Illustrated Physics
physics, drawn
The collection

Every essay — page 29

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.

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

The jump that does not get higher with size

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

5 figures · part 7 on Energy A gas pressed against the wall of a spinning tube. The density of uranium hexafluoride gas in a rotor spinning at a steady rate, relative to its density on the axis, on a logarithmic scale, against the distance from the axis as a fraction of the radius, for wall speeds of 300, 450 and 600 m/s at 300 K. In the turning frame the gas sits in the centrifugal potential −½mω²r² and settles into the same exponential that thins the atmosphere with height: n ∝ exp(mω²r²/2kT). At 600 m/s the density at the wall is e^25.4 — about 10¹¹ — times that on the axis; at 300 m/s, e^6.4. Almost all the gas lies in a thin layer against the wall, and the middle of the rotor is close to vacuum.

The spin that sorts by a difference of mass

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

5 figures · part 7 on Circular motion

Waves

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

Three arrivals, and the one that overtakes. Arrival time against distance from the shot for a 35 km crust at 6 km/s over a mantle at 8 km/s: the direct wave, a line through the origin with slope 1/v₁; the reflection from the base of the crust, a hyperbola; and the head wave, a straight line of slope 1/v₂ that begins at the critical distance, 79 km, where it touches the reflection. Close to the shot the direct wave arrives first. The head wave starts later but travels most of its path at the faster speed, and beyond the crossover distance, 185 km, it arrives first and stays first. At 300 km it beats the direct wave by 4.8 s, though it has travelled 26 km further.

The wave that arrives first by going the long way

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

5 figures · part 8 on Huygens A number of details that frequency cannot change. The number of separately resolvable depths along one line of an ultrasound image, the deepest echo divided by the axial resolution, against frequency, for dynamic ranges of 50, 60 and 80 dB in soft tissue; dashed, the same for a medium that absorbed like water, as the square of the frequency, with 60 dB. In tissue the count is D/(α₀Nc), the same at every frequency: 325 for 50 dB, 390 for 60 dB, 519 for 80 dB. It is a property of the tissue, through α₀, and of the electronics, through D, and changing the frequency only trades depth for sharpness one for one. In a medium with an f² law the count would fall as 1/f, from 88,548 at 1 MHz to 1,771 at 50 MHz. It is the near-linear absorption of tissue that makes the trade exact.

The depth an ultrasound image pays for its sharpness

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

5 figures · part 8 on Attenuation

Optics

Light, and the small number of rules it obeys.

A signal that fades as the fourth power of distance. Received power, in decibels relative to free space at one metre, against distance on a logarithmic scale, for a 900 MHz transmitter 30 m up and a receiver 1.5 m up over flat, perfectly reflecting ground, with the ground's reflection taken as −1; dashed, free space alone, falling 20 dB per decade. Close in, the direct and reflected waves beat as their path difference passes through whole wavelengths. Beyond the break distance 4hₜhᵣ/λ = 540 m, the path difference 2hₜhᵣ/d is less than half a wavelength and shrinking, and the reversed reflection cancels the direct wave more and more completely: the power falls 40 dB per decade, as d⁻⁴, 10 dB below free space at ten times the break and 30 dB below at a hundred times. The ground, which was supposed to add a second path, takes the signal away; raising either antenna, which lengthens the path difference, is the remedy, and is why masts are tall.

The surface that cancels whatever skims it

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

5 figures · part 8 on Refraction Three planes that meet in one line. A thin lens of focal length 50, drawn as a vertical line, and an object plane tilted at 45° to the axis, crossing it 150 units in front of the lens. Four points on the object plane are traced to their images through the lens's centre and along the axis-parallel ray. The images lie on a straight line too — a thin lens maps planes to planes — but one tilted far less steeply, and when the object plane, the lens plane and the image plane are extended they meet in a single line, here 150 units below the axis. That is the Scheimpflug rule. A sensor laid along the tilted image plane records the whole tilted object plane sharp, which no sensor square to the axis can do.

The plane of focus a tilted lens tilts

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

5 figures · part 10 on Imaging

Electromagnetism

Charge, field, and the lines drawn between them.

