True as far as it goes, and silent on what decides — part 2 of 5

“Aberrations come from the surface not having the right shape.”

The paraboloid has exactly the right shape for the axis and the wrong one for everything else, and no single surface has the right shape for both. Abbe's condition is a constraint on the *mapping* between entrance height and exit angle, which one surface has too few degrees of freedom to satisfy alongside stigmatism — which is why an aplanatic system has two.

Tested in The condition a lens must meet · the imaging reading path

“An interferometer measures the diameter of a star.”

It measures one Fourier component of the brightness distribution per baseline. Three sources drawn here — a uniform disc, an equal double star and a Gaussian — lose their fringes at the same separation and are completely different objects; an observer who measured only the first null would report the same size for all three and be wrong about two of them.

Tested in The fringe that measures a star · the coherence reading path

“Optical fibre is used at 1,550 nanometres because that is where it works best.”

That is where the loss is least, at about 0.2 decibels a kilometre. It is not where the dispersion is least: silica's material dispersion crosses zero at 1,273 nm and a standard fibre's total at 1,310, so the operating wavelength has the lowest loss and 18 ps per nanometre per kilometre of dispersion. The two optima are at different wavelengths and every system design is a negotiation between them.

Tested in The wavelength a fibre does not smear · the dispersion reading path

“Gauge freedom is a mathematical convenience with no physical content.”

It has content wherever a gauge-dependent quantity appears in a physical argument. The canonical momentum mv + qA is what Hamiltonian mechanics conserves and what quantum mechanics turns into a derivative, and it is gauge dependent — the two curves drawn here for one trajectory differ completely. What is gauge invariant is its integral round a closed loop, which is the enclosed flux, and that is why the Aharonov–Bohm phase is observable.

Tested in The potentials that are not unique · the maxwell equations reading path

“A thicker conductor carries alternating current better.”

Past about a skin depth it does not. The exact slab calculation gives the resistance ratio as the half-thickness divided by the skin depth once the slab is thick, so the alternating resistance climbs as the square root of the frequency for ever regardless of how much metal is added. That is why heavy radio-frequency conductors are hollow and why litz wire exists.

Tested in How far a field gets into metal · the conductors reading path

“The infinite-wire formula is a good approximation for laboratory wires.”

It is, and the figure says how good: a wire ten times as long as the distance to the field point gives 98 per cent of the infinite answer, one as long as the distance gives 45, and one half as long gives 24. The approximation is excellent where it is used and collapses quickly, and the crossover is at a length ratio of about five rather than at a length.

Tested in The law that is always true and rarely useful · the ampere law reading path

“Interstellar travel is an engineering problem.”

The mass ratio of a photon rocket — the best any rocket can be — is e raised to the total rapidity, which is 38 times the payload for a round trip to Proxima and 10^18 for one to the galactic centre. No improvement in exhaust velocity is possible past a photon rocket and no improvement in anything else changes an exponent, so the difficulty is arithmetic rather than technological.

Tested in The ship that never arrives at c · the accelerated frames reading path

“The experiment confirmed time dilation.”

It confirmed the sum of two effects with opposite signs, which is a sharper test than either alone: a theory getting one term right and the other wrong fails on the eastward flight even while passing on the westward one. What it did not do is reach high precision — the quoted uncertainties are twenty to thirty per cent — and the modern confirmations come from satellite navigation, where the same sum is corrected for continuously.

Tested in The two clocks that flew in opposite directions · the time dilation reading path

“The Chandrasekhar limit is where the star's gravity overcomes the electron pressure.”

There is no such crossing at a finite radius. The limit is where the equation of state's exponent reaches four-thirds, at which the mass of a self-gravitating polytrope is independent of its radius — so the mass is fixed and the radius drops out. Squeezing a star at that exponent buys no additional support at any size, which is a different statement from a force balance failing.

Tested in The mass no cold matter can hold up · the Self-gravity reading path

“A whistler is a descending tone, so low frequencies travel slowest.”

The group velocity is zero at zero frequency and zero again at the gyrofrequency, with a maximum at a quarter of the gyrofrequency — which is found by searching the drawn function here rather than quoted. So the arrival curve has a nose: everything either side of that peak arrives later. The descending tone is the branch below the nose, and the rising branch above it is observed too.

Tested in The whistle that arrives sorted · the plasma oscillation reading path

“4s fills before 3d because 4s is lower in energy.”

It is lower for the neutral atom at the start of the fourth row and not thereafter. The crossing depends on the nuclear charge and on how many electrons are already present, so it moves as the row is filled — which is why the transition metals ionise from 4s rather than 3d, and why a simple ordering rule gets the filling order right and the ionisation order wrong.

Tested in The order the shells fill · the atomic structure reading path

“Nuclear spin is a small correction, since nuclear magnetic moments are a thousandth of electronic ones.”

The magnetic effect is small and the statistical effect is not. Hydrogen's rotational lines alternate three to one in strength, exactly, because the two protons are spin-halves and the molecule's state must change sign when they are swapped — and its low-temperature heat capacity is that of a frozen 3:1 mixture rather than of an equilibrium one, a discrepancy that stood unexplained for fifteen years.

Tested in Four states, and one of them is odd · the spin reading path

“A refractive index below one would let light travel faster than c, so it cannot happen.”

It happens for every material at high enough frequency, and X-ray mirrors are built on it. What travels at the phase speed is a crest, which carries no news; the sum rule fixes the total absorption a medium can have and forces the index to approach one from below above every resonance, and nothing in that arrangement moves a signal faster than light.

Tested in The answer that cannot come first · the attenuation reading path

“Huygens' principle says every point on a wavefront is a source of wavelets, and the next wavefront is their envelope.”

That construction gives the right answer in three dimensions and the wrong one in two, where the disturbance behind the front is not zero and cannot be an envelope of anything. The principle in its strict form — that a disturbance is confined to the front — holds only in odd dimensions above one, and Huygens' own argument had no way to see that.

Tested in The arrival that keeps arriving · the wave motion reading path

“A spectrum determines the shape that produced it.”

It determines the area and the perimeter, and in general not the shape. Pairs of differently shaped drums with identical spectra were constructed in 1992, settling a question open since 1966. What the leading terms of the count give is the area, from how fast the count grows, and the perimeter, from how far it lags.

Tested in How many ways there are to vibrate · the standing waves reading path

“The best angle for a driven golf ball is about thirty degrees, because that is what the drag calculation gives.”

Drag alone gives 38° for a golf ball's ballistic parameter, and the real answer is between ten and fifteen. The difference is lift: a backspinning ball generates an upward force that a drag-only model has no term for, so the calculation here is the right answer to a question that is not quite the one a golfer is asking.

Tested in The angle that drag moves · the projectile reading path

“The pendulum stands up whenever the shaking is fast enough.”

Fast enough and not too violent. The integration finds a band: below aΩ = √(2gL) the inverted position is not an equilibrium at all, and above an upper edge measured here at a fifth to a third of the pendulum's length the fast motion stops being small and the pendulum is thrown out. Stability is a band, not a threshold.

Tested in Held up by a force that averages to nothing · the pendulum reading path

“The principle is an elegant restatement, but the equation of motion is what does the work.”

They are equivalent for a point particle and not interchangeable in practice. The action is a single scalar, so it makes coordinate changes trivial, extends to fields and to relativity unchanged, and connects symmetries to conservation laws directly. And it is the classical limit of the sum over paths, which the equation of motion has no route to.

Tested in Least action, except that it is not least · the least action reading path

“The angle of repose is a property of the material.”

It is a range, not a number. A slope steeper than the angle at which flow stops will keep flowing once started, and one shallower than the angle at which flow starts will not begin — so the same sand has stable slopes anywhere between about 31° and 35°, and where a particular heap sits in that band depends on how it was built.

Tested in The angle that does not know the size of the heap · the granular matter reading path

“The energy needed is the osmotic pressure times the volume produced.”

That is the answer for the first drop only. Taking a fraction of the feed leaves the rest saltier, so the last drop is pushed against a higher pressure than the first: at half recovery the reversible cost is 1.10 kWh/m³ against 0.79, and a single stage held at the final pressure throughout costs 1.58.

Tested in What it costs to take the salt out · the osmosis reading path

“The fountain effect is a curiosity of a laboratory liquid.”

The same relation ΔP = ρSΔT is used as an instrument. Fountain pumps with no moving parts circulate helium in space cryostats, and thermomechanical pressure is how a superfluid's entropy is measured — the effect is the standard way of getting at S for a liquid that no calorimeter can be inserted into without heating it.

Tested in The fountain a lamp can drive · the superfluidity reading path

“The Jeans analysis derives the instability of a uniform self-gravitating gas.”

It perturbs a background that cannot exist. A uniform self-gravitating medium is not static — it is already collapsing — so the unperturbed state does not solve its own equations, which Jeans knew and which is called the Jeans swindle. Redoing it in an expanding background gives the same criterion with a growth that is a power of time rather than an exponential.

Tested in The disturbance that grows instead of travelling · the Self-gravity reading path

“Debye screening means the electric field inside a plasma is zero.”

It means a static excess charge is screened over a length λ_D, which is not the same as no fields. Fields on scales larger than λ_D exist and dominate the dynamics — that is what plasma waves, sheaths and instabilities are made of — and the screening length is precisely the scale below which charge separation is expensive and above which the plasma is neutral.

