supply both. In a plasma they can be split — the ions carry the inertia and the electrons, a thousand times lighter and often a hundred times hotter, carry the pressure — and the result is a sound that travels at a speed set by one particle's temperature and the other's mass. It travels only while the two temperatures stay far apart. Warm the ions to match the electrons and the sound dies within a cycle, with nothing to rub against.

Optics

The spikes a bright star wears

Photographs of bright stars show them with spikes — four in most telescope pictures, eight in the James Webb Space Telescope's, none at all in some amateur images. The spikes are not in the sky and not in the camera. They are the shadows of whatever holds the telescope's secondary mirror, transformed by diffraction into streaks at right angles to themselves, and they can be counted, predicted and removed by reading a telescope's pupil the way diffraction reads it: as a pattern whose Fourier transform is the star's image.

How a hanging chain can swing. The shapes of the first four ways a uniform hanging chain can swing, side by side, against height from its free end: J₀(j₀,ₙ √(y/L)), fixed at the top and largest at the free bottom end. For a 2-metre chain they sound at 0.424, 0.973, 1.525, 2.078 hertz. The first swings as a whole, like a pendulum, with its bottom moving most. The higher modes have nodes, and the nodes are not evenly spaced: they crowd towards the bottom, where the tension is least and waves travel slowest, so a stretch near the free end holds more of each wave than a stretch of the same length near the top.

Waves

The chain that swings in Bessel's notes

Hang a chain from a hook and set it swinging. It can swing as a whole, like a pendulum, or in shapes with still points along it, as a guitar string does — but its higher notes are not two, three and four times its lowest. They are 2.295, 3.598 and 4.903 times it, the zeros of a function that Daniel Bernoulli met for the first time in exactly this problem in 1732, a century before Bessel's name was attached to it. The reason is that a hanging chain is a string whose tension grows from nothing at its free end to its whole weight at the top, and a wave crossing it changes speed all the way.

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

Electromagnetism

The magnet that reverses where its crystal is weakest

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

Two kinds of flicker in one heat bath. The variance of the number of photons in one mode of black-body radiation, against hν/kT, both on logarithmic axes, split into its two parts: n̄, the variance a shower of independent particles would have, and n̄², the variance of the intensity of randomly phased waves; their sum is what thermodynamics demands of Planck's law. Where the photons are rare, at hν much more than kT, the particle term wins; where they are crowded, the wave term wins; they are equal at hν/kT = ln 2 = 0.693, where each mode holds one photon on average. At hν = 5kT the wave term is 0.68 per cent of the particle term; at hν = kT/10 the particle term is 10.5 per cent of the wave term.

Quantum

The flicker that is half wave and half particle

In 1909, sixteen years before quantum mechanics, Einstein asked how much the energy in a small corner of a hot oven should fluctuate, and answered with nothing but thermodynamics and Planck's law. The answer has two terms. One is exactly what a spray of independent particles would give; the other is exactly what randomly overlapping waves would give. Neither theory of light could produce the sum, and the sum is what the oven does. The two terms are equal where each mode of the radiation holds one photon on average — a dividing line that runs through every hot thing, from the microwave sky to the Sun.

Fluids

The sand whose friction depends on how fast it flows

A heap of sand stands at its angle of repose and starts to slide a few degrees steeper; between those angles there are steady avalanches, and their speed is decided by one number. The inertial number compares how long a grain takes to fall into a gap opened under it with how long the flow takes to open the gap. Friction rises with it, from the static value to a ceiling, and that single dependence explains why grains flow steadily on only a twelve-degree window of slopes, why a flowing layer has the velocity profile Bagnold measured, and why a deeper avalanche is faster — but not as a liquid would be.