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

The voltage that bends an electron like glass

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

5 figures · part 5 on Potential One number decides the efficiency. The best efficiency a link of two resonant circuits can reach, with its load chosen to maximise it, against the figure of merit U = k√(Q₁Q₂) on a logarithmic scale: the coupling coefficient times the geometric mean of the two coils' quality factors. The curve is η = U²/(1 + √(1 + U²))², and nothing else about the coils, the frequency or the distance enters: 0.25 per cent at U = 0.1, 17 per cent at U = 1, 52 per cent at U = 3, 82 per cent at U = 10, 94 per cent at U = 30. A weak coupling is rescued by a high Q: k = 0.003 with Q = 1,000 on each side gives U = 3 and 54 per cent, where the same coils used without tuning capacitors, as a plain transformer, deliver under one per cent, with the source driving mostly reactive current that does nothing.

The coupling a resonance rescues

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

5 figures · part 7 on Induction 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.

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.

5 figures · part 7 on Dielectrics

Thermodynamics

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

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

The degree a spoonful of solute buys

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

5 figures · part 9 on Chemical potential How supersaturated a cloud is, to an ice crystal. The supersaturation over ice of air that is just saturated over liquid water, eᵥᵥ/eᵢ − 1, in per cent, against temperature; dashed, the same computed from the chemical potentials, exp(LΔT/RT·Tₘ) − 1, with L the latent heat of melting per mole, ΔT the supercooling and Tₘ the melting point. A supercooled cloud is in balance with its droplets, so to any ice crystal inside it the air is 5.1 per cent supersaturated at −5 °C, 10.5 at −10, 22.0 at −20 and 41.2 at −35. Over the droplets themselves it is exactly saturated, so they neither grow nor shrink by vapour. The excess is the chemical-potential difference between supercooled water and ice, 217 J per mole at −10 °C, expressed as a pressure ratio.

The ice that grows by stealing from the droplets

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

5 figures · part 10 on Chemical potential The highest note the air can carry, with height. The frequency at which a sound wave in the air loses its amplitude by a factor e in a single wavelength, in hertz on a logarithmic scale, against height, with the band of human hearing shaded. Because viscosity does not depend on pressure, the collision time grows as the pressure falls, and the ceiling falls with it. It is about 212 MHz at sea level, 0.17 MHz at 50 km and 3 kHz at 80 km. It passes the top of human hearing, 20 kHz, at about 68 km, and the bottom, 20 Hz, at about 110 km; at 100 km it is 92 Hz. Above about a hundred kilometres the air cannot carry anything a person could hear, at any loudness: the molecules are too far apart to pass a compression on before it has dispersed.

The note too high for thin air

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

5 figures · part 11 on Kinetic theory

Relativity

Space and time, drawn on the same axes.

The neutrino's energy against the pion's, at four angles. The energy of the neutrino from a pion's decay against the pion's energy, for neutrinos going exactly forward and at 1°, 2.5° and 4° to the pion's line of flight. Straight ahead the neutrino takes a fixed share, 0.427, of the pion's energy, so pions of every energy give neutrinos of every energy. Off axis the curve rises, flattens and falls: at 2.5° it peaks at 0.683 GeV for a pion of 3.2 GeV, and every pion between 2 and 10 GeV gives a neutrino between 0.61 and 0.68 GeV. The flat top is where γθ = 1. Its height, p*/θ, depends on the angle alone — the rest-frame momentum of the decay, 29.8 MeV/c, divided by the angle in radians.

The neutrino energy an angle fixes

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

5 figures · part 11 on Relativistic dynamics What the heat shares out, and what it does not. Two averages per particle in the Maxwell–Jüttner gas, in units of kT, against kT/mc² on a logarithmic scale: the momentum times the velocity, p·v, whose average over the three directions gives the pressure, and the kinetic energy. The first is 3 at every temperature, computed here by averaging over the distribution and equal to 3 to four decimal places across the whole figure — so the pressure is exactly nkT, the ideal gas law, from a gas colder than a star to one hotter than the early universe. The kinetic energy is not shared out the same way. It is 1.506 kT when the gas is cold, Boltzmann's 3/2, and rises to 3.00 kT when it is ultra-relativistic, because for a fast particle the energy is p·c and p·v, which is what equipartition hands out, is the whole of it rather than twice it.