Tested in The long-range force that does not reach · the plasma oscillation reading path

“The limit explains why there is a maximum stellar mass.”

It sets the scale and does not do the work alone. A 100-solar-mass star sits at 0.83 of its limit, so radiation pressure dominates its structure, drives a strong wind and makes it unstable — but the observed upper limit near a few hundred solar masses also involves how such stars form and how they shed mass, and none of that is in the ratio.

Tested in The brightness a mass cannot exceed · the radiation pressure reading path

“Supernumeraries are rare because they need unusual conditions.”

They need small drops and nothing else. Their angular spacing goes as the two-thirds power of wavelength over drop radius, so drops larger than about 0.65 mm crowd them closer together than the sun's own half-degree width and the contrast washes out. The physics is always there; what varies is whether a source half a degree wide can show it.

Tested in The fringes below the rainbow · the diffraction reading path

“The phase depends on the sequence of optical elements used to produce it.”

It depends only on the circuit those elements trace on the sphere of states. Six circuits of quite different shapes are computed here and every one gives minus half its own enclosed solid angle, to 10⁻¹⁶ radians — so two entirely different sets of components tracing the same circuit give identical phases.

Tested in The phase that is only a shape · the polarisation reading path

“Isotopes can be separated by effusion because lighter molecules escape faster.”

They do, and the factor is the square root of a mass ratio, which for uranium hexafluoride is 1.00429. Reaching 90 per cent enrichment from natural uranium therefore needs about 1,665 ideal stages — which is why gaseous diffusion plants covered square kilometres, and why centrifuges, separating by mass rather than by its square root, replaced them.

Tested in The gas that leaves is not the gas inside · the kinetic theory reading path

“Classical nucleation theory gives the rate of condensation.”

It gives the exponent well and the prefactor badly, and measured rates depart from it by many orders of magnitude for some substances. The theory survives because the exponent dominates: an error of 10⁵ in the prefactor moves the predicted onset supersaturation by under one per cent, which is smaller than the experimental scatter.

Tested in The barrier a new phase has to climb · the phase change reading path

“The effect has never been observed, so it is speculative.”

It has not been observed directly, and it is a prediction of ordinary quantum field theory in flat spacetime with no new physics in it — the same framework that gives the Lamb shift. What is speculative is not the effect but the interpretations attached to it; the calculation is as standard as any in the subject.

Tested in The temperature of an acceleration · the accelerated frames reading path

“This means relativity is only a convention.”

The conventional part is the synchronisation and nothing else. The invariance of the interval, the round-trip constancy, time dilation between clocks brought back together and every measured prediction are unaffected by the choice — which is why the theory can be written in an anisotropic convention and still make identical predictions, only with uglier equations.

Tested in The speed that cannot be measured one way · the simultaneity reading path

“The theorem is about electrostatics.”

It is about any potential obeying Laplace's equation, which includes the magnetic scalar potential in a current-free region and the gravitational potential in empty space. That is why there is no stable point in the Earth–Moon system's gravity alone except where the rotation supplies something Laplace's equation does not.

Tested in Nothing can be held still by a static field · the potential reading path

“Decoherence needs a measuring apparatus, or at least a macroscopic environment.”

It needs one scattered photon. A single air molecule bouncing off a dust grain carries away enough information to distinguish two positions a micrometre apart, and the coherence is gone. In the emptiest achievable vacuum the microwave background left over from the early universe still does it in about a hundred seconds.

Tested in Where the interference goes · the measurement reading path

“The theorem is a tidy restatement of things already known from the force law.”

It is the only route to most of them. There is no way to read a conserved quantity off a field theory's equations by inspection, and the conserved currents of the Standard Model were all obtained this way; the theorem also delivers quantities nobody would have guessed, including the eccentricity vector of an inverse-square orbit, whose symmetry is a rotation in four dimensions and is invisible in three.

Tested in The conservation law a symmetry hands over · the least action reading path

“Treating the legs as springs fixes the problem.”

It closes the arithmetic and moves the difficulty. The answer then depends on stiffnesses that are rarely known to better than a factor of two, on the floor, which is not four independent springs, and on the manufacturing tolerance of the legs — which the figures here show dominating the load sharing at a shortfall of a hundredth of the leg's own compression.

Tested in The table statics cannot settle · the Free-body reading path

“The equation breaks down at the shock, so the physics stops there.”

The physics is unchanged; the approximation stops. A real front steepens until the gradient is large enough for viscosity and heat conduction to matter, which happens at a thickness of a few mean free paths, and there it settles into a travelling structure of finite width. What is lost is not the physics but the assumption that the medium is a continuum with no dissipation, and the shock thickness is precisely where that assumption is repaid.

Tested in The front that steepens until it cannot · the wave motion reading path

“Russell saw a solitary wave on a canal, so the phenomenon is a curiosity of shallow water.”

The same balance appears wherever a nonlinearity and a dispersion of the right signs meet: in optical fibres, where it carries data at rates that a spreading pulse cannot; in Bose–Einstein condensates; in Josephson junctions; and in the atmosphere as the Morning Glory cloud. The equation is the same equation, and the sech-squared profile with amplitude fixing width is the same profile.

Tested in The pulse two failures keep alive · the wave packets reading path

“A liquid's viscosity follows an Arrhenius law with a single activation energy.”

It follows one over a modest range and nowhere else. Water is measurably non-Arrhenius over a wide interval, and a glass-forming liquid approaching its transition curves away from any straight line so steeply that no single activation energy describes it — the viscosity rises through thirteen decades over a temperature interval in which nothing structural is happening that a diffraction pattern can see.

Tested in The thickness that goes both ways · the viscosity reading path

“Maxwell's counting rule settles when a structure is rigid.”

It gives a necessary condition and not a sufficient one. Counting says a pack is rigid above four contacts per grain in the plane; taking the rank of the actual rigidity matrix shows local regions that are over-braced sitting beside regions that are still floppy, so a pack can satisfy the average and have a mechanism. The rank is what decides, and the count is what predicts the rank when nothing is special about the arrangement.

Tested in The heap that becomes a solid · the granular matter reading path

“The Roche limit is 2.44 planetary radii.”

That is the fluid case with the two densities equal, and both conditions matter. A rigid body of the same density comes apart at 1.26 radii; a body denser than the primary survives closer in; and the drawn curve varies as the cube root of the density ratio, so an iron satellite of a gas giant has a limit less than half an icy one's. Quoting the number without the ratio is quoting a coefficient as if it were a law.

Tested in The distance that forgets the moon · the Self-gravity reading path

“The escape time follows from the mean density and the opacity.”

That gives about two thousand years and the accepted figure is nearer 170,000. The gap is the density profile: the Sun is a hundred times denser at its centre than on average, the free path there is correspondingly shorter, and most of the journey time is spent in the innermost fraction of the radius. The uniform estimate gets the exponent and the mechanism right and the number wrong by two orders, and says which.

Tested in The light that takes a hundred thousand years to leave · the kinetic theory reading path

“The 46° halo is the same effect at the other prism angle, so it should be equally common.”

It is the same effect and it is rare, for a reason the same calculation gives: the 90° prism transmits over a window of incidence only 32° wide against the 60° prism's 76°, so far fewer orientations contribute. Its minimum deviation is also more sensitive to the crystal's orientation out of the plane, which smears it further, and it needs cleaner crystals to be seen at all.

Tested in The ring at twenty-two degrees · the dispersion reading path

“The intensity in a speckle pattern fluctuates about its mean.”

It does not fluctuate about anything. The distribution is a decaying exponential with its maximum at zero, which the histogram here matches to eleven per cent in every well-populated bin with no fitted parameter. There is no typical brightness and no peak near the mean; the most probable intensity at any point is nothing at all, which is why the pattern looks like grain rather than like noise on a grey.

Tested in The grain that is in the light · the coherence reading path

“Grangier's experiment showed g² of zero, so a single emitter is all that is required.”

The observed dip was to about 0.18, not to zero, and the shortfall is mostly the number of atoms in the observation region at once. N independent emitters give exactly 1 − 1/N, so two atoms give 0.5 and ten give 0.9; the experiment needed a beam thin enough that two atoms were almost never present together, and the residual is a measurement of how well that was achieved.

Tested in The experiment a wave cannot pass · the photon reading path

“A system swept slowly through an avoided crossing follows the lower branch.”

It does if the sweep is slow compared with the gap squared over ħ times the rate, and the crossover is sharp in a way worth quantifying. The two-state equation integrated here gives a probability of jumping that matches Landau and Zener's exponential to 0.012 across a hundred-fold range of sweep rates — so doubling the gap makes the system four times harder to shake off the adiabatic path, and that quadratic is why a small gap is far more permeable than it looks.

Tested in The crossing that never happens · the bands reading path

“Exchange effects are small corrections to the Coulomb energy.”

The exchange splitting in helium's lowest excited configuration is 0.8 electronvolts, which is 9,000 kelvin, and the exchange coupling that makes iron ferromagnetic corresponds to 1,000 kelvin — against a magnetic dipole interaction between neighbouring atoms of about 0.1 kelvin. The whole of magnetism above liquid-helium temperature is exchange, and the actual magnetic interaction is four orders of magnitude too weak to matter.

