Illustrated Physics
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What happens at the edge — part 2 of 2

Quantities that live on a surface rather than in a volume, and conditions imposed at an edge that decide what the whole interior may do — a skin that costs energy per unit area, charge that sits entirely on the outside of a conductor, a clamped end that permits some frequencies and forbids the rest, a wall that quantises a box, and the interface where light has to change direction.
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.

 The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly. Fluids

The film that goes black before it bursts

A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

 Three terms, and one distance. The three terms of an oscillating dipole's electric field against distance, in units of a reciprocal wavenumber, both logarithmic. They fall as 1/u³, 1/u² and 1/u, so they are all equal at u = 1 — a distance of λ/2π, which is a property of the frequency and of nothing about the antenna. The heavy curve is the field the three actually add up to, and it is not their sum: the static and radiation terms are in antiphase and partly cancel, which is why the total dips below every one of them just inside the crossing before settling onto the 1/u the far field is made of. Electromagnetism

The distance where a field changes its mind

An oscillating source has three fields around it, falling as the inverse cube, the inverse square and the inverse first power of distance. They are all equal at one radius, and that radius is the wavelength over 2π — a number containing nothing about the source at all. Inside it a source mostly stores energy; outside it, mostly loses it, and the two behaviours are different technologies rather than different strengths.

 A reflection that keeps the angle to gravity and not to the wall. An internal wave beam of frequency 0.5N reflecting from a slope of 12°, with the wavelength ratio for slopes of 12°, 20°, 28° computed beside it. The frequency fixes the angle the energy makes with the horizontal — 30.0° here — because the restoring force is gravity and gravity is vertical, so the reflected beam must leave at that same angle whatever the wall is doing. Incident and reflected rays are therefore not mirror images, and the wavelength changes on reflection by sin(θ+α)/sin(θ−α). A flat floor gives one, checked exactly. A slope approaching the ray's own angle gives infinity, checked as the limit, and that is where a basin's internal tide is compressed until it breaks. Fluids

The reflection that changes the wavelength

An internal wave's frequency fixes the angle its energy makes with gravity, so a sloping wall cannot send it back the way a mirror would. The reflected beam leaves at the same angle to the vertical rather than the same angle to the wall, its wavelength changes by a factor that diverges when the slope matches the ray, and in a closed basin the changes accumulate until every ray in the fluid lies on one line.

 A wave with two directions in it. Light at 60° entering silver, 550 nm, whose index is 0.055 + 3.32i. Phase matching along the boundary fixes the transmitted wave's tangential wavenumber and leaves the normal one to the medium, which supplies a complex answer. The real part decides where the phase goes and the imaginary part where the amplitude does, and they are not the same direction: the surfaces of constant amplitude are parallel to the interface, because the decay is entirely into the metal, while the surfaces of constant phase are tilted by 86.5° from the normal. There is therefore no single refracted angle to quote. The familiar picture, in which one set of parallel planes carries both, requires the absorption to be exactly zero. Optics

The angle that is two angles

Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

 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.

 The electrode that walks itself negative. An electrode driven through a blocking capacitor by a symmetric waveform of amplitude 20 kTe, in an argon-like plasma whose electrons are collected 108 times as readily as its ions at saturation, starting uncharged. Potentials are relative to the plasma, in electron temperatures. On the first positive half-cycle the electrode draws a flood of electrons and nothing on the negative half-cycle can return the charge, because the ions arrive at a fixed, small rate; the electrode walks down, and within 6 cycles its mean potential is within half an electron temperature of the self-bias. That bias, found by integrating the charging, is −22.27 kTe, and the charge balance with a Bessel function gives −22.27. The floating potential of the same wall undriven is −4.68. At 3 eV, a 60 V drive makes a 67 V bias with no direct current supplied anywhere. Astrophysics

The bias no battery supplies

Drive an electrode in a plasma through a capacitor with a perfectly symmetric waveform and it charges to a steady negative voltage almost as large as the waveform's own amplitude. No direct current flows anywhere and no battery is connected. The plasma's sheath lets electrons in far more easily than it lets them out, and every semiconductor wafer etched in the last forty years was held at a voltage made that way.

 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.

 The latitude past which the tide cannot shed its energy by halves. Frequency in cycles per day against latitude. The curve is the inertial frequency, 2Ω sin(latitude), below which no internal wave can oscillate; the shaded region above it is where internal waves exist. The horizontal lines are the semidiurnal and diurnal tides and the frequencies half of each, where a parametric instability would put the waves the tide decays into. Each line ends where it meets the curve, which is its critical latitude: M2, semidiurnal at 1.932 per day, 74.5°; M2 ÷ 2 at 0.966 per day, 28.8°; K1, diurnal at 1.003 per day, 30.0°; K1 ÷ 2 at 0.501 per day, 14.5°. Equatorward of 28.8° the semidiurnal tide can feed waves at half its frequency; poleward of it those waves cannot exist and that route is closed. The diurnal tide's subharmonic is confined within 14.5° of the equator, and the diurnal tide itself cannot propagate as a free internal wave poleward of 30°. 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.

 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.

 Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating. Waves

Two tails that swap everything

Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

 A body at an interface, and the difference that holds it. The net upward force on a ten-centimetre cube straddling the boundary between two fluids, against how far its underside sits below that boundary — each curve scaled by its own largest value so that three very different cases fit on one axis. The displaced weight has to be counted twice, once with each density, so the force is linear in the displacement with a stiffness set by gravity, the cube's cross-section and the difference of the two densities — the difference, and neither of them alone. air over water holds the cube with its underside 60 per cent of the way through, at a stiffness of 97.8 newtons per metre; oil over water holds the cube with its underside 47 per cent of the way through, at a stiffness of 14.5 newtons per metre; water over mercury holds the cube with its underside 24 per cent of the way through, at a stiffness of 1229.4 newtons per metre. Both numbers are read off the drawn curve rather than substituted. The consequence is the one worth the figure: a body at an oil–water interface is held six times less stiffly than the same body at an air–water one, because the density difference is six times smaller, and the ordinary intuition that a denser fluid holds a body more firmly is exactly wrong — what matters is the contrast across the surface the body is sitting in. Fluids

The body that displaces two things

Every earlier argument has a body wetted by one fluid, so the displaced weight is a volume times a density. A body at an interface displaces two, and what holds it is the *difference* between them — so the same block is held six times less stiffly at an oil–water boundary than at an air–water one. In a continuously stratified column the neutral depth becomes stable, which is the exact opposite of the compressible case.

 How large now is, on this planet. By how much a synchronisation carried around a region of the rotating Earth fails to come back to itself, against the size of that region — from a metre to the whole planet, both axes logarithmic. The three horizontal lines are what three kinds of clock can resolve, and where each crosses the curve is where that clock can detect that 'now' is not a global notion: a good wristwatch at 785 thousand km, a quartz oscillator at 25 thousand km, a caesium clock at 785 km. Carried the whole way round the equator the defect is 207 nanoseconds, which is sixty metres of light travel and is the correction every satellite-navigation system applies. The effect is not small and not exotic; it is a routine engineering term, and the reason it was not an engineering term before 1955 is that nothing could measure it. A wristwatch's now is global out past the Moon; a caesium clock's reaches about the width of a large country. Relativity

How big now is

Three earlier arguments have established that a global now is a choice, that part of the choice is convention, and that for a rotating observer no consistent global choice exists at all. What survives is a size. Every observer has a local now, and how local is computable: on the rotating Earth it is 785 kilometres to the nanosecond, and a freely falling frame is inertial over the tolerance times the distance to the centre, divided by two.

 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.

 A thrust that cannot push past the hump. The resistance a hull meets against its speed in knots, in two layers whose fastest interfacial wave travels at 1.02 knots: ordinary friction rising as the square of the speed, plus the drag of the interfacial waves, whose hump sits just below the wave speed. The horizontal lines are 2 steady engine thrusts. A ship settles where its thrust meets the resistance curve. Thrust 0.6 meets it at 0.81 knots; Thrust 1.3 meets it at 0.94, 1.00, 1.82 knots. A ship accelerating from rest reaches the first crossing and stops gaining speed there, below the hump, even when a faster crossing exists beyond it — which is the dead water sailors reported, a ship held to a fraction of its usual speed by a wave it cannot see. Fluids

The wave that holds a ship back

In 1893 the polar ship Fram, which could make four or five knots, was held to about one in a calm Arctic sea with nothing visible in the water. The sea was layered — a metre or two of fresh meltwater over salt — and the ship was making a wave on the boundary between the layers, a wave that travels at about a knot and carries away almost all of a slow ship's power. Below that speed the drag is a hump no steady thrust can climb; above it the wave cannot keep up and the drag falls away.

 The angle at which a total reflection stops being total. Reflectance of light at 633 nm inside a BK7 prism whose face carries a 50 nm gold film with water beyond it, against the angle of incidence, for the two polarisations, from the three-layer Fresnel sum in complex arithmetic. Past the critical angle for glass and water, 61.62°, s-polarised light is reflected almost completely, as total internal reflection requires, apart from what the thin gold absorbs. p-polarised light is reflected too — except near one angle, where the reflectance falls to 0.014 at 71.52°. The dispersion relation of a surface plasmon on gold against water puts its resonance at 71.67°; the difference is the film's finite thickness and the prism on its other side pulling the resonance slightly. Optics

The angle at which total reflection stops being total

Coat the back face of a glass prism with fifty nanometres of gold and total internal reflection still returns almost all the light — except at one angle, where the reflection of one polarisation falls nearly to zero. The light has gone into an electron wave bound to the far surface of the gold, and because that wave's angle depends on whatever touches the gold, the missing reflection has become one of the most sensitive balances in biochemistry.

 The thickness fixes the area, and the design only moves it. How much a metal-backed resistive sheet 7.5 mm from the metal absorbs, measured as ln(1/|Γ|), against wavelength, for sheets of 150, 250, 377, 800 Ω per square. Rozanov's bound says the area under any such curve, over all wavelengths, can be at most 2π² times the thickness: 147.9 mm here. The areas computed are 147.9 mm for 150 Ω, 147.9 mm for 250 Ω, 147.9 mm for 377 Ω, 69.7 mm for 800 Ω. Every sheet up to the impedance of free space meets the bound exactly and only moves its area between wavelengths; the more resistive sheet wastes part of it. Waves

The absorber whose bandwidth is its thickness

A metal plate reflects everything, and a surface that must swallow a radar wave instead has to be a match between free space and a short circuit. A resistive sheet a quarter-wave in front of the metal does it perfectly at one frequency. Widening the band takes more sheets and more depth, and a theorem says why that is not a failure of design: for any absorber backed by metal, the total absorption over all wavelengths is fixed by its thickness, and by nothing else except its permeability.

