Illustrated Physics
physics, drawn
The collection

Every essay — page 28

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Mechanics

Motion, force, and the quantities that refuse to change.

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

The push a millimetre ahead of the axle

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

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

The raindrop that falls at a seventh of g

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

5 figures · part 8 on Momentum

Waves

Oscillation, and everything that turns out to be an oscillation.

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.

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.

5 figures · part 7 on Huygens 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.

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.

5 figures · part 7 on Periodic media A wave that falls with nothing pulling it. The intensity of Berry and Balazs's Airy packet, |Ai(x − t²/4)|², drawn at six moments t = 0 to 5 in units where ħ = m = 1, each profile raised by its time. It is an exact solution of the free Schrödinger equation — no potential, no force — and equally of the equation that governs a light beam in the paraxial approximation, with t then the distance travelled. Its shape never changes and never spreads, and its main lobe, which starts at x = −1.02, moves along the parabola x = t²/4 (dashed): −0.77 at t = 1, −0.02 at t = 2, 1.23 at t = 3, 2.98 at t = 4, 5.23 at t = 5. A uniform acceleration of a half, with nothing accelerating it. The packet carries infinite energy, which is the price of the trick: its oscillating tail stretches to minus infinity, and the tail is what pushes the main lobe along.

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.

5 figures · part 8 on Wave packets Light whose crests wind round its axis. Cross-sections of three beams carrying orbital angular momentum, with phase exp(iℓφ) round the axis, for ℓ = 1, 2 and 3, at one instant: solid lines where the field is at a crest, dashed where it is at a trough. Along the beam the crests form ℓ interleaved helices, and at a fixed plane the pattern turns about the axis once every ℓ optical periods. Looked at from a frame turning about the axis at a rate Ω, every crest arrives ℓΩ/2π times a second more or less often: the field's frequency is shifted by ℓΩ. That is a Doppler shift with no approach and no recession — the angular counterpart of k·v, with the winding number ℓ standing in for the wavenumber and the rotation rate for the speed. Each photon of such a beam carries ℓħ of orbital angular momentum about the axis.

The Doppler shift with nothing coming closer

Every Doppler shift so far has needed something to approach or recede. Light can be shifted in frequency by something that does neither: a surface spinning in place, or a polarising plate turning in a beam that passes straight through it. A beam whose wavefronts wind round its axis, ℓ times per wavelength, comes back from a surface spinning at Ω shifted by exactly ℓΩ, at every distance from the axis. A circularly polarised beam through a spinning half-wave plate comes out shifted by exactly 2Ω, because the plate takes two units of angular momentum from each photon and does work on it. The shift is tiny and perfectly measurable, and satellite navigation has to correct for it.

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

The note a bottle sings whatever its shape

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

5 figures · part 8 on Resonance

Optics

Light, and the small number of rules it obeys.

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.

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.

5 figures · part 8 on Scattering A squeezed disc between crossed polarisers. The dark fringes seen in a 50 mm disc of polycarbonate, 6 mm thick, squeezed across a vertical diameter by 500 N and viewed between crossed circular polarisers in light of 550 nm: each dark line is a contour on which the principal stresses differ by a whole number of fringe values, from the stress-optic law with a coefficient of 78 Brewsters. At the centre the fringe order is 3.61; it climbs steeply towards the two loading points, where the fringes crowd together and the stress grows without limit, and falls to zero at the free edges level with the centre. The pattern is the exact stress field of an elastic disc under diametral load; every fringe can be counted from a dark point, and the count times the fringe value is the difference of the principal stresses at that place.

The stress that polarised light can see

Squeeze a disc of clear plastic and look at it between two crossed polarising filters, and it fills with dark lines and coloured bands that crowd towards the points where it is pressed. Each line is a contour of stress. Glass, plastic and most transparent solids become doubly refracting when they are loaded, in proportion to the difference between their largest and smallest stresses, and polarised light turns that difference into a pattern that can be counted. David Brewster found it in 1816, the same Brewster whose angle polarises reflected light, and for a century the way to learn the stresses in a bridge or a dam was to cut its shape from plastic, load it and photograph the fringes.

