What is taught wrongly, and what settles it

“A swinging rod has the period of a point mass at the same length.”

A uniform rod about one end swings as a point mass at two-thirds of its length, which is a 22 per cent error in period for anybody who measures the rod and trusts the simple formula.

Tested in The pendulum, and the small lie that makes it simple · the pendulum reading path

“Friction depends on the area of contact.”

Surfaces touch only at their high points, so the true contact area is a small fraction of the apparent one and is proportional to the load rather than to the shape. The apparent area cancels, which is why the coefficient works at all.

Tested in The slope, and the two directions that make it easy · the Free-body reading path

“Gauss's law holds only for symmetric charge distributions.”

It holds for any closed surface whatsoever; symmetry is what makes it *useful* for solving, which is a different claim. The figure draws the crumpled surface the argument permits and the flux through it is the enclosed charge, unchanged.

Tested in Counting what comes out, and never looking inside · the Gauss's law reading path

“Entropy is disorder, and there is a law that says disorder increases.”

Every microstate is equally likely and the figure draws them; what changes with the number of coins is how sharply the count peaks. The second law is that statement about counts and nothing else, and it does not exist at four coins — which is drawn too, so the scale at which it starts to hold is visible.

Tested in Entropy is a count, and the arrow of time is arithmetic · the entropy reading path

“A better-designed engine could beat the Carnot efficiency.”

The efficiency of the drawn loop is computed from its enclosed area over the integral of p dV along the hot isotherm, in the figure's own units, and comes out at 1 − Tc/Th — a number the figure never calculates and which contains nothing about the working substance, the fuel or the mechanism.

Tested in The ceiling on every engine, set before it was designed · the heat engines reading path

“A centrifugal force throws a turning object outward.”

The construction that gives the acceleration its size: two velocity arrows of equal length, one swept round by the turn angle, and the difference between them. At every sweep the figure draws, that difference points at the centre. Nothing in the drawing points outward, and what is felt as outward is the seat supplying the inward force.

Tested in Turning is an acceleration, and constant speed does not help · the circular motion reading path

“At resonance the pushes add up because they are in step with the motion.”

If that were right the force would be in phase with the displacement, and at resonance it is a quarter cycle away from it — in phase with the velocity instead, which is exactly the condition for the force to do work on every part of every cycle. The phase is drawn against frequency beside the amplitude.

Tested in The frequency that gets an answer, and the quarter cycle nobody mentions · the resonance reading path

“A microwave oven works by resonating water molecules.”

Water's rotational resonances are in the hundreds of gigahertz and ovens run at 2.45 GHz, far below any of them. The frequency is chosen for penetration depth, not for a peak, and the response curve drawn here shows how far off the peak it sits.

Tested in The frequency that gets an answer, and the quarter cycle nobody mentions · the resonance reading path

“A spherical mirror brings parallel light to a focus.”

The rays are traced and where they cross the axis is measured. The sphere spreads its crossings over 16 units of the drawing; the parabola drawn beside it spreads them over 0. The blur is what the shape does, not a defect in the making of it.

Tested in The mirror that cannot focus, and the shape that can · the imaging reading path

“The electric field inside a conductor is very small.”

Zero, not small, and by an argument one sentence long. The figure solves the induced surface charge rather than sketching it — the solved distribution fits σ ∝ cos θ with R² = 0.99986, which is the analytic answer — and the interior field it produces cancels the applied one everywhere.

Tested in The inside of a conductor, where the field is exactly nothing · the conductors reading path

“Length contraction is an optical illusion caused by light travel time.”

The measurement is defined by local coincidences — the front end passing a marker is an event at one place — and no light travel time enters it. What it actually is, is a disagreement about which two events are simultaneous, which the spacetime diagram draws as a different pair of slicing lines.

Tested in The length that depends on when, and is not really about length · the length contraction reading path

“A heavier body rolls down a slope faster than a light one.”

Hoop, disc and sphere released together at 20 degrees, drawn where each has reached the same moment. Neither mass nor radius survives into the acceleration — only k, the moment of inertia in units of MR squared — so the order is fixed by shape alone, and the race redrawn at 8 degrees gives the same order.

Tested in The mass, and where it sits, which is what decides the race · the rotation reading path

“The sky is blue because it reflects the sea, or because air is blue.”

One number does the whole of it: scattering as the inverse fourth power of wavelength. The same exponent, with no second mechanism added, gives the blue overhead and the red through a long path, and the figure computes both from the same curve.

Tested in Why the sky is blue and the sunset is not, from one exponent · the scattering reading path

“Putting heat into something raises its temperature.”

Boiling a kilogram of water costs 2,260 kJ, which is the heat that would raise it by 541 degrees — a temperature interval that does not exist for liquid water at atmospheric pressure. The plateaus are measured off the drawn curve in kilojoules.

Tested in The heat that changes no temperature, and where it actually goes · the phase change reading path

“Brighter light gives the escaping electrons more energy.”

Two intensities at one frequency, with a retarding voltage pushing escaping electrons back. Both curves reach zero at exactly the same voltage: the brighter light produces more electrons and not faster ones, which is the measurement that isolates the point.

Tested in Light arrives in lumps, and brightness only changes how many · the photon reading path

“The electron orbits the nucleus at the Bohr radius.”

The radial distribution is computed from the Laguerre recurrence, and its most likely radius is the Bohr radius exactly. The radius survives; the orbit does not exist. A model can produce the right number from a picture that is wrong, which is why the number is not evidence for the picture.

Tested in Where the electron probably is · the atomic structure reading path

“A measurement reveals a value the particle already had.”

Three analysers in a chain. The third recovers a component the second had no way of preserving, and it does so at a rate no assignment of pre-existing values to the particle can produce — which is drawn as the chain rather than argued.

Tested in The answer that was not there before · the measurement reading path

“The exclusion principle is a repulsive force between electrons.”

No potential energy term appears anywhere in it. Electrons do repel electrostatically and that is a separate and much weaker effect at atomic scales; what the principle is, is the cost of two objects being genuinely indistinguishable, and it constrains identity rather than pushing anything.

Tested in No two in the same state, and why matter has volume · the exclusion reading path

“A nucleus that has survived a long time is due to decay soon.”

Four hundred nuclei followed for four half-lives, three times over, with every nucleus given its own decay time and no knowledge of its age. The survival ratio is measured over five intervals including two whose start times nothing singles out, and it is the same ratio every time.

Tested in A nucleus with no clock · the decay reading path

“The pressure on the bottom of a vessel is the weight of the water in it, divided by the area.”

Three vessels of the same depth and wildly different contained volume, with the base pressure computed for each. The three pressures are identical and the volumes differ by a factor of 4.7, so the flaring vessel's base carries more force than the water standing over it weighs.

Tested in The pressure that only knows depth · the hydrostatics reading path

“A hydraulic press gets more work out than is put in.”

The two pistons move volumes that are equal, because the liquid between them is very nearly incompressible. So the distance ratio is the inverse of the area ratio, and the figure prints both products: a force sixteen times larger moving a sixteenth of the distance is the same number of joules to the last digit the generator can represent.

Tested in Force multiplied, and nothing gained · the hydrostatics reading path

“Things float because they are lighter than water.”

A steel ship floats and a steel nail does not, and both are made of the same substance. What decides is the mean density of the object including whatever it encloses, which is why the figure draws the submerged fraction as the density RATIO — and a hollow shape can put that ratio anywhere it likes.

Tested in The weight of the water that is not there · the buoyancy reading path

“An object sinks if it is denser than water and floats if it is not, whatever it is resting on.”

A block sitting flat on the bottom with no fluid underneath it has no upward pressure to collect, and stays down whatever its density. Buoyancy requires fluid on the underside, which is why a suction-bedded object stays bedded and why the principle is a statement about a body SURROUNDED by fluid.

Tested in The weight of the water that is not there · the buoyancy reading path

“A heavy keel makes a boat stable because weight at the bottom pulls it upright.”

Weight low down helps by moving the centre of gravity down, and that is only one of the three terms in the stability condition. The other two are set by the shape of the waterplane, and the figure computes all three: a hull can be made stable with no ballast at all by widening it, and no amount of ballast saves a hull whose waterplane is too narrow.

Tested in Why a ship comes back upright · the buoyancy reading path

“Surface tension exists because the surface molecules are pulled inward by the ones below.”

That inward pull is real and it is what makes a liquid hold together at all; it does not produce a tension along the surface. What does is that a molecule at the surface has fewer neighbours than one inside, so it sits at higher energy — making surface costs energy, and the cost per unit area is the tension. The distinction is testable: the first account predicts a pressure and the second predicts an energy, and it is the energy that is measured.

Tested in The skin that is not a skin · the surface tension reading path

“Connect a small bubble to a large one and they equalise into two of the same size.”

The excess pressure is 4γ/R, so the smaller bubble is at the higher pressure and drives air into the larger one — and shrinking raises its pressure further, so the process accelerates. The figure computes both pressures from the two radii and draws the direction of flow rather than asserting it: 72.8 Pa against 24.3 Pa at 4 mm and 12 mm.

Tested in The small bubble blows up the big one · the surface tension reading path

“Water boils when it reaches its boiling point.”

A bubble of vapour has to exist before it can grow, and at a micron the Laplace pressure is a couple of atmospheres — so a perfectly clean liquid can be heated far past its boiling point without boiling. What makes ordinary kettles boil on time is not the temperature but the crevices in the metal, which hold pre-existing gas pockets too large for the curvature term to close.

Tested in The small bubble blows up the big one · the surface tension reading path

“Surface tension holds a stream of water together.”

It holds a DROP together and it pulls a CYLINDER apart, and the difference is which way the two principal curvatures point. Above a wavelength equal to the circumference, pinching the thread lowers its area — so the figure's growth rate is positive there and negative below it, which is the same γ doing opposite things on either side of one wavelength.

Tested in The thread that cannot stay a thread · the surface tension reading path

“Capillary action is water being sucked up by the tube.”

Nothing sucks. The meniscus is curved, so by the Laplace relation the liquid just beneath it is at a pressure BELOW atmospheric, and the column rises until the hydrostatic weight makes up the difference. The figure draws each meniscus as the arc the contact angle forces and computes each height from that same curvature, so the drawn shape and the drawn height are one number.

Tested in How high water will climb · the capillarity reading path

“Capillary action is what lifts water to the top of a tall tree.”

Jurin's law gives the height as 2γcosθ/ρgr. Reaching a hundred metres needs a radius of about 0.15 microns, and xylem vessels are tens of microns across — a hundredfold too wide, giving a rise of centimetres. What actually holds the column is tension in the water itself, sustained by evaporation at the leaves, and capillarity's real role is providing the small pores at the top that can support the meniscus.

Tested in How high water will climb · the capillarity reading path

“A thick liquid is one whose molecules stick together more strongly.”

In a GAS, viscosity rises with temperature — faster molecules carry momentum across more effectively — while in a liquid it falls steeply, because the mechanism there is escaping from neighbouring cages. Two opposite temperature dependences from one word, which no account based on stickiness alone can produce.

Tested in Momentum going sideways · the viscosity reading path

“Doubling a pipe's radius doubles the flow through it, because it doubles the cross-section.”

It quadruples the cross-section and it also doubles the mean speed, because a wider pipe puts its slowest fluid further from the middle. The two effects multiply, so the flow goes as the fourth power: the figure integrates the drawn parabola over the cross-section and prints the factor, which is 16 for a doubling and 81 for a tripling.

Tested in The fourth power in a pipe · the viscosity reading path

“Ketchup will not come out of the bottle because it is very viscous.”

It has a YIELD STRESS, which is a different thing: below a threshold it does not flow at all, and above it flows readily. The figure draws the Bingham curve starting at a non-zero stress with the yield point marked — so the fluid's problem is not that it is slow but that it is, until struck, a solid.

Tested in The fluid that answers back · the rheology reading path

“A tsunami is a very large wave out at sea.”

Its amplitude in deep ocean is often under a metre and ships pass over it unnoticed. What distinguishes it is a wavelength of hundreds of kilometres — vastly greater than the 4 km ocean, so it is a SHALLOW-water wave everywhere, travels at √(gh) ≈ 200 m/s, and carries its energy as one packet because its group and phase speeds are equal.

Tested in The speed that depends on the length · the wave packets reading path

“Molecules in air are packed close together and constantly touching.”

The spacing between neighbours is about 3.3 nm and the free path is 68 nm, so a molecule passes some twenty others' worth of distance between collisions. A gas is mostly empty: at atmospheric pressure the molecules occupy about a thousandth of the volume, which is why a gas compresses easily and a liquid does not.

Tested in How far a molecule gets · the kinetic theory reading path

“Brownian motion is caused by currents in the liquid, or by the particle being alive.”

Both were proposed and both were eliminated by experiment: the motion persists in sealed drops kept at uniform temperature, and it is shown by particles of soot and of rock ground from an Egyptian sphinx. What remains is molecular bombardment, and the figure's measured mean square displacement grows in proportion to TIME, which is what an unbiased random walk gives and a drift does not.

Tested in The jiggle that proved atoms · the diffusion reading path

“Every substance freezes if it is made cold enough.”

Helium does not, at any temperature, under its own vapour pressure. Its zero-point energy — which it has because confining a light atom costs kinetic energy it cannot give up — exceeds the energy that would hold it in a lattice, so it stays liquid to absolute zero and needs about 25 atmospheres to solidify. The exception is not a curiosity; it is what makes everything else on this page observable.

Tested in The liquid that will not slow down · the superfluidity reading path

“Gravity is a force like any other, and a freely falling laboratory is one in which a force is acting.”

A force is detectable from inside the object it acts on: nothing else in physics can be removed by choosing where to stand. Inside a freely falling box every particle is unaccelerated with respect to every other, to the accuracy set by the box's size — which the figure computes as 1.57 µm over a ten-metre fall for two balls a metre apart.

Tested in The floor that cannot be told from gravity · the equivalence principle reading path

“The photon loses energy climbing out of the well, the way a thrown ball does.”

A photon's energy is hν and its speed never changes, so what falls is the frequency — and the fractional fall gh/c² contains nothing about the photon. Treating it as a ball with mass E/c² gets the right answer for the wrong reason: the same shift is measured for gamma rays and for optical light, and a ball's fractional energy loss depends on how fast it was thrown.

Tested in The clock that runs slow lower down · the gravitational redshift reading path

“Newton's theory predicts no bending of light at all, so the 1919 measurement decided between bending and none.”

A corpuscle moving at c past the Sun's limb is deflected by 2GM/bc², which is 0.87 arcseconds — computed here on the same axes as the relativistic 1.75. Cavendish and Soldner had the number in the 1780s and 1801. The measurement decided between two non-zero predictions in the ratio of exactly two, which is why arcsecond precision mattered.

Tested in The bend Newton got half right · the light deflection reading path

“A black hole is enormously dense.”

Density is mass over volume and the radius goes as the mass, so the mean density inside a horizon goes as 1/M². The figure computes the radius for four masses: at four million solar masses the horizon is 12.7 million kilometres across, which encloses a mean density well under that of water. Only the small ones are dense, and the large ones are not dense at all.

Tested in The surface that only lets things in · the horizons reading path

“An object falling into a black hole appears to freeze at the horizon, and stays visible there for ever.”

The image fades exponentially, with an e-folding time of about 2 rs/c fitted off the drawn curve — 0.2 milliseconds for a ten-solar-mass hole. After a millisecond the received frequency is down by a factor of a hundred, and the photon arrival rate with it. What hangs there for ever is a mathematical limit with no photons in it.

Tested in Two clocks that disagree about the fall · the horizons reading path

“An observer falling in sees the entire future of the outside universe at the moment of crossing.”

The blueshift of light falling down and the redshift from the observer's own inward speed very nearly cancel: an observer falling from rest far away sees the outside sky at a redshift of about two on crossing, not an infinite blueshift. The amount of outside history that arrives before the crossing is of order the light-crossing time of the region fallen through — milliseconds for a stellar-mass hole.

Tested in Two clocks that disagree about the fall · the horizons reading path

“The two polarisations are at right angles, like an electromagnetic wave's.”

They are at forty-five degrees. The figure draws both: rotating the plus pattern by 45° gives the cross pattern, and by 90° gives the plus pattern back again. That is the signature of a spin-2 field, and it is why a detector's two arms sit at a right angle — the geometry at which one arm lengthens while the other shortens.

Tested in The wave that stretches one way and squeezes the other · the gravitational waves reading path

“A Crookes radiometer demonstrates radiation pressure.”

Its vanes turn with the black faces retreating, which is the wrong way: light absorbed on a black face pushes it away from the source. The figure computes the radiation pressure at 4.5 µPa and shows that the vanes turn only where the residual gas has a mean free path comparable with the vane — near 0.7 Pa. Pump the tube harder and the radiometer stops, which is the opposite of what a light-driven device would do.

Tested in Light has a pressure · the radiation pressure reading path

“The electron orbits the nucleus.”

An orbiting charge accelerates, an accelerating charge radiates at the rate the Larmor formula gives, and integrating that loss against the orbital energy makes the radius obey r³ = r₀³ − 4k²t/c³. The figure computes the collapse: 1.6 × 10⁻¹¹ seconds from the Bohr radius, through about 200,000 orbits, with a continuously rising frequency. Matter exists and spectra are sharp, so the orbit is not what an electron does.

Tested in A charge that turns must glow · the radiating charge reading path

“A cross-section is the size of the target.”

It is an area with the dimensions of a size and no obligation to equal one. A nucleus of radius 7 fm has a geometrical area of 1.5 × 10⁻²⁸ m² and presents about 10⁻⁴⁷ m² to a low-energy neutrino — nineteen orders of magnitude smaller — while presenting a much larger area than its geometry to a slow neutron at a resonance. The word describes a probability, expressed in units of area.

Tested in How far a neutrino gets · the kinetic theory reading path

“A planet is round for the same reason a raindrop is.”

A drop is round because surface energy costs area and a sphere has the least; a planet is round because its own gravity crushes anything that sticks out. The two mechanisms have completely different size dependence — surface tension dominates below a few millimetres and gravity above a few hundred kilometres — and between them there is a range of sizes where neither enforces anything, which is where every irregular object in the solar system lives.

Tested in The size at which a body becomes round · the Self-gravity reading path

“The twins' situations are symmetric, so the paradox is a genuine contradiction in relativity.”

They are not symmetric, and the asymmetry is visible without any calculation: one twin stays in a single inertial frame throughout and the other does not. The travelling twin feels the turnaround and the stay-at-home never feels anything, which is a locally detectable difference. Anything that both twins can measure separates the two cases.

Tested in The twin who comes back younger · the time dilation reading path

“The Doppler shift depends only on the relative speed of source and observer.”

For sound it does not: a source approaching at speed v gives 1/(1 − v/c) and an observer approaching at the same v gives 1 + v/c, and the figure computes both at β = 0.5 as 2.000 and 1.500. The medium supplies a frame that makes the two situations physically different. For light there is no medium and the claim becomes true — which is a result, not an assumption.

Tested in The shift that survives at right angles · the doppler reading path

“A black hole is the state of maximum disorder, so nothing more can be said about its entropy.”

A great deal can be said and it is quantitative: the entropy is a quarter of the horizon's area in Planck units, which for a solar mass is 1.05 × 10⁷⁷ k against about 10⁵⁸ k for the same mass as gas. The figure computes both. A factor of nearly 10¹⁹ is not a vague statement about disorder; it is a number derived from an area.

Tested in The entropy that lives on a surface · the entropy reading path

“A wider tyre grips better because friction depends on the contact area.”

The coefficient contains no area, and the reason is that the apparent area is not the area in contact. Asperities flatten until they can carry the load, so the real contact area is the load divided by the hardness: 0.0500 mm² under 50 N on a 1,000 MPa surface, whatever the block looks like. The figure draws two faces differing fourfold in apparent area touching over the same 0.05 mm². Wide tyres are used for reasons — heat, wear, the fact that rubber is not a plastic solid — but not for this one.

Tested in The force that takes what it needs · the friction reading path

“The period of a pendulum does not depend on the amplitude.”

The period of a *parabola* does not depend on amplitude; a pendulum is not a parabola. The figure integrates the exact period between turning points found by bisection: at 90° from vertical it is 1.1803 times the small-swing value, which is the tabulated elliptic integral and is not built into the calculation anywhere. The independence is a property of the approximation and it fails at the second order in amplitude, which for a clock is the whole design problem.

Tested in Every minimum is a parabola · the harmonic approximation reading path

“Centrifugal force is what throws a spinning object outward.”

Nothing throws it outward. In an inertial frame the object travels in a straight line and the constraint holding it in is removed, so it departs along a tangent — the figure draws exactly that, as a straight line across a turntable. The outward force exists only in the frame that turns with the object, where it is one of three terms produced by the change of coordinates, and in that frame the object does not move outward either.

Tested in The forces that are not there · the circular motion reading path

“A magnetic field accelerates charged particles.”

It changes their direction and cannot change their speed, because qv×B is perpendicular to v at every instant and a force perpendicular to the velocity does no work. Every accelerator on Earth accelerates with electric fields and steers with magnetic ones; a machine with magnets alone would produce a beam that goes round for ever at the energy it started with.

Tested in The force that does no work · the magnetism reading path

“The field inside a wire is zero, like the field inside a conductor in electrostatics.”

That is the electrostatic result and it does not transfer. A loop of radius r inside a current-carrying wire encloses the fraction r²/a² of the current, so the field rises linearly from zero on the axis to μ₀I/2πa at the surface — 1.000 mT for 10 A in a 2 mm wire — and falls as 1/r outside. The maximum field anywhere is at the surface, and it is a kink rather than a peak.

Tested in The field that wraps a current · the ampere law reading path

“The displacement current is a current — charge really does flow across the gap in a capacitor.”