Tested in The force with no force in it · the exclusion reading path

“The bending is caused by the polarisation charge that appears on the interface.”

That is a correct account and not an alternative one. The surface bound charge is what the discontinuity in the normal E *is*, and computing the field from it reproduces the same answer; the boundary-condition route reaches it without solving for the charge. Both are needed in practice — the conditions give the field and the bound charge says where the force on the dielectric acts.

Tested in A refraction with no wave in it · the dielectrics reading path

“Mutual inductance is symmetric, so a transformer works the same in both directions.”

The mutual inductance is symmetric and a transformer is not. The voltage ratio depends on the two self-inductances as well, so a step-up transformer run backwards steps down; what is symmetric is the coupling coefficient and the power transfer, not the voltage. Confusing the two is the commonest way the theorem is misused.

Tested in The coupling that is the same both ways · the induction reading path

“A superconductor expels magnetic flux from its interior.”

It expels flux from a region a penetration depth thick, which is the whole of a bulk sample and almost none of a thin film. The profiles here give 95 per cent expulsion for a slab forty depths thick and 7.6 per cent for one a single depth thick — same material, same field, same constant. That is why films stay superconducting in fields that destroy the bulk, and why the critical field of a film rises as its thickness falls.

Tested in The field that is pushed out · the superconductivity reading path

“A laser medium is at a negative temperature, which is a metaphor for population inversion.”

It is the definition applied. Putting the measured populations into the Boltzmann ratio returns a negative T, and the same number comes out of the slope of the entropy computed independently — the two agreeing here to a part in a million. What makes it a limited statement is that only the inverted levels are at that temperature; the lattice they sit in is not, which is why the state relaxes and why it can be prepared at all.

Tested in Hotter than any temperature there is · the third law reading path

“Van der Waals's equation predicts the inversion temperature at 6.75 times the critical temperature.”

It does, and real gases give between 4.9 and 8.7. The model gets the mechanism exactly right — the competition between an attraction that costs energy to separate and a volume that costs work to compress — and the coefficient to within a factor well under two. Quoting 6.75 as a prediction rather than as a model's answer is the usual way van der Waals is over-read.

Tested in The gas that cools by being let go · the phase change reading path

“The causal limit on the sound speed is a curiosity with no observable consequences.”

It caps the maximum mass of a neutron star at a little under three solar masses, whatever nuclear physics turns out to be, because a star is held up by stiffness and the stiffness has a ceiling. That bound is what makes a compact object above it a black hole by elimination, and it is how the 2.6-solar-mass companion in GW190814 was argued about before anything else was known.

Tested in Nothing is allowed to be rigid · the relativistic dynamics reading path

“The mass ratios here settle whether interstellar travel is possible.”

They settle the propellant, which is one of four difficulties and not the worst. Stopping at the far end squares the ratio and returning squares it again; the power required to reach a tenth of light speed in a human lifetime is thousands of times humanity's output; and a gram of dust at 0.1c arrives with the energy of a hundred kilograms of explosive. The chart is a lower bound on the difficulty.

Tested in The fuel a starship needs · the Mass-energy reading path

“Chaos is sensitive dependence on initial conditions.”

Sensitive dependence on its own is an instability, and an unstable system runs away. The figures here measure both halves at once on the same orbit: the separation of two trajectories grows at a rate of 0.4188 per step, and every one of the twenty-four thousand points lies inside a box 2.56 by 0.77. A system that does the first without the second flies apart and has no attractor; chaos is the first *subject to* the second, and the fold is what the constraint forces.

Tested in The fold that has to be there · the chaos reading path

“Huygens' cycloidal cheeks solved the pendulum clock's amplitude error.”

They solved it exactly, in theory, and the clocks kept worse time. The construction requires the string to wrap against a shaped surface, which adds friction where a plain suspension has almost none, and a real string has stiffness so it does not lie on the cheek as an ideal one would. The error removed was second order in the amplitude and the errors introduced were first order in the contact, so the trade was a bad one — which is a statement about the mechanism rather than about the mathematics, and the mathematics is exact.

Tested in The curve that does not ask where it started · the pendulum reading path

“On the edge of the geometrical shadow the light is half the unobstructed value, because half the wave has been blocked.”

Half the *amplitude* gets through, and intensity is amplitude squared, so the value on the boundary is a quarter: 0.250000 in these units, exactly, and it follows from the split rather than being measured. The half-and-half intuition is right about what is blocked and wrong about what is being counted, and the same slip turns a straight edge's 25 per cent into 50 wherever it is made.

Tested in The wave that comes from the rim · the huygens reading path

“The surface wave is confined to the surface.”

It is confined to a skin whose thickness is a wavelength, and the wavelength is the wave's own rather than the material's, so a long-period wave reaches deep and a short-period one does not. The vertical displacement at one wavelength down is 19.3 per cent of its value at the surface, and the horizontal displacement changes sign at 0.193 wavelengths — a reversal rather than a node. Because different periods sample different depths, a surface wave is dispersive in a layered Earth even though it is not dispersive in a uniform one, and that dispersion is how the crust is measured.

Tested in The wave a surface is enough to hold · the guided waves reading path

“A minimal surface is one that minimises area subject to its boundary.”

It is one whose area is stationary, which admits saddle points as well as minima. The two rings have two catenoid solutions at every separation below the critical one — one stable and one not — and they are both minimal surfaces in the technical sense. They merge and annihilate at 0.6627 ring radii, which is exactly the shape a fold in a family of solutions has, and which is why the film's disappearance is sudden rather than gradual.

Tested in The angles a film has no choice about · the surface tension reading path

“Polymer solutions climb rotating rods.”

The same solution at the same speed climbs a thin rod and is thrown off a thick one, and the changeover is computable: the hoop tension falls as the fourth power of the distance from the axis and the centrifugal term as the second, so the first wins only inside a radius the fluid's own constants fix. For the solution drawn here that radius is 34.6 millimetres — 10 and 25 millimetre rods climb by 2.21 and 1.16 millimetres, and a 60 millimetre one dips by 4.83. A demonstration that works with a glass stirring rod fails with a broom handle.

Tested in The liquid that climbs the rod · the rheology reading path

“The size of a black hole is its event horizon, at two masses.”

That is the radius nothing returns from, and it is not the radius of anything anybody can see. Light can orbit at three masses, on an unstable circle; and the smallest impact parameter a ray can have and still escape is root twenty-seven, 5.196 masses, which is what fixes the angular size of the dark patch a distant observer sees. The shadow is therefore 2.6 times the horizon radius across, and quoting the horizon as the observed size understates it by that factor.

Tested in The circle light cannot leave · the horizons reading path

“The damping rate is Landau's formula, root pi over eight times the exponential.”

That is the asymptotic form and its accuracy varies enormously across the range it is quoted for. Integrated exactly, the rate at a wavenumber of half a reciprocal Debye length is 0.1534 and the formula gives 0.1514 — within one and a half per cent — while at 0.3 the exact answer is 0.0126 and the formula gives 0.0203, sixty per cent high. The error is invisible unless the exact problem is solved beside it, which is why the figure here draws both.

Tested in The wave that dies with nothing to rub against · the plasma oscillation reading path

“A zone plate has a focal length of r₁²/λ.”

It has that one and an infinite series of others, at every odd fraction of it. At a third of the design distance each open ring spans three half-period zones instead of one, two of the three cancel, and a focus remains; at a half it spans two, the cancellation is complete, and there is nothing. The odd-order peaks reach 100, 99.9, 96.3 and 93.8 per cent of the main peak's on-axis intensity — not a ninth and a twenty-fifth, because only a ninth of the light goes into the third order and its focus is three times tighter, and the two effects cancel.

Tested in The lens that is a set of rings · the diffraction reading path

“Total internal reflection sends all the light back, so nothing happens to it.”

Nothing happens to its amplitude — the coefficients have modulus exactly one past the critical angle — and something happens to its phase, differently for each polarisation, which is the only thing left that can distinguish one angle from another. For a glass-to-air boundary the difference between the two phases rises from zero at the critical angle to 46.53 degrees at 51 degrees of incidence and falls to zero again at grazing. That difference is a retardation, and it is what a device can be built out of.

Tested in The retarder with no crystal in it · the polarisation reading path

“The instrument measures current, so it measures height.”

It cannot, because the current is far too steep a function of height to be read as one: ten picometres — a tenth of an atomic radius — changes it by 24 per cent, and a factor of two in current is nine picometres. The measurement is made the other way round, by a feedback loop that holds the current fixed and records the voltage the piezoelectric scanner needed. The recorded quantity is the correction, and the instrument's precision is the precision of a feedback null rather than of a current reading.

Tested in The last atom does all the seeing · the tunnelling reading path

“Entanglement cannot signal because relativity forbids it.”

It cannot signal because of an arithmetical fact about the state, which the figure computes: summing over the far measurement's outcomes leaves the near half's density matrix exactly unchanged — Bloch length 0.0e+0 and trace one to a part in 10¹⁴ — whatever axis the far analyser is turned to, while the correlation swings by two. Relativity is not used anywhere in that computation. What is worth noticing is the converse: a working copier would break it, since two copies measured along two axes would reveal the far setting.

Tested in The state that cannot be copied · the measurement reading path

“Flux through a superconducting ring is conserved.”

