Illustrated Physics
physics, drawn
Concept

Equilibrium — where it appears

The state a system settles into and stays in, where the quantities describing it stop changing and every competing tendency balances. Which quantity is minimised depends on what is held fixed: energy at fixed entropy, free energy at fixed temperature, and entropy maximised for an isolated system.

Named by 38 essays across 8 fields — each of them below, with the objects they name alongside it.

A heeled hull, and the couple it makes. A rectangular hull of beam 3 m heeled 18°, with the waterline solved so that it displaces the same volume it did upright. The centre of buoyancy has moved 0.244 m to the low side, and weight and buoyancy now act along two lines 0.059 m apart — a couple that turns the hull back upright.

Why a ship comes back upright

Whether a floating body rights itself or rolls over is not decided by its weight, its density or how deep it sits. It is decided by the shape of the slice the water cuts through it, and the number that settles it can be worked out before the vessel is built.

fluids · Buoyancy Where a body can no longer be any shape it likes. The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary.

The size at which a body becomes round

A mountain can be no taller than the height at which the rock beneath it begins to crush, and that height falls as the body gets bigger — so there is a size above which a mountain would have to be taller than the world it stands on. Above it, nothing can be any shape but a sphere.

astrophysics · Self-gravity What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 30.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 22.5 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 22.5 N at the right-hand edge of the axis.

The force that takes what it needs

Static friction has no value of its own. It supplies exactly what equilibrium demands and not a newton more, right up to the moment it cannot — which is the only instant in the whole business at which a coefficient of friction means anything at all.

mechanics · Friction Ways to arrange 12 coins. The number of distinct arrangements giving each number of heads, for 12 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

What a system actually minimises

A ball falls to the bottom of a bowl and a gas fills a room, and neither of those is the rule. A system in contact with a large reservoir minimises U − TS, and the minus sign is the reservoir's own entropy written in the system's variables — which is why a rubber band pulls harder when it is heated.

thermodynamics · Entropy The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

A boiling point is a pressure, not a temperature

Nothing about water names one hundred degrees; the air does. Where a liquid turns to vapour in its bulk is fixed by what pushes on it, which makes the familiar figure a coordinate on a curve — 71 °C on Everest, 119.5 °C in a sealed pot — and one latent heat draws the whole curve.

thermodynamics · Phase change One volume of liquid, several solids. 4 drops of the same 5 µL of liquid, on 4 solids it meets at 20°, 60°, 90°, 140°. Each is the spherical cap that volume and that angle force, so the footprint radius is computed rather than chosen: 20° gives 2.61 mm, 60° gives 1.71 mm, 90° gives 1.34 mm, 140° gives 0.69 mm. At every contact line the three interfacial tensions are drawn to scale, and only their horizontal components balance — γsv = γsl + γlv cos θ. The vertical pull of the liquid surface is taken up by the solid, which is why the angle belongs to three interfaces at once and to no single liquid: change the solid and nothing about the water has changed.

The angle a liquid makes with what it sits on

Everything capillarity does — climbing, beading, wicking, waterproofing — is the sign and size of one cosine, and that cosine belongs to three interfaces at once rather than to the water. Change the solid and nothing about the water has changed, yet the same five microlitres goes from a footprint 2.61 mm across to one of 0.69 mm.

fluids · Capillarity Which failure comes first. The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a paperback, standing, a full filing cabinet, a pint glass.

Slide or topple

Push a wardrobe and it goes over; push a brick and it skids. Both are held by the same friction and both are pushed by the same hand, and which of the two failures arrives first has nothing to do with how hard the push is. The floor decides it, by shifting where it pushes back.

mechanics · Free-body A torque, and no force at all. A loop of 1 turn enclosing 20 cm² and carrying 10 A has a magnetic moment of 0.02 A m². In a uniform field of 0.05 T the torque on it is m B sin θ, drawn here against the angle between the moment and the field: zero when they are aligned, largest at 0.001 N m across, and zero again when they are opposed. The second curve is the energy, −m·B, whose minimum is the aligned position and whose maximum is the opposed one — which is why a compass needle settles one way round and not the other. The net force is zero at every angle on this axis, exactly and not approximately: the force is I dl × B summed round the loop, the sum of dl round any closed path is zero, and a constant B comes outside the sum. The inset shows the four forces on a rectangular loop; the pair across the axis is the couple, and the pair along it cancels.