Nothing crosses the gap; that is the whole point of a capacitor. What the term counts is ε₀ times the rate of change of electric flux, which has the units of a current and is not one. The figure computes both quantities for a 100 cm² capacitor charged at a million volts per second: the conduction current in the wire is 44.271 µA and ε₀dΦ/dt between the plates is 44.271 µA, and the two are equal because both reduce to ε₀A/d times the rate of change of voltage.

Tested in The term that made light · the maxwell equations reading path

“Light takes the quickest path.”

It takes a stationary one, and the distinction is not pedantry. Inside an elliptical mirror every path from one focus to the other has exactly the same length — 2.000000 in units of the semi-major axis, with a spread of 9×10⁻¹⁶ over four hundred sampled points — so no path is quickest and light takes all of them. The rainbow exists at a stationary point of deviation that is a minimum in one variable and nothing special in any other.

Tested in The path that does not change · the fermat reading path

“A better lens resolves finer detail.”

Only up to the point where the aperture takes over, and every decent instrument is already there. The limit is 1.2197λ/D with nothing in it but the wavelength and the width of the hole, and on logarithmic axes it is a line of slope exactly −1. Polishing improves an instrument until it reaches that line and then does nothing at all: a perfect 5 mm pupil resolves 27.7 arcseconds and no amount of care makes it 20.

Tested in How far apart two things have to be · the diffraction reading path

“Light bends round an obstacle because the wavelets from the edge spread out.”

Every point of the front emits, not merely the ones at the edge, and the spreading of a wide front is cancelled by its neighbours: the figure shows the envelope of wavelets from a plane front staying exactly plane. What an edge does is remove the neighbours whose contributions would have cancelled, so the surviving spread is not created by the edge but merely no longer suppressed.

Tested in Every front is a source · the huygens reading path

“A wave reflects off a harder medium and passes into a softer one.”

Hardness is not the quantity. What decides the answer is the impedance Z = √(Tμ), and a medium can be stiffer and lighter in the same proportion with no reflection at all: the coefficients depend on Z₂/Z₁ and on nothing else about either side. At a ratio of one, two media with completely different tensions and densities join invisibly, which is what every matching layer ever designed exploits.

Tested in What happens where the medium changes · the impedance reading path

“A distant sound is quiet because the air absorbs it.”

Over ordinary distances almost all of the loss is geometry and none of it is absorption. At 4 dB per kilometre — a fair figure for a mid-frequency in still air — spreading and absorption are equal only at 21.7 km, computed by bisection on the two drawn curves. At 100 m the spreading has cost 40 dB and the absorption 0.4 dB, a factor of a hundred in the wrong place for the usual explanation.

Tested in How a wave thins out · the wave motion reading path

“Equipartition follows from molecules sharing energy in collisions.”

Sharing is the mechanism by which the distribution is reached, not the reason for the result. The theorem holds for any term in the energy that is quadratic in its variable, whether or not that variable is a velocity, and it holds for a single oscillator in contact with a bath with no collisions at all. What produces the half is the Gaussian integral over that variable, and nothing in the derivation mentions a collision.

Tested in Half a kT for every way of moving · the equipartition reading path

“The rule that a reaction doubles for every 10 °C is a law of chemistry.”

It is one point on a curve and holds for one barrier at one temperature. The figure computes the rise that doubles a 0.35 eV process: 11.1 K starting from 250 K, 16.2 K from 300 K, and 29.3 K from 400 K, growing as the square of the temperature. A different barrier moves the whole curve. The rule is a useful accident of the fact that many biological processes have barriers near half an electronvolt and are used near room temperature.

Tested in The exponential that decides everything · the entropy reading path

“Two events that are simultaneous in one frame can be reordered in another, so cause and effect are relative.”

Only spacelike-separated events can be reordered, and those are exactly the pairs no signal can join. The figure computes the sign for three events: +2.64 for the timelike pair, 0.00 on the cone, −3.25 for the spacelike one. Timelike order is the same in every frame, which is precisely what makes causality safe — the events whose order is negotiable are the ones that cannot influence each other.

Tested in The quantity nobody argues about · the spacetime diagram reading path

“Magnetism is a relativistic effect, so it must be tiny.”

The effect per charge is tiny and the number of charges is not. At a drift speed of 0.1 mm/s, γ − 1 is 5.6×10⁻²⁶ — a correction in the twenty-sixth decimal place. A cubic metre of copper holds 10²⁹ conduction electrons, and the residual is an ordinary force: the figure computes 2.000×10⁻⁸ N/C by the magnetic route and 2.000×10⁻⁸ N/C by the electrostatic one, agreeing to nine decimal places.

Tested in Magnetism is electricity seen sideways · the field transformation reading path

“The centre of mass is a mathematical convenience with no physical content.”

It is the one point in a system whose motion obeys the elementary law. The opening figure has a total momentum of 4.00 kg·m/s and a total mass of 4 kg, so its centre of mass travels at exactly 1.00 m/s before the collision and at exactly 1.00 m/s after — while both individual velocities change, 2.00 to 0.60 and −2.00 to 2.20. No other point drawn in the figure has that property.

Tested in The point that keeps moving as if nothing had happened · the momentum reading path

“Energy travels from one pendulum to the other through the coupling spring.”

Nothing travels. The pair has two motions whose amplitudes never change, and both are standing patterns present everywhere at once. The exchange figure integrates the real motion step by step with velocity Verlet and then measures the beat period off the trace: at a coupling of kc/k = 0.02 it is 51.0 swings of the pair, agreeing with 2π/(ω₂−ω₁) to 2.6×10⁻⁸ per cent. Raise the coupling to 0.25 and the same measurement gives 4.9 swings. The handover rate is a difference of two fixed frequencies, not a transit time for anything crossing the gap.

Tested in The two pendulums that will not stop swapping · the harmonic approximation reading path

“The energy of a charged capacitor is stored in the charge sitting on its plates.”

Both accounts give 1.27 µJ for plates of 200 cm² held 2.00 mm apart at 15 nC, so agreement settles nothing. The force does: at fixed charge and at fixed voltage the stored energy runs opposite ways with separation, and only the density account produces the same attraction of 635 µN from both branches, because Q²/2ε₀A is the energy density 31.8 mJ/m³ read as a stress of 31.8 mPa across the 0.0200 m² the field crosses. A quantity living on the plates has no cubic metres to be a density in.

Tested in Where the energy of a field actually is · the field energy reading path

“A neutral object has no electric field, so nothing can pull on it and it can pull on nothing.”

The flux through any closed surface drawn around a water molecule is exactly zero, because the enclosed charge is exactly zero. The field is not. One nanometre out along the molecule's own axis it is 1.1 × 10⁸ V/m — thirty-seven times the 3 × 10⁶ V/m that ionises dry air. Zero flux constrains the monopole and says nothing whatever about the separation.

Tested in The attraction that needs no charge · the the field concept reading path

“A grating with more lines spreads the colours further apart.”

The grating equation d sinθ = mλ contains no slit count, and the figures are drawn at 2, 20 and 1000 slits with the same spacing of four wavelengths: all three put the first order at 14.48°, to the last digit the drawing carries. What changes is the width of that order — 7.17°, 0.635° and 0.0127° between half-maximum points. Dispersion is set by the pitch; the count buys sharpness and nothing else.

Tested in What a thousand slits buy that two cannot · the diffraction reading path

“Interference happens whenever light from two sources overlaps.”

Overlap is guaranteed and fringes are not. Two independent lamps do produce a fringe pattern, but its phase re-randomises every coherence time — about 100 fs for a source filtered to 10 nm at 550 nm. An eye integrating for 40 ms averages 4 × 10¹¹ independent patterns, and the average of a fringe pattern over all phases is uniform illumination.

Tested in Why two lamps never interfere · the coherence reading path

“A wave goes faster if the string is shaken harder, because each bit of the medium is pushed along more forcefully.”

The amplitude does not appear in T·y″ = µ·ÿ anywhere, so it cannot appear in the answer. On a 1.0 g/m string at 80 N the speed is 283 m/s whether the wave is a millimetre high or a centimetre, and the figure computes 141, 283 and 424 m/s at 20, 80 and 180 N with no amplitude supplied at all. Amplitude enters only through the terms the derivation threw away: at a peak slope of 0.05 the extra stretching raises the tension by about 12 per cent and the speed by 6 per cent, which is a correction to a neglected term rather than the mechanism.

Tested in The equation that lets a shape travel · the wave motion reading path

“A drumhead has a fundamental with overtones at whole-number multiples of it, the way a string does.”

The ratios are computed here from the zeros of Bessel functions, summed from their power series and checked against published values to 4.4 × 10⁻⁷: 1.000, 1.593, 2.136, 2.295, 2.653, 2.917. Not one is a whole number. The nearest approach among the lowest eight partials is mode (1,2) at 2.91730, which is 2.76% below the third harmonic — 48 cents, nearly half a semitone, an interval no ear rounds off.

Tested in The drum that has no harmonics · the standing waves reading path

“A particle with no mass cannot carry momentum, so the momentum of light is a figure of speech.”

Setting m = 0 in E² − (pc)² = (mc²)² gives E = pc, which is finite, non-zero and exactly what a photon has. The Compton figure closes the momentum triangle for a 17.44 keV X-ray scattering through 60°: the photon leaves at 72.31 picometres instead of 71.1, and the 0.29 keV it lost is carried off by the electron as momentum the photon supplied.

Tested in The invariant that survives a boost · the Mass-energy reading path

“The pole cannot both fit and not fit, so one of the two frames has to be wrong.”

Neither is. The two door-closing events are spacelike separated — 10 m apart with no time between them in the barn's frame, so $s^2 = -100\;\mathrm{m}^2$ — and the order of such a pair is frame-dependent. In the pole's frame the far door shuts 57.8 ns before the near one, which is $\beta\gamma L/c$ with $L$ the barn's 10 m rest length. Every local coincidence is identical in both accounts: each door shuts on empty air.

Tested in The pole that fits and does not fit · the length contraction reading path

“Order cannot arise spontaneously, because entropy always increases.”

Entropy of the whole always increases; the entropy of a part need not. A kilogram of water freezing at −10 °C loses 1.22 kJ/K of entropy and hands 334 kJ to its surroundings, whose entropy therefore rises by 334/263 = 1.27 kJ/K. The net is +47 J/K, so a pond crystallising is not an exception to the second law but a consequence of it.

Tested in What a system actually minimises · the entropy reading path

“Water boils at 100 °C.”

The vaporisation curve inverted against a standard atmosphere puts the crossing at 71.1 °C on Everest's 31.4 kPa and at 119.5 °C at two atmospheres — a 48-degree spread with the same water in the pot. The 100.0 °C the figure prints at sea level is a statement about 101 kPa of air.

Tested in A boiling point is a pressure, not a temperature · the phase change reading path

“The contact angle is a property of the liquid — water has a contact angle of about 20 degrees.”

Change the solid and nothing whatever about the water has changed, yet the angle runs from near 0° on clean glass to 110° on paraffin wax. The figure puts one 5 µL drop of the same liquid on four solids and solves each cap from that single volume: the footprint radius falls from 2.61 mm at 20° to 1.34 mm at 90° to 0.69 mm at 140°, with the volume held equal to a part in 10⁹ and the drawn tangent checked against the intended angle to 0.2°.

Tested in The angle a liquid makes with what it sits on · the capillarity reading path

“Gravity is a feeble energy source next to nuclear fusion.”

The energy per unit mass of the innermost stable circular orbit is √(8/9) = 0.94281, so anything that reaches it has radiated 5.719% of its rest mass — a number the figure locates on its own drawn curve by golden section. Hydrogen fusion converts 0.7%. The ratio is 8.2, and for a maximally rotating hole the geometrical figure rises to 42.3%.

Tested in The orbit that cannot be made smaller · the orbit stability reading path

“Measuring one member of an entangled pair changes the other, so information crosses the gap faster than light.”

Each analyser alone is a fair coin whatever the distant one is set to. In the sampled run of 100,000 coincidences there are 25,000 pairs at each of the four settings, so the standard error on a single analyser's rate of plus outcomes is 0.0032, and the figure refuses to draw itself unless every one of the eight rates sits within four of those of one half. The correlation appears only when the two records are compared, and comparing them needs an ordinary channel.

Tested in The correlation no instructions can produce · the entanglement reading path

“A rigid body spun about any of its principal axes will keep spinning about it.”

Linearising Euler's equations about each axis in turn gives a coefficient that is a product of two differences of moments. For the least and the greatest moment that product is negative and a disturbance oscillates; for the middle one it is positive and the disturbance grows as e^(σt). For a 300 × 200 × 30 mm book the growth rate at 4 rad/s is 2.43 s⁻¹, so a departure of four parts in a thousand becomes a full reversal in about two and a half seconds — which the figures obtain by integrating the equations rather than by linearising them, with the energy and the angular momentum holding to eight parts in a million million over the whole run.

Tested in The axis that will not hold · the rotation reading path

“Whether a pushed object slides or tips over depends on how hard it is pushed.”

Both thresholds are proportional to the weight, so the weight divides out of the comparison and the harder push simply arrives at whichever threshold is lower. Sliding needs P/W ≥ μ; tipping needs P/W ≥ b/2y with y the height of the push and b the width of the base. The comparison is μ against b/2y and contains no force at all: a 300 mm wide cabinet pushed at 900 mm goes over at P/W = 0.167 whether it weighs ten kilograms or four hundred.

Tested in Slide or topple · the Free-body reading path

“The image charge is a convenient fiction that gives approximately the right answer.”

It gives exactly the right answer, and the reason is a theorem rather than a coincidence. Laplace's equation with the potential fixed on every boundary has exactly one solution, so any function satisfying both is the field. The two-charge potential vanishes on the plane term by term at every point, and satisfies Laplace's equation everywhere above it because both its sources are below. There is nothing left to check and no error to estimate.

Tested in The charge that has to be somewhere else · the conductors reading path

“A magnet in a magnetic field is pulled along the field.”

In a uniform field it is not pulled at all. The force on a current loop is the sum of I dl × B round the loop, and with B constant that is I(∮dl) × B, in which ∮dl is the sum of the displacement vectors round a closed path and is exactly zero. The figure draws the four forces on a rectangular loop: two cancel outright and two form a couple. What a uniform field supplies is a torque and nothing else, at every angle, exactly.

Tested in The loop that behaves like a needle · the magnetism reading path

“Reflection is weaker for light polarised in the plane of incidence, and at Brewster's angle it is very weak.”

At Brewster's angle it is zero, exactly, and the figure computes it rather than rounding it: r_p = (n₂cos θᵢ − n₁cos θₜ)/(n₂cos θᵢ + n₁cos θₜ) has a numerator that vanishes identically when tan θᵢ = n₂/n₁. The plotted reflectance touches the axis and comes back up; there is no floor beneath it. For air to glass the angle is 56.31° and the reflectance there is 0.000%.

Tested in The angle at which reflection picks a side · the polarisation reading path

“A mirage is light reflecting off a layer of hot air.”

Nothing reflects. The rays in the figure are traced by integrating dz/dx = ±√((n/C)² − 1), and the sign flips where the bracket reaches zero — the ray runs out of height, exactly as a thrown ball does at the top of its arc. There is no discontinuity in the index anywhere in the profile, and the turning point for a ray from an eye 1.5 m up is about half a centimetre above the road, reached after two hundred metres of gradual curvature.

Tested in The ray that bends without a surface · the fermat reading path

“Coherence length is a property of the apparatus — a better interferometer gives fringes over a longer path difference.”

It is a property of the source. The visibility against path difference is the modulus of the Fourier transform of the source's spectrum, computed here by summing over the spectrum rather than taken from a standard result, and the apparatus appears nowhere in it. White light gives 3 µm; a filtered lamp at 10 nm bandwidth gives 30 µm; a sodium lamp gives centimetres; a stabilised laser gives kilometres. The same interferometer produces all four numbers.

Tested in How far a wave can remember · the coherence reading path

“Pumping a swing is driving it at its natural frequency, like pushing it.”

It is done at twice the natural frequency, and the equation is different in kind. A driven oscillator has a force on the right-hand side; the pumped one has a modulated stiffness on the left and nothing on the right, so x = 0 solves it exactly at every modulation depth. The figure maps where that solution stops being stable, and the widest region sits at Ω = 2ω₀ — one rise and one fall of the centre of mass per half swing.

Tested in The swing that is pumped, not pushed · the resonance reading path

“Newton's speed of sound was wrong because his arithmetic contained an error.”

The arithmetic gives √(P/ρ) = √(RT/M) exactly, which is 280.0 m/s for air at 0 °C and is what Newton's assumption entails. The measured value is 331.3. The figure draws both for four gases and the discrepancy is 15.5% for air, 22.5% for helium and argon, and 12.0% for carbon dioxide — not a constant error, but a gas-dependent one, which is the signature of a missing physical quantity rather than of a slip.

Tested in The correction that took a century · the wave motion reading path

“A wave packet travels at the group velocity and keeps its shape.”

It keeps its shape only if ω(k) is straight. Expanding about the carrier, the first derivative moves the packet and the second spreads it: the width goes as σ₀√(1 + (t/τ)²) with τ = σ₀²/|d²ω/dk²|. The figures measure the width off the emitted envelope as a second moment and find the 28%-bandwidth packet spreading more than fourfold in sixty seconds while the 10% one grows by less than a fifth.

Tested in The packet that will not keep its shape · the wave packets reading path

“Moving fast changes the colour of what is seen and not its arrangement.”

The arrangement changes more. The map cos θ = (cos θ′ + β)/(1 + β cos θ′) takes the emitted right angle to 60.0° at 0.5c, 25.8° at 0.9c and 8.1° at 0.99c, and half of everything emitted lies inside that angle. At 0.99c a traveller's whole rear hemisphere has been compressed into a ring near the forward direction, and the stars behind have moved in front.

Tested in The sky that crowds into a cone · the doppler reading path

“Relativistic corrections to satellite navigation are a refinement that improves the accuracy a little.”

Untreated they accumulate at 38.6 µs a day, and a timing error multiplies by the speed of light to become a range error: 11.6 km a day, growing without limit. A system with a metre-scale specification would be useless within the first two minutes of operation. The figure computes both terms from the orbit radius and neither is typed in.

Tested in The clock that is wrong in two directions · the time dilation reading path

“A machine that delivers more heat than the energy it consumes would violate conservation of energy.”

Nothing is created. A heat pump moves heat from outside to inside and adds the work it consumed on top, so the heat delivered is the heat taken plus the work done — and the ratio of delivered heat to work is T_h/(T_h − T_c), which exceeds one whenever the temperature difference is smaller than the absolute temperature. With 21 °C indoors and 7 °C outside the ideal ratio is 21.0. Energy is conserved exactly; what is being paid for is the moving, not the heat.

Tested in The engine that pays back more than it takes · the heat engines reading path

“The surface of a spinning liquid is approximately a paraboloid, for small spin rates.”

It is a paraboloid exactly, at every spin rate at which the liquid stays in the dish, and there is no small quantity anywhere in the derivation. In the rotating frame the surface is a level set of gz − ½ω²r², which rearranges to z = ω²r²/2g with no approximation. The figures draw it at 10, 20 and 40 revolutions a minute and the shape is the same parabola with the vertical axis stretched by ω².

Tested in The surface a spin decides · the hydrostatics reading path

“In a freely falling laboratory, gravity has no observable effect.”

It has an effect proportional to the size of the laboratory and to the square of how long the experiment runs. Two masses released a distance L apart converge at (GM/r³)L across the fall and separate at twice that along it, so after a time t they have moved (GM/r³)Lt²/2 relative to each other. For an instrument resolving a nanometre, the largest undetectably flat box near the Earth's surface is 1.30 mm at one second — and 13 µm at ten seconds, because the size falls as the square of the time.

Tested in The term free fall cannot remove · the equivalence principle reading path

“Critical damping is the fastest way to bring an oscillator back to rest.”

Integrate the equation and measure the time the displacement takes to enter and stay inside a band about zero. The minimum sits below critical damping at every band width tested — at ζ = 0.71 for a band of five per cent, 0.79 for two per cent and 0.91 for two parts in a thousand — because a small overshoot that stays inside the band costs nothing and buys a faster approach. Critical damping is the fastest return that never crosses zero, which is a different requirement and often not the one wanted.

Tested in The three ways of coming to rest · the pendulum reading path

“The copper attracts or repels the magnet, so the pipe must be slightly magnetic.”

Copper's magnetic susceptibility is about −10⁻⁵, so the static force between a magnet and a stationary copper tube is smaller than the magnet's weight by five orders of magnitude and is repulsive rather than attractive. Nothing at all happens while the magnet is held still inside the tube. The retarding force is proportional to the magnet's speed and vanishes at zero speed, which is drawn here as a line through the origin, and no static property of any material can produce that.

Tested in The magnet that falls slowly · the induction reading path

“An achromatic doublet brings all colours to one focus.”

The condition φ₁/V₁ + φ₂/V₂ = 0 sets the derivative of the total power with respect to wavelength to zero, which makes the focal length stationary rather than constant. Evaluating the designed pair at every wavelength from its glasses' own Sellmeier coefficients gives a focus that returns to its corrected value at exactly two wavelengths — 486 and 656 nm, the two the condition was written at — and departs from it everywhere else, by 2.27 mm at 400 nm for a 500 mm lens. That residual is the secondary spectrum and no pair of ordinary glasses removes it.

Tested in Two glasses that cancel a derivative · the dispersion reading path

“A large sphere removes light from a beam in proportion to the area it blocks.”

The scattering efficiency of a large sphere approaches two, not one: it removes twice as much light as it geometrically intercepts, which is the extinction paradox. The Mie sum reaches 2.12 at a size parameter of a hundred and oscillates toward two from above. The extra factor comes from the diffracted light that passes the sphere and is deviated by a small angle — light that never touched it and is still missing from the forward beam.

Tested in When the particle is the size of the wave · the scattering reading path

“An anti-reflection coating absorbs the light that would have been reflected.”