It is conserved and it is also quantised, and the second is the stronger statement. A ring cannot hold an arbitrary flux at all: it holds a whole number of quanta of 2.0678 × 10⁻¹⁵ webers and drives a current to make up whatever the applied field left over. Conservation follows from zero resistance and is a classical statement; quantisation follows from the condensate's wavefunction having to come back to itself round the ring, and no classical argument supplies it.

Tested in The two in the flux quantum · the superconductivity reading path

“The constraints hold automatically.”

One of them does, unconditionally: nothing can change the divergence of B, whatever the fields do. The other holds if and only if charge is conserved, because the rate of change of the divergence of D is minus the divergence of the current. The same integration run twice, with a current whose charge is accounted for and one whose is not, gives residuals of 1.1 × 10⁻¹⁴ and 7.7 — from the same equations, the same grid and the same current. Conservation is a requirement of the theory rather than an extra observation about the world.

Tested in The two equations that are not laws of motion · the maxwell equations reading path

“The number of photons in a cavity depends on how much light was put in.”

In equilibrium it does not depend on it at all. The walls emit and absorb until the free energy is least, and the minimum fixes the number: 16πζ(3)(kT/hc)³ per unit volume, which at the microwave background's 2.7255 kelvin is 4.11 × 10⁸ per cubic metre. Photon number is not conserved, so it is not a variable of state — which is the same statement as the chemical potential being zero, and is where every difference from an ordinary gas comes from.

Tested in The gas that nobody counted · the blackbody reading path

“Onsager's relations say the matrix of transport coefficients is symmetric.”

They say it is symmetric provided nothing breaks the time-reversal symmetry of the underlying motion. A magnetic field does break it, and then the relation becomes L(B) = Lᵀ(−B) — the coefficient measured in one field equals its transpose measured in the reversed field, and the symmetric statement fails. That is not a caveat: the Hall effect and the whole family of thermomagnetic cross-effects live in exactly the part the symmetric version excludes.

Tested in The second experiment that cannot disagree · the diffusion reading path

“The superfluid density falls continuously to zero at the transition, as an order parameter does.”

It falls discontinuously, from a value that is not adjustable. Integrating the renormalisation-group flow gives a stiffness that drops to zero from 0.643 in units where the universal prediction is 2/π = 0.6366, and that number depends on nothing: not on the material, not on the vortex core energy, not on the bare stiffness. Measurements on helium films spanning a wide range of thicknesses all lose their superfluidity on that one line, which is why the prediction is called a universal jump rather than a critical exponent.

Tested in The transition with nothing to order · the phase change reading path

“The effect is a projection, so it tells you nothing about the source.”

It tells you two things at once, and they are the reason it is used. Seeing an apparent speed of β_app requires a true speed of at least β_app/√(1 + β_app²), so an observed 7.02 puts a floor of 0.990 under the true speed and a ceiling of about 16 degrees on the angle to the line of sight. Nothing else about a distant jet gives both a lower bound on its speed and an upper bound on its orientation from one measurement.

Tested in The motion that measures faster than light · the Velocity-addition reading path

“A moving source is Doppler shifted, so its light changes colour.”

It changes colour and it also changes brightness, by a much larger factor, and the two are not separate effects. The one quantity all observers agree about is the specific intensity divided by the cube of the frequency, so a Doppler factor of δ multiplies the intensity by δ³ per unit frequency and δ⁴ in total. For a source at β = 0.99 that is 1.6 × 10⁹ between the approaching and receding directions, which no measurement of a colour would suggest.

Tested in The brightness that is not the same for everyone · the doppler reading path

“The rotating frame's geometry is non-Euclidean, so the disc is curved.”

The disc is not curved and neither is the spacetime it sits in, which is exactly flat. What is non-Euclidean is the three-dimensional geometry a rotating observer builds out of local measurements, and it is non-Euclidean because there is no consistent way to slice a rotating frame into simultaneous surfaces — go once round the rim synchronising clocks and the last one disagrees with the first by an amount proportional to the enclosed area, which is what a Sagnac interferometer measures.

Tested in The disc that cannot be spun · the length contraction reading path

“Lagrangian mechanics is better than Newtonian mechanics because constraint forces drop out.”

They do drop out, and that is a loss as well as a saving. A rope has a breaking strain and a track has a limit on what it can push; neither quantity appears anywhere in the equation of motion written in the angle, so a solution can be perfectly correct and say nothing about whether the apparatus survives. The multiplier drawn here restores the number — 3.000 weights at the bottom of a quarter-circle swing — and it has to be asked for.

Tested in The force a coordinate cannot see · the least action reading path

“A target within range can be hit, so knowing the range tells an artillerist what they need.”

It tells them half. Every point strictly inside the boundary is reached by two launches, not one: the range equation is a quadratic in the tangent of the launch angle, and its discriminant is positive there. The two roots are the flat trajectory and the lobbed one, with different flight times, different impact angles and different clearances — and choosing between them is most of the craft.

Tested in Everywhere a throw can reach · the projectile reading path

“Two modes of a system are orthogonal, so multiplying them together and integrating gives zero.”

Only against the right weight. The modes of the lumpy string here are not orthogonal in the plain sense; multiply two of them and integrate and the answer is not zero. They become orthogonal when each element of the string is counted in proportion to the mass it carries, and the largest off-diagonal overlap against that weight is at the level of the integration's own error. A uniform string hides the distinction, because there the weight is a constant.

Tested in The count that cannot be cheated · the standing waves reading path

“A Doppler radar measures the speed of the target.”

It measures the component along the beam and has no access to the rest. A target crossing at 45° reads 71 per cent of its speed, at 60° it reads half, and at 90° it reads nothing at all — and nothing in a single beam's return says which case it is. Every such instrument therefore reads low, never high, by a factor that cannot be recovered from the measurement it made.

Tested in The shift a mirror gives twice · the doppler reading path

“Reciprocity means a system looks the same from both sides.”

It means the transmission does. The reflections do not: the same stack returns the same fraction from either side — only because nothing here absorbs — and the reflected *phases* differ by 1.6 radians, because a wave meets a different first surface from each side. The theorem constrains the pair of ends and says nothing about what happens between them.

Tested in Swap the ends and nothing changes · the wave motion reading path

“Light refracts because it travels more slowly in the denser medium, so it bends towards the normal.”

That is true and it does not say which angle. The condition is that the component of the wavevector along the surface is unchanged, because the two sides must agree on the phase at every point of the boundary; the magnitude of the wavevector is fixed by the medium at n times the vacuum value, so the direction is whatever makes those two things compatible. Snell's ratio of sines is the projection of two circles of radii n₁ and n₂ onto one line.

Tested in The law that only asks about one component · the refraction reading path

“Intensity interferometry is better than amplitude interferometry because it is immune to the atmosphere.”

It is immune and it is much less sensitive. The correlated signal is smaller than the shot noise by the degeneracy parameter — about 10⁻³ photons per mode for a bright star through a narrow filter — so the measurement is recovered only by averaging, and reaches a few hundred standard deviations after hours where a fringe measurement would take seconds. That is why it works on the brightest few dozen stars and on nothing fainter.

Tested in The correlation that survives what the phase does not · the coherence reading path

“The energy method gives the force, so the mechanism does not matter.”

The energy method gives the total force and says nothing about where it is applied, which is what a designer needs. The stress is concentrated at the slab's leading edge, over a region of order the plate separation, and that is also where the field is largest and where breakdown starts. A part designed on the total force and silent about its distribution fails at the edge.

Tested in The force that lives where the model is not · the dielectrics reading path

“A drop can hold as much charge as is put on it; eventually it will just repel harder.”

It has a limit and the limit is sharp. Electrostatic pressure grows as the square of the charge over the fourth power of the radius while the Laplace pressure holding the drop together goes as one over the radius, so the two are equal along Q² = 64π²ε₀γR³ — 0.635 nC for a one-millimetre water drop. Past that the drop does not merely deform: it draws itself into a cone and throws off a jet of droplets each below its own limit.

Tested in The pressure a charge puts on its own metal · the conductors reading path

“Deep ground is at the annual mean temperature because heat takes a long time to get there.”

It is at the mean because the oscillation has been attenuated, and the attenuation and the delay are the same length. At πδ — seven metres in damp soil — the swing is down to 4.3 per cent of the surface's and the phase is inverted, so that ground is coldest in August. Below about four δ there is no measurable annual signal at all, and the temperature there is set by the geothermal gradient rather than by the sky.

Tested in The summer that reaches the cellar in December · the diffusion reading path

“The triple point is where the three coexistence curves happen to meet.”

It is where they are *forced* to meet and where they can only meet at a point. Counting gives F = C − P + 2: one phase leaves two numbers free and fills an area, two leave one and make a curve, three leave none. Nothing about water enters the count, and the same arithmetic says four phases of one substance would need minus one degree of freedom, which is not a small number but an impossible one.

Tested in Why the triple point is a point · the phase change reading path

“Ehrenfest's theorem shows that quantum averages obey Newton's laws, so classical mechanics is recovered for large objects.”

The theorem gives d⟨p⟩/dt = ⟨−V′(x)⟩, which is the *average of the force over the state*. Newton's law needs −V′(⟨x⟩), the force at the average position. The two agree only when V′ is linear in x, and for a quartic well the computed centroid departs from the classical orbit by more than the starting displacement within a couple of periods — with the theorem holding exactly throughout.