 Stopping down removes one colour error and not the other. The two chromatic errors of a single 50 mm N-BK7 lens with its stop 20 mm in front, at an image point 10° off axis, against the f-number. Axial colour — blue and red focusing at different distances, seen as a coloured blur at the yellow focus — is about 775/N µm across: 387 µm at f/2, 97 at f/8, 35 at f/22. Lateral colour — blue and red images of the point centred at different heights — is 55.0 µm at every f-number, because it is carried by the ray through the centre of the stop. The two are equal near f/14, and past that stopping down leaves only the error it cannot touch. Optics

The colour fringe no aperture can close

A lens bends blue light more than red, and that one fact produces two different faults. One puts the blue focus nearer the lens than the red, and closing the aperture shrinks it. The other puts the blue image of an off-axis point nearer the centre of the frame than the red one, and closing the aperture does nothing at all, because it is carried by the one ray every aperture keeps. It is why photographs have coloured fringes in their corners and not in their middles, and why a lens designer who cannot remove it with glass can sometimes remove it with symmetry, or with a distance.

 Three conditions and the window they leave. For a ray entering a diamond's table straight down, the angle from the normal at which it meets the first pavilion face, the second pavilion face and the top again, against the pavilion angle; diamond's critical angle, 24.4°, is the dashed line. The first two must lie above it, so that the ray is totally reflected; the third must lie below it, so that the ray gets out. The first holds for every pavilion angle drawn. The second fails above 51.9°, where the ray would leak out of the far side of the pavilion. The third fails outside 38.9° to 51.1°, and it is the one that decides. Together they leave a window from 38.9° to 51.1°, and Tolkowsky's 40.75° sits inside it, towards the lower edge. Optics

The window a diamond is cut inside

A well-cut diamond sends back out of its top most of the light that falls on it, and it does so without a single silvered surface. Light entering the top is totally reflected at one sloping face of the pavilion below, again at the opposite face, and then has to leave through the top — where it must not be totally reflected. Each of those three is a condition on one angle, and together they leave a window a dozen degrees wide inside which a stone looks bright and outside which it looks glassy. The window is wide for diamond and nearly closed for glass, which is why a cutter's skill cannot make glass sparkle.

 The melting curve that turns back, then lies flat. The pressure at which liquid and solid helium-3 coexist, against temperature from 0.9 millikelvin to 1 kelvin on a logarithmic axis, from the international provisional low-temperature scale PLTS-2000, which is defined by this curve. Above 315.2 mK the curve behaves like any other: a higher temperature needs a higher pressure to freeze the liquid. Below it the curve turns back — at its minimum, 2.93113 MPa, it takes a higher pressure to freeze colder helium — and it rises to 3.4391 MPa at 1 mK, where it lies almost flat, as the third law requires of every coexistence line. The kinks that mark the liquid becoming superfluid and the solid's spins ordering sit on that flat end, at 2.444, 1.896 and 0.902 mK. Thermodynamics

The melting curve that has to arrive flat

The slope of every line on which two phases coexist is their difference in entropy divided by their difference in volume. The third law says that entropy differences vanish at absolute zero, so every such line — melting, boiling, a superconductor's critical field — must reach the bottom of the temperature scale lying flat. Helium-3 reaches it in the strangest way available: below a third of a kelvin its solid is more disordered than its liquid, the melting curve runs backwards, and squeezing the liquid into solid absorbs heat. That backwards curve was the refrigerator in which superfluid helium-3 was found, and it is now the definition of temperature below one kelvin.

 The pull of a meniscus against the stiffness of a wall. For two neighbouring lines of a wet pattern leaning towards each other, the net Laplace pressure pulling them together (heavy curve) and the bending stiffness pushing them back (straight lines), against the lean as a fraction of the half-gap. The pull grows faster than linearly, because narrowing a gap raises its suction, and it diverges as the gap closes. The stiffness is a straight line whose slope falls as the fourth power of the line's height; the lines drawn are for 0.8 times, 1 times, 1.25 times the critical height. At 0.8 of the critical height the straight state is stable and an unstable balance lies at a lean of 0.77 of the half-gap, a barrier the lines must be pushed over; at the critical height the two start with the same slope and the straight state is marginal; at 1.25 times the straight state is unstable and the pair collapses. Fluids

The walls a rinse pulls together

Every result about capillary rise assumes the walls hold still. Make them thin enough and they do not. The water between two walls is below atmospheric pressure, so it pulls them together, and as the gap narrows the suction grows. Beyond a critical height or length the walls cannot push back and they snap shut. It is why wet hair clumps, why a paintbrush comes out of water as a point, and why a chip factory cannot rinse lines of resist taller than about three times their width without laying them flat.

 A sheet of light a tenth of a micrometre thick. The intensity of the evanescent field in water beyond a glass coverslip (index 1.518 against 1.33), against distance from the glass, for 488 nm light totally reflected at 62°, 65°, 70°, 75°, s-polarised, relative to the light arriving in the glass. The critical angle is 61.2°. At 62° the field at the surface is 3.79 times the incident intensity and falls by e in 234 nm; at 65° the field at the surface is 3.07 times the incident intensity and falls by e in 110 nm; at 70° the field at the surface is 2.01 times the incident intensity and falls by e in 75 nm; at 75° the field at the surface is 1.15 times the incident intensity and falls by e in 63 nm. Ordinary illumination (dashed) lights the whole depth of a cell equally; totally reflected light lights only the layer against the glass, which is where a cell's membrane and the proteins in it are. Optics

The mirror that lights a tenth of a micrometre

Total internal reflection sends back every photon, and it leaves behind a field in the medium beyond that dies away within a fraction of a wavelength. Put a living cell on a coverslip and reflect a laser inside the glass, and that field lights only the hundred nanometres of the cell pressed against the glass: the membrane, and the proteins moving in it. Nothing else glows. The same field works in reverse. A molecule close to the glass sends part of its own light into angles no ray from the water could reach, and how much it sends there says how close it is.