Waves

The long pulse that arrives as a spike

A radar that wants to see far must send a lot of energy, and a transmitter limited in peak power can only do that with a long pulse; a radar that wants to tell close targets apart needs a short one. A chirp — a pulse whose pitch rises steadily from start to finish — is both. Received through a filter that delays its low notes more than its high ones, every part of the echo arrives at the same instant, and fifty microseconds of pulse leave the receiver as a spike one microsecond wide. Resolution turns out to belong to bandwidth, not to duration, and the price of the trick is paid in sidelobes and in a target's speed being mistaken for its range.

Two beams that each stay straight, and wind each other into whirlpools. Phase space of a one-dimensional particle simulation: every dot is an electron, plotted at its position along a periodic box one fastest-growing wavelength long (10.26 v₀/ωₚ) and its velocity, in units of the beam speed. One beam starts at +v₀, the other at −v₀, with a 0.0001 ripple in position. At t = 18/ωₚthe beams are visibly rippled; by 26 the ripple has grown into overturning, and by 40 the two beams have wound round each other into a single vortex, filamented into strands. When growth first stops, at ωₚt = 25.2, the wave's field is 0.56 m v₀ωₚ/e; electrons trapped in it bounce at 0.59 ωₚ, faster than the 0.354 ωₚ at which it was growing, and it traps any electron within ±1.92 v₀ of the wave's speed — zero, by symmetry — so both beams are inside its separatrix; total energy was conserved to 0.31%.

Astrophysics

The two beams that unravel each other

A cold beam of electrons, alone, is perfectly stable: disturb it and it rings at its plasma frequency and carries the ringing downstream. Send a second beam through it the other way and the pair is not stable at all. A ripple of the right length grows by a factor of e in less than half a plasma period, takes a third of the beams' drift energy into an electric field, and then winds both beams into a single whirlpool in which their drift is gone. Neither beam can do it. The growth exists only in the pair, and a beam a millionth as dense as the plasma it crosses is enough to start it.

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

Thermodynamics

The rings a reaction lays down in a gel

Let one salt soak into a tube of jelly that holds another, and where they meet a solid should form, spreading smoothly behind the advancing front. It does not. It forms in sharp, separate bands, close together near the entrance and further and further apart, each one a fixed multiple of the distance of the last. Raphael Liesegang saw the rings in 1896 and they have been found since in rocks, in agate, in gallstones and in microchips. Nothing in the jelly is patterned to begin with. The pattern comes from a solid that will not start until the solution is overloaded, and that then empties the solution around it.

Two fine gratings that make a coarse one. Top: two gratings of dark bars laid over each other, one with a period of 10 units and the other of 11, both with bars half a period wide. Bottom: the average transmission of the pair, smoothed over a few periods. Where the bars of the two gratings coincide, light passes through the gaps between them; where the bars of one fill the gaps of the other, nothing passes. The two drift in and out of step once every 110 units — the moiré period, p₁p₂/|p₁ − p₂| — producing broad dark and light bands 11 times coarser than either grating. Neither grating contains a period of 110; their product does.

Waves

The pattern two gratings make that neither contains

Lay one fine net curtain over another and broad, swirling bands appear that are in neither curtain. Two sets of lines a tenth of a millimetre apart, differing by one part in a hundred, make bands a centimetre apart. This is moiré: the beat of two gratings in space, as two notes a little out of tune beat in time. It magnifies whatever makes the gratings differ — a twist, a stretch, a shift — by the ratio of the bands to the lines, so it is used to measure strain in bridges and to align the layers of microchips. And two sheets of carbon, twisted by just over one degree, make a moiré on which electrons stop moving and the material superconducts.

Where shaking makes the surface unstable. For water 1 cm deep shaken vertically at 50 Hz, the regions of wavenumber and shaking acceleration in which a surface wave grows rather than decays, computed mode by mode from the Floquet multipliers of its damped Mathieu equation (inside the curves, unstable). The largest tongue reaches down to 0.80 m/s² — 0.081 g — at k = 633 m⁻¹, the wave whose own frequency is exactly half the drive's: a wavelength of 9.9 mm. A second, narrower tongue sits at k = 1066 m⁻¹, where the wave's frequency equals the drive's, and needs much harder shaking. Without damping both tongues would reach down to zero; viscosity lifts their tips, most for the shortest waves.