Tested in The average that obeys Newton · the correspondence reading path

“The resonances sit at the energy levels of the well between the barriers.”

They sit near them and not at them. The infinite-well levels of the gap here are 2.467 and 9.870; the resonances are at 6.438 and 13.915. A well with leaky walls has states that are not bound, whose wavefunctions extend into the barriers, and whose energies are correspondingly shifted — and the shift is of the same order as the amount by which the walls are penetrable.

Tested in Two walls that let more through than one · the tunnelling reading path

“A diode conducts one way because current can flow from p to n and not the other way.”

Current flows both ways and the two are wildly unequal. Carriers going uphill over the junction barrier are suppressed by the Boltzmann factor of that barrier, which the applied voltage changes; carriers rolling downhill are not held back by the barrier at all and their supply is set by thermal generation. The ratio at ±0.5 V is 2.5 × 10⁸, and it comes from one exponential and one constant rather than from a passage that opens and closes.

Tested in One level, and the field that bends the bands · the bands reading path

“Special relativity follows from the Lorentz transformation, so understanding it means understanding that transformation.”

The whole theory can be built from one measured ratio and two multiplications. Flashes sent every second by one observer arrive every k seconds at another; the Lorentz factor is (k + 1/k)/2, the speed is (k² − 1)/(k² + 1), and velocities compose because k factors multiply. No coordinates are assigned to any event in the derivation, and the transformation appears at the end as a summary rather than at the start as a postulate.

Tested in Everything from an exchange of pulses · the doppler reading path

“Proper time along a worldline is the integral of dt/γ, which follows from special relativity.”

It follows from special relativity *plus* the clock hypothesis, which is a separate assumption: that an ideal clock's rate depends on its instantaneous speed and on nothing else. Nothing in the two postulates says what an accelerating clock does, because the postulates are about inertial frames. The hypothesis is eminently testable and has been tested to 10¹⁸ gravities, and it remains an input.

Tested in The clock that does not feel the turn · the accelerated frames reading path

“Second sound is a temperature wave, so it is heat behaving like sound.”

It is one of the two wave modes of a two-component fluid, and calling it a temperature wave describes what it carries rather than what it is. In the first mode the two components move together and the total density oscillates; in the second they move oppositely with the density constant, so the *ratio* of the components oscillates — and since that ratio is set by temperature, the wave carries temperature. Both modes come out of the same pair of equations.

Tested in The heat that arrives as a wave · the superfluidity reading path

“The Brazil nut effect is understood: small grains fall through the gaps.”

That is one mechanism of at least three, and it is not always the dominant one. Convection of the whole bed lifts everything and returns only what fits in the narrow downward stream at the walls; interstitial air changes the answer enough that the same experiment in a vacuum can give the opposite result; and at some density ratios a large intruder sinks instead. There is no single criterion that predicts which way a given mixture will go.

Tested in The big one comes to the top · the granular matter reading path

“Ions are pulled into the sheath by its field, so their entry speed is whatever the field gives them.”

They have to arrive at the sheath edge already moving at the ion sound speed or no steady sheath exists at all. Integrating the sheath equation from a sub-sonic entry gives an oscillatory solution that turns back through zero rather than a monotone potential; the threshold is exactly Mach one. So the plasma must accelerate its own ions in a much wider, much gentler presheath before the sheath begins, at a cost of half an electron temperature.

Tested in The wall a plasma builds against itself · the plasma oscillation reading path

“An optical trap holds a particle at the brightest point of the beam.”

It holds it a little past the brightest point. The gradient force vanishes exactly at the waist, where the intensity is stationary, and the scattering force does not — so the equilibrium sits downstream, where the restoring pull has grown to match the push. For the beam computed here that is 0.066 micrometres past the focus, and every optical trap is offset this way.

Tested in The light that pulls rather than pushes · the radiation pressure reading path

“A spinning top precesses at Mgl divided by its spin angular momentum.”

Setting the tilt's second derivative to zero gives a quadratic in the precession rate rather than a value, so there are two steady rates at every spin — a slow one, whose leading term is Mgl/I₃ω₃, and a fast one that rises in proportion to the spin. The quadratic has real roots only above a critical spin, and a top released from rest is at neither of them, because its initial precession rate is zero and zero is not a solution.

Tested in The top that nods before it settles · the rotation reading path

“A Newton's cradle transfers the striking speed to the far ball.”

Integrating the equations of motion for five touching balls with the contact force two steel spheres actually exert gives 0.989 of the striking speed to the far ball, with the remainder distributed among the others. The transfer is exact only when the balls are separated so that each contact is a two-body collision; with them touching, several contacts overlap in time and no two-body argument applies.

Tested in Five balls, and the law that does not choose · the momentum reading path

“Adding turns adds holding power.”

Adding turns multiplies it, and the difference is the whole practical content. Each turn at a coefficient of 0.25 multiplies by 4.8, so the first turn adds 3.8 times the hand's pull and the third adds 87 times it. A rule expressed as an addition would predict that ten turns are twice as good as five; they are twenty-three times as good.

Tested in The part of the wrap that is actually gripping · the friction reading path

“The principle of least action says the true path minimises the action.”

There are two actions and two comparison classes. Hamilton's action compares paths of the same duration and is the time integral of kinetic minus potential energy; Maupertuis' abbreviated action compares paths of the same energy and is the integral of momentum along the path. Both are stationary at the true trajectory and the figures show them rising at different rates from it. A statement naming neither has named neither.

Tested in The principle that fixes the energy instead of the clock · the least action reading path

“A physical system period-doubles because it resembles the logistic map.”

It period-doubles because sampling it once per drive cycle makes it a map, and that requires the dissipation to contract phase-space volume onto an attractor thin enough that one coordinate along it suffices. Where the contraction is weak the sampled dynamics is genuinely two-dimensional, the cascade is modified and the constants shift, so the resemblance is a consequence of the damping rather than an analogy.

Tested in The map a dripping tap turns out to be · the chaos reading path

“A nonlinear oscillator is a linear one with a correction to its amplitude.”

The correction is not to the amplitude but to the set of frequencies present. Driven at one frequency, the oscillator here returns a component at three times it whose size grows as the cube of the drive — a component the linear equation cannot produce at any amplitude. The figures project that component out of the integrated motion rather than assuming it, and its growth exponent comes out at 3.00 against 1.00 for the fundamental.

Tested in The oscillator that answers at three times the question · the harmonic approximation reading path

“The energy stored in an inductor is ½LI².”

It is, and the statement leaves out where it comes from and how it is delivered. Integrating the power that flows against the coil's own back emf during the rise gives ½LI² to four parts in ten thousand, and over the same interval the resistor dissipates the same amount again. Half the energy the supply delivers while the current is arriving never reaches the field.

Tested in The circuit that fights its own change · the induction reading path

“Contact resistance can be reduced by using thicker conductors.”

Ninety-four per cent of a constriction's resistance lies within ten spot radii of the contact, and more than ninety-nine per cent within a hundred. The current has forgotten where it came from long before it arrives, so the size and shape of the bodies either side do not enter the answer as long as they are large. What reduces it is a larger contact spot, or more of them.

Tested in The resistance that is a length · the conductors reading path

“The size of the jumps is set by how disordered the material is.”

The disorder sets the small end. The large end is set by the restoring term — the field a displaced wall sets up against itself — and the figures show the cutoff moving when only that term is changed. That is why the same material in a different shape crackles differently, and why the distribution below the cutoff is the same power law regardless.

Tested in The curve that is really a staircase · the magnetisation reading path

“Maxwell's equations are asymmetric between electricity and magnetism.”

Only the source terms are. With the charges and currents removed, the four equations are unchanged by a rotation taking E into cB and cB into minus E, through any angle. The figure computes the two Lorentz invariants at each angle from the rotated fields and finds them moving on a circle: the pair is conserved and neither member is. The asymmetry is entirely the absence of one kind of source.

Tested in The symmetry one missing charge would complete · the maxwell equations reading path

“A superconductor expels magnetic field.”

Only one kind does. Above a ratio of penetration depth to coherence length of one over root two the energy of a boundary between normal and superconducting material is negative, so the material makes as much boundary as it can: flux enters as a lattice of tubes, each carrying one quantum, with a normal core at the centre of each. The figures compute that surface energy by relaxing the Ginzburg–Landau equations and find it changing sign at 0.72.

Tested in Two lengths, and which one is longer · the superconductivity reading path

“Momentum is conserved because the field carries some.”

For this arrangement the field's momentum alone gives the wrong answer: the system is static and its total momentum must be exactly zero, and the field's is not. The books balance only when the mechanical momentum hidden in the current is included. Conservation is a statement about the total, and leaving out a term that nothing is visibly doing makes it fail.

Tested in The momentum of something that is not moving · the field energy reading path

“Numerical aperture measures how much light a fibre gathers.”

It measures the cone only. The gathering is the etendue — the core's area times π times the square of the aperture — and the two can be traded: a large-core fibre with a small aperture and a small-core fibre with a large one can gather the same light and behave completely differently, in bandwidth, in bending loss and in how many modes they carry.

Tested in The cone a fibre will accept · the etendue reading path

“Stopping down a lens increases depth of field.”