5 figures · part 9 on Polarisation How bright a nebula looks through a 200 mm telescope. The surface brightness of the image of an extended object — a nebula, a galaxy, the Moon — through a lossless 200 mm telescope, as a fraction of its brightness to the naked eye, against magnification on a logarithmic axis, for eye pupils of 7 and 5 mm. It is never above one. For a 7 mm pupil it is one up to a magnification of 28.6, where the exit pupil equals the eye's; below that magnification the exit pupil is wider than the eye's and part of the collected light falls on the iris; above it the image dims as the square of the exit pupil: 0.327 at 50×, 0.082 at 100×, 0.020 at 200×. The telescope makes the nebula larger, not brighter; at the magnification where its exit pupil matches the eye's it makes it as large as it can while keeping it as bright as it is.

The nebula no telescope can brighten

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

5 figures · part 8 on Etendue Four cosines between a lens and the edge of its picture. Light from a direction 30° off the axis passing through a lens and landing at the edge of the image, with the four factors that dim it relative to the centre. The aperture, seen from that direction, is foreshortened to cos θ of its area. The image point is further from the lens, f/cos θ instead of f, and the light spreads as the square of the distance: cos²θ. The light arrives at the image plane at a slant, spreading over 1/cos θ more area: one more cos θ. Together, cos⁴θ: at 30° the edge receives 0.563 of the light at the centre, 0.83 photographic stops darker. Nothing here is a fault of the glass. A perfect thin lens, or a pinhole, does it, because it is geometry.

The four cosines at the edge of a photograph

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

5 figures · part 9 on Imaging

Electromagnetism

Charge, field, and the lines drawn between them.

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.

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.

5 figures · part 6 on Retardation Four plasma columns that need exactly the same current. Left: four pressure profiles across a 1 cm column — flat-topped, parabolic, peaked and hollow — each holding the same 10¹⁹ ions per metre at the same 10 keV, so the same total ∫2πr p dr. Right: the current each needs to hold itself together, from solving the force balance J × B = ∇p with Ampère's law, radius by radius. All four come out at 0.800, 0.800, 0.800, 0.800 MA: Bennett's 0.800 MA. The shape of the column decides where the current flows and how strong the field is inside it, and has no effect at all on how much current there must be.

The current that squeezes what carries it

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

5 figures · part 7 on Ampere law

Thermodynamics

Heat, disorder, and the one law with a direction in it.

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.

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.

5 figures · part 11 on Phase change The race to equilibrium that the hotter start wins. How far each system is from equilibrium with the bath — the relative entropy of its distribution from the bath's, in nats, on a logarithmic axis — against time since it was placed in the bath at 0.30, also logarithmic, for a start at 0.50 (warm, dashed) and one at 0.901 (hot, solid). The hot system starts further away: 1.243 against 0.254. Both close in quickly as their left wells equilibrate internally. Then the warm system must wait for probability to leak between the wells, and stalls, while the hot one, whose wells already had the right shares, carries on falling. The hot curve overtakes the warm one at a time of about 6.9, and from then on the hotter start is always nearer equilibrium.

The hotter start that cools first

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

5 figures · part 9 on Entropy The temperature through growing lake ice. Temperature against depth below the ice surface in a lake under air 10 K below freezing, after 1, 4 and 16 days, from Stefan's law with the ice's top held at the air's temperature. The ice is 11.1, 22.3, 44.6 cm thick; within it the temperature rises in a straight line from −10 °C at the top to 0 °C at the bottom, and below it the water is at 0 °C throughout. Ice forms only at the bottom, where water meets ice at the freezing point, and the latent heat it releases there has to be carried up through all the ice above by conduction. The straight lines get shallower as the ice thickens: the same temperature difference drives heat across a longer path, so less heat escapes and the ice grows more slowly. Sixteen times as long gives four times the thickness.