The loop that behaves like a needle

Far enough away, a current going round in a circle is indistinguishable from a bar magnet, and one number describes both. That number tells a uniform field how to turn the loop and gives it no way to pull on it at all — which is why two magnets attract by the fourth power of the distance and not the second.

electromagnetism · Magnetism Buoyancy that falls away as the body sinks. The net upward force on a body containing a little gas, against how deep it has been taken, for 3 gas fractions. The weight does not change with depth. The buoyancy does, because the gas obeys Boyle's law and the pressure rises by an atmosphere every ten metres, so a body that displaced its own weight at the surface displaces less at depth. Every curve therefore slopes downward, and that slope is the whole point: where a curve crosses zero the body is in equilibrium, and the crossing is always from above, which makes every one of these equilibria unstable. Push the body a little deeper and the force does not push back — it turns downward and grows. The crossings drawn are at 8.2 m, 10.0 m, 13.6 m, and a body sitting at one of them is balanced in the sense that a pencil is balanced on its point. This is why a diver at neutral buoyancy has to keep adjusting, why a submarine's depth is held by hydroplanes and not by ballast alone, and why a fish that loses the use of its swim bladder sinks rather than drifting.

The depth past which it must sink

A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.

fluids · Buoyancy The same block, one of them with no upthrust at all. Two identical blocks 0.8 m tall with their tops 1.2 m under the surface, drawn with the pressure on every wetted face at its true relative size. On the right the block is clear of the floor and the pressure on its underside exceeds that on its top by 7.8 kPa, which is ρgh and is exactly Archimedes' 7.8 kPa. On the left the bedding is perfect and there is no water under it, so nothing pushes up: the resultant is 11.8 kPa downward and the block presses on the floor with more than its own weight. Buoyancy is not something the fluid has. It is what the bottom face is doing, and a face the fluid cannot reach does nothing.

The block the water does not lift

A block bedded flat on the bottom of a tank, with no water underneath it, feels no upthrust at all. It is fully submerged, Archimedes' principle is not suspended, and it presses on the floor with more than its own weight — because buoyancy is not something a fluid has, it is what the bottom face is doing, and a face the water cannot reach does nothing.

fluids · Buoyancy The line that slopes the wrong way. Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does.

The ball of gas that heats up as it cools

Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

astrophysics · Self-gravity The column a dissolved thing holds up. Two arms of one vessel, joined below by a membrane that passes water and not solute. On the right is 10 mol/m³ of dissolved particles at 25 °C; on the left, pure water. Water crosses into the solution until the extra weight of the right-hand column has raised its pressure by the osmotic pressure — 24.8 kPa, which is 2.53 m of water, drawn here to scale. Nothing is pulling. The solvent is at a lower chemical potential where it is mixed, so it moves that way, and it stops when mechanical pressure has made up the difference. A solute a thousand times more dilute than seawater lifts a column taller than a person.

The pressure that comes from counting

Dissolve a teaspoon of anything in a litre of water, put a membrane between it and pure water, and the solution will hold up a column of water two and a half metres tall. Nothing is pulling. The pressure does not depend on what was dissolved, only on how many particles it made — which is the ideal gas law, with the solute in place of the gas.

fluids · Osmosis The most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are.

The brightness no lens can increase

A lens can make an image smaller and therefore hotter, and there is a temperature at which it stops — the temperature of the source. Every arrangement of glass and mirrors ever built obeys a bound that contains no wavelength, no aperture and no material — only the angle the receiver is allowed to accept — and the bound comes from thermodynamics rather than from optics.

optics · Etendue The field near a neutral point. Field lines near a magnetic null, traced by following the local field direction rather than plotted from the closed form. The field is B ∝ (y, k²x) with k = 1, whose lines are the hyperbolae y² − k²x² = constant and whose separatrices are the straight lines y = ±1x. At k = 1 the X is symmetric and the current density is exactly zero: the field is curl-free, and nothing is stored in it beyond the field itself. The four quadrants are four separate flux systems, and which of them a given line belongs to is the quantity a frozen-in field is not allowed to alter.

