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The same equation again — part 2 of 3

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

$Where a small weight is felt, and where it is not. The fractional change in the frequency of harmonic 3 of a stretched string when a point mass of 0.06 of the string's own mass is placed at each position along it. The solid curve is the exact answer, found by solving the string's frequency equation for the loaded string at each position; the dashed curve is the first-order prediction, minus the mass fraction times the square of the mode shape. The two agree to 10.4 per cent at the antinode, where the shift is largest. Every node is a place where the exact shift is zero to better than a part in a thousand million, and that is not an approximation: a mass at a node is never moved by the mode, so it takes no part in the motion and cannot change its rate. Between the nodes the shift follows the square of the displacement, which is the square of the amplitude — the mass is felt in proportion to the kinetic energy the mode was already keeping there.$ Waves

The dent that raises the note

Push a wall of a resonator inwards and the pitch goes up or down depending entirely on where the wall is pushed. A mass added at a node changes nothing at all; the same mass at an antinode changes as much as it can. One rule covers a loaded string, a tuned microwave cavity and a bead drawn through a resonator to read out its field.

$The temperature a planet ought to be. Each body's measured surface temperature against the temperature at which it would radiate away exactly the sunlight it absorbs — computed from two numbers, the sunlight reaching it and the fraction it reflects, with no other property of the body used. Points on the diagonal are bodies the one-line argument gets right. the Moon: balance 270 K, surface 250 K, −20 K with no atmosphere; Mercury: balance 433 K, surface 340 K, −93 K with no atmosphere; Mars: balance 210 K, surface 210 K, +0 K; Earth: balance 255 K, surface 288 K, +33 K; Venus: balance 227 K, surface 737 K, +510 K. The airless bodies fall below the line rather than on it, and the reason is the fourth power: a surface running from noon heat to night cold radiates like its hottest parts and averages like its coldest, so a mean thermometer reading is lower than the temperature that matches the emitted flux. Mars, whose atmosphere is thin and whose surface is nearly isothermal by comparison, sits on the line. Everything with a substantial atmosphere sits above it, by tens of kelvin on Earth and hundreds on Venus, and always in the same direction. Nothing here explains why. What the figure fixes is the size and the sign of what has to be explained.$ Thermodynamics

The height a planet is seen from

A body in sunlight settles where it radiates away what it absorbs, and that takes two numbers and one line of arithmetic. It gets the Moon right and Earth wrong by thirty-three kelvin. The correction is not that the atmosphere traps heat but that it moves the level space sees the planet from, and the rest is done by a lapse rate that is not a radiative quantity at all.

$The height a clock can see. The fractional difference in rate between two clocks against how far apart in height they are, on logarithmic axes — a straight line of slope one, since the shift is gh/c² and comes to 1.09e-16 per metre near the ground. a caesium fountain, good to 1e-16, resolves 91.6 cm; an optical lattice clock, good to 1e-18, resolves 0.9 cm; the best clocks built, good to 8e-19, resolves 0.7 cm. The caesium fountains that define the second reach about a metre. The optical clocks that will replace them reach a centimetre, and the best of them a few millimetres. That is the whole of why this has stopped being a test of relativity and become a way of measuring the ground. A shift once so small that it took a Mössbauer experiment in a tower to see at all is now large enough to be a nuisance: two clocks in the same building disagree, and the disagreement has to be corrected for before either can be used to keep time.$ Astrophysics

The clock that measures a height

A clock a metre higher runs faster by a part in ten thousand million million million. That was once so small it took a tower and a Mössbauer source to see; the best clocks now resolve a centimetre of height, at any distance, without a line of sight. What began as a test of general relativity has become a surveying instrument that measures the quantity surveying actually wants.

$The fringes a flow of water moves. The interference fringe shift against the speed of the water, for two tubes 1.5 m long, an index of 1.333, and light of 526 nm — Fizeau's apparatus. The beam is split, each half goes with the flow in one tube and against it in the other, and the two are recombined; the shift is the difference in transit time counted in wavelengths. Three predictions are drawn and they are not close together. If the water did not affect the light at all the shift would be zero, flat along the bottom. If the water carried the light with it completely the shift would be the steepest line. Fresnel's partial drag is the middle one, and at 7 m/s it gives 0.207 of a fringe — which is what was measured, to the accuracy of an eye reading a fringe pattern in 1851. The experiment therefore did not merely detect an effect; it chose between three quantitative possibilities that differ by factors of two, which is why a fraction of a fringe settled something.$ Relativity

The drag that was only an addition

Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.

The pressure a charged gel develops. The swelling pressure against the gel's fixed charge density, for several salt concentrations, on logarithmic axes. Each curve has two straight parts with different slopes, and both limits are checked against the solution rather than read off the plot. Where the fixed charge is small compared with the salt the pressure goes as the square of the charge divided by four times the salt: the bath's ions screen the fixed ones, and doubling the salt halves the pressure. Where the fixed charge dominates, every counterion it demands is an extra particle in the gel and the pressure goes as the charge itself. At 15 mM salt, a gel with 200 mM of fixed charge develops 427 kPa; At 50 mM salt, a gel with 200 mM of fixed charge develops 306 kPa; At 150 mM salt, a gel with 200 mM of fixed charge develops 150 kPa; At 500 mM salt, a gel with 200 mM of fixed charge develops 49 kPa. Cartilage carries a fixed charge of a couple of hundred millimolar from the sulphated sugars on its proteoglycans, sits in a bath of about 150 mM, and develops a swelling pressure of an atmosphere and a half — which is what holds a joint apart and carries the load across it.

Fluids

The swelling a membrane cannot stop

Give the thing a membrane holds back an electric charge and the small ions that can cross are no longer free to distribute themselves. Two conditions — neutrality on each side, and equal chemical potential for the salt — fix where every ion goes, and they leave the charged side with more particles than the other. That excess is what holds a joint apart, and it is why a cell has to spend a third of its energy pumping.

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

Astrophysics

The same force whichever way the surface faces

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

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

Astrophysics

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

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

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

When the source is not heard all at once

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

Pushed one way, travelling another. A parcel released from rest under a steady pressure-gradient acceleration of 2.0e-4 metres per second squared pointing east, integrated for 36 hours at 60°, 30°, 10°. Without rotation it would accelerate east indefinitely. With it, the motion is an inertial circle about a mean velocity at right angles to the push, and the mean over a whole number of inertial periods is measured off each path and matches the geostrophic value a/f to two per cent. At high latitude the loops are tight and the drift is almost purely along the isobars; near the equator the parcel travels a long way down the gradient before the rotation has had time to turn it, which is why geostrophic balance is a high-latitude statement and the tropics need a different set of approximations.

Mechanics

The ratio that decides whether the planet is turning

Whether the rotating terms matter is not a question about size. It is one dimensionless ratio, U over fL, and it runs from ten thousand in a teacup to a millionth in the Earth's core. Where it is small the pressure gradient stops accelerating the fluid and starts balancing a force on fluid already moving across it — so the flow runs along the pressure contours instead of down them, and a weather map is a streamline plot.

Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

Relativity

Charge and current are one thing

Ask what a boost leaves alone and the answer is charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

Relativity

Six numbers, one object

Three components of E and three of B mix into each other under a boost and never into anything else. Six numbers that transform among themselves are the independent entries of a four-by-four antisymmetric array, and writing them that way is not notation — it turns Maxwell's four equations into two, makes the two invariants the only two there could be, and shows that "electric" is a choice of axes rather than a kind of field.

A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Waves

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

Two pushes that add up to a friction. The force on a sodium-23 atom from each of two counter-propagating laser beams tuned 0.5 linewidths below resonance, at 0.1 of saturation each, and their sum, against the atom's velocity in units of the linewidth over the wavenumber. Each beam pushes along its own direction and is heard loudest by an atom moving towards it, because the Doppler shift brings the red-detuned light up into resonance. At rest the two pushes cancel exactly; moving, the atom scatters more from the beam ahead of it than from the one behind, and the difference points against the motion. Near zero velocity the sum is a straight line through the origin — a friction, with slope −0.0907 ħk² — and it is largest at 3.13 m/s, beyond which the atom has been Doppler-shifted out of resonance with both beams and the grip weakens. The damping time it implies for sodium-23's mass is 17.5 microseconds.