The ice that grows more slowly the thicker it gets

A lake under air ten degrees below freezing grows eleven centimetres of ice on the first day and only another eighteen in the following six. The ice forms at its underside, and every gram of it releases latent heat that must be carried up through the ice already there before more can form: the ice is its own insulation. The thickness therefore grows as the square root of time, and the rule, Stefan's, depends on temperature and time only through their product — the freezing degree-days that lake-ice forecasters count. A few centimetres of snow on top slows it more than any amount of cold speeds it up.

5 figures · part 10 on Diffusion How well a diamond-like crystal conducts heat, from 1 K to 1000 K. The thermal conductivity of a model crystal with diamond's sound speed and Debye temperature, from kinetic theory applied to its phonons, against temperature on logarithmic axes: natural carbon in a sample 1 mm across, isotopically enriched carbon, and natural carbon in a sample 0.1 mm across; copper's room-temperature 400 W/m K is marked for comparison. The model is calibrated to natural diamond's 2,200 W/m K at room temperature. At low temperature the conductivity rises as the cube of the temperature, limited by the sample's walls; at high temperature it falls as umklapp collisions shorten the phonons' paths. Between, it peaks: 3.7·10⁴ W/m K at 100 K for the natural crystal; 7.8·10⁴ W/m K at 94 K for the enriched crystal; 1.2·10⁴ W/m K at 126 K for the thin natural crystal. The model gets the shape — the rise as the cube of the temperature, a peak near 100 K, the fall — but overestimates the heights, because it leaves out the phonon collisions that conserve momentum and every defect but isotopes; measured enriched diamond peaks near 4 × 10⁴ W/m K. An insulator with no free electrons still conducts heat five times better than copper at room temperature.

The gas of sound that carries a diamond's heat

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

5 figures · part 10 on Kinetic theory

Relativity

Space and time, drawn on the same axes.

Two factors of γ between a magnet and an X-ray. An electron of 6 GeV (γ = 11742) crossing an undulator, a row of magnets alternating every λᵤ = 3 cm. Top: the laboratory, where the electron wiggles once per period. Middle: the electron's own frame, where the row is contracted by γ to 2.55 μm and rushes past at nearly the speed of light; to the electron it is a wave of that wavelength, and the electron radiates at it, in the infrared. Bottom: that light, emitted forward and seen back in the laboratory, is Doppler-shortened by a further factor of about 2γ, to λᵤ/2γ² = 1.088 Å; the wiggle's own slowing of the electron adds a factor (1 + K²/2), giving 1.632 Å for a deflection parameter K = 1 — hard X-rays from centimetre magnets. The drawing is schematic: the three periods differ by factors of about twelve thousand and a hundred million, and cannot be drawn to one scale.

The magnet period that comes out as an X-ray

A row of permanent magnets alternating every three centimetres, with a beam of six-gigaelectronvolt electrons sent down the middle, shines X-rays with a wavelength of an ångström and a half — two hundred million times shorter than the spacing of the magnets that made them. The factor is γ squared, and the two γ's come from two different frames. In the electron's frame the row of magnets is contracted by γ and rushes past it as a wave; the electron shakes in step and radiates; and that light, sent forward, comes back to the laboratory shortened by another factor of 2γ. Length contraction is half of every X-ray a modern synchrotron makes.

5 figures · part 7 on Length contraction

Quantum

Where the continuous picture runs out, and what replaces it.

Which atoms in a Fermi sea are allowed to take a photon's kick. A slice through the occupied momentum states of a gas of identical fermions at zero temperature — a filled disc whose radius is the Fermi momentum — with each state marked by whether it can absorb a recoil of 0.8 of the Fermi momentum to the right, the kick a scattered photon gives. A state can scatter only if its kicked momentum lands outside the sea, in an empty state. The kicked sea is the second circle; the atoms in the overlap of the two (grey) land on occupied states and cannot scatter, and only those in the crescent (blue) can. In this slice 50 per cent of the states are allowed; over the whole sphere the allowed share is S(q) = 3q/4 − q³/16 = 0.568. A kick larger than the sea's diameter, twice the Fermi momentum, is never blocked, which is why the effect needs a sea whose Fermi momentum is comparable with the photon's.