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

The barrier the metal cannot choose

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

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

The handedness a mirror cannot keep

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

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

The index that falls below one

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

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

The heat that crosses a gap too narrow for light

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

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

The sand that darkens when it is wet

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

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

The boiling curve that turns back

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

 Rays at a frequency the ionosphere cannot stop head-on. Ray paths at 14 MHz launched at angles from 10° to 75° from the vertical into a Chapman layer peaking at 300 km with a critical frequency of 8 MHz, over a flat Earth. A wave sent straight up at 14 MHz passes through: it is above the layer's plasma frequency everywhere. A ray at φ₀ from the vertical turns back where the plasma frequency equals f cos φ₀ — the secant law — so only rays more oblique than arccos(8/14) = 55.2° return. Those that do land beyond about 1133 km; nothing comes down closer than that: the skip zone, a ring of silence round a transmitter that is audible much further away. Vertical scale exaggerated about six times. Astrophysics

The frequency the sky returns only at a slant

The ionosphere turns back any radio wave below its plasma frequency, about eight megahertz on a summer afternoon, and lets anything above it out into space — if the wave is sent straight up. Send a fourteen-megahertz wave up at a slant and it comes back down, a thousand kilometres away. Only the part of the wave's motion across the layer has to be stopped, and that part is the frequency times the cosine of the angle. The rule turns one frequency into a range of them, sets the highest frequency that can reach any given distance, and leaves a ring of silence round every short-wave transmitter, inside which it cannot be heard at all.

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

The cereal that gathers at the edge of the bowl

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

 The voltage a gas gap can hold, on one curve. The breakdown voltage of a gap in air between flat electrodes against the product of pressure and gap width, on logarithmic scales — Paschen's law, in which pressure and gap enter only as their product because both count how many collisions an electron makes on the way across. Townsend's model, with γ = 0.01 electrons freed per ion at the cathode, has its minimum of 305 V at 1.12 Pa·m — a gap of 11 μm at atmospheric pressure, or 112 Pa across a centimetre. Measured minima in air are about 330 V. To the left the gap is too short, or the gas too thin, for an electron to make enough collisions, and the voltage climbs steeply; below about 0.41 Pa·m there is no gas breakdown at any voltage. To the right, collisions are so frequent that an electron gains too little energy between them, and the voltage rises nearly in proportion to pd. Dashed, the empirical law for air at larger pd: 31.4 kV for a centimetre at one atmosphere, against the model's 35.5. Electromagnetism

The spark that needs room to start

Air is an excellent insulator until it is not, and the usual statement of when — three megavolts a metre — is true only for gaps of about a centimetre. A gap breaks down when one electron's avalanche of ionisation sends back enough ions to the cathode to free one more electron, and that condition depends on the pressure and the gap only through their product. The result is a curve with a floor: about 330 volts in air, below which no gap of any width will spark, and above which both a thinner gas and a denser one insulate better.

 Two stiffnesses, positive together only in a band. The stiffness of the trap for a field-following magnet above a ring base, against height in units of the ring's radius: vertically, proportional to the curvature of the field strength along the axis; sideways, proportional to the curvature of |B| across it, B′²/4B − B″/2. Below half a radius the field's fall is still steepening, the vertical stiffness is negative, and a magnet nudged downward keeps going. Above √(2/5) = 0.632 radii the field strength no longer rises away from the axis, the sideways stiffness is negative, and a magnet drifts off to the side. Only between 0.500 and 0.632 radii are both positive — 6.6 mm of height for a 10 cm ring. For the nominal top, floating at 0.565, the trap's natural frequencies are 1.36 Hz vertically and 0.82 Hz sideways. Electromagnetism

The window a spinning magnet floats in

No arrangement of fixed magnets can hold another magnet still in mid-air, and a spinning top made of a magnet floats above a magnetised base for minutes. The spin does not cancel the theorem; it changes what the top's energy depends on, from one component of the field to the field's strength, which can have a minimum. But the minimum exists only in a band of height a few millimetres thick, for a weight right to a couple of per cent, at a spin neither too slow nor too fast, and the well holding the top is about as deep as the energy of a twenty-gram weight dropped fifty micrometres.

 The speed of sound in dry sand rises with the load on it. The speeds of compressional and shear waves in a dry random packing of quartz grains, porosity 0.36, against the pressure on it, from the Hertz–Mindlin effective-medium theory, for coordination numbers from 6 to 9 (the bands). The pressure axis is logarithmic. Both speeds rise as the sixth root of the pressure. At 1 kPa — the load about six centimetres down in dry sand — compressional waves travel at 316–362 m/s; at 100 kPa, about six metres down, 681–780 m/s; at 10 MPa, 1468–1680. The dashed lines are the speed of sound in air, 343 m/s, and in water, 1,480 m/s. Near the surface dry sand carries sound no faster than air; measured shallow speeds are lower still, by up to a factor of two, because real contacts are fewer and weaker than the theory assumes. Fluids

The sound sand carries faster the deeper it goes

A solid carries sound at a speed set by what it is made of. A heap of quartz grains is made of quartz, and near its surface it carries sound more slowly than the air above it. The grains are not the solid; their contacts are, and a contact between two spheres grows stiffer the harder it is pressed. So the speed of sound in dry sand rises as the sixth root of the load on it — from about air's speed a few centimetres down to twice that at six metres — the paths of sound in a beach curve back up to the surface, and a water table is a mirror to one kind of wave and almost invisible to the other.