Astrophysics

The friction made of light

Two laser beams pointed at each other push an atom both ways at once, and at rest the pushes cancel. Moving, the atom hears the beam ahead of it louder than the one behind, and the difference is a friction. The photons that supply the friction arrive one at a time, so they also kick — and the temperature where the two balance contains the width of a spectral line and nothing else.

Every slope answered by one curve. The boundary of everywhere one throwing speed can reach, drawn about the hand, with straight lines from the hand at −30°, 0°, 20°, 45° running out to it. Distances are in units of v²/g. Because the boundary is a parabola with its focus at the hand, the distance to it along any direction is r = (v²/g)/(1 + sin α), and each drawn length was found separately — by searching every launch angle for the one that lands farthest along that line — and agrees with the formula to ten decimal places. At −30° the greatest reach is 2.000 v²/g, launched at 30.0°; at 0° the greatest reach is 1.000 v²/g, launched at 45.0°; at 20° the greatest reach is 0.745 v²/g, launched at 55.0°; at 45° the greatest reach is 0.586 v²/g, launched at 67.5°. Uphill the reach shrinks and downhill it grows without limit as the line approaches straight down, and the launch that achieves it always bisects the angle between the line and the vertical. The small dots are the foci of those best throws: every trajectory's focus lies on a circle of radius v²/2g about the hand, and the farthest throw along a line is the one whose focus lies on that line.

Mechanics

One curve answers every slope

A throw up a hillside, down one, off a height and into a basket look like four problems with four answers. They are one problem. The edge of everywhere a throw can reach is a parabola with its focus at the hand, and written about that focus it gives the farthest reach in any direction in one line — along with the reason the shot that needs the least effort is the one whose aim matters least.

The pore size the air decides. The largest pore radius that holds condensed water in equilibrium with air at a given relative humidity, for water at 25 °C, on a logarithmic radius axis. A concave meniscus lowers the vapour pressure over it by exp(−2γVₘ/rRT), so a pore whose meniscus would be tighter than a radius set by the humidity is in equilibrium only when full. The upper curve is the emptying condition, through a hemispherical meniscus with two curvatures; the lower is the filling condition, through the cylindrical film that lines a pore before it closes, with one — so the same pore fills at a higher humidity than it empties at. At 50% humidity a pore empties below 1.51 nm and fills below 0.76 nm; at 90% humidity a pore empties below 9.96 nm and fills below 4.98 nm; at 99% humidity a pore empties below 104 nm and fills below 52 nm. The radii run from molecular at low humidity to a tenth of a micrometre at 99 per cent, and every one was checked by putting it back into the vapour-pressure relation.

Fluids

The pore that fills from dry air

Water condenses when the air is saturated — on a flat surface. Over a curved one the vapour pressure is different, higher over a drop and lower over a meniscus, and in a pore a few nanometres across it is low enough that the pore fills with liquid from air at half humidity. The water it holds is under a tension of a hundred megapascals, and the pore empties at a lower humidity than it filled at, for a reason that needs no roughness at all.

A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device.

Optics

The delay that is a random variable

A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

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

Electromagnetism

The pull that grows on the way down

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

The rectangle every cycle is equal to. An ideal Otto cycle for air on a temperature–entropy diagram: compression ratio 9, intake at 300 K, 1400 kJ/kg added at constant volume. Compression takes the charge to 722 K, combustion to 2672 K, expansion back to 1110 K, and the exhaust cools at constant volume. The shaded loop is the work; the region under the lower curve is the heat rejected. Heat enters over a range of temperatures, and its entropy-weighted mean — heat divided by the entropy it brings — is 1491 K; the heat leaves at a mean of 619 K. The dashed rectangle between those two temperatures has exactly the loop's area, and 1 − 619/1491 = 58.5%, which is the Otto efficiency, checked to rounding. A Carnot engine between the coldest and hottest points of the same cycle would reach 88.8%.

Thermodynamics

The temperature an engine really takes its heat at

Carnot's ceiling is set by two temperatures, and no engine that burns fuel takes its heat in at one temperature or gives it out at another. It takes heat over a range, from the moment combustion starts to the moment it ends. For any reversible cycle there is an exact replacement for Carnot's two numbers: the average temperature at which heat arrives and the average at which it leaves, each weighted by the entropy the heat carries. The gap between a real cycle and Carnot is a gap between those averages and the extremes.

$Most of a chain's counterions never leave it. The fraction of a charged rod's counterions lying within a distance r of it, against r in rod radii on a logarithmic axis, for a charge parameter ξ = 4.2 — the Bjerrum length of water, 0.7135 nm, over a charge spacing of 0.17 nm, which is DNA's. Each curve is the Poisson–Boltzmann solution for the rod at the centre of a cell of radius 10², 10⁴, 10⁶ rod radii, with its counterions checked to neutralise it exactly. Diluting the solution widens the cell by four decades at a time, and a counterion free to go anywhere in the cell ought to spread with it; instead each curve keeps a plateau near the rod whose height does not change. At the inflection of every curve the enclosed fraction is 0.762, which is Manning's 1 − 1/ξ, and the plateau sits there: 76 per cent of the counterions stay bound to the chain however dilute the solution, and only 24 per cent spread through it.$ Fluids

The counterions that never leave the chain

Dilute a solution of DNA a million times and its counterions ought to scatter through the whole volume. Three quarters of them do not. A line of charges closer together than the Bjerrum length — 0.71 nm in water — holds on to its counterions however much room they are given, until the chain's charge is cut back to one per Bjerrum length, and every osmotic pressure, swelling gel and packed virus built from such chains is set by that length rather than by the chemistry.

Fluids

The latitude past which a tide cannot split

The ocean's internal tide carries about a terawatt, and somewhere it has to be broken into waves small enough to mix the water. One of the ways it breaks is by pumping waves at half its own frequency, the way a child on a swing pumps at twice the swing's. Those half-frequency waves cannot exist where the planet's rotation forbids oscillations that slow — poleward of 28.8° for the semidiurnal tide — so the route has an edge on the map, fixed by the Moon's period and the Earth's spin.

Long stretches of order, broken without warning. 1800 successive values of the logistic map at r = 3.828427 − 0.00002, a distance of 2.0 × 10⁻⁵ below the setting at which its stable three-cycle is born. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them the orbit bursts through the whole interval with no discernible pattern, and then, at an unpredictable moment, is captured into another calm. In a run of 400,000 iterates at this setting the calms last 173 iterates on average, and the channel the orbit creeps through — measured on the map's own third iterate — has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates. Nothing random is added: the sequence is the same every time it is computed from the same start.

Mechanics

The calm that is the ghost of a cycle

Just before a chaotic system settles into a stable cycle it does something stranger than either: it behaves perfectly periodically for long stretches, and then, at moments nothing in the record predicts, bursts into disorder and back. The calm is a cycle that does not exist yet, creeping through the narrow gap where it is about to be born, and how long each calm lasts is set by the square root of the distance to that birth.

Light that walks through a cloud. Seven photons entering a slab 8 scattering mean free paths thick from above, straight down, and followed until they leave, with every scattering equally likely to send them in any direction; their paths are projected onto the page. Of 20000 photons followed the same way, 82.5 per cent come back out of the top and 17.5 per cent out of the bottom. Diffusion theory predicts 18.2 per cent through the bottom. Only 10 photons cross without scattering at all, where e⁻⁸ predicts 6.7 — within the counting error of so few: nearly all of what is transmitted has walked, and the direction it arrived from is forgotten on the way.

Optics

The cloud light has to walk through

A beam crossing thirty scattering lengths of anything should keep e⁻³⁰ of itself — a ten-millionth of a millionth. A cloud thirty scattering lengths thick lets through nearly a third of the sunlight falling on it. The light has not crossed; it has walked, one scattering at a time, and a walk through a slab obeys a law with the shape of Ohm's rather than of an exponential.

Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength.

Optics

The reflection that needs no surface

Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

$Nitrogen that runs the wrong way. The nitrogen mole fraction in each of two bulbs joined by a capillary, as in Duncan and Toor's experiment: one bulb starts with 0.50086 nitrogen and the rest carbon dioxide, the other with 0.49879 nitrogen and the rest hydrogen, at 35 °C. Solid curves: the capillary solved at each instant from the Maxwell–Stefan equations with the three pairs' diffusivities, 83.8, 68.0 and 16.8 mm²/s. Dashed: Fick's law for nitrogen alone, which can only let the two start values relax together. At the start the nitrogen gradient is 0.00207, yet nitrogen flows at 309 times the rate that gradient would drive. From 0.1 to 6.5 hours it flows from the bulb with less nitrogen into the bulb with more, opening a difference of 0.1468 at 6.5 hours; at 6.6 hours its flux passes through zero with a difference of 0.1468 still in place. Each gas is conserved to a part in 10⁹. The carbon dioxide moving out of the first bulb drags nitrogen with it, because the nitrogen–carbon dioxide pair has by far the smallest diffusivity and so the strongest friction.$ Thermodynamics

The gas that flows towards more of itself

Fick's law says a substance diffuses from where there is more of it to where there is less. In a mixture of three gases, nitrogen can do the opposite for hours — flowing into the bulb that already holds more nitrogen, and then stopping while a difference remains — and in a welded bar of steel, carbon crosses into the side that is already richer. Nothing is wrong with the second law. Diffusion flattens chemical potential, and with more than two components, or a second element changing it, that is not the same as flattening concentration.

$One slit, four distances, one multiplication. The intensity across the beam behind a slit 5 wavelengths wide, at distances of 0.5, 5, 25, 100 wavelengths, each computed by multiplying the slit's plane-wave spectrum by the phase each wave accumulates and transforming back — no approximation about angles. Close to the slit the pattern is the slit's own shape with ripples at its edges; further out the ripples move inwards and the beam develops a bright centre; far away it spreads into the diffraction pattern. The travelling part of the field keeps its power to 10⁻¹⁰, running it back 100 wavelengths recovers it to 5 × 10⁻¹⁴, and at 100 wavelengths the result matches a direct Fresnel integral to 2.75 per cent rms. Near field and far field are not two theories; they are one multiplication at different distances.$ Waves

The fan of plane waves inside every beam

Huygens added up wavelets from every point of a front. The same content can be written as a sum over plane waves travelling in every direction, and then propagation stops being an integral and becomes a multiplication: each plane wave picks up a phase in proportion to the distance. One square root in that phase holds all of diffraction, near field and far field alike — and when the square root turns imaginary, it holds the reason no instrument a wavelength away can see detail finer than half a wavelength.

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

Electromagnetism

What holds a magnet together is not magnetism

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

The stress a temperature change puts into a bar that cannot move. The stress in a member held between supports that will not let it change length, against how much its temperature changes, for four materials. Each line is EαΔT, computed here from a free expansion and the force needed to undo it, and checked by evaluating it for a bar half a metre long and one thirty-seven metres long: the two agree to every figure carried, because neither the length nor the cross-section appears in the answer. steel develops 2.40 MPa for every kelvin and reaches yield at 104 K; aluminium develops 1.59 MPa for every kelvin and reaches yield at 151 K; concrete develops 0.30 MPa for every kelvin and reaches cracking at 10 K; invar develops 0.17 MPa for every kelvin and reaches yield at 1655 K. Concrete reaches its cracking stress after ten kelvin, which is less than a sunny afternoon, and is why every slab has movement joints in it. Invar is in the comparison because it was made to have a small product: it is as stiff as steel and develops a fourteenth of the stress, which is a statement about the expansion coefficient and nothing else.

Mechanics

The load nobody applied

A redundant structure develops forces with nothing on it. Change its temperature and the same extra constraint that made statics unanswerable also refuses the expansion — and a restrained steel member reaches its yield stress after a hundred and four kelvin, a figure that contains no length, no area and no load.

$The modes of a wire, counted. Nyquist's argument of 1928, with its count performed. Two resistors joined by a lossless line are in equilibrium, and the line is a one-dimensional cavity whose standing waves are spaced c/2L apart in frequency. Each mode has an electric and a magnetic energy, both quadratic, so equipartition gives it kT — the same half a kT per quadratic term that a heat capacity counts. The upper points are how many modes fall in a band of a hundred and thirty-seven megahertz, for five line lengths, from 17 on a thirteen-metre line to 8,558 on one of six kilometres. The lower points are the power each end therefore receives, as a fraction of kTΔf, and they converge on one: a longer line has proportionally more modes and takes proportionally longer to deliver them, so the length cancels. The longest line lands within 0.005 per cent and the shortest is 4.5 per cent low, because thirteen metres holds only seventeen whole modes in the band and the remainder is a real granularity rather than an error. What is left in the limit is kTΔf — a noise power with no resistance in it at all, and none of the line's properties either.$ Thermodynamics

Half a kT in a piece of wire

Count the quadratic terms in a molecule's energy and equipartition gives a heat capacity. Count them on a transmission line instead and the same theorem gives a resistor's noise voltage — 4kTRΔf, with nothing in it about what the resistor is made of. A fifty-ohm input at room temperature says 0.91 nanovolts in every root hertz, and no design removes it.

The ripple that sits on top of an absorption edge. The Cu K absorption of copper foil, 293 K through its edge and for seven hundred electronvolts above it, with the smooth atomic background it would have if the absorbing atom were alone drawn beneath it. The difference between the two is the fine structure: a modulation reaching 11 per cent, dying away as the photon energy rises, and entirely absent from a free atom. It is there because the ejected electron is a wave that the neighbouring atoms scatter back onto the atom that emitted it, so the absorption depends on whether the returning wave arrives in step with the outgoing one — which depends on the distance to the neighbour and on nothing else about the sample.

Waves

The ripple that counts the neighbours

An absorption edge is drawn as a step and it is a step with a ripple on it — a modulation of eleven per cent in copper, five in a zinc site buried in a protein. The ripple is the ejected electron's own wave, scattered back onto the atom that emitted it, so its period is a distance. It is the only way of measuring where an atom's neighbours are that does not need a crystal.

The four vortices a standing wave leaves behind. Streamlines of the steady flow that a standing sound wave sets up in a channel, over half an acoustic wavelength, with the horizontal axis in units of the wave's own phase and the vertical axis scaled to the channel. The sound itself is a back-and-forth motion that averages to nothing; this is what does not average to nothing. Four closed cells fill each wavelength, two above the centreline and two below, turning in opposite senses, with the fluid moving along the walls toward the velocity nodes and back along the centre. The boundary layer that generates all of it is 69 micrometres thick, which is 0.7 per cent of the channel and is thinner than the width of a line in this drawing. The cells are not in the layer; they fill the channel.

Fluids

The drift a sound leaves behind

A sound wave moves fluid back and forth and puts it back where it started. Over many cycles it does not: a steady circulation appears, four cells to a wavelength, driven entirely from inside a boundary layer seventy micrometres thick. Its speed contains the sound speed and the amplitude, and it contains no viscosity at all — so making the fluid thinner does not make the drift weaker.

Electromagnetism

A potential that does not come back to itself

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

One dimensionless group between an engine and Carnot. The efficiency of a thermoelectric couple against the temperature of its hot side, with the cold side at 300 kelvin, for 4 values of the figure of merit, and the Carnot ceiling drawn above them. The expression has exactly one material quantity in it — the dimensionless group formed from the Seebeck coefficient squared, the electrical conductivity, the temperature and the thermal conductivity — and everything else is the two temperatures. At 600 kelvin, a figure of merit of one gives 10.8 per cent against a Carnot ceiling of 50.0, and a figure of merit of four gives 22.6. The approach to the ceiling is slow: every doubling of the group buys less than the last, so the difference between a good material and a perfect one is smaller than the difference between a poor material and a good one.