The gas law relativity leaves exact

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

5 figures · part 8 on Relativistic thermodynamics An atom's line and a nucleus's, on one scale. Three energies, in electronvolts on a logarithmic scale, for an atomic transition, sodium's yellow line, and a nuclear one, the 14.4 keV gamma ray of iron-57: the amount by which emission and absorption by free atoms miss each other because of recoil, E²/Mc²; the natural width of the line, ħ over the excited state's lifetime; and the Doppler width from thermal motion at room temperature. For sodium the recoil miss is 2.1·10⁻¹⁰ eV, 197 times smaller than the natural width, so an atom absorbs its neighbour's light. For iron-57 it is 0.00392 eV, 839,015 times its natural width of 4.67·10⁻⁹ eV. The recoil grows as the square of the transition energy and the width does not, so a gamma ray emitted by a free nucleus misses an identical nucleus by nearly a million line widths.

The gamma ray its own kind cannot absorb

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

5 figures · part 8 on Mass-energy

Quantum

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

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

The image made of frequencies

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

5 figures · part 6 on Spin The oldest age a trace of modern carbon allows. The age a radiocarbon measurement would report against the sample's true age, when a fraction of the carbon in it is modern contamination — from groundwater, roots, handling or the chemistry of preparation — of 5, 1 and 0.1 per cent; dashed, no contamination. Young samples are barely affected. Old ones are ruled by the contamination, because their own carbon-14 has dwindled to less than it: with 1 per cent, a sample of any age beyond about 60,000 years reads 38.1 thousand years, and 5 per cent caps it at 24.8 thousand. Even 0.1 per cent, a microgram of modern carbon in a milligram sample, caps it at 57 thousand. The limit of radiocarbon dating, usually quoted as about fifty thousand years, is a limit on cleanliness, not on the half-life.

The clock read by counting atoms

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

5 figures · part 7 on Decay

Fluids

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

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

The rise a joule of heat buys

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

5 figures · part 8 on Hydrostatics A cell swells until its membrane runs out. The volume of a red cell of 90 fL against the salt concentration of the solution round it, in per cent sodium chloride by weight, with 0.9 per cent isotonic. By the Boyle–van 't Hoff law the water inside follows the outside's tonicity: the cell's volume is an inactive part, 40 per cent of the normal volume in haemoglobin and other solids, plus a part inversely proportional to the concentration. Its membrane's 135 μm² can enclose at most 147.5 fL, as a sphere (dashed). The swelling curve reaches that at 0.44 per cent salt: below it the cell has no more shape to give up and bursts. In a 1.5 per cent solution the same cell shrinks to 68 fL and crenates, with membrane to spare.

The cell that bursts by geometry

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

5 figures · part 8 on Osmosis A parcel that leaves the air behind. A parcel of air lifted from the ground at 30 °C with a dew point of 23 °C, against height, beside the air it rises through, which cools at 6.5 K per kilometre to the tropopause at 15 km. Unsaturated, the parcel cools at the dry rate and is soon colder than its surroundings; at 890 m it reaches saturation, a cloud base, and from there cools along the moist adiabat. From 2080 m it is warmer than the air round it and accelerates upward, until at 13.4 km it is cold again. The shaded area, the buoyancy integrated over that climb, is the convective available potential energy: 2543 J/kg. If all of it went into motion the updraft would reach 71 m/s, the square root of twice the CAPE. The 99 J/kg below the level of free convection is the push it needs to get started.

The cloud that cools more slowly than the air

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

5 figures · part 9 on Stratification

Astrophysics

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

Light from a charge that crosses into glass at a steady speed. The transition radiation emitted as an ultrarelativistic charge crosses from vacuum into a dense medium, per unit solid angle, against the angle to the path in units of 1/γ, for photons at three frequencies: one tenth, three tenths and one times γωₚ, where ωₚ is the medium's plasma frequency. The scale is αħγ²/π² per unit frequency. Nothing is emitted straight ahead; the light comes out in a narrow hollow cone with its brightest ring at γθ = 0.98, 0.88, 0.68 for the three. At a tenth of γωₚ the peak is about ten times higher than at γωₚ. The charge never accelerates: it is the field's sudden change of shape at the surface, from the flattened disc it carries in vacuum to the one the medium's electrons allow, that is radiated.