 The share of the reflection that stays a mirror. The fraction of the reflected power that goes into the mirror direction rather than being scattered, which falls exponentially with the square of the phase roughness g = 4πσ sin ψ/λ, against the grazing angle on a logarithmic scale, for surfaces whose height varies with a standard deviation σ of 0.05, 0.5 and 5 wavelengths. A surface rough by a twentieth of a wavelength is a good mirror at every angle: 67 per cent at normal incidence. One rough by half a wavelength keeps only 7·10⁻¹⁸ at normal incidence but 74 per cent at 5°. One rough by five wavelengths reflects nothing specularly until the angle is below about 0.9°, where g = 1, and is a mirror below a few tenths of a degree. The angle at which a surface turns from matt to glossy scales as the wavelength over the roughness; no surface is too rough to be a mirror for light that grazes it closely enough. Optics

The rough surface that is a mirror at a glance

A sheet of office paper is matt from above and shines when sighted along its surface. A choppy sea scatters a radar beam from overhead and returns it like a mirror near the horizon. An X-ray mirror polished to a third of a nanometre is still rough to X-rays except at angles below a degree. All three are one rule: two rays reflected from points at different heights differ in path by twice the height times the sine of the grazing angle, so roughness is measured in wavelengths times that sine — and every surface, however rough, is a mirror for light that skims it closely enough.

 A beaker that empties at a steady rate. The height of liquid helium inside a beaker 1 cm across, held 1 cm above the level of the bath it stands in, against time, as the film carries liquid up the inner wall, over the rim and down into the bath. The film carries a fixed volume per centimetre of rim — 7.5·10⁻⁵ cm³/s at 1.4 K — whatever the difference of levels, so the level falls in a straight line, 180 μm a minute, and the beaker is level with the bath in 56 minutes. At 1.9 K, with less superfluid, it takes 95 minutes; at 2.1 K, 279. Dashed, a flow driven by the difference of levels, like a siphon, starting at the same speed: it slows exponentially as the difference shrinks and never quite finishes. The film's straight line is the signature of a flow set by a critical speed, not by the push. Quantum

The film whose thinnest point sets the flow

Leave a beaker of superfluid helium standing in a bath and it empties itself, the liquid creeping up the inside wall as a film eighty atoms thick, over the rim and down the outside. The film itself is nothing special: ordinary liquid helium coats the wall just as thickly. What is special is that it flows without friction, and so its flow is set not by how hard it is pushed but by how fast it can go. The level falls in a straight line rather than slowing as it nears the bath's, and the whole flow is decided at one place, the rim, where the film is thinnest and has to run fastest.

 What a station hears from one thunderstorm region. The power spectrum of the vertical field from a single region of thunderstorms, received 20° and 90° of arc away, from a sum of the first seven resonant patterns, each with the measured peak frequency and a quality factor of 4 to 6, both taken as inputs. Measured spectra fall off with frequency rather more than this one, because real strokes and the real ionosphere are not as simple. The resonances are so broad that neighbouring peaks run into each other; a quality factor of five means a peak about a fifth as wide as its frequency, and the ringing dies in a few cycles. At 20° every tone is present, the higher ones merging. At 90° the first, third and fifth have vanished, because those patterns have a node a quarter of the world away, and the spectrum is made of the even tones alone. Real stations receive the sum of the world's storms, which move with the time of day, and the heights of the peaks drift with them. Waves

The tone the whole Earth rings at

The ground conducts and so does the ionosphere, eighty kilometres up, and between them lies a waveguide the size of the planet. Two walls, unlike a pipe, carry waves of any length, so the guide takes radio waves forty thousand kilometres long — longer than the world is round — and, closed on itself, it rings. Lightning strikes fifty times a second and keeps it ringing at about 7.8, 14.3 and 20.8 hertz. Perfect walls would put the tones at 10.6, 18.4 and 25.9; the difference is the ionosphere, a ceiling that leaks.

 A drop in a tapered tube. A section through a glass tube tapering from 1.2 mm to 0.2 mm in radius over 60 mm, holding a 2 μL plug of water (contact angle 20°), not to scale across. Both menisci curve into the empty tube, so the water just inside each is below atmospheric pressure — by 269 Pa at the narrow end, radius 0.50 mm, and by 252 Pa at the wide end, radius 0.54 mm. The pressure is lower at the narrow end, so the water flows towards it: a net push of 17 Pa, with no gravity, no pump and no motion of the tube. Fluids

The drop that runs to the narrow end

Put a drop of water into a glass tube that narrows towards one end and it moves, on its own, towards the narrow end — horizontally, with nothing pushing it. Francis Hauksbee reported it to the Royal Society in 1712, and the explanation is two curved surfaces pulling with different strengths: each meniscus lowers the pressure of the water behind it by an amount that grows as the tube narrows, so the end in the narrower part sucks harder. A drop of mercury, which does not wet glass, goes the other way. Which way a drop goes depends on the cosine of one angle, and, for a short drop, on its length.

 How far below the air a surface facing the sky falls. How much colder than the air a surface facing the night sky becomes, when insulated from below and with air at 5 °C, against how strongly the air can warm it, measured by the convection coefficient (3 to 5 W/m²K in still air, 20 or more in a breeze), for sky emissivities of 0.72, 0.82, 0.97 — clear and dry, clear and humid, and overcast. Under a clear dry sky in still air the surface sits 12.3 °C below the air; a breeze of 20 W/m²K halves that and more; under cloud the drop is 1.3 °C. The surface is radiating to something colder than the air: the upper atmosphere and, through the window near ten micrometres, space. Thermodynamics

The cold reservoir overhead

On a clear, still night a car roof can grow frost while the air around it is several degrees above freezing. Nothing in the air is cold enough to freeze water. The roof is radiating its heat upward to a sky whose radiation back is that of something twenty degrees colder than the air — part of it the cold upper atmosphere, part of it space itself, seen through a window in the infrared where the air is transparent. The clear sky is a cold reservoir available everywhere, and it can be used: to make frost, to cool buildings below the air temperature even in sunshine, and in principle to run an engine at night.

