Illustrated Physics
physics, drawn
The collection

Every essay — page 4

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.

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.

5 figures · part 1 on Friction One parabola, several wells. Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

Every minimum is a parabola

A pendulum, a bond between two atoms and a ship rolling in a swell obey the same equation, and the reason is not that they are alike. It is that a function with a minimum has no linear term there, so the first thing every potential well looks like is the same well.

7 figures · part 1 on Harmonic approximation One puck, two frames. A puck slides outward from the centre of a turntable at 1.4 m/s across a table of radius 1 m, which turns through 0.55 of a revolution in the 0.71 seconds the crossing takes. On the left, in the room: a straight line, because no force acts along it. On the right, on the turntable: a spiral, because the coordinates turned underneath it. The two panels are the same numbers with the same clock — the second is the first with the angle rotated by −Ωt — so whatever is bending the path on the right is arithmetic, and it is still worth a name, because on a turning planet the right-hand panel is the one everybody lives in.

The forces that are not there

Writing Newton's law in a frame that is turning produces three extra terms. Nothing was added to the world to make them appear and nothing is removed by calling them fictitious — one of them flattens the planet, one of them turns the weather, and both are computable to four figures.

8 figures · part 2 on Circular motion

Electromagnetism

Charge, field, and the lines drawn between them.

Same field, same charge, three momenta. Electrons entering a 10 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 10.7 mm, 21.3 mm, 32.0 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment.

The force that does no work

A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.

8 figures · part 2 on Magnetism Four paths, one answer. The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

The field that wraps a current

Ampère's law says that going once round a closed path and adding up the field counts the current threaded through it, and nothing else about the path survives into the answer. Four different loops round one wire return the same number to five decimal places — which is exactly why the law is both easy and treacherous.

7 figures · part 1 on Ampere law

Thermodynamics

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

A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

How far a molecule gets

A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot.

4 figures · part 4 on Kinetic theory Seven walks from one point. 7 random walks of 400 steps each, all starting at the same place. None of them goes anywhere in particular and none of them stays put; the typical distance reached after n steps is the square root of n, so quadrupling the time doubles the spread. The tracks are seeded, so this is a property of the figure rather than of any one run.

The jiggle that proved atoms

A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

4 figures · part 2 on Diffusion One mass, two entropies. The entropy of a solar mass as ordinary gas, generously counted at ten Boltzmann constants per proton, against the entropy of a solar-mass horizon. The first is 1.19·10⁵⁸ k and the second 1.05·10⁷⁷ k — a factor of 8.83·10¹⁸. This is why a horizon had to be given an entropy: without one, dropping anything at all through it destroys entropy and the second law fails.

The entropy that lives on a surface

Throw a cup of tea through a horizon and the entropy of the outside world falls. Either the second law is wrong or the horizon has an entropy of its own — and the only quantity available for it turns out to be its area, in units of a length made from gravity, quantum mechanics and the speed of light together.

4 figures · part 2 on Entropy

Relativity

Space and time, drawn on the same axes.

A spacetime diagram at β = 0.6. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.6 of the speed of light.

The twin who comes back younger

If motion slows a clock, and motion is relative, each twin should find the other younger — and yet when they meet, one of them has aged less. The asymmetry is not in the speed and not in the acceleration; it is in which worldline is straight.

4 figures · part 2 on Time dilation Three answers where sound has two, and one where it has none. The factor by which an approaching source's frequency is raised, against its speed as a fraction of the wave speed. For sound it matters which of the two is moving: a moving source gives 1/(1 − β) and a moving observer gives 1 + β, and at 0.5 of the wave speed those are 2.000 and 1.500. For light there is one answer, 1.732 — the geometric mean of the other two, exactly — because there is no medium to be moving with respect to. The fourth curve is the transverse shift, which happens at closest approach when the distance is not changing at all: 0.866, and nothing classical predicts it.

The shift that survives at right angles

For sound it matters which of the two is moving, and the two answers differ. For light there is one answer — their geometric mean — and a term with no classical counterpart at all: a source going past at closest approach, with its distance not changing, is still shifted.

4 figures · part 2 on Doppler

Quantum

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

Two transitions, one shape and one not. The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do.

The liquid that will not slow down

Cool helium below 2.17 kelvin and it starts flowing through gaps no ordinary liquid could enter, climbs out of its own container, and circulates for as long as anyone has been willing to watch. The viscosity is not small. As far as any measurement can tell, it is zero.

4 figures · part 1 on Superfluidity

Fluids

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

Speed against wavelength. Phase and group speed for waves on water 4 m deep. Short waves are dispersive — the speed rises as the square root of the wavelength and the group travels at half the phase speed — and long ones all travel at √(gh) = 6.26 m/s together, which is why a tsunami keeps its shape across an ocean while a wind sea spreads out into swell.

The speed that depends on the length

Long waves on water travel faster than short ones, which is why a distant storm arrives as a slow swell and a tsunami crosses an ocean without spreading. One relation covers both, and the two familiar rules taught separately are its two limits.

4 figures · part 2 on Wave packets

Astrophysics

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

Light crossing an accelerating box. A pulse crosses a box 6 m wide while the box accelerates at 9.81 m/s². The crossing takes 2·10⁻⁸ s, in which the far wall gains 1.96·10⁻⁷ m/s, so the pulse lands 1.96·10⁻¹⁵ m below the height it left at — and the path is a parabola. An observer sealed inside cannot tell that from a beam of light bending in a gravitational field, and the equivalence principle says there is nothing to tell. The sag is drawn 5.6·10¹⁴ times its true size.

The floor that cannot be told from gravity

Seal a laboratory, take away the windows, and no experiment inside it can distinguish standing in a gravitational field from accelerating through empty space. That is not a philosophical remark — it forces light to bend, forces clocks to disagree, and has a size at which it stops being true.