Thermodynamics

An engine with one number in it

A thermoelectric couple has no moving part and no working fluid, and its efficiency is the Carnot value multiplied by a factor containing exactly one dimensionless group of material properties. Sixty years of effort have moved that group from about one to about two, and the reason it is hard is that its three ingredients are not independent: raising the conductivity ruins the coefficient it is squared against, and the only lever that is really free is the heat the lattice carries.

The action at one instant, drawn as a map. Trajectories leaving one point at the same moment, at launch speeds 0.6, 1 and sixteen directions each, in a uniform field pulling downward, drawn up to time 1. Behind them, dashed, are the level curves of the action at that instant, regarded as a function of where a trajectory ends. They are circles, and their common centre is neither the launch point nor anywhere the particles have reached: it is 0.500 above the launch point, while the whole swarm has fallen by the same 0.500. Every arriving velocity points straight out from that centre, so every trajectory crosses the level curves at right angles, and the arriving momentum equals the gradient of the action to one part in ten thousand. The action integrated along each path agrees with the map's value at its end.

Mechanics

The action that knows where every path ends

The action is usually a number attached to one path. Treat it instead as a function of where the true path ends, and a single function of position and time holds every trajectory at once — its slope is the momentum, its rate of change is the energy, and its level curves are wavefronts, drawn about a point that sits above the source while everything falls.

The light of a diode at room temperature is as bright as a surface thousands of kelvin hot. How many photons occupy each mode of the light, on a logarithmic scale, against photon energy. The lowest curve is the thermal glow of a 1.42 eV semiconductor at 300 K with no voltage across it. The solid curve above it is the same device with 1.3 V across it, emitting only above its gap. The dashed curve is a blackbody at 2571 K, the temperature whose light has the same occupation as the diode's at 1.472 eV, just above the gap. They cross there and nowhere else: the diode's occupation falls a factor e every 25.9 meV, as its lattice's temperature requires, and the blackbody's every 222 meV. No single temperature describes the diode's light. At each photon energy it has a brightness temperature, and that temperature is 300 K multiplied by ε/(ε − qV).

Thermodynamics

The glow that carries a voltage

Thermal radiation has no chemical potential, because walls make and destroy photons freely. A light-emitting diode is a body that glows at room temperature with a voltage written into its light — Planck's law with the voltage as the photons' chemical potential — which is why its light can be as bright as a surface thousands of kelvin hot, why at low voltage it can put out more light than the power it draws and cool itself doing so, and why a reverse voltage makes a surface look colder than it is.

Two paths through a neutron interferometer at two heights. A neutron interferometer cut from one silicon crystal, tilted by 30 degrees about its incoming beam so that one path runs higher than the other. Left: the two paths, split at the first slab, turned at the second and recombined at the third, enclosing 10.1 square centimetres; the heights are drawn exaggerated. Taken as a rectangle of the same area, the upper path runs 15.8 mm higher for 3.2 cm. There a neutron of wavelength 1.445 Å, moving at 2738 m/s, is slower by 56.5 micrometres per second, so its wavelength is longer by 20.6 parts per thousand million. Over 3.2 cm that accumulates 28.7 radians less phase than the lower leg — 4.6 whole fringes — computed by integrating the local wavenumber along both legs and checked against 2πm²gλA sin α / h².

Quantum

The fall that leaves the mass in the phase

Every body falls the same way whatever its mass, and a neutron is no exception. But a neutron is also a wave, and the phase that wave accumulates while falling depends on the mass — as its square, at a fixed wavelength. Tilt a neutron interferometer so that one path runs a centimetre higher than the other and the neutrons swing between its two detectors, which in 1975 was the first measurement in which gravity and quantum mechanics both had to be right at once.

A hump that is not a soliton comes apart into solitons. A single smooth hump of height 6, shaped as the square of a hyperbolic secant, released into the Korteweg–de Vries equation and followed by a pseudo-spectral integration, drawn at times 0.00, 0.15, 0.35, 0.60, each snapshot raised above the last. The hump is too tall for its width to be a soliton, and it separates: by the last time there are 2 crests, of heights 8.00 and 2.00, running apart at different speeds, with a small ripple left behind. Read as a potential well, the same hump holds 2 bound states, at κ = 2.000 and 1.000, and a soliton of height 2κ² belongs to each: 8.00 and 2.00. The integration conserved the hump's area to 3.1·10⁻¹⁵.

Waves

The solitons a hump already contains

A soliton is one height for one width. Release a hump of any other shape and it does not keep that shape or simply spread — it comes apart into a fixed number of solitons of fixed heights, running off in order of size, with a ripple left behind. The number and the heights can be read off before anything moves, by treating the hump upside down as a well and counting the levels it holds.

$The light follows one supermode through the crossing. Inside a tapered coupler 800 µm long at 1550 nm. Above: the fraction of the light in the first guide along the coupler, from integrating the coupled-mode equations, and dashed, the share of the first guide in the local supermode the light was launched into. They agree to within 8.3 percentage points along the whole length: the light does not beat between the guides but follows the supermode as that supermode changes from being in the first guide to being in the second. Below: the two supermodes' propagation constants relative to their average, which approach each other and repel across a gap of twice the coupling, 0.084 per micrometre, at the point where the guides are equally wide; dashed, the two single guides' constants, which cross.$ Waves

The coupler that does not care about the colour

Two identical guides side by side swap their light back and forth, and a coupler cut to the length of one swap works perfectly at one wavelength and badly at every other. Make the guides unequal, and sweep the inequality from one sign to the other along their length, and the light no longer swaps — it follows a single mode of the pair as that mode moves from one guide to the other. The transfer is then nearly complete across hundreds of nanometres of wavelength and immune to the widths being made wrong, and it costs length.

A repeat in space gaps the frequencies; a repeat in time gaps the wavenumbers. Two media with the same modulation depth, 0.2, computed the same way. Left: permittivity repeating in space, a stack of layers. For each frequency the wave equation is integrated across one spatial period, and the Bloch wavenumber drawn against frequency; between frequencies 0.478 and 0.526 (in units of c over the period) there is no real wavenumber, so light of those frequencies cannot travel and is reflected. Right: permittivity repeating in time. For each wavenumber the equation is integrated across one period, and the frequency drawn against wavenumber; between wavenumbers 0.474 and 0.518 there is no real frequency. The axes of the two panels are swapped, and so is everything else: the spatial gap is a band of frequencies that decays in space, the temporal gap is a band of wavenumbers that grows in time.

Optics

The crystal made of moments

A stack of layers that repeats in space refuses a band of frequencies and reflects them. A medium that repeats in time — its refractive index swung up and down everywhere at once — refuses a band of wavenumbers instead, and a wave with a wavenumber in that band does not reflect. It grows, exponentially, drawing on whatever is swinging the index. The construction is exact, the gap is computable from one period of the modulation, and the obstacle to building one for light is how fast a material would have to change.

With interference kept, transmission falls exponentially; without it, only as one over the thickness. Light through stacks of randomly thick transparent layers, alternating indices 1 and 2.6, with thicknesses scattered by ±50 per cent about a quarter wave, on a logarithmic scale against the number of layers. The falling line is the transmission with the waves' interference kept, computed exactly by multiplying transfer matrices and averaged as the logarithm over 40 random stacks and five wavelengths. It falls in a straight line: the transmission drops by a factor of e every 17 layers, however thick the stack, which is exponential decay — localisation. The upper curve is the same stacks with every surface's reflection and transmission added as intensities, so that no interference survives. It falls only as one over the thickness, checked to grow in exact proportion, which is the diffusion of light through a cloud: at 400 layers it still transmits 1.0 per cent where the coherent stack typically transmits 4.5·10⁻¹¹.

Optics

The walk that interference can stop

Light scattered many times walks through a cloud, and a walk always gets through eventually — a slab twice as thick lets through half as much. Keep the waves' interference instead of adding intensities, and in one dimension the same disorder does something a walk cannot: it stops the light exponentially, traps it in modes with nothing special about where they sit, and turns transmission from a number into a spread over powers of ten. Whether the same can happen to light in three dimensions has been claimed, retracted and argued for thirty years.