Waves

The ripples that answer at half the shaking

Shake a dish of water up and down fifty times a second and nothing happens — until the shaking passes a threshold, when the whole surface erupts into a pattern of standing ripples, squares or stripes, each crest rising and falling twenty-five times a second, at exactly half the frequency of the shaking. Michael Faraday saw it in 1831. The size of the ripples is not set by the dish. It is chosen by the water itself, as the one wavelength whose natural frequency is half the drive's, and the threshold is set by how quickly the water's own viscosity would kill that wave.

A cloud that grows wider instead of denser. The density of rubidium atoms in a magneto-optical trap against distance from its centre, for 10⁵, 10⁶, 10⁷, 10⁸ atoms, computed from the trap's spring (200 Hz), the atoms' temperature (100 μK) and the repulsion of rescattered light, with σᵣ/σₗ = 1.30. The smallest cloud is the trap's Gaussian, 78 μm in radius, with a peak of 0.91 × 10¹⁰ per cm³. Larger clouds stop getting denser: each fills a sphere at the same 2.5 × 10¹⁰ atoms per cm³ (dashed), out to 0.21, 0.46, 0.99 mm, with a soft edge about a Debye length, 45 μm, wide. More atoms make a bigger ball, not a denser one.

Astrophysics

The light that makes a cold cloud repel itself

Load more atoms into a magneto-optical trap and the cloud does not get denser. It gets bigger, at a density that stays fixed whatever the number. The cause is the trap's own light. A photon one atom scatters can be absorbed by another, which is pushed away, and the push falls off as the inverse square of the distance, like the force between two charges. A cloud of laser-cooled atoms is therefore a plasma of neutral particles. Each behaves as if it carried a ten-thousandth of an electron's charge, and the trap fills with them as a harmonic well fills with charge: at one density, out to whatever radius the number requires.

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

Electromagnetism

The atom whose nucleus is a reflection

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

How a dark ridge fades into the sky. The apparent contrast against the horizon sky of a black ridge at a given distance, seen in green light (550 nm) along a level path at sea level, on a logarithmic scale: e^(−βx), the light from the ridge scattered out of the path and the sky's light scattered in. The dashed line is the eye's threshold, a contrast of 2%. Clean air: β = 0.0116 per km, visible to 337 km; light haze, 0.05 per km: β = 0.06 per km, visible to 64 km; haze, 0.3 per km: β = 0.31 per km, visible to 13 km. Pure air sets a limit of a few hundred kilometres to how far anything can be seen; in practice haze shortens it to tens.

Optics

How far the air lets a mountain be seen

A dark mountain on the horizon fades not because its light grows too faint to see but because the air between it and the eye glows. Sunlight scattered into the line of sight fills the dark ridge in until it matches the sky behind it, and its contrast falls as a simple exponential of the distance. With the eye's threshold at two per cent, that gives a visual range of 3.912 divided by the air's scattering coefficient: about 337 kilometres for perfectly clean air in green light, far less in the hazes real air carries. The same arithmetic makes near ridges blue and far ones pale.

$The power a cavity holds depends on where its mirror is. The power circulating in a cavity of finesse 10,000, a metre long and lit at 1064 nm, as a fraction of its value on resonance, against the detuning of the laser from the cavity's resonance in units of the cavity's half-linewidth (7.5 kHz). Lengthening the cavity lowers its resonance and moves every point of the curve to the right: 38 half-linewidths for each nanometre of mirror travel. On the blue side (laser above resonance, right) that motion loses power, so the radiation force on the mirror falls as the mirror moves out — a restoring spring, steepest at δ/γ = 0.58 (dashed tangent). On the red side the same motion gains power and pushes harder: a spring of the opposite sign.$ Astrophysics

The mirror held on a spring of light

Light bouncing between two mirrors pushes them apart, and how hard it pushes depends on how long the gap is, because the gap decides how much light the cavity holds. A mirror free to move therefore sits on a spring made of light: with a kilowatt stored in a metre-long cavity it is four-fifths as stiff as a steel wire a millimetre thick. The spring has a flaw no steel spring has. Stored light takes time to rise and fall, so the push always arrives a little late, and a late spring either winds the mirror up or calms it, depending only on which side of the cavity's resonance the laser sits.