The light a full sea will not scatter

An atom that scatters a photon recoils, and the recoil has to put it into a new state of motion. In a cold, dense gas of identical fermions most of the states it could recoil into are already taken, and the exclusion principle forbids the scattering outright. Nothing about the atom has changed — not its electrons, not its transition, not its cross-section — yet the gas becomes dimmer than the same number of atoms scattered one at a time. The effect was predicted in the 1990s and seen in 2021, and it turns transparency into a measurement of how full a quantum gas is.

5 figures · part 6 on Exclusion The heights a neutron is allowed to bounce to. A neutron above a horizontal mirror in the Earth's gravity: the potential energy mgz (straight line), the first four allowed energies and, drawn on each, the probability density of the neutron's height. The natural length is z₀ = (ħ²/2m²g)^(1/3) = 5.87 μm and the natural energy mgz₀ = 0.602 peV. The allowed energies are 1.41, 2.46, 3.32, 4.08 peV — pico-electronvolts — and a classical neutron with each energy would rise to 13.7, 24.0, 32.4, 39.8 μm. Each density is an Airy function, with n − 1 nodes, leaking a little above the classical turning height and vanishing at the mirror. The lowest state never touches the mirror and never rises above about 25 μm.

The bounce that comes in fixed heights

A ball dropped on a floor can bounce to any height. A neutron cannot. Slow enough, and resting on a flat mirror in the Earth's gravity, it has a lowest state about fifteen micrometres high, with an energy of 1.41 pico-electronvolts, and a staircase of higher states above it — the first quantum states ever seen in a gravitational field, measured in 2002 with a slit narrower than a hair. Shaking the mirror at 255 hertz drives a neutron from the first state to the second. The staircase is now a laboratory for gravity at distances where nothing else can test it.

5 figures · part 6 on Matter waves

Fluids

Matter that will not hold a shape, and the forces that act in it anyway.

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.

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.

5 figures · part 8 on Capillarity The lift of a zero-pressure balloon as it climbs. The gross lift, in tonnes, of a zero-pressure balloon whose envelope holds at most 1.1 million cubic metres, against height, for launches with 3,500 and 5,000 cubic metres of helium; the dashed line is its 3 tonnes of envelope, payload and ballast. At launch the envelope is mostly empty. As the balloon climbs the gas expands exactly as fast as the air thins, so the mass of air it displaces stays the same and the lift stays at 3.69 or 5.28 tonnes all the way up. The envelope fills at 37.5 km with the larger fill and 39.9 km with the smaller; above that, gas vents through the open base, the volume is fixed and the lift falls with the air's density. Both reach the same float, 41.3 km, where the lift has fallen to the weight. More helium only makes the climb faster; the ceiling is set by the envelope's size and the mass it carries.

The balloon that floats at a density

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

5 figures · part 8 on Buoyancy

Astrophysics

Gravity read as geometry, and the laws carried where no laboratory can follow.

The cone a charge leaves behind in water. A charged particle crossing water at 0.99 of the speed of light in vacuum, which is 1.317 times the speed of light in water, drawn at one instant. From each point it passed, its field set off a spherical wavelet travelling at c/n; the wavelets from earlier points have grown larger. Because the particle outruns them, they all touch one cone with its tip at the particle, at an angle θ = arccos(1/nβ) = 40.6° to the path — the light of Cherenkov radiation, which travels out perpendicular to the cone. No point on the path is special and the particle never accelerates: the radiation is a property of the field moving faster than the medium can carry it, the electromagnetic counterpart of a sonic boom or a boat's wake.

Why the glow of a fast charge is blue

A charge moving in a straight line at constant speed does not radiate — in a vacuum. In water, if it moves faster than light does in water, it glows, and the glow is blue. The cone of the light is the familiar half of the story. The other half is a count: Frank and Tamm's formula says each centimetre of water yields about two hundred visible photons, in numbers that rise as the inverse square of the wavelength, which is why a reactor pool shines blue and why the light keeps getting stronger into the ultraviolet until water stops being transparent. The charge never accelerates. What radiates is the water.

5 figures · part 7 on Radiating charge 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.

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.

5 figures · part 8 on Plasma oscillation

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