The knot the field cannot untie

A perfectly conducting fluid cannot change which field line joins which piece of it. So two flux systems pushed together may be squashed indefinitely and can never merge, and the energy of the squashing accumulates with nowhere to go. The release happens only where the perfect conductivity locally fails — in a sheet three metres thick inside a structure ten thousand kilometres across — and the rate that follows is a hundred thousand times too slow for the flares that are observed.

astrophysics · Flux freezing Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives.

The staircase that never reaches the floor

Absolute zero is unreachable, and the reason is not that the apparatus is not good enough. Every stage of cooling removes a fixed fraction of what is left rather than a fixed amount, so the steps shrink in proportion to the distance remaining — and the fixed fraction cannot be made one, because the entropy curves at two field strengths are required to meet where the axis is.

thermodynamics · Third law Everything is decided against one line at 9.76 K per kilometre. Temperature against height for five environments, with the dry adiabat drawn heavy. A parcel lifted from the ground cools along the adiabat, at g/c_p = 9.76 K/km — a number with no meteorology in it, only gravity and the heat capacity of air. If the environment cools faster than that, a lifted parcel finds itself warmer than its surroundings and keeps going; if it cools more slowly, the parcel finds itself colder and sinks back. -5 K/km gives N² = 5.02e-4 s⁻², a period of 4.7 min; 0 K/km gives N² = 3.32e-4 s⁻², a period of 5.7 min; 6.5 K/km gives N² = 1.11e-4 s⁻², a period of 9.9 min; 9.8 K/km gives N² = -1.43e-6 s⁻², an e-folding time of 835 s; 12 K/km gives N² = -7.63e-5 s⁻², an e-folding time of 114 s. The classification is a comparison of two slopes and nothing else: no density appears in it, and the same cold air is stable under one profile and unstable under another.

The layer a parcel cannot leave

Whether a column of air overturns is not decided by its density but by a difference of two gradients — the rate the environment cools with height, and the rate a lifted parcel cools on its own. Subtract one from the other and what is left is a restoring force per unit displacement, so a stable atmosphere rings at a period of minutes and an unstable one has no period at all.

fluids · Stratification The potential a shaken pivot creates. The effective potential of a pendulum whose pivot is shaken vertically, against the angle from hanging, for shaking rates of 8, 14, 20, 30 times the pendulum's own frequency at an amplitude of 0.12 of its length. The shaking averages to no force at all — it is up as often as down — and yet it adds a term to the potential, because the pendulum's position is correlated with the phase of the shake rather than independent of it. The added term is proportional to sin²θ, so it is largest sideways and zero at both the hanging and the inverted positions, and it turns the maximum at 180° into a minimum once 14× and 20× and 30× the natural frequency is reached. The criterion is (aΩ)² > 2gL: the shake speed must beat the speed a fall through the pendulum's own length would give. Upside down then becomes a stable equilibrium, with a restoring force and a period of its own.

Held up by a force that averages to nothing

Shake a pendulum's pivot up and down fast enough and the pendulum will stand upside down, balanced, and push back if it is nudged. The shaking supplies no average force at all — it is up as often as it is down — and the reason it nevertheless holds is that the pendulum's position and the phase of the shake are not independent.

mechanics · Pendulum Every stationary point has a way out. The potential along four lines through the most symmetric point of a symmetric arrangement of 4 equal charges at the corners of a square — the one place a trap might be expected. In the plane of the square the potential rises in every direction; out of the plane it falls. The point is stationary and it is a saddle, which is what Laplace's equation forces: the three second derivatives must sum to zero — computed here as 1.4e-5 against the individual values of order 2e+0 — so if two of them are positive the third must be negative. A particle released here rolls away along the direction that falls. No amount of ingenuity in placing the charges changes this, because the constraint is on the equation rather than on the arrangement, and it is why every real trap for a charged particle either uses time-varying fields, or a magnetic field with a velocity, or a material with a negative response.