 The width a self-focusing beam is forced to. The root-mean-square width of a collimated Gaussian beam in a Kerr medium, relative to its starting width, against distance in Rayleigh ranges, at powers of 0, 0.5, 1, 1.5 and 3 times the Gaussian threshold, the wavelength squared over 2π n0 n2. This curve is exact, not a model: the mean square width of any beam obeyed by the two-dimensional cubic wave equation is a parabola in distance, its coefficient set by one minus the power over the threshold, whatever happens to its shape. At low power the beam spreads by diffraction. At exactly the threshold its width never changes. Above it the parabola reaches zero at a finite distance — 1.41 Rayleigh ranges at 1.5 times, 0.71 at three times — which a width cannot do, so the beam must have collapsed to a point at or before it. The equation stops being true there; the glass usually breaks first. Optics

The beam that focuses itself

In glass the refractive index rises very slightly with the intensity of the light, so a beam that is brightest in the middle writes a converging lens into the glass it is crossing. Diffraction spreads it at the same time. Both effects grow in the same proportion as the beam is narrowed, so neither can win by a change of size — only by a change of power. Above a critical power, 4.3 megawatts in fused silica, the beam is forced to collapse within a finite distance, and every high-power laser is built around not letting that happen where it is not wanted.

 Two ways to spin, against angular momentum. The spin Ω²/πGρ of the Maclaurin spheroids (blue) and the Jacobi ellipsoids (red) against their angular momentum, in units of √(GM³R) with R the radius of a sphere of the same volume. The two families meet at the bifurcation, angular momentum 0.304. Below it only spheroids exist. Above it a body of given angular momentum could be either, and the two differ: the ellipsoid spins more slowly, its spin falling as its angular momentum grows, because it gets longer and its moment of inertia climbs faster than its angular momentum. Every Maclaurin spheroid past the bifurcation is a shape a slightly viscous body will leave. Astrophysics

The spinning fluid that becomes a cigar

A ball of fluid held together by its own gravity flattens when it spins, as the Earth bulges at the equator. Spin it faster and it flattens further, into a lens — but only up to a point, and long before that point something unexpected happens. In 1834 Jacobi found that a fast-spinning fluid can also take a shape that is not symmetric about its axis at all: an ellipsoid with three different axes, stretched along its equator, tumbling end over end. Past a definite spin the stretched shape has less energy than the flattened one, and a body that can shed energy will choose it.

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

The radio that only heavier electrons can make

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

 How a layer of grains moves down a slope. The velocity through a steady layer of 0.5 mm glass beads 20 grains deep flowing down slopes of 22°, 25° and 28°, against height above the rough base in grain diameters; the dashed curve is a Newtonian liquid's parabola, scaled to the 25° surface speed. Because every depth must have friction tan θ, every depth has the same inertial number, and the shear rate grows with the square root of the pressure above it: Bagnold's profile, u ∝ h^(3/2) − (h − z)^(3/2). Its surface speeds are 8.0, 41.0, 113.7 cm/s; compared with a liquid's parabola the shear is spread more evenly through the depth, because the shear rate falls off only as the square root of the depth below the surface rather than in proportion to it. Fluids

The sand whose friction depends on how fast it flows

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

 The field round a thin wire. The electric field against distance from the axis, on a logarithmic axis, for a wire inside a coaxial cylinder of radius 10 cm at 60 kV, for wire radii of 0.2, 1 and 5 mm; the dashed line is 30 kV/cm, the field at which air breaks down across a gap of a few centimetres, and each curve starts at its wire's surface. The field falls as 1/ρ, so it is concentrated at a thin wire: 483 kV/cm at the surface of the 0.2 mm wire, 130 at the 1 mm wire and 40.1 at the 5 mm one. The field exceeds 30 kV/cm only within 3.2 mm of the axis for the thinnest wire, 4.3 mm for the middle one and 6.7 mm for the thickest; across the remaining nine centimetres the air is far below breakdown, so ionisation can start at the wire and has nowhere to go. Electromagnetism

The glow a thin wire wears before it sparks

Raise the voltage on a thin wire inside a metal tube and, long before anything sparks across, the wire starts to glow faintly violet and to hiss. The air right next to it is ionising, and the air a millimetre further out is not. The field round a wire falls away so fast that an avalanche of ionisation can start at the surface and find nowhere to go. That partial breakdown, corona, costs power lines energy in the rain, sets the size of the wires they are made from, cleans the smoke from chimneys and counts particles in detectors — and the field it needs is not a property of air but of the wire's radius.