The layer is transparent and its absorption is negligible; the energy is not removed but redirected. With the matching layer in place the transmitted fraction rises to exactly one at the design wavelength, which is computed here from the characteristic matrix and comes out as unity to round-off. The two reflected waves — one from each surface of the layer — are equal in amplitude and half a cycle apart, so they cancel each other and the energy they would have carried goes forward instead.

Tested in The layer that makes a reflection vanish · the impedance reading path

“A superluminal group velocity means information has travelled faster than light.”

The emergent pulse is reconstructed from the leading edge of the input, which arrived in good time. A smooth pulse is analytic: its whole shape is determined by any arbitrarily early piece of it, so nothing about the peak is news when the peak arrives. Cutting the input off sharply — giving it a genuine front, a moment before which it is exactly zero — removes the reconstruction, and the front then travels at exactly c in every medium, because the front is carried by the infinite-frequency limit of the index and that limit is one.

Tested in The speed that carries no signal · the wave packets reading path

“The entropy of mixing measures how thoroughly two gases have interpenetrated, so it should depend on how different they are.”

It contains no property of either gas. For ideal gases the entropy gained per particle is −[x ln x + (1−x) ln(1−x)] in units of k, which is a function of the proportions alone: two gases mixed half and half gain k ln 2 per particle whether they are helium and argon, or two isotopes of neon, or two nuclear spin states of the same molecule. Mixing the same gas with itself gains nothing at all. The function does not interpolate between those cases; it takes one value until the species become identical and then takes the other.

Tested in Mixing what is already mixed · the entropy reading path

“The loop in a van der Waals isotherm is a mistake in the equation, which a better equation would not have.”

Every analytic equation of state that describes both a liquid and a gas has one. An analytic function cannot have a flat segment followed by a return to curvature, so any single expression covering both phases must pass through the coexistence region somehow, and the only smooth way through is a loop. The loop is what an equation of state looks like when it is asked about states that do not occur, and the construction is the instruction for reading the answer.

Tested in The part of the curve no fluid follows · the phase change reading path

“Zero-point energy is residual thermal motion that a cold enough refrigerator would remove.”

It is present at absolute zero by construction: the energy minimised here contains no temperature at all, and the minimum over the width of the wavefunction is ħω/2 whatever the surroundings. Removing it would mean localising the particle at the bottom of the well, and the figure shows the kinetic cost of doing so rising as the inverse square of the width — faster than the potential saving falls. The floor is a consequence of confinement, not of warmth.

Tested in The motion that cannot be stopped · the uncertainty reading path

“Nuclear energy comes from converting matter into energy, which chemical energy does not do.”

Both do, and the equation applies identically to each. Burning a kilogram of coal releases 32.8 MJ, which is 3.6 × 10⁻¹⁰ of its rest energy; fission of uranium-235 releases 8.2 × 10¹³ J/kg, which is 9.1 × 10⁻⁴. The difference is ten million times in magnitude and nothing at all in kind. What makes the chemical case invisible is that a mass change of one part in three billion is far below what any balance can weigh, and it was for two centuries taken as evidence that mass is conserved in chemistry.

Tested in The mass that is missing · the Mass-energy reading path

“A faster-than-light signal would merely be fast; nothing about it is logically impossible.”

A signal at 2c from the origin to a point three light-seconds away arrives at t = 1.5 s in the frame it was sent in and at t′ = −0.375 s in a frame moving at 0.6c — before it left. The reversal happens for any boost above the reciprocal of the signal speed. Two such signals arranged head to tail, one in each of two frames, deliver a message to the sender's own past, and the impossibility is a contradiction rather than an inconvenience.

Tested in Which came first, and who decides · the simultaneity reading path

“The instability is a consequence of the gas pocket being compressible, so a solid body of the same density would be neutrally stable at every depth.”

It is a consequence of the body being MORE compressible than the fluid, which is a comparison rather than a property. Water's compressibility is 4.5 × 10⁻¹⁰ per pascal and is not zero: a body slightly less compressible than water is stable, returning to its depth after a nudge, and this is why an oil-filled float holds station in the deep ocean while a gas-filled one cannot. Both signs occur and the sign is what decides.

Tested in The depth past which it must sink · the buoyancy reading path

“Surface tension is a property of a liquid, so a table of values for water settles the matter.”

It is a property of an interface and depends on both sides and on the temperature. Water's falls by 0.15 mN/m for every degree of warming, and two per cent of ethanol by mole fraction takes it from 72 to 56 mN/m — a fall of twenty-two per cent for a change in composition small enough to taste and not to see. The steepness of that dependence at low concentration is the whole reason the effect exists: a trace of the second component covers most of the surface.

Tested in The surface that pulls toward the stronger side · the surface tension reading path

“Crossing a black hole's event horizon means being crushed or torn apart.”

The tidal stretch at a horizon is ℓc⁶/4G²M², which falls as the square of the mass. For a ten-solar-mass hole it is 1.9 × 10⁷ times Earth's gravity across a person and is lethal long before arrival; for a 10¹⁰-solar-mass hole it is 1.9 × 10⁻¹¹, which is a hundred billion times smaller than the tide the Moon raises on a human body. The two curves cross at about 10⁴ solar masses, above which a horizon can be entered whole.

Tested in The horizon that nothing marks · the horizons reading path

“Newton's inverse square was chosen because it fits the planets, and a slightly different exponent would fit them nearly as well.”

The apse motion is linear in the departure with a slope measured here as −3.14 radians per unit of exponent, so a departure of one part in a hundred would turn the line of apsides of the innermost planet through 720 degrees a century. Run backwards, the observed unexplained precession of 43 arcseconds a century corresponds to a departure of 1.6 × 10⁻⁷ in the exponent. The lever between the exponent and the angle is enormous, which is what makes a small residual a severe constraint rather than a tolerance.

Tested in The orbit that does not come back to itself · the orbit stability reading path

“Kinetic energy is conserved when a spinning body changes shape with nothing touching it.”

Kinetic energy is L²/2I, so at fixed L it goes as the inverse of the moment of inertia and cannot stay constant unless the shape does. Applying energy conservation to the skater instead of momentum conservation predicts that pulling the arms in leaves the rate unchanged, which is not what happens and not what the figures compute.

Tested in The quantity that survives a change of shape · the rotation reading path

“A spinning gyroscope resists gravity, or is held up by its spin.”

Nothing supports it: the weight is exactly balanced by the pivot's reaction, as it is for a body that is not spinning at all, and the net torque about the pivot is the same in both cases. What the spin changes is the *response* to that torque. A stationary body answers a torque by falling; a spinning one answers it by turning sideways, because the torque adds to an angular momentum that already exists instead of creating one from nothing.

Tested in The push that comes out sideways · the rotation reading path

“At the exact edge of a geometrical shadow, half the light gets through.”

A quarter does. The straight edge blocks half the wavefront, so it removes half of the *amplitude*, and the intensity is the square of the amplitude — the chord of the spiral from one eye to the centre is exactly half the chord from eye to eye, and 0.5² is 0.25. The figure computes the two chords and refuses to draw if their ratio is not exactly a half.

Tested in The spiral that says how much light arrives · the huygens reading path

“A vibration absorber works by absorbing the energy of the vibration.”

An undamped absorber dissipates nothing at all, and it is the undamped one that gives perfect protection. At the tuned frequency the protected mass does not move, so no work is being done on it and there is no energy to absorb — the absorber's spring is delivering a force equal and opposite to the applied one, and the applied force therefore does no work either. Adding damping to the absorber makes it dissipate and makes the protection worse at the tuned frequency, which is the opposite of what the name suggests.

Tested in The mass that makes another stand still · the resonance reading path

“A birefringent crystal splits light into two beams because it has two refractive indices.”

Two indices alone do not split anything: at normal incidence Snell's law gives a zero angle of refraction for any index whatever, so both wave normals go straight through. The splitting is a separate effect — in a crystal the electric displacement and the electric field are not parallel, so the extraordinary wave's energy travels at an angle to its own wave normal. In calcite at 45° to the optic axis that walk-off is 6.22°, which is 1.09 mm of separation through a 10 mm slab, and it is exactly zero along the optic axis and across it, where the two indices are still different.

Tested in The crystal that answers twice · the polarisation reading path

“Nothing is on the far side of the boundary in total internal reflection.”

The reflected beam emerges displaced along the surface, by 350 nm for s-polarised light at 45° in glass and 560 nm for p — measured by Goos and Hänchen in 1947 by accumulating the offset over many reflections. A displacement of the outgoing beam from the incoming one has no interpretation in which the light stayed on the near side of the surface.

Tested in The reflection that happens where the glass is not · the total internal reflection reading path

“A thinner gas is less viscous.”

It is not. The viscosity is a third of the density times the mean speed times the free path, and the free path is inversely proportional to the density, so the two cancel exactly. The figures compute nitrogen, helium and argon over eight decades of pressure and the lines are flat. Air at a hundredth of an atmosphere damps a moving vane as strongly as air at one, which is the measurement Maxwell made and the reason a rough vacuum is no help at all in a delicate instrument.

Tested in The viscosity that does not care how much gas there is · the kinetic theory reading path

“The electrons in a metal are a gas at the temperature of the metal.”

Their Fermi energy is 7.04 eV in copper, which corresponds to 81,700 K. A classical gas at 300 K has a mean energy of 0.039 eV. The electrons are not thermal at all — at room temperature only 1.6 per cent of the way down the sea is affected by the temperature, and everything below that is in exactly the state it would be in at absolute zero.

Tested in The pressure that is not a temperature · the exclusion reading path

“A fast-moving object looks squashed along its direction of travel.”

It looks turned. Light from the far side left earlier than light from the near side, by the time needed to cross the difference in distance, and the object moved in that interval — so the far face is recorded displaced backwards and the side comes into view. The contracted width and the displacement are cos α and sin α of the same angle: at β = 0.8 a cube photographs as one turned 53.1°, and the figure computes the apparent corners from the light delay and refuses to draw unless they coincide with a rotated cube's to a part in a million million.

Tested in The contraction no photograph shows · the length contraction reading path

“Two Lorentz boosts compose to a Lorentz boost.”

Only if they are parallel. Composed as matrices and decomposed again, the product of two boosts of equal size at right angles is a boost together with a rotation — 2.70° at β = 0.3, 36.1° at β = 0.85, 63.2° at β = 0.95. The figures compute the rotation by multiplying the matrices and pulling the rotation out of the product, and check the answer against the closed form for perpendicular boosts.

Tested in The turn that two pushes leave behind · the velocity addition reading path

“A coffee stain has a ring because the coffee particles are carried to the edge as the drop shrinks.”

The drop does not shrink. Its contact line is pinned — it stays at the same radius for almost the whole of the drying — and what shrinks is its height. The particles are carried outward by a flow that exists precisely because the edge cannot retreat: liquid lost to evaporation at the rim has to be replaced from the interior, and everything suspended in that liquid goes with it.

Tested in The ring the drop leaves behind · the surface tension reading path

“Water cannot be pulled — a liquid cannot sustain a negative pressure.”

It can, and by a great deal. Water in careful experiments has been held at tensions of tens of megapascals, hundreds of times more than the tenth of a megapascal the atmosphere supplies, and the theoretical limit from the intermolecular forces is higher still. What breaks a real column is not the water but a nucleation site — a dissolved bubble, a crevice on a wall, a speck of dust — which is why degassing and smooth tubes change the answer so completely.

Tested in The height a siphon cannot pass · the hydrostatics reading path

“The delay is accumulated near closest approach, where the field is strongest.”

Half of it is accumulated beyond twenty-two solar radii, in a field a few hundred times weaker than at the limb. The integrand falls as 1/r, so every factor of ten in distance contributes about the same amount — which is a logarithm, and a logarithm has no scale. That is also why the total depends on where the two endpoints are and not only on how close the path came.

Tested in The delay that is not a bend · the light deflection reading path

“A block that has stuck is at rest.”

It creeps. A contact obeying a logarithmic rate law has no velocity at which the friction reaches a fixed static value, so what looks like sticking is sliding at nanometres a second, and the figures follow it through six decades of speed within one cycle. The distinction matters because that creep is what lets the contact heal, and the healing is what sets the next event's size.

Tested in The chatter a stiffer holder removes · the friction reading path

“A body has a definite kinetic energy.”

It has one per observer, and the figures compute six of them for the same collision, ranging over a factor of five. Nothing distinguishes the observers and nothing makes one of the answers the real one. What every observer does agree on is the energy the collision destroyed, which comes out identical to fifteen decimal places across the whole set.

Tested in The energy that depends on the observer · the energy reading path

“A hollow pipe with nothing in it cannot disperse a pulse, because there is no medium to disperse it.”

It disperses strongly, and the dispersion has no material in it at all. Confinement across the pipe fixes one component of the wavevector, so the remaining component and the frequency are related by ω² = ω_c² + c²k² rather than by ω = ck — a hyperbola, not a line. The figures measure the phase and group speeds off that curve and find their product equal to c² at every frequency, which no non-dispersive relation can produce.

Tested in The pipe that will not carry a low note · the guided waves reading path

“Light takes the path of least time.”

Not on a concave mirror more curved than the ellipse through the source and the image. The figures compute the total path length against where the ray strikes, for three mirrors sharing one tangent point, and on the most curved of them the actual ray sits at a maximum: every neighbouring path is shorter. The ray is the same ray in all three, because the law of reflection holds at the tangent point whatever the curvature, so nothing about the light has changed — only what is around it.

Tested in The path that takes the longest time · the fermat reading path

“Alpha decay is the nucleus falling apart when it becomes unstable.”

Every one of these nuclei is unstable in the sense that the products weigh less, and that says nothing about when. Thorium-232 has been unstable for the whole history of the solar system and is still here. What separates it from polonium is not stability but a barrier: the alpha is energetically free to leave and geometrically unable to, and the rate is the rate at which it gets through something it cannot climb.

Tested in A wall that a factor of two makes impassable · the tunnelling reading path

“The field inside a dielectric is the applied field divided by the relative permittivity.”

Only for a slab lying across the field, and that is the one geometry the textbook picture draws. The figures compute the internal field for three shapes of the same substance and get 1/(1 + Nχ), where N is the shape's depolarising factor — one for a slab, a third for a sphere, essentially zero for a needle along the field. At a susceptibility of a thousand the needle keeps the whole applied field and the slab keeps a thousandth of it, and nothing about the material distinguishes the two cases.

Tested in The field the matter takes away · the dielectrics reading path

“The strength of a magnet is a property of the material it is made of.”

The material fixes the remanence, which is the flux density a magnet would have if it were infinitely long. Every real magnet sits below that, at the point where its own demagnetising field meets its material curve, and the figures compute working points from 98 per cent of remanence for a long thin rod down to 30 per cent for a squat one — the same material, cut differently. A horseshoe is a shape that reduces its own demagnetising factor, and it works for that reason and not because of anything about the iron.

Tested in The magnet that has to fight its own field · the magnetisation reading path

“A solid expands when heated because its atoms vibrate more.”

Vibrating more is not moving further out, and in a symmetric well it is exactly not. The figures compute the Boltzmann-weighted mean position of three oscillators against temperature and the symmetric one returns zero at every temperature to the precision of the quadrature, while the amplitude of its motion grows the whole time. What produces expansion is the asymmetry of the well — the outward side being the shallower one — and a solid with a symmetric interatomic potential would vibrate at any amplitude one liked and never grow.

Tested in Why heating a perfect spring changes nothing · the harmonic approximation reading path

“At the critical point the liquid boils away.”

Nothing boils. Boiling requires two phases with different densities and a surface between them, and what happens on the way to the critical point is that the two densities converge: the figures solve van der Waals' coexistence condition at ninety temperatures and the difference between the liquid and vapour densities goes continuously to zero. The meniscus does not rise or fall out of the tube — it loses contrast and then is not there, because there is no longer anything on either side of it.

Tested in The point at which the two become one · the phase change reading path

“F = ma holds in relativity with a mass that increases with speed.”

No single mass makes it hold. The figures compute the angle between an applied force and the resulting acceleration and find it non-zero at every angle except along and across the motion: at 0.99c a push at forty-five degrees produces an acceleration thirty-four degrees away from it. A scalar mass, however it varies, cannot rotate a vector. What is true is F = dp/dt, and the rotation comes from differentiating γv rather than from any property of the body.

Tested in The push that does not point where the body goes · the relativistic dynamics reading path

“A magnetic field can always be transformed away by moving with the charges.”

Only when the field is magnetic-dominated. The condition is that E² − c²B² be negative, and that quantity is the same for every observer — the figures recompute it along a boost from −0.98c to +0.98c and find it drifting by parts in 10¹⁵. A field with E² above c²B² has no frame in which the electric part vanishes, and one with them exactly equal has neither a pure-electric nor a pure-magnetic frame at any speed.

Tested in The field nobody can transform away · the field transformation reading path

“A submerged body always experiences an upthrust equal to the weight of the fluid it displaces.”

Not if the fluid cannot reach underneath it. The figures compute the pressure on every wetted face of two identical blocks: the one held clear of the floor has an underside pressure exceeding its top pressure by ρgh, which is exactly Archimedes' value, and the one bedded flat has no underside pressure at all and a net downward resultant of 11.8 kPa. The principle is a theorem about a closed surface, and a body sealed to the floor is not enclosed by fluid.

Tested in The block the water does not lift · the buoyancy reading path

“A substance is either a solid or a liquid.”

It is one or the other with respect to a duration. The figures place seven materials on a logarithmic axis of relaxation time spanning thirty-seven decades and draw the observation as a vertical line: everything to the right of the line behaves as a solid and everything to the left as a liquid. Move the line and materials change sides. Pitch is a solid on an afternoon and a liquid on a decade, and neither answer is the correction of the other.

Tested in The liquid that remembers · the rheology reading path

“A binary radiates at its orbital frequency.”

It radiates at twice it, and nothing at all at its own. Gravitational radiation is quadrupolar, and a quadrupole returns to the same configuration twice per revolution — the two masses swapped is the same mass distribution. The figures draw the chirp for three pairs and every frequency on them is double the orbital one, which is why the frequency a detector reports has to be halved before it means an orbit.

Tested in The orbit that has to shrink · the gravitational waves reading path

“A star radiating energy into space is cooling down.”

It is heating up. The virial theorem gives 2K + U = 0 for a bounded system held by an inverse square, so the total energy is minus the kinetic energy — and the figures compute both against radius and find the two curves are mirror images. Losing energy makes the kinetic energy rise, and the kinetic energy is the temperature. The measured heat capacity read off the drawing is negative, at −4.1 × 10³⁴ J/K for a solar mass, which is exactly the value the theorem predicts.

Tested in The ball of gas that heats up as it cools · the Self-gravity reading path

“The pressure at the bottom of a silo is the weight of the grain divided by the floor area.”

It is not, and not by a small margin. The figures compute the stress at the base of a half-metre silo of wheat at 14.7 kilopascals, against the 117.7 the same depth of a liquid of the same bulk density would deliver — 88 per cent of the weight is standing on the walls. Adding twenty metres more grain changes the floor's reading by less than a pascal, because the stress saturated four metres down.

Tested in The silo that does not weigh what it holds · the granular matter reading path

“The dissolved substance attracts water through the membrane.”

It exerts no force on the water on the far side and never touches it. The figures count particles arriving at the membrane from each side: on the pure side every arrival is water, on the solution side only a fraction is, and the imbalance in the *rate* is the whole mechanism. Nothing about the solute's chemistry enters the answer — the pressure depends on how many particles it dissolved into and on nothing else, which is why sugar and salt at the same particle concentration give the same number to three figures.

Tested in The pressure that comes from counting · the osmosis reading path

“A dielectric mirror reflects because each layer reflects a little and the reflections add up.”

That account predicts a reflectance approaching one only as the number of layers grows, with no special frequencies. What actually happens is a band of frequencies at which no propagating solution exists at all, entered as soon as the medium is periodic, and outside which the same stack transmits almost everything. The figures compute the dispersion relation of the infinite medium: inside the band the Bloch wavenumber is complex and there is no wave, at any length of stack.

Tested in The gap a repeat opens · the periodic media reading path

“An optical fibre works because the light bounces off the inside of the glass.”

There is nothing for it to bounce off — the boundary between core and cladding is a change of index of about half a per cent, and both sides are transparent. What happens is that beyond the critical angle the transmitted wave has an imaginary wavenumber and carries no energy away, so the reflection is total by conservation rather than by obstruction. The figures compute how far the field reaches into the cladding, and at a wavelength from the boundary it is still a substantial fraction of its peak.

Tested in The channel with no walls · the guided waves reading path

“A big enough lens can make sunlight as hot as you like.”

It cannot exceed 5,770 K, and the figures compute that number from the Sun's angular radius, the solar constant and the Stefan constant, with no optics in the calculation at all. The greatest concentration a receiver accepting light out to an angle θ can be given is n²/sin²θ, which for the Sun's 0.267° is 46,165 — and 46,165 suns on a black plate delivers exactly the temperature of the Sun's surface. A lens that beat the bound would let an engine take work from a single temperature.

Tested in The brightness no lens can increase · the etendue reading path

“Optical rotation happens because the light is twisted as it goes through.”

Nothing twists. A plane-polarised wave is the sum of a left- and a right-handed circular wave in equal parts, and if the medium has different refractive indices for the two, they emerge with a phase difference and their sum is a plane at a different angle. The figures compute the rotation as a path length times a difference of two indices, linear in the path, and the beam's amplitude, ellipticity and energy are unchanged throughout.

Tested in The rotation a return trip doubles · the polarisation reading path

“Spin is the electron rotating about its own axis.”

Give the electron a radius and ask how fast the equator of a uniform sphere would have to move to carry ħ/2. The answer is 5ħ/4mr, and the figures plot it: at the classical electron radius the equator moves at 170 times the speed of light, and at the experimental upper bound on the electron's size, 10⁻¹⁸ m, at 480,000 times. There is no radius at which the picture is even nearly legal.

Tested in The angular momentum that is not a rotation · the spin reading path

“Every quantum measurement disturbs what it measures.”