It increases the geometric depth in proportion to the f-number and shrinks the aperture, so the diffraction blur grows in proportion too. Past the aperture where the two are equal, stopping down makes everything worse at once. The wave-optical depth goes as one over the square of the numerical aperture, so the exchange rate between sharpness and depth is fixed and cannot be improved by any design.

Tested in The focus that is a slab, not a plane · the imaging reading path

“A rotating polariser measures the direction of polarisation.”

It measures the depth of a modulation as well, and that depth is the degree of polarisation. Every beam gives the same average — a polariser passes half of anything over a whole turn — and the four beams in the figure differ only in how deeply the transmitted power swings. A beam with no preferred direction gives a flat line and no angle at all.

Tested in The light with no direction of shaking · the polarisation reading path

“The refractive index is a property of a medium and scattering is a property of particles in it.”

Both are the same amplitude. The transmitted wave is the incident wave plus everything scattered forward; the imaginary part of that scattered amplitude removes energy and gives the extinction, and the real part retards the sum and gives the index. Computing the index of air from a molecule's forward scattering gives the measured value to a per cent.

Tested in Everything a scatterer removes, from one direction · the scattering reading path

“A sphere is a poor imaging surface.”

It is a perfect one for exactly two points, at R/n and nR from its centre. Rays from the first extrapolate back through the second with no spread the arithmetic can find, at any angle. Those aplanatic points are what the front element of an oil-immersion objective is used at, and they are the reason such a lens can collect a numerical aperture above one without introducing the aberration a strongly curved surface normally brings.

Tested in The surface that images one point exactly · the fermat reading path

“A spectrometer separates wavelengths.”

A dispersive one does. An interferometer measures the same spectrum with nothing that separates anything: it records the total power against path difference, which is the cosine transform of the spectrum, and the wavelengths are separated arithmetically afterwards. The two panels of the first figure carry identical information and neither is more fundamental.

Tested in The fringe and the spectrum are one measurement · the coherence reading path

“Interference always redistributes energy rather than removing it.”

That holds when the sources cannot feel each other. Two sources closer together than half a wavelength and driven in opposition radiate essentially nothing, and the missing power is not redistributed anywhere: it is never emitted. Each source finds itself working against a load the other has changed, and the effect is measurable at the driver rather than in the field.

Tested in What adding does to the energy · the superposition reading path

“A resonance is a peak.”

It is a peak when the resonant path is the only one. Where a smooth path exists alongside it the two amplitudes add before squaring, and the response is Fano's profile: asymmetric, with a maximum on one side and an exact zero on the other, at positions whose product is minus one. A symmetric peak is the special case in which the resonant path overwhelms the other.

Tested in The resonance with a zero in it · the resonance reading path

“Huygens' construction explains reflection, refraction and diffraction.”

It gets all three right and predicts a fourth thing that does not happen. A spherical secondary wavelet radiates backwards as strongly as forwards, so summing the wavelets over a plane gives a wave going back the way the original came. The figure does that sum and finds it as large as the forward one for an isotropic wavelet, and essentially nothing for Kirchhoff's.

Tested in The backward wave Huygens had to remove · the huygens reading path

“A band gap is opened by the strength of the periodic modulation.”

It is opened by the repeat. A chain of alternating masses has a gap whose edges are the frequencies of each mass oscillating alone against fixed neighbours, and the gap closes as the two masses become equal — at which point the repeating unit is really half as long and the second branch is the first folded back. The springs are unchanged throughout.

Tested in The frequency a lattice cannot carry · the periodic media reading path

“A pattern with deep minima means a bad match.”

It means a large reflection, which is a bad match only if power transfer is the object. A deliberately large reflection is how a resonant cavity, an antenna trap and a quarter-wave transformer all work, and in each the depth of the pattern is a design target rather than a fault. What a deep pattern always means is that the two waves are of comparable size.

Tested in The node that is not standing still · the standing waves reading path

“A bend loses a fixed fraction of the light.”

It loses a fraction that depends strongly on which mode is carrying it. A higher-order mode is held less tightly, its tail reaches further, and its caustic matters at a much larger radius — so a bend removes the higher modes first and leaves the fundamental. A coil of a few turns is therefore a mode filter, and is used as one.

Tested in The mode that will not turn a corner · the guided waves reading path

“Temperature is a measure of the average kinetic energy of the particles.”

It fixes the average and also fixes how much the total wanders about it, by kT² times the heat capacity — a relation between a fluctuation and a response that is not contained in the statement about averages. For a mole the wandering is a part in 10¹² and can be ignored; for a nanoparticle it is per cent and is measured; for a molecule it is of order the energy itself.

Tested in The temperature a molecule does not have · the equipartition reading path

“Pressure melting is irrelevant.”

It is irrelevant to a skate and decisive under a glacier. Three kilometres of ice produces about 270 atmospheres and depresses the melting point by two kelvin, which is why the base of an ice sheet is at its pressure melting point and wet, and why a glacier slides on water it made itself.

Tested in The melting curve that leans the wrong way · the phase change reading path

“The Soret effect is too small to be useful.”

A single stage separates by a few per cent, and stages compound. A thermogravitational column stacks the effect using convection, so a fraction of a per cent per gap becomes a complete separation over a few metres — which is how Clusius and Dickel separated chlorine's isotopes in 1938 with a glass tube and a hot wire.

Tested in The gradient that drives the other thing · the diffusion reading path

“The entropy of an isolated system never decreases.”

It never decreases on average, and for a small system it decreases on individual runs with a probability that is computable. Crooks' theorem gives the ratio of the probability of a trajectory producing entropy σ to that of its reverse producing −σ as e raised to σ over k, so an entropy decrease of ten k has a probability of about one in twenty thousand — small, non-zero, and observed.

Tested in The second law, with a probability attached · the entropy reading path

“Van der Waals' equation explains the departures from ideality.”

It reproduces the first correction with two fitted constants. The second virial coefficient is computable from the pair potential alone by one integral, with no fitting, and its temperature dependence is a curve rather than the straight line van der Waals' constants imply. Measuring the coefficient over a range of temperature is therefore a measurement of the intermolecular potential, which no two-parameter equation can be.

Tested in The first correction to the gas law · the kinetic theory reading path

“Two observables fail to commute because measuring one disturbs the other.”

Disturbance is a consequence and not the reason. The commutator is a property of the two operators computed with no apparatus anywhere in the calculation, and it is what decides whether a state with definite values for both exists at all. A relation about which states exist cannot be explained by what an instrument does to one.

Tested in The questions that can be asked together · the uncertainty reading path

“The shift is caused by a residual field leaking out of the solenoid.”

A leak would displace the fringes by an amount depending on how far out the paths run, and it would not be periodic. The measured displacement is periodic in the enclosed flux with period h/e exactly, and the definitive experiments used a closed toroidal magnet with no ends for the flux to leave by, and a superconducting shield thicker than the penetration depth.

Tested in The phase a magnet leaves on a path it never touched · the matter waves reading path

“A classical oscillation has a definite amplitude, so it has a definite energy.”

A coherent state of mean quantum number n has a spread of √n in that number, which is a definite classical amplitude and an indefinite quantum energy at the same time. The relative spread falls as 1/√n, so the two descriptions agree in the limit without ever agreeing exactly.

Tested in The state that swings like a pendulum · the correspondence reading path

“Attosecond ionisation experiments have measured the tunnelling time.”

They measure a delay between an ionising field and an electron's appearance, which requires a model to convert into a time spent under a barrier — and the answer depends on which of the definitions above the model implements. Different groups analysing similar data have reported both a substantial delay and a delay consistent with zero.

Tested in How long the crossing takes · the tunnelling reading path

“Sharing an entangled state among many parties gives each pair a share of it.”

The pairwise concurrence of an N-party W state is 2/N, computed here through Wootters' formula and checked against the closed form. Past about six parties no pair can violate a Bell inequality at all, although the state as a whole remains fully entangled.

Tested in What two have they cannot give a third · the entanglement reading path

“The size of an atom is a property of the electron.”

It is a property of the reduced mass of the pair. Positronium is twice the size of hydrogen and muonic hydrogen is a hundred and eighty-six times smaller, both computed here from the same expression with only the masses changed, and the muonic value matches the 285 femtometres the proton-radius experiments are built around.

Tested in Why an atom is the size it is · the atomic structure reading path

“A spacetime diagram shows what is happening.”

It shows one frame's coordinates on a Euclidean sheet, and that choice makes one worldline vertical and the others tilted. The symmetric drawing puts two observers at equal and opposite tilt with identical page units — the same physics, and none of the impression that one of them is at rest.

Tested in The diagram a ruler cannot read · the spacetime diagram reading path

“A spinning body has a centre, even if finding it needs care.”

Its centre of energy is displaced by S/Mc times the frame's speed, perpendicular to both the spin and the motion, so sweeping over all frames sweeps the centre over a disc. For an electron that disc is half a reduced Compton wavelength across, which is larger than any structure the electron is known to have.

Tested in The centre that is not a place · the relativistic dynamics reading path

“The neutrality of a current-carrying wire is the evidence for charge invariance.”