 The innermost level dives towards the sea. The energy of an atom's innermost (1s) electron, in units of the electron's rest energy mc², against the nuclear charge Z: dashed, Dirac's formula for a point nucleus, √(1 − (Zα)²); solid, the same equation solved for a nucleus of realistic size. Above +mc² lie the free electrons; below −mc², the filled sea of negative-energy states that Dirac's theory requires. For a point nucleus the level stops at Z = 137, where the formula fails. A real nucleus keeps it: it crosses zero near Z = 148 and reaches −mc² at Z ≈ 170, binding the electron by twice its own rest energy. Beyond that the level has nowhere left to go but into the sea. Quantum

The nucleus that would make positrons from nothing

Dirac's equation gives the innermost electron of an atom an energy that falls as the nuclear charge rises, and at 137 protons the formula fails outright. Real nuclei have a size, and with that correction the level does not stop: it falls through zero and on towards minus the electron's rest energy, the edge of the sea of states that Dirac's theory says are already full. It gets there at about 170 protons. Past that point an empty innermost level could be filled from the sea, and the vacuum round the nucleus would emit positrons of its own accord. No nucleus that large exists — but two uranium nuclei, for an instant, come close enough to be one.

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

The atom a light wave opens like a wall

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

 An ice sheet resting on a bed below the sea. A section along the flow of an ice sheet on the MISMIP benchmark bed, which lies below sea level beyond 480 km from the divide and deepens inland between 974 and 1265 km. Dashed: the height the ice surface would stand at if the ice were just thick enough to touch the bed — 1.11 times the water's depth thick, a ninth of it above the sea. At the rate factor drawn, the balance of snowfall against export has three solutions: grounding lines at 800, 1124, 1376 km. The outer two are stable; the middle one, on the stretch where the bed deepens inland, is not. The ice drawn — its surface a plastic profile, a sketch rather than a solution — rests on the bed out to 1376 km, where it must be 803 m thick to touch it, and floats beyond as a shelf. Fluids

The grounding line with nowhere to stop

Where an ice sheet flowing into the sea lifts off its bed and begins to float, buoyancy fixes how thick it is: just thick enough to touch the rock, about 1.1 times the depth of the water. The ice crossing that line goes as the 19/4 power of the thickness, so twice the depth sends out twenty-seven times the ice. On a bed that deepens inland, a grounding line that retreats a little moves into deeper water, exports more, and retreats further. No steady state can sit there, and an ice sheet whose edge reaches such a slope crosses the whole of it.

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

The salt that dissolves in the air's own water

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

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

The atom whose nucleus is a reflection

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

 The boiling point in a column of water, and the depth heating reaches it first. The temperature at which water boils against depth in a geyser's conduit 2240 m above sea level, where the air's pressure is 77.6 kPa (solid): 92.7 °C at the surface, 103.6 °C at 4 m, 115.5 °C at 10 m and 131.8 °C at 22 m — rising steeply near the top and more gently below, because each extra metre adds the same pressure to water whose boiling point responds less as it gets hotter. Dashed: the water as it heats from beneath, cooled to 80 °C at the top by the pool and below that warming by only 0.8 °C per metre, mixed by convection; 6 °C and 3 °C short of the curve at 4 m, and then touching it. It touches at 4 m, not at the bottom: there the water is still 13.7 °C below boiling. Thermodynamics

The column that boils from the middle

Water at the bottom of a geyser's tube is hotter than water anywhere else in it, and it is not the water that boils first. The weight of the column raises the boiling point with depth faster than the heat from below raises the temperature, so the column reaches its boiling curve partway down, under a cooler surface layer. A little boiling there lifts the water above. Everything beneath is then at a lower pressure than the one it was heated under, so it is superheated and flashes, and the eruption feeds itself: past about fourteen centimetres of water thrown out, each slug of steam makes more room than it was given.

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

The building that breathes through its doors

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

 The sideways shaking that arrives longest waves first. The transverse ground motion 3000 km from a sudden shear source, synthesised from the computed dispersion of the fundamental Love wave in a 35 km crust, with periods from about 8 to 120 seconds. Nothing arrives before 11.0 minutes, the travel time at the mantle's shear speed. Then the longest periods come in first — 80 s waves at their group velocity of 4.23 km/s, after 11.8 min — and the period shortens steadily through the train, 30 s waves arriving at 14.3 min. The strongest shaking comes last, near 14.8 min, where the group velocity passes through its minimum and waves over a band of periods arrive together. Read off a real seismogram, the arrival time of each period is the crust's dispersion curve, and the crust's thickness with it. Waves

The shaking a uniform Earth could not carry

Early seismographs recorded a large sideways shaking that arrived after the body waves and before the Rayleigh wave, and no theory of a uniform Earth allowed it: a horizontal shear wave at a free surface reflects as itself and nothing holds it there. Augustus Love showed in 1911 that a layer does. A crust slower than the rock beneath traps the shaking between a perfect mirror at the top and total reflection at the bottom, and because the long waves reach deeper and run faster, the order in which the periods arrive is a measurement of how thick the crust is.

 The same grains as a crystal and as a liquid. 160 grains repelling one another by a screened Coulomb force, held together by a bowl-shaped confinement as a levitated layer is, and jostled by gas collisions at two temperatures; each picture is the end of a simulation of their motion. Grains with exactly six close neighbours are coloured. Left: strongly coupled — thermal energy about a thousandth of the interaction at the spacing — the grains settle into a hexagonal lattice, 79 per cent of those away from the edge with six neighbours. Right: the same grains with their thermal energy raised seventy-fivefold, the lattice melted, 9 per cent. In the laboratory the grains are a hundred micrometres apart and big enough to film with an ordinary camera, so the melting of a crystal can be watched particle by particle. Astrophysics

The dust a plasma freezes into a crystal

Sprinkle microscopic plastic spheres into a glowing discharge and each one charges itself to tens of thousands of electrons, because the plasma's light, fast electrons reach it far more often than its heavy ions. The charge holds the grains up against gravity in the sheath above the electrode, and it makes their mutual repulsion so strong compared with their jiggling that, at room temperature, they freeze into a hexagonal crystal with a lattice spacing a camera can resolve. A plasma, the hottest and most disordered state of matter, is where the most strongly ordered one was found to form most easily.