The claim is a rule of thumb about a common case, not a theorem. The figures compute the outcomes of a balanced interferometer with an opaque object in one arm: a quarter of the photons arrive at a detector that receives nothing when the arm is clear, and each of those has established that the object is there while leaving it unabsorbed and unlit. The object's state is unchanged, its energy is unchanged, and it has been detected.

Tested in The measurement that never touched it · the measurement reading path

“Because the field takes time to arrive, it points at where the charge used to be.”

For a charge in uniform motion it points at the present position, checked in the figures on every drawn line to zero pixels. Nothing has travelled faster than light: the field at each point was set by the charge as it was a distance over c ago, and for unaccelerated motion that retarded solution happens to extrapolate, because the charge's future was predictable from its past. The field lines miss the retarded position by a hundred pixels on the drawing.

Tested in The field that points where the charge is now · the retardation reading path

“Angular momentum is a property of moving matter.”

The apparatus at the start of this essay has nothing moving in it and holds 3.2 × 10⁻¹⁰ kg m²/s. The figures run the solenoid's switch-off and add the two ledgers at every instant: the field's share, computed as ε₀∫r × (E × B), falls; the matter's share, integrated independently from the torque the induced field exerts on the ring, rises; and the sum is constant to a part in 10¹⁶. A quantity that has to be somewhere before the switch, and is not in the matter, is in the field.

Tested in The angular momentum that is in nothing at all · the field energy reading path

“Maxwell's demon is impossible because measuring the molecules costs energy.”

Measurement can be made arbitrarily cheap and in principle free: there is no thermodynamic lower bound on finding out which side a molecule is, because the operation is logically reversible. The cost is at erasure, which is not reversible — two states are mapped to one — and the figures compute it as kT·ln2, exactly the work the engine's expansion delivers. The demon fails at forgetting, not at looking, and the distinction was eighty years in the making.

Tested in The bit that has to be paid for · the entropy reading path

“Length contraction is something that happens to a moving object.”

Two rockets with identical acceleration programmes never change their separation in the laboratory, and a string between them breaks. Nothing has happened to the string except that the events at its two ends which its own frame calls simultaneous have moved apart. The figures measure that separation on the rockets' own slices and it grows; the laboratory separation, measured on horizontal slices of the same diagram, does not change at all.

Tested in The string that breaks between two rockets · the length contraction reading path

“A massive body focuses light like a lens.”

It focuses each ray somewhere and no two rays at the same place. A glass lens deflects by an angle proportional to the distance off axis and gravity deflects by 4GM/c²b, which falls as that distance grows — so the crossing distance goes as b², verified on the drawn curve to a part in 10¹⁶. For the Sun, a grazing ray crosses the axis at 548 astronomical units and one passing at twelve solar radii crosses at 78,857.

Tested in The lens with no focal length · the light deflection reading path

“Diamagnetism is the classical effect and paramagnetism is the quantum one — orbits induced by the field oppose it, which is Lenz's law applied to an atom.”

The classical partition function is computed here at six field strengths and does not move by more than three parts in ten thousand million million. Every classical magnetic effect, of either sign, is zero: the induced orbital moments of the interior are cancelled exactly by the skipping orbits at the boundary, and the cancellation is not approximate. Diamagnetism is as quantum as paramagnetism.

Tested in The magnetism classical physics forbids · the magnetisation reading path

“n = √ε_r fails for water, which shows that the relation is only approximate.”

It does not fail. The figures compute the permittivity of water from zero frequency to the ultraviolet: it is 80.1 at the bottom and 1.777 at 589 nanometres, and the square root of 1.777 is 1.3330, which is water's refractive index to four figures. The relation is an identity between two numbers taken at the same frequency, and quoting the static value in it is asking one question and reading the answer to another.

Tested in The constant that depends on how fast it is asked · the dielectrics reading path

“A microwave oven works at 2.45 GHz because that is the resonant frequency of the water molecule.”

The computed permittivity of water has no resonance anywhere near 2.45 gigahertz. The nearest feature is the Debye relaxation, whose loss peak is at about nineteen gigahertz — nearly an octave and a half away — and 2.45 sits on its low-frequency shoulder, where the loss is a third of its peak value. That is a choice and not a coincidence: at the peak the absorption length in water would be a few millimetres and a joint of meat would be cooked on its surface and raw inside.

Tested in The constant that depends on how fast it is asked · the dielectrics reading path

“The correlations are explained by a signal from one measurement to the other.”

A signal would have to arrive in time, and the two-particle experiments have been done with the settings chosen while the particles were in flight and far enough apart that light could not cross. What the three-particle argument adds is that no timing is involved at all: the contradiction is between four statements about outcomes, and it survives if every measurement is made at leisure and compared afterwards by post.

Tested in The disagreement that one run settles · the entanglement reading path

“A hotter gas is always further from collapse, because pressure rises with temperature.”

It rises, and what decides stability is not the pressure but how the pressure answers a squeeze. The figures follow a gas across the relativistic crossing: 1 + P/u falls from 5/3 to 4/3, and at 4/3 the total energy of a self-gravitating sphere is exactly zero, so squeezing it releases nothing to resist with. Making a supporting gas hotter than its own rest energy makes it worse at supporting.

Tested in The share that is not half a kT · the equipartition reading path

“A thermal camera measures temperature.”

It measures the radiance leaving a surface and divides by the Stefan–Boltzmann constant. The figure computes what that gives for a surface at 473 K in a room at 293 K: 471 K for skin, 468 for matt black paint, and 312 for polished aluminium. The last is not an error bar. It is what the surface is really emitting plus what it is really reflecting, and no instrument looking only at the light can separate them.

Tested in The glow that says nothing about the surface · the blackbody reading path

“A superfluid rotates with its container like any other liquid, only without friction.”

It cannot. Its velocity is the gradient of a phase, so its curl is zero everywhere the fluid exists, and rigid rotation has a curl of 2Ω everywhere. The figures compute the threshold below which it stays still in the laboratory while the bucket turns — 0.256 radians per second for a bucket a millimetre across — and above it the circulation is a staircase of whole quanta rather than a continuous function of speed.

Tested in The whirlpool that comes in one size · the superfluidity reading path

“A faster projectile hits harder, so raising a beam's energy raises what a collision can make in proportion.”

Only if the target is moving too. The figures compute the invariant mass of the pair: against a stationary proton it goes as the square root of the beam energy, so a hundredfold rise buys a factor of ten. The LHC's beams give 13,000 GeV head-on and 110 GeV against a stationary target — the same particles, at the same speed, with a factor of 118 between what the two arrangements can produce.

Tested in The collision that wastes most of the energy · the relativistic dynamics reading path

“Mass adds: the mass of a system is the sum of the masses of its parts.”

The figure computes the invariant mass of two photons of equal energy against the angle between them. Both parts are massless at every angle and the system's mass runs from zero to 2E/c². The sum of the parts is zero throughout, and it is the answer only when the two are parallel.

Tested in The box of light that weighs something · the Mass-energy reading path

“The gravitational redshift is a photon losing energy as it climbs out of a potential well.”

A photon's frequency is not something it carries; it is what a receiver measures, and the measurement is a comparison of the photon's phase against a local clock. The figure draws the two events at each end and their light paths: what differs between the top and the bottom is the rate of the clocks, and the photon's own worldline is the same either way. The energy account gets the right number and describes the wrong thing.

Tested in The parallelogram that will not close · the gravitational redshift reading path

“A better conductor releases magnetic energy faster, because the currents flow more freely.”

The Sweet–Parker time is the geometric mean of the Alfvén and the diffusion times, so it grows as the square root of the conductivity. The figure plots it against the Lundquist number over fourteen decades with a slope of exactly one half: every hundredfold rise in conductivity slows the release tenfold. A better conductor makes the diffusion layer thinner and the throughput through it smaller.

Tested in The knot the field cannot untie · the flux freezing reading path

“Magnification and resolution are the same thing — magnify enough and any detail appears.”

The image reconstructed from the zeroth order alone is exactly uniform: its visibility is zero to the last bit of the arithmetic, and no magnification recovers a structure from a flat field. Information about a spatial frequency is present in the image only if the corresponding order got through the aperture, and magnification is applied afterwards.

Tested in The image that is a diffraction pattern twice · the imaging reading path

“Forming an image requires a lens, a mirror or a pinhole.”

It requires a periodic object and a distance. The figure computes the intensity behind a grating at nine planes and finds that the plane at 2d²/λ reproduces the object to a part in ten thousand million million, with nothing between the two but empty space. Free space performs the imaging because propagation multiplies each order by a phase that is a whole number of turns at exactly that distance.

Tested in The grating that photographs itself · the diffraction reading path

“A rocket works by pushing against the air, so it could not work in space.”

Thrust is ṁv_e + (p_e − p_a)A_e, and the second term rises as the ambient pressure falls. The figure computes it for a real engine: 6.78 meganewtons at sea level and 7.77 in vacuum, with the line falling at exactly the nozzle's exit area in newtons per pascal. The wrong account predicts thrust falling to zero in vacuum. The measurement has it rising by fifteen per cent.

Tested in The push that needs nothing to push against · the momentum reading path

“A defect in a periodic structure degrades it, so purity is what a photonic crystal or a semiconductor needs.”

The defect is the device. A laser cavity in a photonic crystal, a donor level in silicon and a resonant filter in a dielectric stack are all one flaw in a repeat, and the perfect structure does nothing at all. What the figures show is that the quality of the state depends on the *purity of everything around it* — the trap is better the more perfect the mirrors are, and the flaw has to be the only one.

Tested in The mode that lives in the mistake · the periodic media reading path

“The current in a metal is carried by electrons, so the Hall voltage tells which way they are going.”

Sodium, potassium, silver and copper give the sign an electron model predicts. Aluminium, indium and beryllium give the opposite sign, and aluminium's measured coefficient of +1.02 × 10⁻¹⁰ m³/C is one *positive* carrier per atom to within 1.5 per cent of the value computed from its own atomic density. A carrier density cannot be negative, so no choice of electron count reproduces a positive coefficient.

Tested in The mass a curve decides · the bands reading path

“The long-time tail is a small correction that could in principle be absorbed into a slightly different decay constant.”

It is a change of functional form, not of constant. Beyond the crossing at 23 lifetimes the computed survival falls as t^−2.003 while the exponential goes on falling by a factor of e per lifetime; at 2,000 lifetimes the true answer is 4 × 10⁻¹⁴ and the exponential is below the smallest number a double-precision float can hold. No decay constant makes an exponential into a power law.

Tested in The exponential that is only true in the middle · the decay reading path

“The field at the mouth of a solenoid is half the field at the middle.”

That is the infinite-solenoid limit, approached from above. Summed on the axis, the end-plane value is 0.632 of the centre for a winding as long as it is wide, 0.528 at 2.5:1, 0.507 at 5:1 and 0.502 at 10:1. The half is exact nowhere and useful almost everywhere, which is a different claim.

Tested in The field outside the solenoid, which is not zero · the ampere law reading path

“Water is blue because it scatters blue light, like the sky but denser.”

At 550 nm pure water absorbs 0.0638 per metre and scatters 0.00186 — a ratio of 34 — and the absorption rises ninety-four-fold across the visible while the scattering falls as λ⁻⁴. Over ten metres, 91 per cent of 450 nm light survives and 3 per cent of 650 nm does. The colour is red being removed, which is the opposite mechanism to the sky's.

Tested in Why a litre of water is not blue for the reason the sky is · the scattering reading path

“The third law says the entropy of everything is zero at absolute zero.”

The law that does the work says the entropy change of any process goes to zero as the temperature does, which is a statement about differences and is what makes the two field curves meet. Where a system has a genuinely degenerate ground state — the residual entropy of ice is 3.4 J/K/mol, about R ln(3/2) — the entropy does not go to zero, and cooling is still no easier, because both curves carry the same residue and it cancels out of the step.

Tested in The staircase that never reaches the floor · the third law reading path

“The self-force is a small correction, so the classical theory is fine as long as it is treated perturbatively.”

The ratio of reaction to applied force is τ divided by the time the force takes to change, and it reaches one only for a force varying over 1.9 femtometres — two thirds of the classical electron radius, and 10⁻¹² of the region a real electron's charge would have to occupy for that radius to mean anything. The correction is never large where the theory applies and the theory does not apply where it would be.

Tested in The force a charge exerts on itself · the radiating charge reading path

“A metal is shiny because its electrons absorb the light and re-emit it.”

Nothing is absorbed. With no damping in the model at all the reflectance below the plasma frequency is exactly one — checked to a part in 10¹², and exact in the arithmetic because the refractive index is purely imaginary and (1 − iκ)/(1 + iκ) has modulus one. The four per cent aluminium actually loses is damping, added on top of a reflection that would otherwise be perfect.

Tested in The frequency below which nothing gets in · the plasma oscillation reading path

“Sound is absorbed in air by viscosity, which is why the loss goes as the square of the frequency.”

The classical Stokes–Kirchhoff expression computed from air's own viscosity and thermal conductivity gives 0.12 dB per kilometre at 1 kHz. The measured value at 20 °C and 50 per cent humidity is 3.7 — thirty times larger — and at 16 kHz the factor is 211. The mechanism is vibrational relaxation of nitrogen and oxygen at a rate the water vapour sets, and it has nothing to do with either transport coefficient.

Tested in The distance that takes the treble out · the attenuation reading path

“The sonic boom happens when an aircraft breaks the sound barrier, so it is heard once, at that moment.”

The cone exists for as long as the aircraft is supersonic and sweeps the ground at the aircraft's own speed. Its intersection with the ground is a hyperbola crossing the flight track 15.0 km behind a Mach 1.6 aircraft at 12 km altitude — 32 seconds after it passed overhead — and a listener anywhere along a flight hundreds of kilometres long hears it once as that curve passes.

Tested in The cone the source leaves behind · the doppler reading path

“The friction coefficient is a property of two materials.”

It is a property of two materials and the state of their surfaces. A single smooth elastic contact between the same pair gives A ∝ W^(2/3) and μ falling as W^(−1/3) — a factor of 22 across four decades of load, computed here. A polished sphere on a polished flat is exactly the case where the number in the table is not the number measured.

Tested in The grip that is not a coefficient · the friction reading path

“A chaotic system is one with a random element in it.”

There is none. Two double pendulums are integrated from the same equations with a fourth-order Runge–Kutta step, differing only in a release angle of 120° against 120.00000001°, and they end up on opposite sides of the apparatus. The energy of both is held to 2.6 × 10⁻¹⁰ of itself over the whole run, so nothing has been added and nothing has leaked.

Tested in The error that doubles on a schedule · the chaos reading path

“Water rises up a tree by capillary action.”

A capillary lifts 2γcosθ/ρgr, so a 20 µm vessel — which is what a tree's conducting cells are — lifts 70 centimetres, and reaching a hundred metres needs a radius of 0.1 µm. A tube that fine carries essentially no flow, since the throughput goes as the fourth power of the radius. The rise and the conduction cannot be done by the same pore.

Tested in The column that is pulled, not pushed · the capillarity reading path

“Warm air rises, so a column with warm air at the bottom overturns.”

Every column has warm air at the bottom — the standard atmosphere cools at 6.5 K per kilometre — and almost none of them overturns. What decides is whether the environment cools faster than g/c_p = 9.76 K/km, because a lifted parcel cools at that rate whatever the environment does. At 6.5 K/km a displaced parcel comes back, with a period of 9.9 minutes.

Tested in The layer a parcel cannot leave · the stratification reading path

“Clocks on a rotating platform can be synchronised with each other, since each pair is close enough that special relativity applies.”

Each pair can, and the set cannot. Synchronising neighbours all the way round a rim of radius 0.5 m at 12.566 rad/s accumulates 0.037 femtoseconds per step over twelve stations and arrives back at the first clock 0.439 femtoseconds out — which is 4AΩ/c², the same quantity the two beams measure. The procedure that works locally everywhere fails to close.

Tested in The ring where the two beams disagree · the simultaneity reading path

“A horizon is a feature of a black hole, so it requires a strong gravitational field.”

It requires an observer who never stops accelerating, and nothing else. The worldline x² − c²t² = (c²/a)² is asymptotic to the ray x = ct, so every signal released at or behind that ray chases the rocket for ever without arriving. The boundary is at c²/a — 0.97 light years for one gravity — in flat spacetime with no mass anywhere in the problem.

Tested in The wall of silence behind a rocket that never stops · the accelerated frames reading path

“A system is either regular or chaotic, and turning up the coupling flips it from one to the other.”

Both behaviours are present at every coupling above zero. The phase portraits here show regular islands and a chaotic sea side by side at every value drawn, and the transition the number 0.971635 marks is not a change in any orbit's character — it is the disappearance of the last barrier that kept the chaotic orbits local.

Tested in The last curve to go · the chaos reading path

“A scale reads the weight of whatever is resting on it.”

Not while anything is arriving. A chain dropped from rest with its lower end touching the pan reads exactly three times the weight of the length that has landed, throughout the fall — one part pile and two parts momentum flux, because v² = 2gx makes the second term exactly twice the first at every instant. The reading falls discontinuously to one the moment the last link lands.

Tested in The pile that lands heavier than it weighs · the momentum reading path

“The quality factor measured from a ringdown and the one measured from a resonance curve are two different quantities that happen to be close.”

They are one quantity. The figure takes a decay, transforms exactly those samples, and reads the half-power width off the result: 25.0 from the decay and 25.0 from the width, differing by two hundredths of a per cent, which is the resolution of the frequency grid. Any disagreement larger than that is evidence that the system is not one damped mode.

Tested in The width that is a lifetime · the resonance reading path

“A graded transition is a matching layer with a wider bandwidth.”

It is a different kind of device. A quarter-wave layer has a design wavelength and a band around it; a taper has no design wavelength at all and a cutoff below which it does nothing, above which it works for ever. The computed curves cross zero in the first case and are flat in the second, and no amount of widening turns one shape into the other.

Tested in The taper that matches every note · the impedance reading path

“A diffusing particle explores its whole neighbourhood given enough time.”

Only in one and two dimensions. In three the expected number of returns to any given site is finite, so a walk visits a vanishing fraction of the sites within its own range: the fraction of walks that have ever come home flattens at 0.3405 in the simulation and stays there however long the run.

Tested in The walk that comes home · the diffusion reading path

“The third law says the entropy of everything goes to zero at absolute zero.”

It says the entropy approaches a constant independent of the other variables, and that the constant is zero for a perfect crystal. Ice, carbon monoxide and nitrous oxide are measured to keep 3.41, 4.6 and 4.8 joules per kelvin per mole, and each number is the logarithm of a count of arrangements the substance froze into before it could reach its ground state.

Tested in The entropy that is still there at zero · the third law reading path

“A yield stress is just a very large viscosity at low shear rate.”

A very thick Newtonian liquid pushed gently flows slowly; a yield-stress fluid pushed gently does not flow at all. In a pipe below the threshold pressure gradient the computed velocity profile is identically zero rather than small, and above it there is a core moving as a rigid plug — a shape no viscosity, however large or however rate-dependent, produces.

Tested in The paste that holds up its own hill · the rheology reading path

“A dispersion relation tells you how the speed of a wave depends on its wavelength.”

This one contains no wavelength at all. ω = N cos φ relates the frequency to the direction of the wavevector and says nothing about its magnitude, so beams of any thickness travel at the same angle and a wave of any wavelength at a given frequency goes the same way. The frequency selects a direction, which is a kind of dispersion relation nothing else in this collection has.

Tested in The wave that picks an angle · the stratification reading path

“A mirror that focuses parallel light to a point is a good imaging mirror.”

It is a good imaging mirror for one point. The traced paraboloid here is exactly stigmatic on axis and its blur grows in proportion to the field angle immediately off it, so a Newtonian telescope's field of good images is a few minutes of arc wide however perfectly the mirror was figured. The quality of the axial image says nothing at all about the field.

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

“Coherence is a property of a light source — a laser is coherent and a lamp is not.”

Transverse coherence is a property of the source's angular size seen from where the light arrives, and of nothing else. The coherence radius is λ/πθ for every source in the figure, from the Sun at nineteen microns to a Sun-like star at ten parsecs at thirty-nine metres. The same lamp is coherent across a bench and incoherent across a room, and starlight is coherent over a whole telescope because the star is small.

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

“Dispersion means different colours travel at different speeds, so a pulse of one colour is unaffected.”

A pulse of one colour does not exist. A pulse of duration τ occupies a band of frequencies of width at least 1/τ, and the shorter it is the wider that band — so the pulses most damaged by dispersion are exactly the short ones a fast system needs. The broadening is the product of the dispersion, the length and the spectral width, and the third factor cannot be reduced below what the first two make necessary.

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

“The vector potential is the field's underlying reality and the field is derived from it.”

Three different vector potentials are drawn here for one uniform magnetic field, and their arrow patterns share no feature at all — one circulates, two are unidirectional in perpendicular directions. The curl recovered from each is 1.000000 tesla. A quantity that can be changed arbitrarily without changing anything measurable is not more fundamental than what it produces.

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

“There is no field inside a conductor.”

There is no *static* field inside a conductor, because the charges have had time to arrange themselves. An alternating field enters to a depth √(2/μσω), which for copper is 9.2 mm at fifty hertz and 2.1 microns at a gigahertz. Whether a conductor is thick or thin is a statement about that length rather than about millimetres.

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

“The circulation of B round a circle enclosing a current is μ₀I.”

For a finite segment it is μ₀I cos α, with α the half-angle the segment subtends — 0.2425 of μ₀I for a wire half as long as the distance. The remainder is supplied by the displacement current of the charge accumulating at the segment's ends, and the two terms, both integrated here, sum to unity at every length to better than a ten-thousandth of a per cent.

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

“Relativity limits how far anybody can travel in a human lifetime, because nothing goes faster than light.”

It limits the distance measured in *home* years and not in shipboard years. At one gravity the galactic centre is 19.8 years of the traveller's own time away and Andromeda is 28.6, both computed here. Nothing exceeds c at any point: the speed is the hyperbolic tangent of the rapidity, and it is the rapidity that grows without limit.