4 figures · part 1 on Equivalence principle Where a clock gains, and where it loses. The rate of a clock in a circular orbit against one on the ground, in microseconds per day, plotted against altitude. Height makes it gain and speed makes it lose, and the two cancel exactly at 3186 km — where a satellite keeps the same time as the ground for two reasons that have nothing to do with each other. At 20200 km the total is 38.5 µs a day, which is about ten kilometres of position error if it is ignored.

The clock that runs slow lower down

Two identical clocks, one on the floor and one on a shelf, do not keep the same time — and the difference is large enough that a satellite navigation system which ignored it would be useless within a morning. The derivation needs nothing but a photon and a conservation law.

4 figures · part 1 on Gravitational redshift Two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit.

The bend Newton got half right

A photon treated as a falling body passes a mass and comes out deflected. So does a photon treated as a straight line in curved spacetime — by exactly twice as much. The factor of two is not a refinement; it is the sharpest available statement that gravity is geometry.

4 figures · part 1 on Light deflection How small each mass would have to be. The Schwarzschild radius of 4 masses, on a logarithmic scale spanning 35 orders of magnitude. A horizon is not something a mass has; it is a size a mass would have to be squeezed inside. For the Sun it is 2.95 km against a real radius of 696,000 km, a factor of 2.36·10⁵. One row has no real size to set beside the number, which is the one case where the horizon is not hypothetical.

The surface that only lets things in

An eighteenth-century calculation asking where the escape speed reaches the speed of light gives exactly the right radius, by reasoning that is wrong in every step. What is actually there is not a surface in space at all, and nothing local happens when it is crossed.

5 figures · part 1 on Horizons What the distant observer actually receives. The frequency of a signal from a clock falling into a horizon, as received far away, against the receiver's own time. It is a straight line on a logarithmic axis, which means the fading is exponential: the e-folding time fitted to the drawn curve is 2.01 rs/c, which for a 10-solar-mass hole is 198 microseconds. Nothing hovers. The image reddens, the photons arrive at an exponentially falling rate, and within a millisecond there is nothing left to see.

Two clocks that disagree about the fall

A clock falling into a horizon crosses it in a few milliseconds by its own reckoning and never crosses it at all by a distant one. Both accounts are right, and the thing everybody remembers about the second — that the image hangs there for ever — is wrong.

4 figures · part 2 on Horizons What a passing wave does to a ring. A ring of 8 free masses at 4 phases of a passing gravitational wave, in both polarisations. The upper row is the + mode: one diameter lengthens while the perpendicular one shortens, and half a cycle later they swap. The lower row is the × mode, which is the same pattern rotated by forty-five degrees rather than ninety — the signature of a spin-2 field, and the reason a detector is built as two arms at a right angle. The drawn strain is 0.42; a real one is 10⁻²¹, so the deformation is exaggerated 4.2·10²⁰ times. At that true strain a four-kilometre arm changes length by 4·10⁻¹⁸ m.

The wave that stretches one way and squeezes the other

A gravitational wave passing through a ring of free masses lengthens one diameter while shortening the perpendicular one, then swaps. The two conservation laws that forbid anything simpler are why the effect is a part in a thousand million million million.

4 figures · part 1 on Gravitational waves Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

Light has a pressure

Sunlight pushes on a square metre with about the weight of a grain of sand, which sounds like a curiosity until the object being pushed is small enough. The demonstration in every school cupboard turns the wrong way, and the reason it does is more interesting than the effect it is supposed to show.

4 figures · part 1 on Radiation pressure The classical atom, and how long it lasts. An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion.

A charge that turns must glow

An accelerating charge radiates, and a charge going round in a circle is accelerating. Apply that to an electron orbiting a nucleus and classical physics predicts that every atom collapses in sixteen picoseconds — a calculation with nothing wrong in it except its conclusion.

6 figures · part 1 on Radiating charge The same law, across twenty-eight decades. The mean free path 1/nσ against cross-section, for a target density of 6.83·10³⁰ targets per cubic metre — solid lead. It is a straight line of slope minus one, because there is only one thing in the law. At 10⁻²⁸ m² the path is 1.46 mm; at 10⁻⁴⁷ m² the path is 1.55 light-years. Nothing about the physics changes between those ends. Only the area does.

How far a neutrino gets

A mean free path is one over the number density times the cross-section, and nothing else. Change only the cross-section — by twenty-eight powers of ten — and the same arithmetic that gives a molecule seventy nanometres in air gives a neutrino a light-year of solid lead.

4 figures · part 5 on Kinetic theory What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

The field that cannot get out

Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.

4 figures · part 1 on Flux freezing 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.

4 figures · part 1 on Self-gravity The two lengths every mass has. The Compton wavelength and the Schwarzschild radius of the same mass, against mass, on logarithmic axes. One falls and the other rises, so they cross exactly once — here at 1.539·10⁻⁸ kg and 2.286·10⁻³⁵ m, found by bisecting the difference rather than by writing down √(ħG/c³). The conventional Planck values are 2.176·10⁻⁸ kg and 1.616·10⁻³⁵ m; the crossing sits a factor of 1.414 away from them, which is exactly √2 and is the factor of two in the Schwarzschild radius coming through a square root. That is the whole precision this argument has, and it is worth saying, because a number written to four figures invites a reader to believe the definition is doing more work than it is. Nothing in physics is known at that length.

Where every model runs out at once

Every mass carries two lengths — one below which quantum mechanics will not let it be located, one below which gravity will not let anything escape. One falls with mass and the other rises, so they cross exactly once, at a length nothing in physics has ever probed.

4 figures · part 1 on Planck scale

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