Nothing can be held still by a static field

However many charges are arranged, however cleverly, a charge placed among them has somewhere to fall. The reason is one line of arithmetic — the potential in empty space satisfies Laplace's equation, and a solution of that equation has no interior maximum or minimum — and the consequence is that every real trap for a charged particle works by breaking one of the assumptions rather than by being cleverer.

electromagnetism · Potential Two angles a film is not allowed to depart from. The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn.

The angles a film has no choice about

A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number.

fluids · Surface tension Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up.

The gas that nobody counted

A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

thermodynamics · Blackbody Three curves that can only meet at a point. Water's phase diagram within a kelvin of its triple point, with each boundary drawn at the slope Clausius and Clapeyron give it from measured latent heats and densities: 44.4 Pa/K for boiling, 50.3 for sublimation, and -135 bar per kelvin for melting — negative, and so steep that it is drawn vertical here, because ice is the less dense phase and pressure therefore melts it. Two things follow that no amount of measurement could adjust. The three slopes are not independent: the sublimation and vaporisation curves differ in slope by 5.92 Pa/K, which is exactly what the fusion latent heat predicts, because going solid → gas directly and solid → liquid → gas must cost the same enthalpy. And the sublimation curve is the steeper of the two, so the two cross rather than touch — which is why the triple point is a crossing and not a tangency, and why ice sublimes below it and melts above it.

Why the triple point is a point

Three phases of one substance coexist at one temperature and one pressure and nowhere else, and the reason is arithmetic rather than chemistry: count the numbers that describe the state, count the conditions equilibrium imposes, and subtract. The same subtraction says four phases of one substance are impossible, and it says so without knowing what the substance is.

thermodynamics · Phase change How much a wrap holds, against how many turns it is. The ratio of the two tensions a rope can hold across, against the number of turns it is wrapped, for coefficients of 0.1, 0.25, 0.5. The axis is logarithmic because the law is exponential, so each line is straight and its slope is the coefficient. At µ = 0.1 one turn multiplies by 1.9, two turns by 4 and three by 7 — so a person pulling with the strength of one arm holds a load that a small crane would be needed to lift. The practical consequence is the one a sailor states as a rule: turns are cheap and each is worth as much as the one before it, which is a statement about a constant factor rather than a constant force.

The part of the wrap that is actually gripping

The capstan equation gives the largest tension ratio a wrap can hold, and almost nothing spends its life at that limit. Below it the wrap divides in two: an idle arc doing nothing at all and an active arc creeping and carrying the whole exponential — and the division explains a belt's speed loss and its squeal.

mechanics · Friction Melting curves, and the one that leans the wrong way. Melting temperature against pressure for water, benzene, naphthalene, each measured from its own melting point at one atmosphere, with pressure in bars. The slope of every coexistence line is the latent heat divided by the temperature and the change in volume, and the latent heat of melting is positive for everything — so the sign of the slope is the sign of the volume change, and nothing else. Almost everything expands on melting and its line leans forwards. Water's solid is less dense than its liquid, so its line leans backwards at 135 bars a kelvin: pressing on ice at just below zero melts it, and it takes 135 atmospheres to gain a single degree. The anomaly is not in the thermodynamics; it is in the fact that ice floats.

The melting curve that leans the wrong way

The slope of any coexistence line is the latent heat divided by the temperature and the change in volume. Latent heat is always positive, so the sign of the slope is the sign of the volume change — and for water the volume change is negative, which is the whole of why ice floats and why the melting curve leans backwards.

thermodynamics · Phase change Effusion rate against molecular mass. The rate at which a gas escapes through a small hole, against its molar mass, normalised to hydrogen. The rate is a quarter of the number density times the mean speed times the area, and the mean speed goes as the inverse square root of the mass — so the rate does too, which is Graham's law of 1848. Hydrogen escapes four times faster than oxygen and 13.3 times faster than uranium hexafluoride. The practical consequence is isotope separation, and its difficulty is on this chart. The two uranium hexafluorides differ by 3 out of 352 in mass, so a single stage enriches by a factor of only 1.00429 — four parts in a thousand. Reaching 90 per cent from natural uranium's 0.72 per cent therefore takes about 1665 ideal stages, and a real cascade needs more because each stage is imperfect. That number is why gaseous-diffusion plants were among the largest industrial structures ever built, and why centrifuges — which separate by mass directly rather than by the square root of it — replaced them.