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

“A clock flown at altitude runs fast, because it is higher up.”

It depends which way it went. Height gains 3.5 nanoseconds an hour at nine kilometres, and motion loses a quantity quadratic in a speed that changes sign with the direction, so an eastward flight comes out at −61 nanoseconds and a westward one at +304 in this model, against measurements of −59 and +273. Both terms act on the same clock and only their sum is observable.

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

“A heavier star is a bigger star.”

Not when it is held up by degeneracy. The integrated structure here gives a radius falling as the mass rises — about the Earth's radius at a solar mass and a few thousand kilometres near the limit — because adding mass compresses the electron gas and the pressure it gains grows more slowly than the weight it must carry.

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

“A wave below the plasma frequency cannot get into a plasma.”

That holds for an unmagnetised plasma. Add a magnetic field and a wave circularly polarised the same way the electrons gyrate travels happily below both the plasma frequency and the gyrofrequency — 4 kilohertz through a medium whose plasma frequency is 90 kilohertz, in the figure here. The magnetic field gives the electrons a second frequency of their own and the response is no longer a simple cutoff.

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

“An electron's energy depends on its principal quantum number.”

Only in hydrogen. The solved levels here for a nucleus of charge 19 come out 3s at −10.500, 3p at −10.346 and 3d at −10.032 hartree, ordered by angular momentum, and the ordering appears the moment the potential is anything other than a bare Coulomb one. Hydrogen's degeneracy is a special property of the 1/r potential and not a general feature of atoms.

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

“Two spin-halves can be up-up, up-down, down-up or down-down, so there are four equivalent states.”

There are four states and two of them are not those. One spin up and the other down does not specify a total spin, because it does not say which spin is which; the states that do are the sum and the difference, and applying S² as a matrix gives 2ħ² for one and 0 for the other. Three of the four are one kind of object and one is another.

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

“How much a material absorbs and how much it bends light are two independent properties, to be measured separately.”

They are one property looked at twice. Handed nothing but the absorption at every frequency, the Kramers–Kronig integral returns the refractive index at every frequency, and the computation here reproduces the model's own dispersion curve to two parts in ten million. No parameter of the material is used and none is fitted.

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

“A wave in two dimensions is a wave in three with one direction ignored, so it behaves the same way with a different spreading law.”

The spreading law is the smaller difference. A three-dimensional impulse arrives as the pulse that was sent; a two-dimensional one arrives and then keeps arriving, with a tail falling as one over the time that is still at a tenth of its peak twelve pulse-lengths after the front. Both are exact solutions of the same wave equation with the same source.

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

“The ultraviolet catastrophe came from a mistake about how much energy each mode holds.”

The energy per mode was the standard classical result and was not the error. What made the integral diverge was the number of modes: it grows as ω² in three dimensions, without limit, so any fixed energy per mode gives an infinite total. Planck's constant enters as a bound on the energy of the high-frequency modes, not as a bound on how many there are.

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

“A trajectory with drag is a parabola with a smaller range.”

It is not a parabola and not any closed-form curve. The horizontal and vertical equations are coupled through the total speed, and at a drag of 2.4 weights the launch spends 45 per cent of its flight going up, arrives at 64° after leaving at 45°, and lands at 48 per cent of the speed it started with.

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

“A force that averages to zero over a cycle has no average effect, so shaking the pivot cannot hold the pendulum anywhere.”

The average of a force is not the force on the average when the two are correlated. The fast force is proportional to sin θ and the fast displacement it produces is proportional to the same thing, so their product has a non-zero average — which comes out as a term ¼(a/L)²(Ω/ω₀)² sin²θ added to the potential, and turns the maximum at 180° into a minimum.

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

“A particle takes the path of least action.”

It takes the path of stationary action, and past the kinetic focus that path is a saddle. For a harmonic oscillator the second variation of the action under a deformation sin(πt/T) is (λ²T/4)[(π/T)² − ω²], which is negative for every trip lasting more than half an oscillation — so nearby paths with the same endpoints have less action, and they are drawn here.

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

“A bigger heap of the same sand is steeper, because there is more weight pressing down.”

There is more weight and it acts on both sides of the comparison. The driving stress on a surface layer is ρgh sin θ and the frictional resistance is μρgh cos θ, so the thickness, the density and gravity all cancel and the criterion is tan θ = μ. Checked over three decades of layer thickness the ratio of the two stresses differs by zero to machine precision.

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

“A reversible separation is a theoretical idealisation with no bearing on how a plant is built.”

It is the target every stage is added to approach. One stage at half recovery costs 1.584 kWh/m³, two cost 1.320, eight cost 1.149, against a reversible 1.098 — the excess falling as one over the number of stages. That arithmetic is why real plants run two or three passes and not one, and why they stop at three.

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

“Helium II is an extraordinarily good thermal conductor.”

It does not conduct at all. Heat is carried by the normal component physically flowing toward the cold end while the superfluid flows back underneath it, and the transport is convective rather than diffusive — which is why its effective conductivity depends on the heat flux and on the channel width, and why it fails abruptly above a critical velocity.

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

“The instability is a property of the particular cloud — its size, its shape, how much gas it holds.”

None of those appears in it. Measured in each gas's own units, the dispersion relations of four densities six decades apart lie exactly on top of one another to 10⁻¹⁶: the curve is the same for a diffuse cloud and a protostellar core, with different numbers written on the axes. What the gas supplies is one sound speed and one density.

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

“A plasma is any ionised gas.”

It is an ionised gas in which the collective response works, and the test is the number of particles inside a Debye sphere. That number is 10⁹ for interstellar gas, 10⁵ for the ionosphere and about 8 in the core of the Sun — which is why the solar core is on the edge of being strongly coupled and the ordinary theory is applied to it with care.

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

“The Eddington limit is a hard ceiling that nothing can exceed.”

It is a ceiling on a spherical flow of fully ionised hydrogen, and both conditions can be broken. A flow confined to a disc lets radiation escape along the poles while matter arrives in the plane; and where the gas is dense enough for radiation to be trapped and carried inward, the limit is evaded outright. Observed sources exceeding it by factors of hundreds are known.

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

“The rainbow is brightest exactly at the rainbow angle, where the deviation is stationary and the rays pile up.”

The ray calculation gives an infinite intensity there, which is the signature of an approximation failing rather than of anything bright. The wave answer is the squared Airy function: finite, and peaking at ζ = −1.0188 in the scaled variable — displaced inside the geometric angle by an amount that depends on the drop size and the wavelength.

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

“The geometric phase is a quantum effect, discovered by Berry in 1984.”

Pancharatnam published the optical case in 1956, and it is elementary interference: the phase of the product of three overlaps between polarisation states. Berry's contribution was to recognise the same structure in adiabatic quantum transport, and the optical result had been sitting unread for twenty-eight years.

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

“The gas coming out of a small hole is a fair sample of the gas inside.”

It is biased by one factor of speed, because a molecule's chance of reaching the hole is proportional to how fast it is moving. The beam's distribution goes as v³ rather than v², its mean speed is 3π/8 times larger, and its mean kinetic energy is 2kT against 3/2 kT — a third higher, with no heating anywhere.

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

“Nucleation switches on at a critical supersaturation.”

The rate is smooth, finite and computable everywhere. It spans eighty decades over the range drawn here, so a ten per cent change in supersaturation moves it by ten orders of magnitude — which is why every experiment sees a sharp onset. Nothing switches; a continuous function is simply very steep.

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

“The Unruh effect is an analogy with Hawking radiation.”

It is the same formula. T = ħa/2πck for an accelerated observer and T = ħκ/2πck for a horizon of surface gravity κ, and evaluating the first at a black hole's surface gravity reproduces the second to a part in 10¹² — which is not an analogy but the equivalence principle applied to a temperature.

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

“Two clocks can be synchronised by moving one slowly from the other, which measures the one-way speed without assuming it.”

Slow transport gives the same answer as the radar convention only if the moving clock's rate change is assumed isotropic — which is the assumption being tested. The procedure is self-consistent and not independent: it reproduces whatever anisotropy was built into the description of the transported clock's timekeeping.

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

“A clever enough arrangement of fixed charges could hold another charge in stable equilibrium.”

No arrangement can, and the obstruction is not difficulty. In a charge-free region ∇²V = 0, so the average of V over any sphere equals its value at the centre — checked here to five parts in a hundred thousand — and a maximum or a minimum would require the average to differ from the centre. Every stationary point is a saddle.

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

“Decoherence explains why a measurement has a single definite outcome.”

It explains why the outcomes stop interfering and why one particular set of states is picked out, and it does not remove any branch. The state after decoherence is still a superposition — of system-plus-environment — and the step from that to one outcome is exactly the measurement problem, untouched. Every serious account says so; the claim persists anyway.

Tested in Where the interference goes · the measurement reading path

“Conservation of energy is an experimental fact, established by never having seen it fail.”

It is a consequence of the laws not mentioning the time, and it fails precisely when they do. A pendulum whose length is being changed by hand has no conserved energy, and the rate at which its energy changes is the explicit time-dependence of its Lagrangian — computable in advance rather than discovered by measurement. In an expanding universe the same statement applies to light, whose energy falls as it redshifts, with nowhere for it to go.

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

“A statically indeterminate problem is one that is hard to solve.”

It is one that has no solution as posed. Four reactions under a rigid top satisfy three equations, and the solutions form a line: every member of that line balances every force and every moment to machine precision, which is drawn here at five points along it. No amount of cleverness selects one, because nothing in the statement distinguishes them.

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

“A sound wave in air is described by the wave equation.”

It is described by the wave equation to first order in the amplitude and by nothing so simple beyond that. The speed of sound rises with the local temperature and the medium is itself moving with the wave, so a crest travels faster than a trough by (γ+1)/2 times the particle velocity. At an ordinary conversational level that correction takes kilometres to matter; at 194 decibels it takes metres, and every sonic boom is the correction having finished.

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

“A soliton is a wave that does not spread because the medium happens to be non-dispersive.”

The medium is strongly dispersive and that is the point. Switching the dispersive term off in the integration here leaves a pulse that steepens towards a shock; switching the nonlinear term off leaves one that spreads and sheds a tail. Only with both present does the profile survive, and it survives to a part in ten thousand billion of its own height.

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

“Viscosity is internal friction, so heating a substance always makes it flow more easily.”

Every gas gets more viscous as it warms. Air's viscosity rises by 55 per cent between 250 and 900 kelvin, and the exponent measured on the drawn curve is +0.77, where glycerol's over a comparable range is −23. Friction is the wrong picture for a gas: what is transported is momentum carried across a plane by molecules in flight, and a hotter molecule crosses more often.

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

“A granular material becomes rigid when it becomes dense enough.”

Density is a proxy and a poor one. The transition is in the contact count, and two packs of identical density can differ by a contact per grain if one was tapped and the other poured. The measurement here holds the grains, their positions and their density fixed and varies only which contacts are present: at a mean of three contacts each the pack has fifteen ways to rearrange for nothing, and at four it has none.

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

“A moon breaks up inside the Roche limit because the tidal force there exceeds the planet's pull on it.”

The tide is a *difference* of the planet's pull across the satellite and is far smaller than the pull itself everywhere. What it is compared against is the satellite's own surface gravity, which is what holds the satellite together; the drawn comparison crosses zero at 1.15 planetary radii for ice around a planet of Saturn's density, where the planet's own attraction is thousands of times larger than either.

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

“The sunlight arriving now was made in the core a hundred thousand years ago.”

No photon survives the journey. Energy is absorbed and re-emitted at every step — 10²⁰ of them on this estimate — and at each absorption the identity of the quantum is destroyed. What crosses the Sun is energy, and it arrives as a great many low-energy photons rather than as the gamma ray that started; the temperature falls from 15 million kelvin to 5,800 along the way, and the photon count rises to match.

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

“A halo is a rainbow made of ice instead of water.”

The mechanisms are different and the two look different in the way the mechanisms predict. A rainbow is refraction, one internal reflection, refraction, with the light coming back towards the sun and the deviation having a *maximum*; a halo is refraction through a 60° prism with no reflection at all, in the forward direction, with the deviation having a *minimum*. The consequence is visible: a rainbow's red is on the outside and a halo's is on the inside, and the two are drawn here from the same dispersion.

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

“Speckle is a texture of the illuminated surface, brought out by coherent light.”

The grain size is set by the aperture and not by the surface. Measured here over five apertures spanning a factor of four, the grain width goes as the aperture to the power −0.994 against an exact −1; a rougher surface, or one made of a different material, gives grains of exactly the same size through the same aperture. What the surface decides is where the grains fall, not how big they are.

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

“The photoelectric effect proves that light comes in quanta.”

A classical electromagnetic wave falling on an atom whose energy levels are quantised reproduces the threshold frequency, the instantaneous onset and the intensity-independent stopping voltage — the whole of the observed phenomenology. That calculation was done by Wentzel and Beck in 1926 and the effect has not been a proof of light quantisation since. What the effect demonstrates is that *matter* is quantised.

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

“Two levels avoid crossing because they repel one another, like two charges.”

Nothing repels. The eigenvalues of a two-by-two matrix with an off-diagonal element are plus and minus half the square root of the squared detuning plus four times the squared coupling, and that expression cannot vanish unless the coupling does. The separation at closest approach is exactly twice the coupling — measured off the drawn curves at 0.10, 0.30 and 0.70 eV for couplings of 0.05, 0.15 and 0.35 — and calling it a repulsion attributes to a force what belongs to an algebraic identity.

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

“The exchange interaction is a force that keeps identical fermions apart.”

There is no term in the Hamiltonian for it and no momentum carried between the particles. The pair correlation computed here has no interaction anywhere in it: the only difference between the three cases drawn is whether the two-particle state is symmetric, antisymmetric or neither, and that difference alone produces a hole reaching exactly zero at contact and a pile reaching exactly two.

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

“Field lines refract at a dielectric boundary in the same way light does, so the same law applies.”

The form is different and the difference is measurable. Light obeys a ratio of sines and a static field obeys a ratio of tangents, which agree only for small angles; at 68° incidence into a medium four times as permittive the tangent law gives 84° and a sine law would give an angle that does not exist at all. The two also bend in opposite senses relative to the normal for the same ratio.

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

“A large coil couples more strongly to a small one than the small one does to the large, because it intercepts more flux.”

It intercepts more flux and it also produces a weaker field per unit current over the region where the small one sits, and the two effects cancel exactly. The two couplings are computed here by separate surface integrals over a disc and a tilted rectangle differing by a factor of twenty-one in area, and they agree to 0.075 per cent across a range of separations.

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

“A superconductor is a metal whose resistance has fallen to zero, so its magnetic behaviour follows from that.”

Zero resistance predicts that whatever field was present when the resistance vanished stays trapped, because any change of flux would drive a current that never decays. A superconductor cooled *in* a field expels it instead. Those are different final states from the same history, so the magnetic behaviour is an independent property and not a consequence — which is what Meissner and Ochsenfeld found in 1933, twenty-two years after the resistance was measured.

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

“A negative temperature is colder than absolute zero.”

It is hotter than any positive temperature. Bring a negative-temperature system into contact with anything at all and energy flows out of it, because that is what raises the total entropy — the same criterion that decides every other direction of heat flow. The ordering is continuous in 1/T, which runs from large positive through zero to negative; the temperature itself has the discontinuity, and reading the ordering off T rather than 1/T is what produces the confusion.

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

“A gas cools when it expands, because it does work against its surroundings.”

In a throttling expansion no work is done on anything and no heat crosses any boundary; the enthalpy is what stays constant, and an *ideal* gas throttled through a plug does not change temperature at all. Whatever cooling occurs is entirely the departure from ideality, and it can be a warming: helium, hydrogen and neon all warm when throttled from room temperature.

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

“A rigid body is an idealisation that a sufficiently hard material approaches.”

It is a signal at infinite speed, and the light cone is a floor no material approaches. A 3-metre steel rod's far end waits 600 microseconds, against a light-crossing time of 10 nanoseconds — a factor of sixty thousand — and the stiffest material imaginable would still wait 10 nanoseconds. What separates steel from rigidity is not a large number but an infinite one.

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

“Relativity makes interstellar travel harder, because the mass increases as the ship speeds up.”

The relativistic mass ratio is *smaller* than the Newtonian one at every speed, because rapidity grows more slowly than velocity would need to. The difficulty is not relativistic at all — it is Tsiolkovsky's logarithm, which was there in 1903, and the exhaust speed of chemical propellant. A chemical drive needs 10^2968 even without relativity, and relativity reduces that number.

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

“The attractor of a chaotic system is a curve too complicated to draw.”

It is not a curve at all, and a number says so rather than an adjective. Counting boxes gives a dimension of 1.29, and the two Lyapunov exponents — measured separately, by a method that never counts anything — give 1.26 through Kaplan and Yorke's formula. A curve would give one and a filled region two. The two routes agree to 0.03, and neither can return a non-integer for anything a finite list of curves could describe.

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

“A pendulum is isochronous, so its period does not depend on the amplitude.”

It is isochronous only in the limit of no amplitude, and the same quadrature applied to a circular arc and to a cycloid separates the two cases completely: the cycloid's four release points give 1.003205 seconds each, agreeing to seven parts in ten million million, while the circle's spread over the same range of releases is eighteen per cent. The flat start of the circle's curve is what makes the error survivable and also what made it hard to notice — it is second order in the amplitude, so halving the swing quarters the error.

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

“Diffraction is what happens when the wave nature of light smears out the edge of a shadow.”

Smearing is a description and not a mechanism, and it predicts nothing. The split computed here is exact: the field behind a straight edge is the unblocked incident wave plus one further wave, and the two add back to the directly computed field to a part in 10¹². The second term is not a correction to the first. In the shadow it is the entire field, and in the light it is the thing the incident wave beats against — which is where the fringes come from, since a smooth decay has nothing to interfere with.

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

“A surface wave is a bulk wave trapped by repeated total internal reflection at the surface.”

Total internal reflection needs two media and an angle; a free surface has one medium and the surface wave has no angle at all. It is a separate solution of the same equations, with its own speed — 0.9194 shear speeds at Poisson's ratio 0.25 — obtained here by finding where the free-surface traction vanishes, and that speed is *slower* than every bulk wave in the material. A trapped bulk wave would have to travel at a bulk speed along the surface and cannot travel slower than one.

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

“A soap film always takes the shape of least area, so it breaks as soon as a smaller-area alternative exists.”

It takes a shape of *locally* least area, which is a weaker requirement. The film between two rings has a larger area than the two flat discs once the half-separation exceeds 0.5293 ring radii, and it survives to 0.6627 — 25 per cent further — because there is no continuous deformation that gets it to the discs without passing through larger areas. What ends it is not losing the competition but running out of solutions: past 0.6627 the equation fixing the catenoid has no root at all.

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

“A polymer solution climbs a stirring rod because it is very viscous and drags itself up.”

Viscosity produces one stress in simple shear and only one, at every rate, for any value of the viscosity: the shear stress. Glycerol at a thousand times water's viscosity produces exactly the same dip as water at the same rotation rate, because the surface shape depends on stresses the viscosity does not generate. What climbs the rod is the first normal stress difference, which grows as the square of the shear rate where the shear stress grows in proportion, and which is identically zero for every Newtonian liquid however thick.

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

“Light orbits the photon sphere, so a black hole is surrounded by a shell of trapped light.”

The effective potential for light has a maximum there and no minimum anywhere, so the circular orbit exists and is unstable: any ray on it leaves, inward or outward, on the smallest perturbation. Nothing accumulates there. What the photon sphere produces instead is the infinite sequence of ever fainter images at the shadow's rim, each from light that went round one more time before escaping, and the traced rays here wrap 1.06 times at an impact parameter two per cent above critical.

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

“A wave that damps must be losing energy to something dissipative.”

The equation integrated here has no collision term, no viscosity and no resistivity; the only operator acting on the distribution is a rotation of phase whose rate depends on the particle's speed. The field falls by four decades over forty plasma periods, at 0.1534 per period against a published 0.1534, while the free energy of the perturbation is constant to 2.4 × 10⁻¹⁰. Nothing is dissipated. The energy is in the particles' ordered motion and can, in principle, be given back.

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

“Blocking half of an aperture can only make the image dimmer.”

Blocking the right half makes the axis brighter, and the arithmetic says by how much. The open rings all contribute with the same phase, so the amplitude grows in proportion to how many there are and the intensity as the square — measured here to within nought point one per cent of a parabola across twelve zone counts. The same apertures with nothing blocked do not grow at all: they oscillate between roughly four times the free intensity and almost nothing, according to whether the outermost zone they happen to contain is a contributing one.

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

“Making a quarter-wave plate requires a birefringent material.”

It requires a means of delaying one polarisation relative to the other by a quarter of a cycle, and a total internal reflection provides one with no material property involved beyond the index. Two reflections at 47.70 degrees inside a block of ordinary borosilicate give 90.0 degrees of retardation, and the block is not birefringent at all. Fresnel built one in 1817, before the wave theory that explains it was accepted.

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

“The image is a map of the surface's atoms.”

It is a map of the surface's density of electronic states near the Fermi level, at the energy the bias selects, convolved with the tip's own states. On graphite only half the carbon atoms appear, because the two sublattice sites have different densities of states there and one of them is nearly invisible; on some semiconductor surfaces the image inverts entirely when the bias is reversed, because filled and empty states sit in different places. What is measured is where the electrons are available, not where the nuclei are.

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

“A quantum state cannot be copied because any copier would disturb it, and a better-designed one might not.”

No design enters the argument. A copier is a linear machine, so a machine that copies two states perfectly has already had its behaviour on every superposition of them decided; the figure computes what it then does to an equal superposition, and the answer is an entangled pair whose overlap with the two copies wanted is exactly 0.500. There is nothing to improve, because there is no free parameter left by the time the contradiction appears.