The light a charge makes by changing medium

A charge moving at constant speed in a straight line radiates nothing in a vacuum, and nothing in glass either if it is slower than light there. Let it cross from one into the other and it radiates at the boundary, though it neither turns nor slows. The field it carries has one shape in each medium and has to be rebuilt as it crosses, and what is shed in the rebuilding is light. The energy shed grows in proportion to the particle's Lorentz factor, the one property of a fast particle that almost nothing else measures at high energy, and this is why detectors stack hundreds of plastic foils in a particle's path to collect one or two X-rays.

5 figures · part 8 on Radiating charge Which way the atoms leave, against the bias. A toy version of the 2023 ALPHA-g measurement: the fraction of trapped antihydrogen atoms that leave through the bottom mirror as both mirrors are lowered over twenty seconds, against the magnetic bias, in units of the gravitational tilt, for three assumed accelerations. Each atom is given an energy between 0.02 and 0.5 K and a random starting phase, bounces between the mirrors, and leaves through the first one it reaches that has dropped below its energy. With no bias, 100 per cent leave downward if antihydrogen falls normally, 49 per cent if it does not fall and 0 per cent if it falls up. Each curve crosses one half exactly where the bias cancels the assumed gravity, and the crossing point is what the experiment measures. The curves are not steps — each falls from nine-tenths to one-tenth over about 0.6 g of bias — because the mirrors keep falling during the few milliseconds an atom takes to cross the trap: an atom that happens to arrive at the higher mirror just after it has dropped below its energy leaves that way. ALPHA-g's measured balance gave an acceleration of 0.75 g, with an uncertainty of about 0.2 g, ruling out falling upwards.

The antiatom that had to be weighed anyway

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

5 figures · part 7 on Equivalence principle The redshift of a white dwarf weighs it. The gravitational redshift at the surface of a cold white dwarf, expressed as the velocity that would give the same Doppler shift, against the star's mass; computed from the radius each mass is forced to have. The Sun's surface redshift is 636 m/s. A white dwarf of 0.6 solar masses shifts its light by 30.1 km/s, one of 1.0 by 77.6, of 1.2 by 128 and of 1.35 by 223. Because the radius is not free, the curve is single-valued and steepening: a measured shift returns one mass. The cold model puts Sirius B, at 1.018 solar masses, at 80.8 km/s; its measured mass and radius together give 80.0.

The redshift that weighs a dead star

Light leaving the surface of an ordinary star is reddened by gravity, but the shift alone says little, because a star's radius is free and the shift depends on mass over radius. A white dwarf has no free radius. Degenerate electrons fix its size from its mass, so one measured line shift returns one mass — and the gauge grows sharper the closer the star is to collapse. The first measurement of it, in 1925, was hailed as a confirmation of general relativity and was wrong by a factor of four, in agreement with a forecast that was wrong by the same factor.

5 figures · part 6 on Gravitational redshift A line carved into the X-rays of a magnetised star. A model X-ray spectrum of the kind fitted to accreting pulsars, drawn as energy per logarithmic interval on a logarithmic scale: a hard continuum that bends over above about 25 keV, with the photons near the cyclotron energy removed by resonant scattering. Dashed, the same continuum without the field. The fundamental is placed at 38 keV, where lines of this kind are seen; thermal motion of the scattering electrons along the field at kT = 8 keV broadens it to a width of about 4.8 keV. The second harmonic falls at 73.0 keV, a little below twice the fundamental. Read through ħeB/m with the relativistic level spacing and the redshift of a 1.4 solar-mass, 12 km star, z = 0.24, the fundamental says the field where the line forms is 4.24 × 10⁸ T. The depths and widths here are illustrative, not fitted to a star.

The circling that comes in quanta

An electron circling a magnetic field radiates at the frequency of its circling, and classically that is all: a continuous glow at whatever energy the circling has. On a neutron star the field is a hundred million tesla, the quantum of circling is a hard X-ray, and an electron lifted one step up falls back in a quarter of a femtosecond. So every electron there sits on the bottom step, and the X-rays that match the step are scattered out of the beam. The gap they leave is a line, and its energy is the field.

5 figures · part 9 on Radiating charge

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