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

The plume that stops below the surface

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

 Where the charge goes on a conducting egg. The surface charge on three conducting spheroids of the same length, a sphere and spheroids 3 and 10 times as long as they are wide, drawn as a band whose thickness is proportional to the charge per unit area at each point. On the sphere it is uniform. On an ellipsoid the density is exactly proportional to the fourth root of the surface's Gaussian curvature, and so it is 3 and 10 times as dense at the tips as at the waist — not the 9 and 100 times by which the tips' curvature exceeds the waist's, and not the ratio of their mean curvatures either. The charge does crowd to the sharp ends, but by how much is set by the whole shape, not by any property of the surface at the point. Electromagnetism

The charge that follows no local rule

Charge on a conductor crowds where the surface curves sharply, and every high-voltage engineer rounds the edges for that reason. But curves sharply by what measure? On a conducting egg the density is exactly the fourth root of the surface's Gaussian curvature, so a tip a hundred times more curved than the waist carries only ten times the charge. On two spheres joined by a wire it is the square root, and on a flat disc the density goes without limit at the rim, where no curvature is defined. No local rule can be right, because where charge sits is decided by the whole shape at once.

 The resistance a pan offers the coil. The surface resistance — the square root of πfμρ — of five pan materials against frequency, on logarithmic axes, with the band where induction hobs work, 20 to 100 kHz, shaded. All five rise as the square root of the frequency. At 25 kHz carbon steel offers 9.9·10⁻⁴ Ω and ferritic stainless 0.0024 Ω, while austenitic stainless, which is not magnetic, offers 2.7·10⁻⁴ Ω, aluminium 5.1·10⁻⁵ Ω and copper 4.1·10⁻⁵ Ω. For the same coil current the heating goes as this resistance, so steel heats 19 times as hard as aluminium. Matching steel with aluminium by frequency alone would need 377 times the frequency — 9 MHz, where the hob's own electronics and coil would be the problem. Electromagnetism

The pan a better conductor will not heat

An induction hob heats a pan with currents the pan carries itself, driven by a field oscillating twenty-five thousand times a second, and it will heat a cast-iron skillet furiously while an aluminium saucepan, which conducts electricity six times better, barely warms. The better conductor is the worse load. What a pan presents to the coil is a surface resistance that goes as the square root of permeability times resistivity, and on that measure iron beats aluminium twenty times over. The same arithmetic explains why kitchen foil on the same hob can catch fire, and why a steel tip heated this way switches itself off at a fixed temperature.

 The angle at which water lets the light out. The fraction of light reflected inside a windscreen of index 1.52 at its outer surface, unpolarised, against the angle of incidence, when the outer surface is dry (glass to air) and when it is wet (glass to water). Beyond 41.2° all the light is reflected from a dry surface; under water total reflection does not begin until 61.3°. Between the two angles, shaded, the surface is a perfect mirror where it is dry and a window where it is wet. A sensor working at 45° sits in that band: at a wet spot only 1.2% of its light is reflected and the rest leaves the glass into the water. The difference between a mirror and a window is the refractive index of what is touching the glass — a property of contact, not of what can be seen through it. Optics

The windscreen that feels the rain

Behind the rear-view mirror of most modern cars a small box decides when the wipers should run, and it does so without looking at the rain. It shines infrared light into the windscreen at forty-five degrees, steep enough to be trapped inside the glass by total internal reflection, and counts how much comes back out of the far end of the glass a few centimetres away. A raindrop on the outside changes the medium the glass is touching from air to water, and water's critical angle against glass is sixty-one degrees, not forty-one: at forty-five the light is no longer trapped, and nearly all of it leaves into the drop. The sensor measures the area the rain is touching, and the rain rate from how fast that area grows.

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

The eye that loses its lens in water

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

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

The pipe that carries heat as vapour

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

 A lifetime that oscillates with a distance. The rate at which an excited atom emits a photon, relative to its rate in empty space, against its distance from a perfect mirror in wavelengths, for its dipole pointing across the mirror and along it. On the mirror itself the rate doubles for the first and vanishes for the second. As the distance grows both swing about 1 with a period of half a wavelength and a shrinking size: at half a wavelength the along-mirror rate is 0.96, and at a wavelength 0.99. Nothing in the atom has changed; its excited electron and its transition are the same in every position. An emission rate is a property of an atom and of the room it is in. Quantum

The lifetime that depends on where the mirror is

A half-life is treated as the most private fact about an atom or a nucleus, and for a nucleus that is nearly true. An excited atom beside a mirror decays faster or slower depending on how many half-wavelengths away the mirror stands, and between two mirrors closer than half a wavelength it does not emit at all. The atom has not changed. A decay is a transition to a state with a photon in it, and the room decides whether that state exists.

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

The rings no ray can reach

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

 Waves that turn to face the beach. The paths of crests' motion for a 10 s swell arriving at 50° on a plane beach of slope 1 in 50, seen from above, with depth contours dashed. Snell's law, sin α over speed is constant, holds along the ray because the depth changes only in the offshore direction, and the integrated paths were checked against it. The end of each ray is nearly normal to the shore: 5° from square-on by the point where it is about to break. In this sense all surf comes in nearly straight whatever direction the storm was in. The bend is complete because the speed falls almost to zero at the shore, whereas a prism's angles are bounded by its index. What remains of the angle drives the current along a beach. Waves

The waves that always arrive square to the beach

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

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