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

“The carriers in a superconductor are electrons, so the flux quantum is h/e.”

h/e is 4.1357 × 10⁻¹⁵ webers and the measured quantum is 2.0678 — a clean factor of two, which no experimental accuracy is required to distinguish. Three candidate carrier charges give 4.1357, 2.0678 and 1.3786, and only the second matches. The experiment therefore weighs the carrier's charge without measuring a charge, and it did so in 1961, before any independent confirmation that pairing was what the microscopic theory required.

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

“The absence of magnetic monopoles is an experimental fact that the equations record.”

It is an experimental fact, and what the equations record is stronger and different: the divergence of B cannot change under Faraday's law whatever happens, so if it is zero once it is zero for ever. A monopole cannot be created by any process the equations describe. Adding one therefore means changing the equations, not adding a particle to them — which is why a monopole would force a magnetic current term into Faraday's law and would make the vector potential impossible to define globally.

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

“Radiation is a gas of photons, so it obeys the gas law with the photons as the particles.”

It obeys a different law, and the reason is that the number of particles is not an independent variable. The pressure of blackbody radiation is one third of its energy density; a molecular gas at the same energy density pushes twice as hard. The difference is that a photon of energy E carries momentum E/c while a slow molecule carries the square root of 2mE, and no amount of counting photons repairs it.

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

“The Seebeck and Peltier coefficients are two independent properties of a material, to be measured separately.”

They are one property measured two ways, related by a factor of the temperature. The figure computes both for the same conductor by two computations that share no step — one nulls a thermally driven current with a voltage, the other takes the ratio of two currents driven by a voltage at uniform temperature — and they agree to 3.8 × 10⁻⁷ of the sweep's own scale, through a sign change. A laboratory that measures both and finds them different has made an error, not a discovery.

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

“A phase transition is the breaking of a symmetry, so a system that cannot break one has no transition.”

A two-dimensional system with a continuous symmetry cannot develop long-range order at any temperature above zero, and it has a sharp transition all the same. What changes is topological: below the transition vortices exist only in bound pairs of opposite circulation, above it they are free. The free energy of a single vortex is (πJ − 2kT) ln(L/a), so what changes at kT = πJ/2 is the sign of a coefficient, and nothing anywhere becomes non-zero.

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

“A source whose components separate faster than light across the sky is moving faster than light.”

The apparent transverse speed is β sinθ/(1 − β cosθ), which exceeds one for any β above 1/√2 at the right angle, with the true speed below one throughout. At β = 0.99 the largest apparent speed is 7.02, reached at 8.1 degrees from the line of sight, and the maximum is γβ at arccos β — found here by searching the drawn curves rather than assumed. The excess comes from the source having closed on the observer between the two observations, so the light of the second had less far to go.

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

“Motion towards an observer makes a source brighter, so motion in any direction makes it brighter than at rest.”

Seen at right angles a moving source is *fainter* than the same source at rest, by γ⁴ — a factor of 5.8 × 10⁵ at β = 0.99. The Doppler factor there is exactly 1/γ, which is less than one, because the transverse redshift is a pure time-dilation effect with no line-of-sight component to cancel it. Only sources within about one over γ of the line of sight are brightened at all, and everything outside that cone is dimmed.

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

“A rigid body can be set spinning, since rigidity is an idealisation that only fails at extreme accelerations.”

It fails at every acceleration, because the requirement is geometrical rather than dynamical. Each circle of material must end shorter in its own terms than it began, by the local Lorentz factor, while its distance from the axis is unchanged — 8.3 per cent at a rim reaching four-tenths of light speed and 56.4 per cent at nine-tenths, and zero on the axis in every case. A body that keeps every internal distance fixed as measured by itself admits no change of rotation rate at all, which is Born's definition and Herglotz and Noether's theorem about it.

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

“A ball rolling off a sphere leaves the surface when the component of gravity along the radius can no longer supply the centripetal force, which happens at 45°.”

It leaves at 48.19°, where cos θ = ⅔, and the condition is not about gravity's component but about the sign of the surface's own push. The reaction is mg(3cos θ − 2cos θ₀) and it reaches zero there; past that point the surface would have to pull inward, which a surface cannot do. Forty-five degrees is where gravity's radial and tangential components are equal, which is not a condition on anything.

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

“The boundary of everywhere a projectile can reach is the trajectory launched at 45°.”

The 45° launch reaches the greatest range along the ground and is nowhere near the boundary anywhere else. The boundary is the envelope of the whole family — a parabola of height v²/2g and half-width v²/g — and the 45° trajectory touches it at exactly one point, its own landing point, as every other launch does at exactly one point of its own.

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

“The nth mode of a drumhead divides it into n regions, as the nth mode of a string divides it into n pieces.”

It divides it into at most n, and usually fewer. Counting the sign regions of the computed modes of a circular membrane by flood fill gives 1, 2, 4, 2, 6, 4, 8 against ranks of 1, 2, 4, 6, 7, 9, 11 — the second radial mode is two rings where its rank would allow six. Courant proved the inequality in 1923 and Pleijel proved in 1956 that equality fails for all but finitely many modes of any shape.

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

“An echo from a moving object is Doppler shifted twice, so the shift is the square of the one-way shift.”

The two shifts are different functions. A moving observer multiplies the frequency by (c + v)/c because it runs into the fronts; a moving source divides by (c − v)/c because it chases the ones it has made. Their product is (c + v)/(c − v), which equals the square of either only in the limit v/c → 0. At a tenth of the sound speed the round trip is 1.2222 and the square of the one-way shift is 1.2100.

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

“A cleverly shaped structure — a funnel, a horn, a wedge of absorber — can let a wave through in one direction and not the other.”

No passive, linear, stationary arrangement of materials can. A stack of nine slabs of random thickness and random wavenumber, with no symmetry in it anywhere, transmits 0.669424094 from the left and 0.669424094 from the right, in modulus and in phase. The result depends on the wave equation being unchanged under reversing the sign of time, and no amount of shaping alters that.

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

“In total internal reflection no light enters the second medium.”

A field is there and it is not small. What is absent is a *direction*: the conserved tangential component exceeds anything the far medium's circle can supply, so the normal component is imaginary and the field decays instead of travelling — at 50° from glass to air it falls by a factor e in 0.21 vacuum wavelengths. Bringing a second piece of glass within that distance recovers a transmitted beam, which no theory of a light that stops at the surface could explain.

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

“Interference requires the light to be coherent, so an instrument that measures a source's size must preserve the phase.”

Correlating the *intensities* at two apertures measures |γ|², which contains the same information about the source as the |γ| a fringe contrast measures — one curve determines the other, and both vanish at the same baseline. No phase is preserved anywhere in the apparatus, and the two detectors need not even be looking at the same wavefront to better than a few metres.

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

“At constant voltage the stored energy increases as the slab enters, so the system is moving to a state of higher energy and the force must be outward.”

The force is inward in both cases and the same size. At fixed voltage the battery supplies V²ΔC while the field keeps only half of it; the other half is the work done on the slab. The quantity a system at fixed voltage minimises is not the field energy but the field energy minus the battery's contribution, and reading a force off the wrong potential is exactly how the sign is lost.

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

“The force per unit area on a charged surface is σE, the charge density times the field it sits in.”

It is σE/2. The field is σ/ε₀ just outside and exactly zero just inside, and the surface layer feels neither — a charge exerts no force on itself, so what acts on the layer is the field the rest of the charge makes, which is the mean of the two sides. Using the full external field doubles the answer, and the same factor appears wherever a surface distribution is asked what force it feels.

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

“The three latent heats at a triple point are three independent properties of the substance.”

Two of them fix the third, because enthalpy is a state function and the three paths between three phases must close. For water at 273.16 K the measured values are 333.5, 2500.9 and 2834.4 kJ/kg, and the first two sum to the third within four parts in ten thousand. The same closure makes the sublimation curve steeper than the vaporisation curve by exactly the amount the fusion latent heat predicts — 5.92 Pa/K on water's numbers.

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

“The classical limit is what happens as Planck's constant becomes negligible.”

Planck's constant is a constant and does not become anything. What has to be small is the spread of the state compared with the distance over which the force changes appreciably — a condition involving the potential's third derivative and the packet's width, both of which are properties of the situation. A macroscopic object in a sufficiently curved potential loses the correspondence, and a microscopic one in a harmonic well never does.

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

“Two barriers in series transmit the product of their individual transmissions.”

That would be true if probabilities composed, and they do not. Two barriers of width 0.9 that each pass 3.9 per cent transmit 1.000000 at the resonant energy — 640 times more than the product, and 26 times more than either barrier alone. What compose are amplitudes, and the amplitudes for the many bounces between the barriers add in phase at particular energies.

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

“A silicon diode has a forward voltage drop of 0.7 volts.”

It has no fixed drop. The current is exponential in the voltage, rising a decade every 59.5 millivolts at room temperature, so any current a circuit is likely to pass lands in a window a couple of tenths of a volt wide. The constancy is an artefact of reading a very steep curve on a linear axis, and the same diode drops 0.4 V at a microamp and 0.9 V at an amp.

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

“The travelling twin ages less because of the acceleration at the turnaround.”

Make the turn gentler and gentler and its contribution goes to nothing while the age difference does not. At a tenth of a gravity the turnaround accounts for about half the difference; at a thousand gravities it accounts for a hundredth of a per cent, and the total tends to 2.67 years for the four-light-year trip drawn — which is exactly what the instantaneous-turnaround idealisation gives. The acceleration decides which twin's path is bent; the legs decide by how much.

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

“Liquid helium below the lambda point is an extraordinarily good heat conductor.”

It does not conduct heat at all in the ordinary sense. Heat is carried by a counterflow — the normal component moving one way and the superfluid the other — and disturbances in it propagate as a wave with a definite speed rather than diffusing. Measured as a conductivity the answer is not a large number but a meaningless one: it depends on the geometry, the heat flux and the length of the sample, because there is no conductivity there to measure.

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

“Large particles rise because they are lighter, or because buoyancy carries them up.”

The effect works with an intruder denser than the bed as well as lighter, and the standard demonstration uses a large steel ball in fine sand, which is denser than what surrounds it. Density enters the problem and does not drive it: the ordinary version of the effect is driven by small grains percolating *down* through gaps too small for large ones, and the large one rises because the material beneath it has been replaced.

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

“A surface immersed in a plasma sits at the plasma's potential.”

It floats several electron temperatures below it. The electron flux to an uncharged surface exceeds the ion flux by the square root of the mass ratio, so the surface charges negative until the Boltzmann factor of its own potential has cut the electron flux down to match — which for hydrogen is 2.84 kTe/e and for xenon 4.68. The dependence is logarithmic in the mass ratio, which is why the whole periodic table lands between three and five.

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

“Light can only push, since it carries momentum in the direction it travels.”

The momentum flux gives a push and is not the only force light exerts. A non-uniform field polarises a particle and pulls the induced dipole toward the strong field, which for a focused beam is *back toward the waist* — against the direction of travel, for a particle downstream of the focus. The two forces are computed from different quantities: one from the intensity, the other from its gradient.

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

“The nod is a small imperfection in a real top and disappears in an ideal one.”

The nod is a property of the ideal equations, not of a real top's defects: the figures here have no friction in them and every path but one nods. What removes it in a real top is exactly the imperfection — friction at the pivot damps the fastest motion first, and the nod is the fastest motion. An ideal top released from rest nods for ever.

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

“Momentum and kinetic energy conservation require that one ball leaves a cradle when one ball strikes it.”

Three equal balls with two conservation laws leave three unknown final velocities constrained to a curve, drawn here as an ellipse. Every point of it conserves both quantities exactly, including one where the middle ball rebounds at two thirds of the striking speed. Restricting to outcomes in which the balls do not pass through one another leaves an arc, and an arc is still a continuum. The two laws are two equations in three unknowns and cannot have a unique solution.

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

“A belt drive slips a little all the time, and the speed it loses is wear or stretch.”

The loss is creep over the active arc, and it is present in a perfect belt at any load below capacity. The figures compute how the wrap divides: below capacity part of it rides at constant tension and does nothing, and the rest creeps at the full exponential rate. The belt genuinely leaves the driver slower than it arrived, and no improvement in the belt removes it.

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

“The two versions differ only in notation.”

They differ in what is varied. A deformed path at fixed duration carries a different energy from the true one, and the figure shows that energy changing at first order in the deformation; a deformed path at fixed energy takes a different time, and that time changes at first order too. Each principle is stationary in its own quantity precisely because it lets the other one move.

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

“The irregular values a chaotic map visits are noise, and a longer run would average them out.”

The map here has no random number in it and the same starting value gives the same sequence every time, to every digit the arithmetic carries. What is drawn past the accumulation point is the set of values a single deterministic orbit visits, and it is a band with structure — windows of stable cycles inside it, each with a cascade of its own, at every magnification examined.

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

“The steady response of a driven oscillator is a function of the drive frequency.”

Over a range of frequencies this oscillator has two stable amplitudes and which one it is in depends on where the drive has been. Swept up, the amplitude climbs the leaning peak and falls off its edge; swept down over the same frequencies it follows the low branch and jumps up somewhere else. Both sweeps are the same equation with the same coefficients.

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

“An inductor opposes current.”

It opposes change. The figures show the voltage across a coil falling from the full supply to nothing while the current rises to its steady value: at the end there is a large current and no voltage at all. A coil carrying a steady current is a piece of wire, and the whole of its behaviour is in the derivative.

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

“Pressing two flat surfaces together makes contact over their apparent area.”

Surfaces are rough, so contact happens at the highest asperities and the real area is a small fraction of the apparent one. The measured resistance says how small: it is set by the number and size of the spots, and it flattens onto a floor decided by the size of the patch they are scattered over. That measurement is one of the two independent routes to the real contact area, the other being friction, and they agree.

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

“The jumps are noise, and a better sample would not have them.”

Nothing in the model here is random in time: the pinning landscape is fixed, and the same crystal gives the same sequence of jumps every time it is swept. The jumps are a deterministic consequence of a wall having to cross a landscape with local maxima in it, and a sample with less disorder has larger jumps rather than fewer.

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

“A magnetic monopole would need a new law.”

It would need a source term in two equations that presently have none, and nothing else. The difficulty is not with the field, which is the simplest field there is, but with the potential: the figure shows that any vector potential producing a radial field must be infinite along a line, because a potential regular everywhere would give zero net flux. That line is the whole of the theoretical trouble.

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

“A better superconductor carries more current.”

A perfect type-II superconductor carries almost none. A current exerts a force on the flux tubes, moving tubes induce a voltage, and a voltage with a current is dissipation — so a defect-free type-II material has resistance at any field above the first critical one. What makes a wire useful is defects strong enough to pin the lattice, which is why practical conductors are deliberately made dirty and cold-worked.

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

“An object at rest has no momentum.”

A current loop at rest in an electric field carries mechanical momentum of magnitude mE/c², where m is its magnetic moment. Its carriers are moving, and those on the high-potential side are more energetic than those on the low, so the same number of them per second carries unequal momentum round the two halves. Nothing about the loop as a whole moves, and the loop's momentum is not zero.

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

“A fibre with a bigger core accepts light from a wider angle.”

The acceptance angle is the square root of the difference of the squared core and cladding indices, and the core's diameter appears nowhere in it. A fibre a hundred times thicker accepts exactly the same cone; what it gains is area, and area times solid angle is the quantity that decides how much light gets in.

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

“Depth of field is a property of a lens.”

It is a property of a lens, a distance, an aperture and a decision about how closely the result will be examined. The figures compute the blur circle for one lens at three apertures focused at one distance: the acceptable range runs from a fraction of a metre to infinity depending only on which horizontal line is drawn, and the line is a choice about the viewer rather than about the optics.

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

“Unpolarised light is light whose direction of vibration changes rapidly and at random.”

That description is a model of one way of arriving at the same measurements, and there are others which are experimentally identical — a beam that is half horizontal and half vertical with no correlation between them behaves the same way and has no changing direction anywhere. What is measurable is the set of Stokes parameters, and every arrangement giving the same four is indistinguishable by any experiment.

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

“A large opaque particle removes exactly the light its shadow blocks.”

It removes twice as much. Half is the shadow; the other half is light diffracted out of the beam at the shadow's edge, and it exists for any obstacle large compared with the wavelength. The figure computes the extinction efficiency against size and it approaches two, through an oscillation that reaches beyond three for particles a few wavelengths across.

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

“Spherical aberration is unavoidable in a refracting surface.”

A Cartesian oval has none: the figures trace rays over a half-angle of twenty-six degrees and every one arrives at the image point, with the optical path lengths across the drawn surface agreeing to one part in a billion. The surface is generated by the equal-path condition and is then checked to refract by Snell's law, which is Fermat's principle tested rather than assumed.

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

“The resolution of a Fourier-transform spectrometer is set by the detector or the light.”

It is set by how far the mirror travels. A scan of length L convolves the recovered spectrum with the transform of the window, giving lines about 0.6/L wide whatever the source or the detector. To resolve a tenth of a reciprocal centimetre the mirror must move six, and no improvement to anything else substitutes for the distance.

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

“At a dark fringe the two waves' energy is destroyed.”

The cross term that makes the fringes averages to exactly nothing over a whole cycle of phase and over a whole number of fringes, so the total power on a screen is the same whether the sources interfere or not. The figure integrates the pattern and finds the excess above the no-interference line equal to the deficit below it. Nothing is destroyed; it has been moved from the dark places to the bright ones.

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

“The disturbance at a point comes mostly from the part of the wavefront directly in line with it.”

Each half-period zone contributes the same amount, because a zone's area grows exactly as fast as its distance. What makes the sum converge is the obliquity factor and the finite extent of the wavefront, and the total comes to half the first zone's contribution. Blocking every second zone leaves the remainder in phase and gives a far brighter result than the unobstructed wave, which is what a zone plate is.

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

“A wave driven above the cutoff is absorbed.”

There is no damping anywhere in the model. Driven above the cutoff, the chain has no travelling solution at all, and the only bounded response is one that falls by a fixed factor from each mass to the next while alternating in sign. The decay is the chain's own dispersion relation continued past the point where it has a real solution, and the energy is reflected rather than taken up.

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

“A standing wave has nodes where nothing happens.”

Only a completely reflected one does. With a partial reflection the minima sit at one minus the reflection magnitude rather than at zero, and the power crossing every point of the line — including the minima — is the same, equal to the fraction transmitted. Energy passes steadily through the places the amplitude is smallest.

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

“Bend loss sets in gradually as the radius is reduced.”

It falls exponentially with the radius, so over a millimetre or two it goes from unmeasurable to fatal. The figures compute the loss per turn against radius and it crosses six decades in a few millimetres, which is why a fibre tolerates a coil of one size and not one slightly smaller.

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

“Fluctuations are experimental noise and vanish for an ideal system.”

They are a property of equilibrium and are largest for the most ideal systems, since a system with fewer degrees of freedom fluctuates more. The mean square fluctuation is fixed by the heat capacity, which is a bulk thermodynamic quantity, and the relation runs both ways: measuring how much a small system's energy wanders is a measurement of its heat capacity.

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

“Skating works because the pressure of the blade melts the ice.”

A seventy-kilogram skater on a blade a millimetre wide and twenty centimetres long produces about thirty-five atmospheres, and the melting point falls by 7.4 millikelvin per atmosphere — a quarter of a kelvin in total. Nobody skates at a quarter of a degree below zero. The lubricating layer is there because a crystal's surface is disordered whether or not anything is pressing on it.

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

“A mixture at uniform temperature and uniform composition is at equilibrium, and one at uniform composition with a temperature gradient will simply conduct heat.”

It will also separate. A temperature gradient drives a flux of each component, so the composition drifts until the ordinary diffusion back down the concentration gradient it has created cancels it. The steady state is a balance of two flows rather than the absence of any, and the separation is measurable — a few per cent across a hundred degrees in a liquid.

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

“A free energy can only be measured by a reversible process.”

The two work distributions from a fast process cross at exactly the free-energy change, whatever the dissipation, and the exponential average of the work is exactly e to the minus the free energy. Both relations hold arbitrarily far from equilibrium, and both are used routinely on single molecules pulled far too quickly for any quasi-static argument.

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

“Real gases depart from ideality because their molecules take up room.”

That is one of two effects and it has the opposite sign from the other. The repulsive core makes the second virial coefficient positive and the attractive tail makes it negative, and at ordinary temperatures the attraction usually wins — most gases are more compressible than an ideal gas, not less. Which effect dominates is a matter of temperature, and the figures compute the crossover.

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

“The uncertainty relation is a fixed number attached to a pair of observables.”

Robertson's bound is half the size of the average commutator, and an average is taken in a state. The figures follow the same pair of spin components round two great circles of states: on one the bound is an equality everywhere, and on the other it is exactly zero while the product it bounds is not.

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

“A charged particle responds only to the fields at its own position.”

Both paths run where the field is smaller than a twentieth of a per cent of the winding's, and the fringe pattern moves by a whole fringe for every flux quantum enclosed. A response that depends on what is inside a region the particle never enters is not a response to the local field.

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

“Quantum wave packets always spread.”

In a harmonic well one width does not. A Gaussian of exactly the ground state's width, displaced, is propagated here for more than two full periods and its width moves by a part in ten thousand, while packets sixty per cent and seventy per cent wider breathe visibly at twice the swing frequency.

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

“There is a well-defined time a particle spends inside a barrier.”

Four defensible definitions are computed here from the same exact amplitude, and they disagree without limit as the barrier thickens: the phase time saturates, one Larmor time grows in proportion to the thickness, the other saturates at a different value, and the free-flight comparison grows too. A quantity with four different values is not one quantity being measured badly.

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

“A particle can be strongly entangled with several others at once.”

The CHSH values a party shares with two others obey S² + S² ≤ 8, so the most either can be while the other also exceeds the classical value of 2 is exactly 2 — no violation at all. The trade-off region is drawn and the best simultaneous value is found on it numerically.

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

“Heavier nuclei make smaller atoms, since the attraction is stronger.”