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

Astrophysics

The nineteen decades a slow coupling buys

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

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

The stripes a superconductor makes of a field

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

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

Relativity

The throw a uniform field bends into a chain

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

Two columns of air side by side, falling at different rates. The air pressure against height inside a 100 m building kept at 21 °C (solid) and outside it at −10 °C (dashed), relative to the pressure outside at the ground, for a building whose openings are equal at top and bottom. Both fall by about 12 Pa a metre, the weight of a metre of air on each square metre, but the cold outside air is 12 per cent denser and its pressure falls faster. The two lines cross at the neutral plane, half-way up (dot). Below it the outside pressure is higher and pushes air in at every crack; above it the inside pressure is higher and pushes air out — 69 Pa at the roof, 69 Pa at the ground. The building is a chimney.

Fluids

The building that breathes through its doors

In winter the lobby doors of a tall building are hard to push open and, once open, let in a gale, though there is no wind outside. The building is a chimney. Its warm air and the cold air outside are two columns of fluid standing side by side, and the lighter one loses pressure more slowly with height, so the two pressures can be equal at only one level. Below it the outside pushes in through every crack; above it the inside pushes out. The difference is a few tens of pascals, the weight of a few millimetres of water, and over the area of a door it is the push of a strong hand — set entirely by hydrostatics, and running backwards in summer.

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

The voltage a moving magnet adds to an insulator

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

Relativity

The wave a row of moving clocks leaves on an instant

Line up identical clocks, synchronise them where they stand, and send the whole row past a laboratory at a steady speed. On any one laboratory instant the clocks do not agree: the ones further ahead read less, by an amount that grows steadily along the row. The readings form a wave laid out in space, its crests are the moving row's instants, and they sweep along faster than light. Give each clock the frequency of its rest energy and the wavelength is exactly h/p. De Broglie's matter wave began, in 1923, as an argument about the relativity of simultaneity.

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

$A plume rising into a stratified fluid, until it cannot. A turbulent plume from a small steady source of buoyancy, rising through a fluid whose density increases steadily downward, integrated from the Morton–Taylor–Turner equations with an entrainment coefficient of 0.1; height across, in units of F₀^(1/4)N^(−3/4), the only length the source's buoyancy flux and the stratification can make. Up: the buoyancy flux, as a fraction of the source's (solid); the momentum flux, as a fraction of its peak (dashed); and the plume's radius, as a fraction of its radius at the neutral level (dotted). The buoyancy flux falls from the start, because the fluid the plume entrains came from lower down and is denser than the fluid it is carried up into; it reaches zero at 4.37 — the level at which the plume will spread sideways — and goes negative. The momentum flux, built up by the buoyancy, peaks at 4.37 and is then spent carrying the now-heavy plume on to 5.75, where it stops. The radius grows almost in proportion to height until near the top, where the model's top-hat radius runs away as the momentum vanishes — the point at which the plume stops being a plume and becomes a spreading intrusion the equations do not describe.$ Fluids

The plume that stops below the surface

Hot smoke rises, oil and gas from a seabed leak rise, and in a fluid of uniform density they would go on rising for ever. Real oceans and atmospheres are stratified, denser lower down, and a rising plume carries its own downfall with it: it rises by drawing in the fluid around it, and the fluid it draws in near the bottom is denser than the fluid it is carried up into. Its buoyancy runs out, it overshoots, and it spreads sideways at a height set only by how buoyant its source is and how strongly the surroundings are layered — a quarter power of one and a minus three-quarters power of the other.