The gradient that drives the other thing

A concentration gradient drives a flow of matter and a temperature gradient drives a flow of heat. Each also drives the other, by coefficients that are equal — a relation nobody could have guessed and which follows from the fact that the underlying motion runs the same forwards and backwards in time.

thermodynamics · Diffusion The wedge that proves it. A wedge of water 4 mm on its vertical side, at a depth of 3 m, with the pressure on each of its three faces as an unknown. The two force balances decide them. Horizontally, the sloping face's push has a component that must exactly cancel the vertical face's, and since the sloping face is longer by exactly the factor its slope reduces the component by, the two pressures are equal — the geometry cancels, at every angle, for every size. Vertically the same cancellation happens except for the wedge's own weight, which needs the bottom face to carry 0.0667 per cent more. That excess falls in proportion to the size of the wedge, so at a point it is nothing and the three pressures are one number. Pressure being the same in every direction is the conclusion of that argument, not an assumption in it.

The push that has no direction

That the pressure at a point in a still fluid is the same whichever way the surface faces is not a definition. It is a theorem, and its proof is an argument about how two kinds of force scale with size — which is also the exact statement of when it stops being true.

fluids · Hydrostatics The fraction of the pressure a membrane can hold. The osmotic pressure actually developed across a membrane against a 300 mol/m³ solution at 298 K, as the reflection coefficient runs from a membrane the solute passes freely to one it cannot pass at all. The line is straight with the van 't Hoff pressure 743.69 kPa as its slope — checked against the drawn line — because the coefficient enters as a simple factor. At σ = 1 the pressure is 743.69 kPa; At σ = 0.6 the pressure is 446.21 kPa; At σ = 0.2 the pressure is 148.74 kPa. Van 't Hoff's law is the ceiling rather than the answer, and a membrane's coefficient against a given solute is as much a property of the pair as the concentration is of the solution.

The membrane that almost holds

Van 't Hoff's law gives the osmotic pressure a perfectly selective membrane would develop, and no membrane is. What a real one develops is a fraction of it — a number between zero and one that belongs to the membrane and the solute together, and that decides whether a solution is isotonic in effect or only on paper.

fluids · Osmosis The energy a spin is allowed, and the only direction it can go. A body with principal moments 1.000, 3.000, 4.000 spinning with a fixed angular momentum. Every possible motion has an energy somewhere in the band drawn here, and the three marks are the three principal-axis spins: energy L²/2I, so the largest moment of inertia gives the smallest energy. The band runs from 0.1250 to 0.5000 in units where the momentum is one — a ratio of 4.0. Anything inside the body that flexes and warms takes energy out and leaves the momentum untouched, so the state can only move leftwards along this band, and there is exactly one place for it to stop. A spin about the axis of least inertia is at the far right: it is a perfectly good solution of the equations of motion, stable against small disturbances in a perfectly rigid body, and it sits at the top of a hill the smallest leak will roll it off.

The axis a leak of energy chooses

A body spinning with nothing pushing on it keeps its angular momentum exactly, and keeps its kinetic energy only while nothing inside it flexes. At fixed momentum the energy is least for a spin about the axis of greatest inertia — so any leak, however small, has a destination. The first American satellite found this out in orbit.

mechanics · Rotation 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.

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.

fluids · Osmosis 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.

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.

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

The same force whichever way the surface faces

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

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

The ratio that decides whether the planet is turning

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

mechanics · Circular motion Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore.

The chain that runs at its slowest member's rate

Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives.

quantum · Decay The entropy a substance keeps depends on how fast it was cooled. The entropy of a supercooled liquid in excess of its crystal's, against temperature, for a substance melting at 305 K with an entropy of fusion of 43 joules per kelvin per mole and a liquid heat capacity exceeding the crystal's by 62. The equilibrium curve — the one the liquid follows while it can still relax — falls steadily and would reach the crystal's entropy at 152.4 kelvin. It never gets there, because the liquid falls out of equilibrium first, at a temperature that depends on how long it is given: 0.01 s/K freezes at 196.1 K with 15.61 left, 1 s/K freezes at 188.1 K with 13.03 left, 100 s/K freezes at 182.1 K with 11.03 left. Ice's residual entropy is a count and does not move; a glass's is whatever it happened to have when it stopped being able to change, and that is a property of the experiment.