Across thirty-four main-group elements the one-electron prediction a₀/Z spans a factor of 83 and correlates with the measured radii at −0.50. The measured radii span 10 and repeat rather than fall. Every added proton arrives with an added electron, and what the outermost electron feels is a screened charge that barely grows across a row.

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

“The relativistic centre of mass is the ordinary centre of mass, with relativistic masses substituted.”

Weighting positions by rest mass gives a point that does not move uniformly through an elastic collision, because the rest masses are unchanged while the energies are redistributed. Weighting by energy gives a point whose worldline is straight to a part in a million million, at exactly the total momentum divided by the total energy.

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

“Charge transforms like energy, growing with the Lorentz factor.”

If it did, the inner electrons of an atom of atomic number Z — moving at about Zα — would leave the atom with a charge imbalance of order (Zα)²/2. For helium that is a hundred-millionth of an elementary charge, and the measured bound is 10⁻²¹, excluding it by thirteen orders of magnitude.

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

“Velocity space is the interior of a ball of radius c, which is an ordinary flat region with a boundary.”

Its boundary is infinitely far away in its own metric, so it is not a region with an edge — it is a complete plane. Equal steps of rapidity crowd towards the drawn edge without ever reaching it, and the crowding is checked ring by ring.

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

“The travelling twin ages less because of the acceleration at the turnaround.”

Every route drawn here has the same two endpoints and none of them has any special event: the proper time falls off smoothly with how far the route wanders, and the fall is second order in the wandering. The deficit is an integral over the whole route, and acceleration enters only as the reason a route can differ from the straight one at all.

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

“The drag on an oscillating surface is in phase with its velocity, like any drag.”

The stress leads the velocity by exactly forty-five degrees, located on the drawn trace. That splits it into equal halves: one that dissipates energy and one that is an added mass and does not. Neither half exists without the other, and no fluid or frequency moves the split.

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

“The pressure at a point in any fluid is the same in all directions.”

Only at rest. Shearing the same fluid opens the single point into a circle of stresses whose radius is the viscous stress: some planes then carry shear and the normal stress depends on the orientation. What survives is the average normal stress, and calling that the pressure is a convention rather than a theorem.

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

“A hopper empties faster when it is full, like any container.”

The fill height does not appear in the discharge law. The walls carry the weight of everything above a depth of a couple of silo diameters, so the grains at the opening are pushed only by their immediate neighbours. A liquid through the same hole falls to a third of its rate by the time a tenth of the head is left; the grain rate is flat until the surface reaches the outlet.

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

“Two solutions with the same osmolarity are equivalent to a cell.”

Osmolarity counts particles; tonicity asks what the membrane does about them. At the same concentration a red-cell membrane develops the full pressure against sucrose and essentially none against ethanol — a factor of a hundred between the extremes drawn, with the same particle count.

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

“An unbounded capillary rise would need an unbounded amount of liquid.”

The filament's width falls as one over the height, so its cross-section falls as the square, and the integral converges. The volume held above any height is that height times the cross-section there — a fraction of a microlitre for a millimetre corner — checked against a numerical sum over four thousand times the starting height.

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

“Mercury's anomalous precession is a special case needing a special explanation.”

One formula with no free parameters — six pi times the gravitational radius over the semi-latus rectum, per orbit — reproduces the quoted residual advance for Mercury, Venus, Earth, Mars and Icarus alike. Mercury is the largest because it is closest and quickest, and both enter.

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

“An interferometer is sensitive to waves from anywhere in the sky.”

The plus pattern vanishes along four lines and the cross pattern along others, and the sky average of each is well under one — their squares add to two fifths over the whole sky, summed over a hundred and sixty thousand directions. A detector's quoted range is set by that average and is far below its reach for a source directly overhead.

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

“Evaporation is a gradual fading away.”

The mass falls as the cube root of the time remaining, so it is nearly constant for almost the whole life and then collapses. When one second of life is left the hole weighs about 230 tonnes whatever it started as, and it releases some 2 × 10²² joules in that second — a burst, not a fade.

Tested in The hole that outlives everything and then does not · the horizons reading path

“The paradox could be settled by measuring a charge held stationary in a gravitational field.”

The Larmor power for an electron at one gravity is 5 × 10⁻⁵² watts. Held for the whole age of the universe it would radiate 2 × 10⁻³⁴ joules — nine orders of magnitude below a single quantum at any frequency it could be emitted in. The question has never been experimental and could not be.

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

“All the Planck quantities are absurdly far from anything human.”

The Planck mass is twenty-two micrograms — a visible speck — and the Planck energy is the chemical energy of about fifty-seven litres of petrol, computed here rather than quoted. That one quantity sitting in the human range is the statement that gravity is weak, and it is the reason the length and the time are so extreme.

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

“A neutron looks different after one full turn, so an object can be distinguishable from itself rotated by 360 degrees.”

Every prediction for a single spin comes from the density matrix, which is built from the state and its conjugate and is therefore blind to an overall sign. The figure measures both distances across one turn: the state moves the full width of the space it lives in, and the largest change in any component of the spin's direction is at the level of the integrator's rounding. No measurement on one neutron distinguishes before from after.

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

“Making a resonator smaller always raises its frequency.”

The map of a cavity wall computed here is positive over part of its area and negative over the rest, with a contour between them where a dent does nothing. Pushing the wall in at the middle, where the electric field peaks, lowers the frequency; pushing it in near the side walls raises it. The whole-cavity result is the sum of these, and the sum being positive for a uniform squeeze says nothing about any one place.

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

“The stress tensor is a way of writing Coulomb's law and adds nothing to it.”

Coulomb's law is a statement about two point charges in vacuum. The surface integral computes the force on the contents of a region whatever they are — a polarised dielectric, a conductor with unknown surface charge, a magnetised body — and it is how such forces are actually computed. The figures here recover Coulomb's law only as a check that the method is right.

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

“Heavier particles drift more slowly, so an electric field separates the species in a magnetised plasma.”

Three species with masses differing by four orders of magnitude and charges of both signs are integrated here in the same crossed fields. Their orbits differ in size and period by those same factors and their guiding centres move together to within a fraction of a per cent. An electric force is itself proportional to charge, so the charge cancels out of the drift and no current is produced.

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

“A rolling wheel has no relative motion at the contact, so a driven wheel rolls without slipping until it breaks away.”

Rolling without slipping is the zero-force case. The moment a torque is applied, the tread ahead of the contact is elastically stretched relative to the carcass, and the integration here shows the trailing part of the patch sliding at every non-zero force. Two per cent slip is normal cruising, and it is sliding over about a quarter of the footprint.

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

“The body flips because something is exerting a torque on it.”

The damping used here is constructed to be orthogonal to the angular momentum, so the momentum is conserved identically rather than approximately, and the integration reports its drift over the whole run. Nothing acts from outside; the momentum vector in space never moves. What changes is which body axis is lined up with it.

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

“Field curvature is an aberration, so a better-made lens has less of it.”

The Petzval sum depends only on the powers and refractive indices of the elements. Bending a surface, moving the stop, adding an aspheric profile and improving the polish change every other aberration and leave this one identically unchanged. It is removed only by adding elements of the opposite sign in a different glass, which is a change of design rather than of quality.

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

“The sky is completely polarised at ninety degrees from the sun.”

Ideal single scattering by isotropic dipoles gives a hundred per cent there. Air molecules are not isotropic, which costs a few per cent that no sky can recover, and light scattered more than once arrives having forgotten the geometry, which costs much more. A clear high sky reaches about eighty per cent and a hazy one about half of that.

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

“A ratchet in a gas turns slowly in one direction, because collisions in the allowed direction are kept and the others are refused.”

The forward rate needs a collision able to lift the pawl and is exp(−ε/kT); the backward rate needs the pawl to lift by its own fluctuation and is exp(−ε/kT) with the same ε and the same T. At one temperature they are identical for every notch depth, which the figures verify, so the net rate is exactly zero and not merely small.

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

“The recurrence shows equipartition failing.”

It shows equipartition being slow. The threshold figure runs the same chain across three decades of energy density and the spread rises steadily with it; below the knee the sharing time is longer than the run, above it the sharing happens while the run is going on. The original calculation sat well below the knee and ran for a few thousand periods.

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

“The airless bodies confirm the balance argument exactly.”

The Moon's mean surface is twenty kelvin below its balance temperature and Mercury's ninety below, because radiated flux goes as the fourth power: a surface running from noon heat to night cold radiates like its hottest parts and averages like its coldest. Only a body near isothermal sits on the line, and the deviation for an airless body is in the opposite direction from an atmosphere's.

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

“A black hole's mass can only increase, since nothing gets out.”

The merger computed here loses three solar masses to gravitational radiation, and a spinning hole can in principle give up twenty-nine per cent of its mass without anything crossing the horizon outwards. What cannot decrease is the area, and the two statements are different because area depends on the spin as well as the mass.

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

“Clock comparisons over long distances are limited by the length of the link.”

The uncertainty of an optical-fibre frequency comparison does not grow with distance the way a levelled line's does, because there are no accumulating set-ups in between. The two methods cross at around a hundred kilometres, beyond which the clock is the only one that does not degrade — which is the case for chronometric levelling.

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

“The satellite test worked because a spacecraft has a better accelerometer.”

Its differential resolution is within an order of magnitude of the best torsion balance's, as the figures here show. What it has is a driving acceleration five hundred times larger, because a body in free fall around the Earth is pulled by the whole field rather than by the thousandth of it that the Earth's spin leaves pointing sideways.

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

“Fizeau's experiment measured a partial dragging of the ether.”

It measured how fast light travels in moving water, which is a fact about light and water. The interpretation as ether drag was the framework of the time and required a coefficient depending on the refractive index that no ether model produced. Composing c/n with v relativistically gives the same coefficient in two lines and mentions no medium beyond the water.

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

“A fast enough parent sends all its decay products forward.”

Only those slower than the parent. A product whose own centre-of-mass speed exceeds the parent's can still be found going backwards, at any parent speed, and the figures compute the largest lab angle for both cases. The threshold is a comparison of two speeds, not a matter of the parent being fast in absolute terms.

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

“The four assumptions rule out Galilean relativity.”

They do not. The zero value of the constant satisfies every one of them, has an associative composition law, and gives absolute simultaneity — which the figures verify. It is the physically wrong answer and not the logically excluded one, and knowing which is which is the point of doing the derivation this way.

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

“A membrane potential requires a pump.”

The Donnan potential computed here is tens of millivolts and arises at equilibrium, with no current flowing and no work being done. A nerve cell's resting potential is a different thing of a similar size — maintained against a leak by pumps, and therefore not an equilibrium at all.

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

“A black patch in a soap film is a hole, or a film about to break.”

The reflectance computed here goes to exactly zero as the thickness does, because the two surfaces reflect equally and half a cycle out of step, so a film thinner than about a tenth of a wavelength is black while entirely present. A black film is also the most stable state available: it is where the thinning stops, and a film that has reached it can last for hours.

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

“A column of grains settles to its densest packing after enough shaking.”

The approach is logarithmic in the number of taps, so each further slice of the remaining gap costs a factor rather than an amount. The figures compute the number of taps to close successive fractions: eighty per cent takes tens of thousands, ninety-five per cent would take longer than the age of the Earth at one tap a second.

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

“A flux tube in pressure balance with its surroundings has the same density as them, so it stays where it is put.”

Total pressure balance is p_in + B²/2μ₀ = p_out, so the gas pressure inside is lower by exactly the magnetic pressure. At equal temperature the density follows, and the deficit is 1/(1+β) — one half at β = 1 and one part in ten thousand at β = 10⁴. The curve drawn has no root anywhere. A magnetised tube is never neutrally buoyant; it is only ever slowly buoyant.

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

“A wave in a plasma is a sound wave with magnetic corrections.”

One of the three is not. The shear wave's speed is B/√(μ₀ρ) and contains no temperature, no pressure and no sound speed, so it is unchanged if the gas is made ten times hotter or replaced with a different gas at the same mass density. The polar diagram drawn here shows it as a figure-of-eight sitting between the two magnetosonic branches, and it is the branch whose formula has no thermodynamics in it at all.

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

“The near field is where the far-field formula has not become accurate yet, so it is a matter of a few per cent.”

The three terms fall as 1/r³, 1/r² and 1/r, so at a tenth of the crossing radius the near terms are a hundred and a thousand times the radiation term. That is not a correction; it is a different field with a different distance law, a different phase and a different energy budget. The figures show the total dipping below every individual term just inside the crossing, because the static and radiation terms are in antiphase and partly cancel — behaviour no small correction produces.

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

“The dipole approximation holds when the antenna is small compared with the distance to the observer.”

The distance to the observer does not appear in the condition. What matters is the extra path from one end of the source to the other, compared with a wavelength: a source a tenth of a wavelength long is down 1.6 per cent at the worst angle, one of 0.6 wavelengths is down 36 per cent, and one of 1.2 wavelengths has nulls in its pattern. All three statements hold at any observation distance in the far field.

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

“Maxwell's equations say that fields propagate outward from their sources.”

They say a field satisfies a wave equation, and a wave equation is second order in time and therefore symmetric under reversing it. The converging solution f(r+ct)/r is drawn here beside the diverging one and both are checked by differencing the same samples the figure draws, twice in space and twice in time: the residual vanishes for both. Retardation is imposed on the solutions afterwards, and choosing it is choosing a boundary condition.

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

“The wavelength of a mountain wave is set by the width of the mountain.”

The vertical wavelength is 2πU/N, in which no dimension of the ridge appears. What the ridge's width decides is which horizontal wavenumbers are excited and therefore how much of the response is a propagating wave rather than a local disturbance — an amplitude question rather than a wavelength one. The figures compute the pattern for two different ridges at the same wind and stability and the vertical spacing is identical.

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

“A wave reflecting from a boundary leaves at the angle it arrived at, measured from the boundary.”

That is true of every wave whose speed is a scalar, and false of an internal wave. Its frequency fixes the angle its energy makes with the horizontal, because gravity supplies the restoring force and gravity is vertical, so a reflection must preserve that angle and not the angle to the wall. The figures draw incident and reflected beams at the same angle to the horizontal off a slope, which for a mirror would be impossible, and the wavelength changes by sin(θ+α)/sin(θ−α) rather than staying fixed.

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

“The inertial period is twelve hours.”

It is π over Ω sin φ, so it is twelve hours at the pole and longer everywhere else: 13.8 hours at 60°, 16.9 at 45°, 23.9 at 30°, and 69 at ten degrees of latitude. It is exactly half a pendulum day, checked at five latitudes against an independently computed Foucault period, and it diverges at the equator where f vanishes.

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

“Water drains out of a bath in opposite directions in the two hemispheres.”

The Rossby number of a draining bath is about four thousand, computed from its own speed, size and latitude. The Coriolis acceleration is four thousand times smaller than the accelerations the flow already has, so it is beaten by the shape of the basin, the residual swirl from filling, and the asymmetry of the plug. The experiment can be made to work — it has been, in a symmetric tank left still for a day — and the conditions required are exactly the conditions a bath does not meet.

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

“Snell's law with a complex index gives the refraction angle into an absorbing medium.”

It gives a complex number, and a complex angle is not the direction of anything. What the boundary condition actually fixes is the tangential wavenumber, which stays real; the normal component is complex, and its real and imaginary parts point along different directions. Surfaces of constant amplitude lie parallel to the interface and surfaces of constant phase are tilted, so there are two angles rather than one — and the construction reduces to Snell's law exactly, to twelve decimal places, when the absorption is set to zero.

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

“A flat slab cannot focus, because it has no curvature.”

A flat slab of index −1 focuses twice: once inside at the source's own distance from the front face, and once beyond it at the remaining thickness. Both foci are read off the traced rays here and are identical for every ray angle drawn, so there is no aberration of any kind — because the refraction is exact rather than paraxial. What the slab has instead of a focal length is a working distance that moves with the source.

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

“The continuous beta spectrum shows that energy is not exactly conserved in nuclear decay.”

That was the live hypothesis for four years and it was wrong. A continuum is what three bodies in the final state produce, because the energy can be divided in a continuum of ways; a line is what two produce. The shape is a phase-space count with no free parameters beyond a normalisation, and it fits — so the observation is evidence for an unseen third body rather than against a conservation law.

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

“In secular equilibrium the members of a decay chain are present in equal amounts.”

Their activities are equal and their amounts are not. Equal activity means λN is the same for every member, so N is inversely proportional to λ — the abundance ratio is the half-life ratio. In the uranium series that spans twelve orders of magnitude, from a whole gram of ²³⁸U to 2.3 × 10⁻¹² grams of ²²²Rn, and each ratio here is computed from half-lives alone and agrees with tabulated abundances to two per cent.

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

“A half-life is a fixed property of a nuclide and nothing outside the nucleus can change it.”

Electron capture is a nuclear process whose rate is proportional to the electron density at the nucleus, and that density is a chemical quantity. Beryllium-7 decays 0.83 per cent faster as the metal than as the fluoride, and its rate changes with pressure and with what cage it is put inside. Those are small numbers and they are not zero, and the mechanism is not in dispute.

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

“Conservation of charge is a separate experimental law, additional to the field equations.”

Taking the divergence of the Ampère–Maxwell equation gives ∂ρ/∂t + ∇·J = 0 with no assumption at all, so the field equations already contain it. Written covariantly it is the vanishing four-divergence of the four-current — a single scalar equation, the same in every frame — and if it failed the four-current would not be a four-vector, so the transformation the previous rung established would have nothing to act on.

Tested in Charge and current are one thing · the field transformation reading path

“Writing Maxwell's equations covariantly is a notational convenience.”

It changes the count. Four vector equations become two tensor ones: the inhomogeneous pair collapses into a single equation with a free index and the homogeneous pair into a cyclic identity on F, which is the statement that F is a curl of the four-potential. The second pair stops being a law at all and becomes an identity, which is a change in what kind of statement they are rather than in how they are written.

Tested in Six numbers, one object · the field transformation reading path

“The Kauzmann paradox is that a supercooled liquid's entropy falls below its crystal's.”

No measured entropy does that. The paradox is that the *extrapolation* of the equilibrium liquid's entropy crosses zero excess at a finite temperature — 152 K for the substance drawn — and the liquid always falls out of equilibrium first. What is paradoxical is that the intervention appears to be needed, not that the crossing is observed.

Tested in The entropy that depends on how fast it was cooled · the third law reading path

“A sample at a microkelvin has essentially zero entropy.”

Its lattice does. Its nuclear spins carry R ln(2I+1) — 11.5 joules per kelvin per mole for copper's spin-3/2 nuclei — down to the temperature at which their own dipolar interactions order them, which for copper is about 58 nanokelvin. A sample at a microkelvin is three orders of magnitude above that, and its nuclear spin system is as disordered as it was at room temperature.

Tested in A law about spectra, not about heat · the third law reading path

“Acoustic streaming is a nonlinear effect that only matters at high amplitude.”

It is nonlinear in the sense that it comes from a second-order term, and its speed is linear in the intensity while the oscillation itself goes as the square root. So streaming grows faster than the motion that drives it: doubling the drive doubles the flow and increases the oscillation by forty per cent, and at ordinary diagnostic intensities the flow in water is millimetres per second.

Tested in Where the loudness goes · the attenuation reading path

“Absorption falls smoothly with photon energy, because a harder photon interacts less.”

It falls steeply and it is not smooth. At the binding energy of an inner shell the coefficient jumps upward — by a factor of 4.4 for iodine at 33.17 keV and 4.0 for lead at 88, computed here from tabulated values on either side. A material becoming more opaque as the photon gets harder is a discontinuity in a material property as a function of frequency, and nothing in the classical account of absorption produces one.

Tested in The steps in an absorption curve · the attenuation reading path

“Light absorbed by a gas can only heat it.”

Each absorption from the beam an atom is moving towards takes a photon's momentum off its motion, and the light it scatters leaves bluer than it arrived by the Doppler shift that motion gave it — so the kinetic energy goes out with the light. An ensemble followed photon by photon here falls from 1.10 millikelvin to 240 microkelvin in a few hundred microseconds, after about 1,200 scatterings per atom.

Tested in The friction made of light · the radiation pressure reading path

“Nothing that scatters light can cool an atom below the Doppler limit ħΓ/2kB.”

That limit is exact for an atom with one ground state and says nothing about one with several. In crossed-polarised beams a spin-½ atom is cooled by the light shift rather than by the Doppler shift, and the stochastic model here settles caesium at 6.96 µK in wells a hundred recoil energies deep, against a Doppler limit of 125 µK. The first measurement of a sodium molasses found 43 µK against a limit of 240.

Tested in The limit that belonged to a simpler atom · the radiation pressure reading path

“Throwing uphill and downhill are the same problem with the range reduced or increased a little.”

The reach along a line at elevation α is (v²/g)/(1 + sin α), which is not symmetric about level ground and diverges as the line turns straight down. A 45° uphill line gets 0.586 of the level range and a 45° downhill line 3.41 times it — nearly six times as far — from the same arm at the same speed.

Tested in One curve answers every slope · the projectile reading path

“Thrown from a moving platform, the best angle is still 45° — the platform just adds distance.”

The platform shifts the whole circle of available launches, and the tangency moves. At a platform speed of half the throwing speed the best aim is 53.6° relative to the platform, the ball actually leaves at 36.4° to the ground, and it goes 1.76 times as far as a standing throw. Searching every angle and solving 2cos²θ + u cos θ − 1 = 0 give the same answer.

Tested in The best throw is a tangency · the projectile reading path

“Water vapour only condenses into liquid when the air is saturated.”

That is true over a flat surface. A concave meniscus lowers the vapour pressure it is in equilibrium with by exp(−2γVₘ/rRT), so a pore narrow enough is full of liquid in unsaturated air: at 50 per cent humidity every pore below 1.51 nm radius holds condensed water, and at 90 per cent every pore below 9.96 nm.

Tested in The pore that fills from dry air · the capillarity reading path

“The contact angle a liquid makes on a solid is fixed once the solid, the liquid and the surrounding medium are chosen.”