It is evidence, and it is weak. Conduction electrons drift at a fraction of a millimetre a second, the effect goes as the square of that, and the predicted field beside a 40 A wire is a hundredth of a microvolt per metre. An atom's inner electron moves ten million times faster, which makes an atom some 10¹⁴ times the instrument a wire is.

Tested in The one quantity a boost leaves alone · the field transformation reading path

“Rapidity is a convenient substitution that makes boosts add.”

It is the arc length of a genuine geometry. The circumference of a circle of rapidity radius r is 2π sinh r, and reading the curvature off the small-radius behaviour gives exactly −1 — which is why the rotation left behind by two boosts equals the triangle's area rather than some multiple of it.

Tested in The space that speeds live in · the Velocity-addition reading path

“A thrown ball follows a parabola because gravity pulls it down.”

Trial paths that join the same two events are compared by the proper time along them, and the maximum is at the real trajectory to better than a part in five hundred — found by searching, not assumed. Climbing higher runs the clock faster and moving faster runs it slower, and the balance point is Newton's parabola.

Tested in The longest way round is the shortest clock · the time dilation reading path

“Flow in a pipe has a parabolic profile.”

Only when viscosity reaches the axis within a cycle. Above a Womersley number of about five the core moves as a plug and the fastest fluid is not on the axis but near the wall — at 0.85 of the radius at a Womersley number of 12, computed from Bessel functions of complex argument summed from their series.

Tested in The shear that only reaches so far · the viscosity reading path

“Treating pressure as uniform around a small body is an approximation that introduces error.”

It introduces an error of about ρgh over the body's own height, which is a part in ten million for a cell at three metres and a part in a few thousand for a fish. The second is not an error to be minimised — it is the buoyancy, and the whole reason the fish floats.

Tested in The push that has no direction · the hydrostatics reading path

“Beverloo's law is a power law with exponent five halves.”

The local slope is measured on the drawn curve and reaches five halves only for wide openings. Near the jamming end the subtracted annulus makes the apparent exponent four or five, which is what a measurement over a narrow range returns — and the historical disagreements about the exponent are disagreements about which part of this curve was measured.

Tested in The hourglass that keeps time · the granular matter reading path

“The osmotic pressure of a solution is given by van 't Hoff's law.”

That is the pressure a perfectly selective membrane would develop. A real one develops σ times it, with σ between zero and one and dependent on the solute as much as on the membrane. The figures compute the pressure that stops the flow, which is σ times the ideal value, and it is how σ is measured.

Tested in The membrane that almost holds · the osmosis reading path

“A square tube and a round tube of the same size behave alike.”

Only above a contact angle of forty-five degrees. Below it the square tube's corners wick without limit while the round tube stops at Jurin's height, and the filament reaches a metre before it has narrowed to a micrometre where the round column has risen twenty millimetres.

Tested in The corner a liquid never stops climbing · the surface tension reading path

“An orbit is determined by its energy and its angular momentum.”

Those two fix the semi-major axis and the eccentricity and leave the orientation free — every rotation of an orbit about the centre has the same energy and the same angular momentum. What fixes the orientation is a third conserved quantity, and it exists only for an exactly inverse-square force.

Tested in The arrow that says which way the orbit points · the orbit stability reading path

“Two detectors confirm a detection, and a third improves it.”

Two give a ring on the sky, not a point: an arrival-time difference fixes only the angle to the baseline, and the figure computes the ring for a given delay. Three narrow it to two patches and a fourth chooses between them, so a network is what makes a detection pointable rather than merely credible.

Tested in What the instrument actually hears · the gravitational waves reading path

“A freely falling charge does not radiate, because it is locally inertial.”

Local inertiality settles what a co-falling detector finds and nothing else. A detector held stationary nearby is accelerating relative to the charge and does register radiation, and the two statements are consistent because they are statements about different detectors — the same structure as the Unruh effect, where one observer finds a thermal bath and another finds a vacuum.

Tested in Whether a charge on a table glows · the radiating charge reading path

“The Planck length is where quantum gravity becomes important.”

It is also where the standard procedure for measuring a length stops returning an answer. Adding a gravitational term to the uncertainty relation gives a sum with a minimum, and scanning the drawn curve over four hundred thousand momenta finds it at ℓ_P√2 — a floor rather than a regime boundary.

Tested in The length no experiment can resolve · the planck scale reading path

“The 4π period was measured, so a rotation of a spin has been shown to require 720 degrees.”

What was measured is the period of an interference pattern against a magnetic field. Reading that as a rotation angle requires the identification of Larmor precession with the action of the rotation group on the state, which follows from the Hamiltonian being minus the magnetic moment dotted into the field. That identification is well tested elsewhere and is an input here rather than an output.

Tested in The turn that has to be made twice · the spin reading path

“A bead-pull measures the field in a cavity.”

It measures a frequency shift proportional to the square of the field at the bead, so it recovers the magnitude and not the sign. It also depends on the bead: a dielectric bead reports the electric term alone, while a metal bead reports the electric term minus half the magnetic one and changes sign along the resonator. Which quantity has been measured is a property of the bead, not of the cavity.

Tested in The dent that raises the note · the standing waves reading path

“The force on the contents of a surface is the integral of the stress over it.”

That holds for static fields. In general the integral equals the force plus the rate of change of the electromagnetic momentum stored inside, and the second term is not always small: it is what makes a static-looking system carry momentum, and it is why an antenna recoils. The static statement is a special case in which one term happens to vanish.

Tested in The force read off a surface that touches nothing · the field energy reading path

“A guiding centre is where the particle really is.”

It is the leading term of an expansion in the Larmor radius over the length the field changes across. The figures here work at ratios of a few parts in ten thousand to a few per cent, and the last one measures the ratio explicitly. Near a magnetic null the ratio is unbounded and the guiding centre is not defined at all, which is why reconnection layers and shock fronts need the full orbit.

Tested in The drift that does not care what the charge is · the magnetism reading path

“Anti-lock braking works by keeping the wheel rolling.”

It works by holding the wheel near the maximum of the force-slip curve, which is at ten per cent slip or so rather than at zero. A wheel held at zero slip delivers no force at all. The controller is a maximum-seeking device operating on a curve whose right-hand branch is unstable, and the instability rather than the loss of force is what makes the problem hard.

Tested in The grip that needs a little slipping · the friction reading path

“A small leak makes a spin about the long axis impossible.”

It makes it temporary. The time scales as the reciprocal of the leak, which the figures verify across a decade, so a smaller leak buys proportionally more time and never buys immunity. That is the reason the failure mode is dangerous: an object can be spun on the ground, tested, launched, and only then flip.

Tested in The axis a leak of energy chooses · the rotation reading path

“There is a curved surface of best focus.”

There are three. Astigmatism splits the off-axis focus into two line images at different distances, lying on a sagittal and a tangential surface, with the Petzval surface between them at a fixed ratio: the tangential departure is exactly three times the sagittal. A lens can be made to place either one on a flat sensor and never both.

Tested in The flat scene that comes back curved · the imaging reading path

“The sky is unpolarised only towards the sun and away from it.”

Those are the only zeros the single-scattering model has, and three more were found by measurement: Arago above the antisolar point in 1809, Babinet above the sun and Brewster below it. They sit fifteen to twenty degrees away and move with the sun's height and the dust in the air, which is what makes them a measurement rather than a curiosity.

Tested in The pattern the sky is written in · the polarisation reading path

“The ratchet reaches Carnot efficiency at stall, so it is a reversible engine.”

The rate balance does give exactly 1 − T₂/T₁ at the stall load, which is computed rather than quoted here. But the pawl and the paddle are joined by a shaft, so the two baths are in permanent thermal contact, and that leak runs whether or not the wheel turns. With any leak at all the efficiency peaks below Carnot at a finite load and falls to zero at stall.

Tested in The engine a fluctuation cannot run · the heat engines reading path

“The recurrence is explained by the chain being close to an integrable system.”

That is one of two explanations and they describe different regimes. Near-integrability accounts for the trajectories staying on tori at low energy density; the solitary-wave picture accounts for the recurrence itself, since pulses that pass through one another unchanged bring the initial shape back. Both are right, and neither predicts the threshold's position without a calculation.

Tested in The energy that refuses to be shared · the equipartition reading path

“One opaque layer explains the thirty-three kelvin.”

It gives forty-eight, which overshoots. The atmosphere is not opaque across the whole infrared: about a third of the surface's radiation falls in a band where it leaves more or less directly, computed here from the Planck curve. A model with a fractional number of layers reproduces the number by being tuned to it, which is a different thing from explaining it.

Tested in The height a planet is seen from · the blackbody reading path

“The area increase in GW150914 confirms the theorem.”

It is a test rather than a confirmation, and its strength is a number. The areas before and after are measured from different parts of one signal — the inspiral fixes the masses, the ringdown fixes the remnant — and the increase is a few times the uncertainty on the difference. That is enough to have failed and not enough to be called a precision measurement.

Tested in The area that is not allowed to shrink · the horizons reading path

“The shift is a test of general relativity.”

It is a test of the equivalence principle and of little else. The rate difference follows from a photon losing energy climbing, which requires only that gravity acts on energy; every metric theory of gravity that respects the equivalence principle predicts the same first-order result. Distinguishing general relativity from its rivals needs the deflection of light or the perihelion advance, not this.

Tested in The clock that measures a height · the gravitational redshift reading path

“A null result means nothing was learned.”