The entropy that depends on how fast it was cooled

Ice's residual entropy is a count, and it comes out the same whoever measures it. A glass's does not. A glass keeps whatever entropy it happened to have when its own relaxation time crossed the experiment's, so cooling ten times more slowly leaves less behind — and extrapolating the equilibrium liquid below that point takes its entropy under the crystal's at a finite temperature, which cannot happen and does not, for a reason that is still argued about.

thermodynamics · Third law The entropy each degree of freedom has not yet given up. The entropy carried by each kind of degree of freedom, drawn against the temperature at which it orders and hands that entropy over. Lattice vibrations freeze out around room temperature; electron spins in a paramagnetic salt order in the millikelvin range, which is what makes adiabatic demagnetisation work; and nuclear spins hold R ln(2I+1) — 11.5 joules per kelvin per mole for copper — down to some tens of nanokelvin, where their own dipolar interactions finally sort them out. A copper sample at a microkelvin therefore has a large entropy and violates nothing: its nuclear spin system has not reached its ground state, and the third law is a statement about ground states rather than about thermometers.

A law about spectra, not about heat

The third law is usually met as a statement about cooling. Its statistical form is a statement about a spectrum: the entropy of a system in its ground state is k ln g, and it vanishes only when the ground state is unique. Every apparent exception is a degeneracy or a system that never reached its ground state — and copper nuclei carry eleven joules per kelvin per mole down to a hundred nanokelvin without violating anything.

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

The gas that flows towards more of itself

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

thermodynamics · Diffusion 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.

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.

fluids · Buoyancy A blackbody in every direction, at a different temperature in each. The spectrum of a blackbody at 100 kelvin in its own frame, seen by an observer it is moving past at 0.5 of the speed of light, in 5 directions. Each curve is a Planck spectrum exactly — the Planck form survives a Doppler shift, with the temperature multiplied by the shift — and the temperatures run from 57.74 kelvin looking one way to 173.21 looking the other. So the body is a perfect blackbody in each direction and has no single temperature. A thermometer placed in the radiation reads something between, and what it reads depends on where it is put and on how much of the sky it sees — which is the reason a transformation law for temperature was argued about for sixty years without being found.

The body that has no temperature when it moves

Energy, momentum, length, duration and field strength all change when the observer moves. Temperature was argued about for sixty years, with three transformation laws proposed and each defended by people making no mistake. The resolution is that a moving blackbody is a perfect blackbody in every direction at a different temperature in each — so a thermometer's reading depends on where it is put, and the quantity the law was for is not there.

relativity · Relativistic thermodynamics What a boost leaves alone, and what it does not. How each quantity of a box of blackbody radiation changes when the observer moves at 0.8 of the speed of light, a Lorentz factor of 1.667, on a logarithmic axis with one at the centre. The top four do not change at all, and the reason is the same in each case: they are counts, or logarithms of counts, or invariants built from four-vectors. A number of photons is a number, and every observer arrives at the same number. The rest change, by powers of the Lorentz factor that follow from the first four. And the entry that matters is the last two together: the energy density rises as the square of the factor while the entropy density rises as the factor itself, so the ratio between them that would define a temperature does not stay fixed — which is why the boosted radiation cannot be a blackbody at any temperature at all.

The count that no observer can disagree about

A moving body, it turns out, has no temperature. What it does have is an entropy, and every observer agrees about it — because entropy is the logarithm of a count of arrangements, and a count is a number. That one invariant, with energy and momentum being parts of one object, is enough to compute everything a temperature could not: what happens to the energy density, the entropy density, and the relation between them that having a temperature consists of.

relativity · Relativistic thermodynamics

Named alongside it

The objects these essays reach for when they reach for this one.

StabilityEntropyBuoyancySurface tensionTemperatureChemical potentialDensityDiffusionHydrostaticsLatent heatPressureScaling

All concepts