Astrophysics

The half of the bend a slow body never feels

Light passing the Sun is bent by twice Newton's angle, and the factor of two is usually read as a fact about light. It is the end point of a curve every body lies on. A body at speed v is bent by (2GM/bv²)(1 + v²/c²) — the 1 from the curvature of time, which a slow body feels in full, and the v²/c² from the curvature of space, which is the same absolute angle for a comet as for a photon and is simply swamped when the body is slow.

A rosette: the orbit of a body whose inertia is its energy. An orbit of eccentricity 0.45 under inertia from energy, Newton's pull, integrated for 7 revolutions from the closest approach of a body with k/Lc = 0.35. The dashed curve is the Newtonian ellipse from the same starting position and momentum, which closes on itself. The integrated orbit does not: its closest approach moves forward by 24.3° each revolution, against the exact 24.3° of 2π(1/Γ − 1) with Γ = √(1 − k²/L²c²). The dots are the successive closest approaches.

Astrophysics

The orbit special relativity cannot close

An inverse-square orbit closes on itself because of a conserved vector that nothing else has. Give the orbiting body an inertia that grows with its speed, as special relativity requires, and the vector turns — the orbit becomes a rosette, advancing by a sixth of Mercury's famous 43 arcseconds a century. A different theory that respects special relativity just as well turns the rosette backwards by the same amount. The 43 is not special relativity plus a correction; it is a measurement of what gravity pulls on.

$How full a surface is, against the pressure of the gas above it. The fraction of adsorption sites occupied on a surface that binds a molecule with 0.35 eV, against the gas pressure on a logarithmic axis, at 250, 300, 350 K. Each curve is θ = p/(p + p½) — Langmuir's isotherm — and each is the same S-shape moved along the axis: on a logarithmic pressure scale the coverage is a function of ln p alone, shifted by the binding energy over kT. The half-full pressure is 5.58 Pa at 250 K, 132 Pa at 300 K, 1341 Pa at 350 K. Warming the surface does not change the shape; it slides the curve to higher pressure by a factor that is a Boltzmann factor, e^(ε/kT), which is how a binding energy is read off a family of isotherms.$ Thermodynamics

The site that fills like an electron level

A patch of catalyst holding a gas molecule, a haem in a muscle cell holding oxygen, an energy level in a metal holding an electron: each is a place that can hold one particle or none, in contact with a reservoir it trades particles with. Each fills according to the same function of one variable — how far the reservoir's chemical potential lies above the energy of the site. Langmuir's isotherm and the Fermi function are not analogous. They are the same law, and haemoglobin works because its sites break it.

Electrons and holes in silicon against the donors added. The concentrations of free electrons and holes in silicon at 300 K against the concentration of donor atoms added, on logarithmic axes, solved from charge neutrality and the product law n·p = nᵢ². Below the intrinsic concentration of 10¹⁰ per cubic centimetre the donors change almost nothing. Above it the electrons follow the donors one for one and the holes fall in exact proportion: a hundred times more donors, a hundred times fewer holes. At 10¹⁶ donors the electrons outnumber the holes by 10¹² to one. The dashed line is √(n·p), which does not move at all across ten decades of doping — checked to a part in a billion.

Thermodynamics

The product doping cannot move

Add one phosphorus atom for every five million silicon atoms and the crystal's free electrons multiply a millionfold — and its holes fall a millionfold, although nothing was done to them. Add a drop of acid to water and the hydroxide ions vanish in exactly the proportion the hydrogen ions grow. Both are one law: when a reaction makes two species together, equilibrium fixes the sum of their chemical potentials, and so the product of their concentrations. Every semiconductor device is built on the minority carriers that law leaves behind.

$The innermost electron of an atom with Z = 79, with and without relativity. The probability of finding the innermost electron at each distance from a nucleus of charge 79 (Au), with the distance in units of a₀/Z so that the non-relativistic curve is the same for every element. The dashed curve is Schrödinger's, r² e^(−2Zr/a₀). The solid curve is Dirac's exact ground state, r^(2γ) e^(−2Zr/a₀) with γ = √(1 − (Zα)²) = 0.817. It is pulled in towards the nucleus: its mean radius is 0.878 of Schrödinger's, a contraction of 12 per cent, and it has a sharper peak close in. Nothing about the nucleus has changed; the electron, moving at an appreciable fraction of the speed of light near it, simply behaves as a heavier particle, and a heavier particle is bound more tightly and more closely.$ Quantum

The metal that is yellow because it is heavy

Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.

A relativistic mass on a spring, at four amplitudes, scaled to one size. The displacement of a mass on an ideal spring against time, over two periods, with the mass's inertia growing with its energy as relativity requires, at amplitudes of 0.2, 1, 3, 10 times c/ω₀ — the distance light travels in one radian of the slow oscillation. Each curve is divided by its own amplitude and its own period so that shape alone is compared. The small swing is the familiar sine. The large one is a triangle: the mass spends almost the whole cycle moving at nearly the speed of light, turning round abruptly at each end. Periods: 1.007, 1.175, 2.132, 6.431 times the slow period.

Relativity

The spring that becomes a light-clock

A mass on an ideal spring keeps the same period however far it swings — that is what makes it a clock. Give the mass the inertia relativity demands, pull it back far enough, and the clock breaks. The speed can never pass c, so a large swing spends almost all its time moving at nearly the speed of light, turning round abruptly at each end: the sine becomes a triangle, the period grows until it is simply the time light takes to cross the swing four times, and a clock riding on the mass ages a small fraction of what one beside it does.

$Modes added in step: a train of pulses. The intensity of the sum of N waves of equal amplitude at equally spaced frequencies, all starting in phase, against time in units of the round trip — the inverse of the spacing — for N = 5, 10, 20. The sum is a train of pulses, one per round trip, each of peak intensity N² times one wave's and width about 1/N of the round trip. Between the pulses the waves cancel almost completely. The average intensity is N, as it must be — 5.01 for N = 5, 10.04 for N = 10, 20.19 for N = 20 — so the pulses do not create energy; they move it all into a fraction 1/N of the time, where it is N times more intense than it would be if spread evenly.$ Waves

The phases that turn a glow into pulses

A laser oscillates on many frequencies at once, spaced by the time light takes to go round its cavity. Add those waves with random phases and the output is a steady glow with noise in it. Add exactly the same waves with their phases equal and the output is a train of pulses, each N² times brighter than one wave and a fraction 1/N of the round trip long. Nothing about the frequencies, the amplitudes or the average power changes — only the phases, and they decide everything a detector fast enough would see.

Waves

The lattice no phase can bend

Cross a few laser beams of one colour and they paint a crystal of light in the space where they overlap. The beams' phases drift whenever a mirror trembles, and it would seem the pattern must tremble with them. Whether it does depends on a count. Three beams in a plane, or four in space, give a lattice whose shape no change of phase can alter — the phases can only slide it. One beam more and the phases decide the shape, and must be held still.

$One turn of a circling charge, as it arrives. The radiation field reaching a distant observer in the plane of a charge's circular orbit, over one full turn, for speeds of 0.2c, 0.6c, 0.9c, each scaled to its own peak and plotted against arrival time as a fraction of the orbital period. At 0.2c the flash arrives 20.9 per cent of a turn wide at half height, having been emitted over 25.4 per cent; at 0.6c the flash arrives 6.0 per cent of a turn wide at half height, having been emitted over 14.2 per cent; at 0.9c the flash arrives 0.7 per cent of a turn wide at half height, having been emitted over 6.3 per cent. The slowest charge sends something close to a sine wave, with a trough 0.44 of its peak; the fastest sends a spike whose peak, divided by its speed, is 100 times a slow charge's, and a trough of 0.0028 of that peak. The field integrates to zero over every turn, as a radiation field must.$ Astrophysics

The flash a circling charge sends once a turn

A slow electron going round a magnet broadcasts one radio frequency. The same electron at nearly the speed of light, on a circle of the same kind, sends X-rays. Nothing about the mechanism changes between the two. The charge chases its own light, and the chase compresses one power of γ worth of emission into three.