Each limit excludes a class of theories that would have produced a departure at that level — a long-range force coupling to baryon number, a scalar partner to the graviton, a violation arising at a particular energy scale. The measurement is worth repeating because it has always come out the same way, which is exactly what would make any departure decisive.

Tested in The fall that does not depend on what is falling · the equivalence principle reading path

“The drag coefficient is 1 − 1/n².”

There is a further term, of a few per cent, because the medium sees the light Doppler-shifted and therefore responds with a slightly different index. Lorentz predicted it and Zeeman measured it. It is the part of the result that cannot be obtained from any argument about a dragged medium, because it is a statement about how two frames are related.

Tested in The drag that was only an addition · the Velocity-addition reading path

“The invariant mass identifies the parent.”

It identifies the four-momentum of whatever was combined. Combining two products that came from different decays gives a value too, distributed smoothly — the combinatorial background every such measurement fights. A peak is evidence of a parent; a single event's invariant mass is evidence of nothing.

Tested in The cone a decay cannot leave · the relativistic dynamics reading path

“The negative case is mathematically impossible.”

It is perfectly consistent: the transformations form a group, the relativity principle holds, and the geometry is Euclidean. What it does is let a chain of ordinary boosts reverse the time order of any pair of events, which the figure computes explicitly. It is excluded by an argument about causality, and that argument is an additional physical assumption rather than a consequence of the four.

Tested in The transformation that never mentions light · the spacetime diagram reading path

“A cell reaches osmotic balance by letting water in until the concentrations match.”

It cannot. The mobile-ion excess is positive for every positive fixed charge, so diluting the charge by swelling reduces the excess without reaching zero, and the cell swells until it bursts. Balance requires a second impermeant species outside, which is what a sodium pump manufactures by throwing sodium out as fast as it leaks in.

Tested in The swelling a membrane cannot stop · the osmosis reading path

“The coloured bands of a draining film are its equilibrium thickness profile.”

The equilibrium profile computed here spans seventy-five to fifty-eight nanometres over three centimetres, which is black throughout. The colours belong to a film thicker than equilibrium and still draining, and describing them needs the flow rather than the balance. The two are different calculations of different things.

Tested in The film that goes black before it bursts · the surface tension reading path

“The packing a column settles at is a function of how hard it is tapped.”

It is a function of how hard it is tapped and of what the column has been through. A freshly poured column follows one curve; once shaken hard it moves onto another, which it then retraces in either direction. The two differ by several per cent in packing fraction at the same tap intensity.

Tested in The pile that is never finished settling · the granular matter reading path

“A magnetic field exerts a pressure B²/2μ₀ across its lines and a tension B²/μ₀ along them.”

The two numbers are right and the sentence hides the structure. The traction on a surface is T·n, and computing it for a normal at any angle to the field gives a vector of magnitude B²/2μ₀ — the same at 0°, at 37°, at 90°, checked to machine precision at seven angles. What changes is the direction, and it changes by reflection in the field rather than by rotation with the surface. The tension quoted as B²/μ₀ is the difference between pulling out at one face and pushing in at the opposite one, not the traction on either.

Tested in The same force whichever way the surface faces · the flux freezing reading path

“Alfvén waves heat the corona.”

The flux is there and the dissipation is the open question. Measured transverse amplitudes of twenty to thirty kilometres a second, at coronal densities and a ten-gauss field, give 460 to 1,040 watts per square metre — enough for the quiet Sun's 300 and a coronal hole's 800, short of an active region's 10,000 by a factor of ten. And a wave that is not damped deposits nothing: the Alfvén wave's own dissipation length at coronal resistivity is longer than the solar system, so every candidate mechanism is a way of getting energy out of it that is not resistivity.

Tested in The wave that does not know what the gas is made of · the flux freezing reading path

“A short antenna is inefficient because it radiates poorly.”

It radiates poorly and that is not usually the binding constraint. At a hundredth of a wavelength the radiation resistance is 0.02 Ω and the capacitive reactance is tens of thousands of ohms, so almost all of the applied current goes into charging and discharging the near field rather than into the radiation resistance at all. The problem is a match between a resistor and a capacitor differing by six orders of magnitude, and it is narrowband for the same reason it is difficult.

Tested in The distance where a field changes its mind · the retardation reading path

“A phased array steers its beam by changing the phase of each element.”

True and it omits the failure. The steering law sin θ₀ = −α/kd puts the beam where the phase says, and the same law puts extra beams wherever the phase advances by another whole turn. For elements more than half a wavelength apart those extra beams land in real angles: the figure here counts three for a spacing of 1.5 wavelengths, which is 2⌊d/λ⌋+1, and they are why array elements are spaced at half a wavelength however inconvenient that is.

Tested in When the source is not heard all at once · the retardation reading path

“An excited atom decays at a rate fixed by the atom.”

The rate is a property of the atom and of what its field can couple into. In a resonator of quality factor Q and mode volume V the rate is multiplied by 3Q/4π²V in cubic wavelengths — ten thousandfold in a photonic-crystal defect — and between two mirrors closer than half a wavelength it falls to nothing, because there is no mode to emit into. Hulet, Hilfer and Kleppner measured that suppression in 1985. Whether a source radiates is not decided by the source.

Tested in The solution that is thrown away · the retardation reading path

“A mountain slows the wind that passes over it.”

It slows the atmosphere, mostly somewhere else. The wave carries a momentum flux of (π/4)ρNUh² upward without depositing any of it until it breaks, which for the largest ranges is in the stratosphere tens of kilometres up. Models that left the term out had stratospheric westerlies too strong by tens of metres per second, and the force is exerted on air that never came near the mountain.

Tested in The wave that is required to stand still · the stratification reading path

“Internal waves in a closed basin form standing modes, like sound in a room.”

For a rectangular basin they do. Tilt one wall and they do not: the reflection law contracts the ray on each circuit, so a trajectory traced through 260 reflections collapses from a spread of a third of the basin to one part in a million, measured off the traced path. The energy piles onto a limit cycle rather than filling a mode. The prediction is from 1995 and it was photographed in a tank two years later.

Tested in The reflection that changes the wavelength · the stratification reading path

“The Coriolis force deflects a moving body to the right in the northern hemisphere.”

It deflects it to the right continuously, which means the path is a closed circle rather than a bent line. Integrating the two-line system with no other force gives a trajectory that returns to its starting point after one inertial period, checked here to a millionth of its own radius. A description that stops at 'deflects' leaves out that the deflection has a period and that the net displacement over that period is zero.

Tested in The deflection that closes on itself · the circular motion reading path

“A pressure gradient pushes air from high pressure to low.”

It does, and in a rotating frame at small Rossby number the air ends up moving at right angles to the push. The trajectory here is integrated from rest under a steady eastward gradient: the parcel oscillates about a mean velocity that is northward, and the mean over a whole number of inertial periods matches the geostrophic value a/f to two per cent, measured off the path. Isobars on a weather map are streamlines, not directions of travel.

Tested in The ratio that decides whether the planet is turning · the circular motion reading path

“The refracted wave in a metal is absorbed after travelling a short distance.”

It is absorbed before travelling any distance worth the name. The ratio of the decay depth to the wave's own wavelength inside the material is 0.03 at most across the metals drawn, so the amplitude has fallen by a factor of e before a single oscillation is complete. Calling the result a wave is a courtesy, and the reflection is decided by a layer a few tens of nanometres thick.

Tested in The angle that is two angles · the refraction reading path

“Snell's law says the refracted ray is on the opposite side of the normal from the incident one.”

Snell's law says sin θ₂ = n₁ sin θ₁ / n₂, and the sign of n₂ decides which side. The construction has two intersections and the physical one is the one whose energy flows away from the interface, which for a negative index is the lower — so the refracted ray is on the same side, at an angle the signed law predicts and the geometry reproduces to nine decimal places.

Tested in The ray on the wrong side of the normal · the refraction reading path

“A neutrino mass would shift the endpoint of the beta spectrum down.”

It ends the spectrum early by exactly the mass and also bends the approach to it, and the bend is what is actually measured — an endpoint's absolute position is not known to a fraction of an electronvolt from any other source. The Kurie plot is exactly straight for a massless neutrino, verified here as a residual against a fitted line at a part in 10⁹, and a mass makes it curve down over the last few electronvolts.

Tested in The energy that did not all arrive · the decay reading path

“A daughter builds up until it reaches its parent's activity.”

Only if it is the shorter-lived. Three regimes follow from one ratio: a much shorter-lived daughter reaches exactly the parent's activity and stays there; a comparably short-lived one settles at λ₂/(λ₂−λ₁) times it, which is measurably above one; and a longer-lived daughter never reaches equilibrium at all, accumulating while the parent lasts and then decaying on its own schedule.

Tested in The chain that runs at its slowest member's rate · the decay reading path

“Stripping the electrons from an atom slows electron capture, because there is nothing left to capture.”

It does, and the more startling effect goes the other way. A beta decay whose available energy is too small to put an electron into the continuum can put one into an empty bound orbital instead — a channel that exists only when the orbital is empty. Rhenium-187 goes from 41.6 billion years as an atom to 32.9 as a bare nucleus, and dysprosium-163 is stable as an atom and decays in 47 days when stripped.

Tested in The half-life that chemistry can change · the decay reading path