The height at which hot air stops being light enough. Lift per cubic metre of hot air against height in the standard atmosphere, with the inside at 100 °C and 120 °C, and (dashed) what a 2800 m³ balloon carrying 550 kg plus its envelope needs per cubic metre, 0.241 kg. As the balloon climbs, the outside air thins, and although it also cools — which improves the temperature ratio — the density wins: lift per cubic metre falls by about a tenth for every kilometre. The balloon can climb only to where the curve crosses its need: 6.0 km with the envelope at 120 °C. Above the tropopause the outside temperature stops falling and the lift falls faster still; a hot-air balloon that reaches the stratosphere needs an envelope enormous compared with its load.

Fluids

The balloon a cold morning lifts

A hot-air balloon is open at the bottom, so the air inside is at the same pressure as the air outside and lighter only because it is hotter. Its lift is the outside air's density times one minus the ratio of the two absolute temperatures — a small number, a third of a kilogram per cubic metre at best — and since the fabric cannot be heated much above a hundred degrees, the lift depends as much on the cold outside as on the heat inside. That is why balloons are the size of houses, why they fly at dawn, why each has a ceiling, and why the pilot must burn for where the balloon will be half a minute from now.

The force a rider feels on the way up a loop. The force of the seat on a rider, in multiples of the rider's weight, against how far the track has turned on the way from the bottom (0) to the top of the loop (1), for the 20 m circle and clothoid of the previous figure, each entered at the speed that leaves 0.5 g at the top. On the circle the force jumps the moment the car enters the loop, from 1 g on the straight to 6.5 g, and falls steadily to the top. On the clothoid it starts at 1 g, because the curvature starts at zero, and rises smoothly to 4.3 g partway round before falling. The clothoid's car also enters more slowly — 22.7 m/s against 23.2 — because its tighter top needs less speed to hold the car on.

Mechanics

The loop that is not a circle

A roller coaster's vertical loop looks circular from a distance and is not. A circular loop, entered fast enough for the car to stay on the track at the top, presses its riders into their seats at six times their weight at the bottom — six, whatever the circle's size — and slams that force on in an instant at the entry. The fix, found for railways a century earlier, is a curve whose curvature grows steadily with distance, the Euler spiral: the same curve that describes how light spreads round the edge of a shadow. Tight at the top, gentle at the bottom, it lets the force build instead of arriving all at once.

The chirp a staircase returns. The pitch of the echo of a clap from the staircase, heard by a listener 10 m from its foot, against time since the clap: the strongest frequency in successive 24 ms slices of the synthesised echo (dots), and the rate at which the reflections arrive, together with twice that rate (lines). The fundamental falls from about 767 Hz to 476 Hz over a sixth of a second, with its second harmonic an octave above — a descending chirp, which to many listeners sounds like a bird. It is Bragg reflection: the stair is a grating, the first-order echo is the frequency whose half-wavelength equals the extra path from one step to the next, and as the line of sight climbs, that extra path grows and the reflected pitch falls.

Waves

The chirp a staircase sings back

A hand clap at the foot of the great stair of El Castillo at Chichén Itzá is answered by a chirp — a falling whistle that some have compared to the call of the quetzal bird. Nothing in the stone is tuned. Each of the ninety-one steps returns its own copy of the clap, the copies arrive as a train of clicks, and the ear hears a regular train of clicks as a pitch. The pitch falls because the steps are not evenly spaced as the listener sees them: low steps are seen almost level, a tread apart, and high ones almost along the slope, a whole step's diagonal apart. It is Bragg reflection from a grating whose period, seen from one spot, changes along its length.