Every colour a bound electron can meet. The light-scattering cross-section of one electron bound with a resonance at 2.105 eV (589 nm) and damped only by its own radiation, in units of the Thomson cross-section, against the frequency of the light in units of the resonance, on logarithmic axes spanning nine decades of frequency and twenty-five of cross-section. Far below the resonance it rises with slope 4.001, Rayleigh's fourth power: at a hundredth of the resonance it is 1.0 × 10⁻⁸ times the Thomson value. At the resonance it spikes to 2.5 × 10¹⁵ times the Thomson value, which is 3λ²/2π = 1.66 × 10⁻¹³ m² exactly, over a width of 2.0 × 10⁻⁸ of the resonant frequency. Above it the cross-section settles to the Thomson value, 6.652 × 10⁻²⁹ m² — 1.0002 times it at a hundred times the resonance — and stays within 10 per cent of it from 4.6 times the resonant frequency, 9.8 eV, to 29 keV, 3.5 decades, until the photon energy approaches the electron's rest energy, where the Klein–Nishina factor for a free electron takes it down to 0.431 of the Thomson value at 511 keV and 0.077 at 10 MeV.

Astrophysics

The cross-section that forgets the colour

An electron on a spring, shaken by light, scatters it — weakly and very blue below its resonance, enormously at it, and above it by exactly the same amount whatever the colour. That flat stretch is the Thomson cross-section, 6.65 × 10⁻²⁹ m², and it is why an X-ray picture of a crystal is a map of electrons and why the corona is white.

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

Fluids

The sea that stands lower because water gives

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

Quantum

The atom the size of a bacterium

Excite hydrogen's electron to the hundredth level and the atom is a micrometre across, bound by a thousandth of an electronvolt, pulled apart by the field of a torch battery, and — in its most circular state — able to live for nearly a second. Every one of those numbers is a power of n, and together they make the most exaggerated atom there is into one of the most useful.

A charge going past, as the field it delivers. The electric field across the line of motion at a point a distance b from the path of a passing charge, against time in units of b/c, on a logarithmic field axis in units of q/4πε₀b², for Lorentz factors of 1.2, 3, 10, 30. The peak is γ times the Coulomb field at rest and the width at half height is 2.311 b/c at γ = 1.2, 0.542 b/c at γ = 3, 0.154 b/c at γ = 10, 0.0511 b/c at γ = 30: taller by γ and shorter by γ, so the area under each curve — the sideways impulse the passing charge delivers, 2q²/4πε₀bv — is the same for all of them to within the integration's precision. A fast charge's field arrives as a thin, intense flash.

Relativity

The charge that passes as a flash of light

A charge at rest has a field that no change of frame can turn into light. A charge going past at nearly the speed of light delivers a field that is, for every practical purpose, a pulse of light anyway: thin, perpendicular to the motion, with its electric and magnetic parts equal and crossed. Enrico Fermi saw in 1924 that such a field could be counted as photons, and the Large Hadron Collider now collides lead nuclei's photons with each other.

Fifty-nine millivolts for every factor of ten. The voltage that holds a singly charged ion at one concentration on one side of a boundary and another concentration on the other, in equilibrium, against the ratio of the concentrations on a logarithmic axis: (kT/e) ln(c₁/c₂), from requiring the electrochemical potential to be the same on both sides. At 0 °C, 54.20 mV per decade; at 25 °C, 59.16 mV per decade; at 37 °C, 61.54 mV per decade; at 100 °C, 74.04 mV per decade. The voltage depends on the ratio and the temperature and on nothing about the ion, the solvent or the boundary: it is the entropy of dilution, expressed per unit charge.

Thermodynamics

The voltage that is a chemical potential

A battery, a nerve cell and a pH meter all turn a difference in chemical potential into a voltage, and the conversion is the same in each: fifty-nine millivolts for every factor of ten in concentration, at room temperature, whatever the ion. The voltage a battery holds as it discharges is the chemical potential of the atoms stored in its electrodes — and a battery whose voltage barely changes is one whose electrode is going through a phase transition.

$Hydrogen comes apart far below its binding temperature. The fraction of hydrogen that is ionised in equilibrium, against temperature on a logarithmic axis, for densities of hydrogen nuclei of 10²³, 10¹⁸, 10¹², 10⁶ per cubic metre, from Saha's equation. Hydrogen's binding energy, 13.6 eV, corresponds to 157,803 K. It is half ionised at 13,980 K at 10²³ per cubic metre, 7,235 K at 10¹⁸ per cubic metre, 4,519 K at 10¹² per cubic metre, 3,270 K at 10⁶ per cubic metre — at a temperature where kT is between an eleventh and a fiftieth of the energy needed to free the electron. The thinner the gas, the cooler the temperature at which it comes apart, because a free electron in a thin gas has vastly more room to be in than a bound one.$ Thermodynamics

The electron that leaves because there is room

It takes 13.6 electronvolts to pull the electron off a hydrogen atom, the energy of a particle at a temperature of 158,000 kelvin. Hydrogen is half ionised at the surface of a star at 14,000 kelvin, in a thin nebula at 7,000, and in the early universe at under 4,000 — a fortieth of its binding energy. The electron leaves not because it has enough energy but because a free electron has so much more room to be in than a bound one, and at equilibrium room counts as much as energy. Saha's equation is that trade written down, and it turned out to be the key to what stars are made of.

Waves

The field that turns with nothing turning

A coil fed with alternating current makes a magnetic field that grows, shrinks and reverses along the coil's axis and never leaves it. Put three such coils at 120° to one another and feed them currents a third of a cycle apart, and their fields add to one of constant strength that turns steadily round the centre — a rotating magnet made of nothing that moves. The trick is superposition read backwards: a field pulsing along a line is already two fields turning opposite ways, and the three coils are arranged so that one pair cancels and the other adds. Everything that goes wrong afterwards is a field turning the wrong way.

The deepest water reads warmer than the water above it. A column of seawater 10.9 km deep whose potential temperature — the temperature each parcel would have if brought to the surface without gaining or losing heat — falls from 20 °C at the top to 1 °C in the deep, and the temperature a thermometer lowered into it would read. The two agree near the surface and part company with depth: the thermometer reads a minimum of 1.38 °C near 3.9 km and then rises to 2.35 °C at the bottom, although the water there is the same water, with the same potential temperature, 1.00 °C, as the water two kilometres higher. The warming is compression alone; no heat reaches the deep trench from anywhere.

Fluids

The water that warms on the way down

Lower a thermometer into the Mariana Trench and it reads colder and colder for four kilometres and then starts to warm, reaching about two and a half degrees at the bottom, eleven kilometres down. No heat is coming up from the sea floor to do it. The water at the bottom is the same water as the water two kilometres higher, carried down and squeezed, and squeezing warms it. In fresh water near four degrees the same squeezing does nothing at all, and below four degrees it cools — so a parcel of cold lake water pushed deep enough keeps sinking on its own.

The orbits of an electron in a travelling wave. The possible motions of an electron in a plasma wave whose phase moves at γₚ = 10 and whose field is 0.2 of the cold wave-breaking field, drawn as its energy against its position in the wave over two wavelengths — the level curves of H = γ − βₚ u − ε cos ψ, which the electron's motion conserves. Inside the separatrix (thick) an electron is trapped: it slides back and forth through the wave, gaining energy while it outruns the wave and losing it while the wave outruns it. The separatrix reaches γ = 99 at its top, near 4γₚ²ε = 80, starting from electrons moving at the wave's own speed. Outside it, electrons that are too slow are overtaken by the wave and those that are fast enough outrun it; neither gains energy on average.

Relativity

The wave an electron rides to a gigaelectronvolt

A laser pulse fired into a thin plasma pushes its electrons aside and leaves them oscillating behind it, a wave of charge travelling at nearly the speed of light. The field in that wave is a thousand times stronger than any metal accelerator can hold, and an electron caught in it at the right moment is carried along like a surfer, gaining a gigaelectronvolt in a few centimetres. Then it outruns the wave. The distance it can ride, and the energy it can gain, are both set by how nearly the wave keeps pace with it — which is to say, by relativity.

Thermodynamics

The speed limit that does not care how big the molecules are

Two molecules in water cannot react faster than they can find each other, and how fast they find each other is set by diffusion. A bigger molecule is an easier target, so it seems a reaction between big molecules should be faster. It is not: a molecule twice as big is twice as easy to hit and wanders half as fast, and the two cancel exactly. Every reaction that happens at the first touch runs at the same rate, about seven billion per molar per second in water, whether the partners are ions or proteins — a speed limit written by the viscosity of water and the temperature, and nothing else.

$Hydrogen ionised by heat at one end and by crowding at the other. The ionised fraction of hydrogen against the density of nuclei, at 5,000 K, 10,000 K, 20,000 K, from Saha's equation with the binding energy lowered by the neighbouring charges (solid) and without the lowering (dashed). Thin gas is ionised by the room a free electron gains; compression takes the room away and the gas recombines, as the dashed curves continue to do for ever. The solid curves turn round: at 5,000 K the gas is least ionised, 2.5·10⁻⁴ per cent, near 10²⁶ per cubic metre; at 10,000 K the gas is least ionised, 0.41 per cent, near 10²⁷ per cubic metre; at 20,000 K the gas is least ionised, 12 per cent, near 10²⁸ per cubic metre, and by 6·10²⁸ per cubic metre — 0.10 g/cm³ — every curve is fully ionised, because the ground state no longer exists. That density does not depend on the temperature at all.$ Thermodynamics

The pressure that ionises

Squeeze a hot gas of hydrogen and Saha's equation says it recombines: a free electron has less room to be in, and room was what paid for its freedom. Squeeze it far enough and it comes apart again, completely, at any temperature — not because the room has grown but because the atom has nowhere left to be. Crowded to within a few atomic radii of each other, hydrogen atoms have no bound state, and their electrons, freed by pressure alone, form a metal. Deep inside Jupiter most of the planet's hydrogen is in that state.

Waves

The clock that is pulled into step

In 1665, ill in bed, Christiaan Huygens noticed that two of his pendulum clocks hanging from the same beam were swinging in exact opposition, and that however he disturbed them they returned to it within half an hour. A clock is not a passive resonator waiting to be pushed; it keeps its own time, by its own mechanism. What a weak push can do to it is not change how strongly it swings but pull its phase — and within a range of frequencies set by the push's strength, pull it all the way into step. The same arithmetic locks lasers, pacemaker cells, power stations and the body's daily clock.

$The proton fraction a cold star settles at. The fraction of nucleons that are protons in cold matter in beta equilibrium, where the neutron's chemical potential equals the proton's plus the electron's, against density on logarithmic axes, for free neutrons, protons and electrons with no nuclear forces. Below 1.22·10⁷ g/cm³ the electrons cannot pay the 1.29 MeV difference between a neutron and a proton and the matter is all protons. Above it the fraction collapses, to 0.014 per cent at its lowest near 7.9·10¹¹ g/cm³, and at nuclear density, 2.66·10¹⁴ g/cm³, it is 0.49 per cent. At higher densities it climbs slowly towards one ninth, 11 per cent, which it never reaches. The dashed part lies where real matter is made of nuclei rather than free nucleons.$ Thermodynamics

The protons a star cannot afford

A free neutron decays in fifteen minutes into a proton, an electron and an antineutrino. Inside a neutron star it cannot, and the star is made of neutrons because of it. Beta decay is a reaction like any other, and at equilibrium the neutron's chemical potential must equal the proton's plus the electron's. In a crowded star an electron can only be added at the top of a sea tens of megaelectronvolts deep. So the matter settles where protons are rare: half a per cent of the nucleons at nuclear density, if nucleons were free, and never more than one in nine at any density.

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

Fluids

The pressure that stops counting molecules

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

$A seabed that reflects one wavelength. The fraction of a surface wave's amplitude reflected by a patch of sinusoidal sandbars 50 m apart and 0.8 m in amplitude, in 4 m of water, against the wavelength of the incoming wave, for 4 bars and 8 bars, from the shallow-water equations integrated through the patch. 4 bars reflect 56 per cent of the amplitude at 99.8 m; 8 bars reflect 86 per cent of the amplitude at 100.4 m, close to twice the bar spacing, where each bar's small reflection returns in step with the others. Away from that wavelength the reflections cancel, and the more bars there are, the narrower the band they reflect.$ Waves

The sandbars that reflect the sea

A stack of alternating layers reflects the one colour whose half-wavelength matches the repeat. A row of sandbars on a gently sloping seabed is the same arrangement for ocean waves, and it does the same thing. Waves twice as long as the bar spacing are sent back out to sea, one small reflection per bar, all adding in step. A patch of eight bars can return most of the swell that meets it. The reflected wave, added to the incoming one, makes a standing pattern whose nodes are spaced exactly like the bars. Sand gathers in that pattern, so a field of bars can build its own next bar.

Sound cones that tip over. Time against position for sound in a fluid that speeds up from 0.55 to 1.3 times the speed of sound, drawn as the cone of places a sound made at each point can reach: its edges move at v − c and v + c. Upstream, where the flow is slow, the cones straddle the vertical and sound can go either way. At x = 0.20, where the flow reaches the speed of sound, the upstream edge stands vertical. Beyond it both edges point downstream: sound made there can no longer reach anywhere upstream of where it was made. That is a horizon, drawn in the same way as the light cones of a black hole, and it arises from nothing but the ordinary addition of a sound's speed to the speed of the water that carries it.

Relativity

The river that sound cannot swim up

Sound travels at a fixed speed through the air or water carrying it, so a flowing medium adds its own speed to the sound's: faster downstream, slower upstream. That is the plainest velocity addition there is. Where the flow speeds up past the speed of sound, upstream-going sound is carried backwards, and nothing made beyond that point can be heard before it. The point is a horizon. Sound cones tip over there exactly as light cones do at a black hole, and the rate at which the flow speeds up plays the part of the surface gravity. A kitchen sink under a running tap makes the reverse kind of horizon, a white hole for ripples.

Relativity

The contraction the forces work out for themselves

Relativity usually introduces length contraction as a consequence of how moving observers disagree about simultaneity, not as something that happens to matter. There is a second way to reach the same number, and it is worth knowing because it is the way the idea was first found. Take an atom, accelerate its nucleus gently, and compute what the electron does using only Maxwell's field of a moving charge and the law of motion. Nobody tells the orbit to contract, and it does, by exactly 1/γ, while each turn takes exactly γ times as long. The forces holding matter together work out the contraction for themselves.

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

Electromagnetism

The force between pieces of a current

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

Relativity

The heat hidden in a Doppler shift

An observer who accelerates without stopping sees empty space as warm. The usual derivation takes quantum field theory and several pages. There is a shorter one, and it needs no quanta at all. Watch a single wave of a single frequency from a steadily accelerating rocket: it is redshifted, and because the rocket keeps speeding away, the redshift grows exponentially. Take the spectrum of that fading signal, and it is exactly a Planck spectrum, at a temperature set by the acceleration and nothing else — the same whatever frequency the wave started with. The thermal distribution was hidden in the geometry of the horizon all along.

The melting line of water, through five ices. The pressure at which water freezes, in megapascals on a logarithmic axis, against temperature, along each of the five solid phases that meet the liquid, from the international standard equations. Along ordinary ice, Ih, the melting pressure rises as the temperature falls, from 611 Pa at 273.16 K to 209 MPa at 251.165 K — the backwards slope of the familiar melting line. There the liquid meets ice III, and from then on every ice melts at a higher pressure as the temperature rises: ice V, ice VI, and ice VII, which reaches 20.6 GPa at 715 K. The dots are the four triple points where two ices and the liquid coexist. The lowest temperature at which liquid water can exist in equilibrium, 251.165 K, is at the first of them.