$One note becomes a comb. The spectrum of a steady tone heard through the turning horn, level with it, against frequency offset from the tone, for tones of 250 Hz (top), 1000 Hz and 4 kHz (bottom): the power in each line as a fraction of the whole, each row on its own baseline with the height of a row standing for all of the power, computed from the received sound over one turn. A pitch that swings periodically is not a shifted line but a comb of lines spaced by the rotation rate, 6.7 Hz, with powers Jₙ(β)², where β = 2πfr/c is 0.69, 2.75 and 11.0 for the three tones; the 1000 Hz comb matches the Bessel functions line for line. The comb is narrow for the low tone, which keeps 78% of its power in the central line, and broad for the high one, whose power is spread over about ±80 Hz — so the turning horn blurs the upper partials of an organ chord into a shimmer while its fundamentals stay nearly pure.$ Waves

The loudspeaker that goes round

The swirling sound of a Hammond organ comes from a loudspeaker cabinet in which the speaker's horn turns, six or seven times a second, and the swirl is the Doppler effect — the horn's mouth swings towards and away from the listener at about six metres a second, and every note it plays rises and falls by a third of a semitone. But a pitch that rises and falls periodically is not a note being shifted: it is a note split into a comb of new frequencies, spaced by the rotation rate and spread in proportion to the note itself, so the turning horn leaves the bass nearly pure and blurs the treble into a shimmer. In a room each wall returns its own copy, swinging out of step with the others, and the copies beat against each other several times a turn.

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

The push that makes an electron swing

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

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

Mechanics

The throw a spinning station bends

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

The pipe that out-conducts any solid. The temperature difference needed to carry 30 W along a rod 300 mm long and 6 mm across, on a logarithmic scale: for solid aluminium 1343 K, for copper 796 K, for diamond, the best conductor of heat at room temperature, 159 K; and for a water heat pipe of the same size 0.8 K, almost all of it across the thin wet wicks where the heat goes in and comes out. Solid metal could not do it at all: the hot end of the copper rod would have to be hundreds of degrees above the cold. The heat pipe carries the heat as latent heat in moving vapour, and the vapour needs almost no temperature difference to flow. As an effective conductivity that is 1000 times copper's — which is why every laptop and many satellites move their heat in pipes rather than through metal.

Thermodynamics

The pipe that carries heat as vapour

A copper rod six millimetres across and thirty centimetres long, asked to carry thirty watts from a laptop's processor to its fan, would need its hot end eight hundred degrees above its cold end. The flattened copper tube that actually does the job manages it with less than a degree. It is a heat pipe: sealed, emptied of air, and holding a few drops of water and their vapour. The water boils at the hot end, the vapour rushes to the cold end and condenses, and a wick of sintered copper powder drags the liquid back by capillarity. The heat goes as latent heat, and vapour needs almost no temperature difference to flow — which makes the pipe a thousand times better than copper, as long as its wick can keep up.

Waves

The echo hidden in random noise

A seismometer that is not recording an earthquake is not recording silence. It hears the Earth's hum: the pounding of ocean waves on every coast, arriving from all directions as surface waves with periods near ten seconds, indistinguishable from random noise. Two seismometers sixty kilometres apart hear different noise. Yet if their records are correlated and averaged over hours, a pulse appears at a lag of twenty seconds — the time a wave takes to travel from one to the other — exactly as if an earthquake had happened under one station and been recorded at the other. Noise from everywhere contains the echo between any two points, and since 2005 seismologists have mapped the Earth's crust with it.

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.

Electromagnetism

Two dipoles that agree everywhere but inside

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

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

Mechanics

The bicycle that steers itself upright

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

Waves

The waves that always arrive square to the beach

A storm a thousand kilometres off sends out swell in whatever direction it pleases, and that swell arrives on the beach running almost straight in. It does so whichever way the coast faces. Waves slow down in shallow water, so the part of a crest nearest the shore lags and the whole line swings round, as light does in glass. It also explains why waves shrink before they grow, and why they gather on a headland and leave the bay alone.

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