Thermodynamics

The melting line that turns round

Squeeze water and it freezes at a lower temperature — the backwards slope of its melting line, and the reason ice floats. Keep squeezing and the slope reverses. Past two thousand atmospheres ordinary ice gives way to a denser ice, then another, and from there on every ice melts at a higher temperature under more pressure, exactly as a normal substance's does. At twenty thousand atmospheres water freezes into an ice that is still solid at a hundred degrees Celsius; at two hundred thousand, one still solid at four hundred. The famous anomaly belongs to one ice out of five, and the same relation that explained it explains its end.

$A lever that makes air and water meet. The fraction of sound power crossing from air into water through an ideal lossless transformer that multiplies pressure by a factor n, against n on a logarithmic axis. With no transformer, n = 1, it is 0.11 per cent. It rises as the transformer makes the water look less stiff and air look stiffer, reaching everything at n = 60, and falls again beyond, where the water is made to look softer than air. The middle ear's 22 gives 43 per cent in this picture — a gain of 26 dB over no transformer at all. The curve is flat near its top: a lever within a factor of two of the ideal still passes about two-thirds of the power, which is why a mechanism need not be precise to be effective.$ Waves

The lever that lets air speak to water

Sound in air striking a surface of water is almost entirely reflected: 99.9 per cent bounces off, which is why a swimmer underwater hears the world above as a muffled murmur. The inner ear is filled with fluid, and the eardrum faces air, so every land animal has the problem of getting sound across that boundary. The answer is three tiny bones and a difference in area, which together multiply the pressure about twenty-fold and turn a thousandth of the power into nearly half. The middle ear is an impedance transformer, the mechanical twin of the one in every radio, and it works best over just the band where speech carries its meaning.

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

The wobble that radiating makes larger

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

$Only a slow grating sends everything one way. The largest fraction of particles, arriving at Bragg's angle, that a standing light wave can put into the first diffracted order, choosing the wave's depth freely, against how long it acts, in units of ħ divided by the two-kick recoil energy, on a logarithmic axis. A brief pulse acts as a thin phase grating and can put at most 33.9 per cent into that order (dashed), the Raman–Nath limit, spreading the rest over the others. A pulse lasting longer than a few units has time to notice that only one exchange conserves energy, and can put nearly all of it there: 99.9 per cent at the longest time drawn. The same distinction between thin and thick gratings decides whether sound in a crystal diffracts light into many orders or one.$ Quantum

The grating made of light

A diffraction grating is ordinarily made of matter and diffracts light. In 1933 Pyotr Kapitza and Paul Dirac proposed the opposite: a standing wave of light, whose bright and dark bands repeat every half wavelength, should diffract a beam of electrons. It took fifty-three years to see the effect with atoms and sixty-eight to see it with electrons, because a free electron barely notices light. The grating works by momentum: it can only change a particle's motion in whole pairs of photon kicks. A brief flash scatters particles into a spread of orders in proportions fixed by Bessel functions; a slow, gentle one sends every particle one way, which is how laser light became the mirrors and beam splitters of atom interferometers.

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

Mechanics

The ball a turntable will not throw off

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

Five square logs of different densities, as they float. Cross-sections of five long square logs in their stable floating positions, with relative densities 0.1, 0.25, 0.5, 0.75 and 0.9, each drawn with its water line and its submerged part shaded, and a dot at its centre. The lightest floats flat and high; at 0.25 it has tilted to about 27°; at 0.5 it floats on its corner with a diagonal vertical and the water line across the middle; at 0.75 it tilts to the same angle as at 0.25, and at 0.9 it floats flat again, low in the water. The heavy logs are the light ones turned upside down, with air and water exchanged.

Fluids

The log that floats on its corner

A long square beam of very light wood floats flat, one face up, as anyone would expect. A beam of wood half as dense as water does not: it rolls onto a corner and floats with a diagonal vertical, like a diamond. Between the two, at a quarter of water's density, it floats tilted at twenty-seven degrees, neither flat nor on edge. Heavier beams repeat the sequence in reverse, and the whole pattern is symmetric: a beam of density s floats at exactly the angle of one of density 1 − s. None of it depends on anything but a square, a density and Archimedes, and the same arithmetic decides which way an iceberg rolls when it breaks.

$A scan into a layered sample, with a broad source and a narrower one. The envelope of the interference signal against depth in a sample with five reflecting boundaries (marked), at 60, 72, 150, 158 and 260 µm, scanned with a 50 nm source (solid) and a 10 nm source (dashed), both centred at 840 nm. The broad source, with a 6.2 µm resolution, separates every boundary, including the two that are only 8 µm apart. The narrow one, with 31 µm, merges the pairs into single blurred echoes. Every boundary in a real eye or skin sample is a small change of refractive index that reflects a tiny fraction of the light; the scan is a map of where those changes are.$ Optics

The depth a broad spectrum can see

Interference is usually spoiled by a broad spectrum: many colours drift out of step, and fringes wash out a few wavelengths from where two paths are equal. Turned round, that failure is a ruler. Light whose colours stay in step only when two paths match to within a few micrometres can report the depth of every reflecting layer inside something translucent — a retina, a skin, a coat of paint — to that precision, without cutting it open. Optical coherence tomography reads depth off the width of a spectrum, and it has become one of the most used imaging methods in medicine; its resolution is set by how many colours the light contains, and the finest instruments use sources whose spectra span hundreds of nanometres.

The focus that gets sharper when the wave is scattered. The refocused amplitude against distance across the source, in wavelengths, for the same one-sided mirror and pulse, with the space between empty (dashed) and with a slab of 200 lossless point scatterers, 15 wavelengths thick, filling the space between source and mirror (solid). Through the slab the wave reaching the mirror is a jumble — scattered many times, arriving from every direction the slab can send it — yet time-reversed and sent back through the same slab, it refocuses on the source, in a spot 1.5 wavelengths wide against 4.8 in empty space. The scatterers, bouncing waves in from a wide range of angles, act as a lens much wider than the mirror: disorder that would ruin an ordinary lens makes a time-reversal mirror sharper.

Waves

The mirror that sends a wave back to where it began

Record a wave as it arrives at a row of microphones, then play each recording backwards through the loudspeaker beside it, and the sound converges on the place it came from — however twisted the route it took. Huygens said every point on a wavefront acts as a source; a time-reversal mirror takes him literally, making every point of a recorded front re-emit its own history in reverse. The result has a property no lens has. Put a thicket of scatterers between the source and the mirror, so that the arriving wave is an unrecognisable jumble, and the reversed wave still finds its way back — to a sharper focus than it could reach in empty space, because the disorder acts as a lens wider than the mirror.

$Light from one guide, spreading through a row of them. The power in each of a row of identical coupled waveguides after light launched into the central one has travelled 1.5, 3, 5 and 7 coupling lengths (1/C) along the row, drawn as stacked bars with the distance increasing upward: |Jₙ(2Cz)|², where n counts guides from the centre. The light does not spread into a single widening hump as it would in a uniform slab. It forms two beams that run outward at a constant rate, carrying most of the power in their leading edges near n = ±2Cz, and leaves the centre nearly dark: after 7 coupling lengths the central guide holds 2.9 per cent and the brightest guides, at ±12, hold 8.1 per cent each. This is discrete diffraction: the spreading of light that can only step between neighbouring sites.$ Waves

The light that spreads in two beams

Shine light into one of a row of identical optical fibres laid side by side and close enough to leak into each other, and it does not spread out into a widening glow as it would in a slab of glass. It splits into two beams that run outward at a fixed speed, bright at their leading edges, leaving the middle almost dark. The reason is that light hopping between neighbours has a fastest possible sideways speed, and piles up at it. The same periodicity gives the row two tricks no uniform medium has: a direction in which a beam travels without spreading at all, and a direction in which diffraction runs backwards, so that a spread beam can be gathered up again.

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

Electromagnetism

The force that forgets Planck's constant

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

Waves

The packet that bends with nothing pushing it

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

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