A voltage across a thin insulator under the drop lowers the effective solid–liquid energy by ε₀εᵣV²/2d without changing any of the three. A 2 µL water drop on a 1 µm fluoropolymer coating goes from 115° to 50° between 0 and 95 V, and its footprint from 1.53 mm to 2.72 mm across, at constant volume — which the figure checks.

Tested in The angle a voltage can set · the capillarity reading path

“A light-emitting diode that turns every electron into a photon emits all of that light.”

Only the light inside the escape cone crosses a flat face. Generated in every direction inside gallium arsenide, 1.97 per cent of it lies within that cone and 1.31 per cent survives the reflection at the face; for gallium nitride the figures are 4.35 and 3.41 per cent. The rest is reflected back inside, and every gain in extraction has come from changing where that light goes next.

Tested in The cone light has to find to get out · the etendue reading path

“A fibre has a polarisation delay that can be measured once and compensated with a fixed device.”

The delay is a random variable of the fibre's condition, not a constant of its construction. One simulated 100 km fibre built from a thousand randomly oriented sections has a delay that wanders between 0.6 and 9.8 ps across a 4.4 THz band, and the same fibre watched over days samples a Maxwell distribution as temperature and handling reshuffle its sections.

Tested in The delay that is a random variable · the dispersion reading path

“Gravity gets weaker as one goes down into the Earth.”

It gets stronger through the whole mantle. Gauss's law applied to the reference model's density gives 9.82 m/s² at the surface and 10.69 m/s² at the core–mantle boundary, 2,891 km down — 8.8 per cent more — and gravity only starts falling once the core itself is being entered.

Tested in The pull that grows on the way down · the Gauss's law reading path

“A steady voltage across a junction between two superconductors drives a steady current through it.”

It drives an alternating supercurrent, because a voltage makes the phase difference advance at 2eV/ħ and the supercurrent is the sine of that phase. In a shunted junction biased at 1.3 times its critical current the voltage is a train of pulses between 0.3 and 2.3 in units of IcR, each one phase slip, with a mean of 0.831 — √(i² − 1) exactly, as the integration checks.

Tested in The voltage that is a frequency · the superconductivity reading path

“A wave packet that has spread out has lost its initial shape for good.”

In an oscillator whose levels carry a small quadratic term, a coherent state of nine quanta collapses in about nine periods, reassembles whole on the opposite side at 120 periods and exactly as it began at 240, and at an eighth and a quarter of that time it is four and two whole copies of itself. A classical ensemble started from the same distribution collapses the same way and never regroups.

Tested in The return a classical cloud never makes · the correspondence reading path

“Any quantum state of a particle can be pictured as a cloud of classical possibilities in position and momentum, blurred by the uncertainty principle.”

The phase-space function whose shadows are the true position and momentum distributions is the Wigner function, and for a superposition of two packets it runs from 0.289 down to −0.289, with its interference stripes taller than the packets themselves at 0.159. No cloud has a negative density, so no cloud reproduces the state; the momentum density's zeros are exactly where the negative stripes cancel the positive ones.

Tested in The probability that goes below zero · the correspondence reading path

“At full power a car's speed rises steadily, like a body under a constant force.”

Constant power means the kinetic energy rises steadily, so the speed rises as the square root of the time: without resistance the second 50 km/h takes three times as long as the first would at the same power. The integrated run is a straight line for 0.86 s while the grip limits it and a square root afterwards, matching the closed form to better than a millimetre per second.

Tested in The speed at which grip hands over to power · the energy reading path

“The causal structure of spacetime sets the length of a second.”

A uniform stretch of spacetime about any event keeps every cone and every causal relation, and multiplies every squared interval by the square of the stretch: by 4 for a stretch of 2, checked on 500 pairs. Nothing about what can influence what distinguishes the two, so the unit of time has to come from a clock.

Tested in What the light cones alone can decide · the spacetime diagram reading path

“A drawing of the whole of an infinite spacetime has to distort its light cones.”

Applying arctan separately to u = t − x and v = t + x maps the infinite plane into a diamond, keeps every light line at 45° to a part in a billion, and changes the causal relation of none of 4,000 sampled pairs of events. What it gives up is distance: the same half-unit of proper time looks 0.426, 0.094 and 0.010 times its size at the origin when it starts at t = 1, 3 and 10.

Tested in The five places infinity turns out to be · the spacetime diagram reading path

“Burning the fuel hotter is what makes an ideal Otto engine more efficient.”

Its ideal efficiency depends only on the compression ratio. Cut into twelve thin Carnot strips, every strip of the cycle has the same efficiency, 58.48% at a ratio of 9, because the curves bounding it are one exponential scaled by the compression's temperature ratio; adding more heat stretches the loop along the entropy axis without changing any strip's efficiency.

Tested in The temperature an engine really takes its heat at · the heat engines reading path

“An engine running between a kilogram of water at 90 °C and a kilogram at 10 °C can convert 22% of the heat it takes.”

22% is the Carnot efficiency at the start, and it falls as the hot water cools and the cold water warms. The best possible engine draws 177.8 kJ from the hot water and delivers 20.8 kJ of work, 11.70% of the heat — exactly 1 − √(283.15/363.15) — before the two meet and it stops.

Tested in The work left in two buckets of water · the heat engines reading path

“A state inside a band gap needs a defect or an impurity to hold it.”

A perfectly regular chain with alternating couplings v and w holds two states at zero energy, in the middle of its gap, whenever v < w. For a chain of 20 cells at v/w = 0.5 both lie within a millionth of the coupling of zero; at v/w = 1.5 the same chain has none, and its nearest level is 0.527 from zero. Nothing was broken — only where the chain ends.

Tested in The end that knows how the middle was cut · the periodic media reading path

“A structure that repeats in only one direction cannot reflect light arriving from every direction; that needs a three-dimensional photonic crystal.”

A quarter-wave stack of indices 4.6 and 1.6 forbids both polarisations at every angle of incidence from air between 0.848 and 1.321 of its design frequency, a window 43.6% wide, and ten pairs reflect more than 99.99% at every angle from normal to 89° in the middle of it. What a three-dimensional crystal adds is a gap for light arriving from inside a dense medium.

Tested in The mirror that works from every direction · the periodic media reading path

“Gibbs's entropy, which never gives a negative temperature, shows that negative temperatures are an artefact of a bad definition.”

Two systems of two-level units, 1,000 and 200 of them, with 600 and 145 excited, have equal Gibbs temperatures. In contact the energy still moves: the pair has a few hundred times more arrangements with 21 quanta passed from the smaller system to the larger, so that is where the energy goes. It moves from the system with the lower Boltzmann inverse temperature, −0.964 against −0.405, and stops where the Boltzmann temperatures agree, to 0.006.

Tested in The count that decides which entropy is right · the third law reading path

“A charge accelerating steadily loses energy to radiation reaction at the rate it radiates.”

The reaction force is proportional to the rate of change of the acceleration and is exactly zero while the acceleration is steady, while the radiated power is the full Larmor rate. Through a push that rises, holds and stops, the reaction force takes nothing during the steady part and 0.775 units while the push stops, against 0.750 radiated over the whole push.

Tested in The bill that arrives when the pushing stops · the radiating charge reading path

“A difference in how the Earth and the Moon fall towards the Sun would be too small to see in the Moon's orbit.”

The Sun pulls at 5.93 mm/s², the two bodies' self-energies differ by 4.3 × 10⁻¹⁰, and the push that difference would produce turns once a synodic month, close enough to the orbital frequency that the orbit magnifies it about twenty-two times. The result is a displacement of 13.1 metres per unit η, and the Earth–Moon distance has been ranged to centimetres and then millimetres.

Tested in The binding energy that has to fall too · the equivalence principle reading path

“An electrode driven by a symmetric alternating voltage through a capacitor sits at zero average voltage.”

In a plasma it charges negative until one cycle's electron charge matches one cycle's ion charge. With an argon plasma at 3 eV, a 100 V drive leaves the electrode at a mean of −106 V, and integrating the charging cycle by cycle gives the same number as the charge balance, which involves a Bessel function and nothing else about the circuit.

Tested in The bias no battery supplies · the plasma oscillation reading path

“Charge spreads evenly over a conducting surface, because like charges repel equally in every direction.”

It collects where a walker arriving from far away is most likely to touch first. On a thin charged strip the density follows 1/π√(1 − x²), and walkers from outside land in the outermost bins 6.4 times as often as in the same width at the centre; near a square's corner the density rises as distance to the −⅓ and near a knife edge as −½.

Tested in The potential is where the wanderers stop · the potential reading path

“A polymer chain stretches in any fast enough flow.”

In simple shear at λγ̇ = 1 a dumbbell's mean square stretch settles at 1.67 times its resting value, because the flow's rotation turns each chain away from the stretching direction as fast as it is stretched. In uniaxial extension at the same rate nothing rotates it, and by eight relaxation times the stretch is nearly two thousand times its resting value and still growing.

Tested in The stretch a chain cannot outrun · the rheology reading path

“Diluting a charged polymer frees its counterions, since each is then surrounded by more space to explore.”

Only while the charges along the chain are farther apart than the Bjerrum length. For DNA, a charge every 0.17 nm against a Bjerrum length of 0.71, the Poisson–Boltzmann solution keeps 76.2 per cent of the counterions inside the inflection of the charge profile in cells of 10², 10⁴ and 10⁶ rod radii alike. Diluting a millionfold does not release them.

Tested in The counterions that never leave the chain · the osmosis reading path

“A critical latitude is a sharp line past which nothing happens.”

The instability is a tongue of finite width, so a subharmonic detuned slightly by the inertial frequency still grows. The fastest available growth stops at 30.0° with a modulation strength of 0.1 and at 31.5° with 0.3, and a wave integrated at 29.5° grows at 0.145 e-folds a period, nearly as fast as a resonant one.

Tested in The latitude past which a tide cannot split · the stratification reading path

“A chaotic system is irregular all the time.”

Two hundred-thousandths below the setting where its three-cycle is born, the logistic map repeats itself to within 0.004 every third step for calms averaging 173 iterates, between bursts that cross the whole interval. Nothing random is added: the same start gives the same sequence of calms and bursts every time.

Tested in The calm that is the ghost of a cycle · the chaos reading path

“Light passing through a scattering medium falls off exponentially with thickness.”

Only the light that crosses without being scattered does. Thirty mean free paths of cloud droplets keep 9.4 × 10⁻¹⁴ of a beam unscattered, and let through 29 per cent of the light by diffusion. Beyond a few transport lengths the diffuse transmission falls as one over the thickness — 0.180, 0.060 through 8 and 30 mean free paths of an evenly scattering medium — which is Ohm's law, not an exponential.

Tested in The cloud light has to walk through · the scattering reading path

“The frequency of light cannot change in a linear medium unless something moves.”

A change of the medium everywhere at once keeps the wavenumber and therefore changes the frequency. Switched from index 1 to 1.5, the grid's period goes from 30.04 to 45.07 — a ratio of 1.500 — in a linear medium in which nothing moves and nothing is emitted.

Tested in The reflection that needs no surface · the refraction reading path

“Measuring a particle's polarisation from its decay requires knowing the direction the particle was travelling.”

Along the line of flight it needs only the parent's energy. For taus from Z decays, the fraction of the tau's energy carried by the pion is spread as 1 + P(2x − 1), and 100,000 decays drawn with P = −0.14 read back −0.150 ± 0.0055. The curve moves by 0.085 per cent at ten thousand times the energy, so the neutrino hiding the tau's direction does not matter.

Tested in The slope a spin leaves in a spectrum · the relativistic dynamics reading path

“Determining a particle's spin requires measuring the spins of the particles it decays into.”

For a resonance decaying to two spinless particles, the spin is written along its band: at fixed pair mass the other mass squared is a straight line in the decay angle, and 30,000 decays each through spins 0, 1 and 2 go dark at zero, one and two places, following the squared Legendre polynomials at χ² of 18, 17 and 13 over 20 bins.

Tested in The plane in which three bodies are flat · the relativistic dynamics reading path

“A sail that reflects nearly all the light stays cool.”

In a 100 GW beam, reflecting all but one part in 100,000 still leaves a megawatt absorbed over a few square metres. Radiating it from both faces, a 4 m sail settles at 915 K with an emissivity of 1 and 1936 K with an emissivity of 0.05; the temperature falls only as the fourth root of the absorption, so a thousand times better mirror is only 5.6 times cooler.

Tested in The rocket that leaves its fuel at home · the Mass-energy reading path

“Energy levels shift in straight lines as a magnetic field is increased.”

Only levels that nothing else shares a quantum number with. Of sodium's six 3p states, the two with mⱼ = ±3/2 are straight lines; the other four bend, and each level's moment changes from its Landé value, such as 2/3, to its strong-field value, such as 1, over about two decades of field. States with equal mⱼ never cross — the mⱼ = −½ pair comes no closer than √2ζ, 486.0 GHz, at 12.3 T.

Tested in The field an atom calls strong · the atomic spectra reading path

“Faster computation needs more energy per operation.”

It needs more energy per operation per second. Margolus and Levitin's theorem caps the rate of orthogonal steps at 2⟨E − E₀⟩/πħ, 6.04 × 10³³ per second for each joule above the ground state — 5.43 × 10⁵⁰ a second for a kilogram of rest energy — and says nothing about what each step must dissipate, which can be made as small as desired by going slowly.

Tested in The fastest a state can stop being itself · the uncertainty reading path

“A diffusion flux stops when the concentration gradient of that species is zero, and not before.”

Nitrogen's flux in the two-bulb experiment is 309 times what its own starting gradient of 0.00207 would drive, and 6.6 hours in it passes through zero with a gradient of 0.1468 still in place. A species' flux depends on every species' gradient, so it can flow with no gradient of its own and stop with a large one.

Tested in The gas that flows towards more of itself · the diffusion reading path

“The sharpest resonance of a structure is the one its transmission spectrum shows most clearly.”

At a bound state in the continuum it is the one the spectrum does not show at all. Far from the bound state the slower mode's feature has a width of 0.029; nearer, 0.0099 and 0.0018; at the bound state it is gone, because a mode that cannot leak into the channel cannot be excited from it either, while its quality factor is infinite.

Tested in The resonance that refuses to leak · the resonance reading path

“With enough matching elements any load can be matched to any source over any band.”

Only a load that stores no energy. A resistance shunted by a capacitance with RCω = 2 is held across the band to 0.707, 0.392, 0.320, 0.289, 0.275 and 0.270 by optimised networks of none to five elements, and can never go below exp(−π/RCω) = 0.208 however many are added. A plain resistance, by contrast, is matched perfectly at every frequency by an ideal transformer alone.

Tested in The mismatch no network can remove · the impedance reading path

“The Fresnel approximation is accurate once the observer is in the near field of a wide aperture.”

Its accuracy depends on how much light goes out at wide angles, not on the regime's name. At a Fresnel number of one, a slit two wavelengths wide differs from the exact result by 17.5 per cent rms and one sixteen wide by 2.4 per cent, falling roughly as the inverse of the width — the rate at which a sharp edge's wide-angle share of the power falls.

Tested in The fan of plane waves inside every beam · the huygens reading path

“A ferromagnet is a set of little magnets that have lined themselves up with each other.”

The magnetic energy of two neighbouring moments in iron is 0.20 K in temperature units, computed here from the moment and the lattice spacing. Iron orders at 1,043 K, five thousand times higher, so an arrangement held together by that interaction would be destroyed by any temperature at which it could be handled. Four other ferromagnets are computed in the same figure and the smallest ratio is 112.

Tested in What holds a magnet together is not magnetism · the magnetisation reading path

“A permanent magnet keeps its magnetisation indefinitely if nothing is done to it.”

It keeps it for a time that is an exponential of a barrier, and every barrier is finite. The lifetime figure here reads the barrier needed for each of four lifetimes off its own curve: one second needs 20.7 thermal energies, ten years needs 40.3. A magnet is stable in the same sense a supercooled liquid is — long-lived, not permanent — and the creep figure shows a sample drifting steadily through eight decades of time with nothing applied to it.

Tested in Nothing keeps a magnetisation for ever · the magnetisation reading path

“A structure with no load on it has no forces in it.”

Only if it is determinate. Two steel bars and an aluminium bar between rigid crossheads, heated uniformly by a hundred kelvin with nothing applied, carry 77.7 kN of compression and 38.8 kN of tension each — computed here, and summing to zero at every temperature to twelve figures, because there is nothing outside the assembly for them to balance against. Remove one bar and the forces vanish.

Tested in The load nobody applied · the Free-body reading path

“A prestressed beam is at its most stressed when it is fully loaded.”

The top fibre is in tension at 1.7 MPa with the prestress applied and nothing whatever opposing it, because a force below the centroid bends the beam upwards. The beam's own weight is what rescues it, bringing the top back to 0.3 MPa of compression the moment it is lifted. The condition that decides how much eccentricity is permitted is the empty one, not the loaded one, which is the upper bound in the tendon-zone figure.

Tested in A state no load could reach · the Free-body reading path

“A structure fails when its most heavily loaded member reaches its yield stress.”

It fails when enough members have yielded to leave it a mechanism, which for the propped cantilever computed here is twelve and a half per cent later, and for the four-legged table is anywhere from a third later to four times later depending on a manufacturing error. Between first yield and collapse the structure is redistributing: the yielded member holds a constant force and the others take the increment.

Tested in The one number the tolerances cannot touch · the Free-body reading path

“In a graded fibre the rays near the axis arrive first, because they have the shortest path.”

They have the shortest path and the lowest speed, because the index is highest on the axis, and in a parabolic profile the two cancel. Four rays launched between three and a hundred and twenty-five milliradians are traced here and cross the axis within 0.36 per cent of a pitch of each other; the pitch itself, 1.11 millimetres, contains no launch angle at all.

Tested in The same cone, and a different arrival · the etendue reading path

“A nonlinear coupling between two modes and a third lets energy pass between them.”

It lets energy pass and come back. Two oscillators coupled with strength V and mismatched in frequency by Δ exchange a fraction 1/(1 + (Δ/2V)²) of their energy and no more, periodically — 5.9 per cent at a mismatch of eight couplings, integrated here and checked against the closed form. Accumulation needs the mismatch to be zero, and on a mass chain it never is.

Tested in The condition three modes never meet · the equipartition reading path

“A larger resistor delivers more noise power.”

It delivers a larger voltage into a proportionally larger impedance and the same watts. The available power is kTΔf — 4.14 × 10⁻²¹ W per hertz at room temperature, −174 dBm in a hertz — with no resistance in it at all, identical at every point on every curve of the third figure. A 50 Ω input and a 10 GΩ electrometer differ by four decades in noise voltage and not at all in noise power.

Tested in Half a kT in a piece of wire · the equipartition reading path

“A mass estimated from velocities is a model-dependent number and could be wrong by a large factor.”

The mass goes as the square of the velocity dispersion, so a dispersion wrong by thirty per cent moves the mass by less than a factor of two. Coma's virial mass is 10¹⁵ solar masses against 2 × 10¹³ in stars, and no plausible error in the dispersion, the radius or the geometric factor closes a ratio of fifty. What the estimate does depend on is whether the cluster has settled, which is the third figure and is checkable separately.

Tested in Weighing what cannot be put on a scale · the equipartition reading path

“The estimate is too large because the cutoff was put absurdly high, and a reasonable cutoff fixes it.”

The density goes as the fourth power of the cutoff, so thirty decades of cutoff buys a hundred and twenty of density and a hundred and twenty is what has to be found. The cutoff that would give the measured answer is 8 millielectronvolts — a wavelength of a tenth of a millimetre — which corresponds to no known physics, and which is a strange enough number to be a clue rather than a resolution.

Tested in The estimate that misses by a hundred and twenty · the planck scale reading path

“Tabletop gravity experiments are gradually closing in on ever shorter distances.”

The departure from an inverse square is an exponential of the separation divided by a range, so a measurement at a given distance excludes ranges above it almost completely and ranges an order of magnitude below it not at all — 4.0 and 1.3 × 10⁻⁴ respectively, computed here at the fifty-two micrometres reached. The programme is a series of definite exclusions rather than a tightening limit, and pushing to ten micrometres would test ten micrometres and little shorter.

Tested in The scale that may not be where it looks · the planck scale reading path

“Entanglement swapping sends something from the middle to the ends.”

The four outcomes of the middle measurement leave the outer pair in four different Bell states, all maximally entangled and all equally likely, and their correlations average to exactly zero at every analyser angle — checked here at seven hundred and twenty angles to twelve figures. Until two classical bits arrive saying which occurred, the outer parties' records are indistinguishable from noise. The entanglement appears at once; nothing usable does.

Tested in A link between two that never met · the entanglement reading path

“Ground states are typical states of their Hilbert space.”

They are extraordinarily atypical. A randomly chosen state's block entanglement is the block's length times ln 2 — a volume law — while a gapped ground state's saturates, varying by less than a twentieth of a per cent from a block of eight sites to one of forty, and a gapless one grows only as the logarithm with a coefficient measured here as 0.333 against the third conformal field theory gives. At forty sites the random state's entanglement is more than eight times either.

Tested in The corner of Hilbert space that is ever visited · the entanglement reading path

“A stronger laser will reach the nonlinear regime soon.”

The strongest focus achieved, 10²² W/cm², corresponds to 2.7 × 10¹⁴ V/m, which is 2 × 10⁻⁴ of the Schwinger field — so the correction is of order 4 × 10⁻⁸. The departure goes as the square of the field and the intensity as the square of the field too, so closing four decades of field means eight decades of intensity. No projected facility is within six.

Tested in The one medium that was supposed to add exactly · the superposition reading path

“The transfer is partial, so a long enough coupler moves all the power across.”

It is complete at the coupling length and complete back at twice it, and goes on alternating for as long as the guides run together. A coupler that divides power in a chosen ratio is therefore made by cutting the interaction region to a length — half the power at half the coupling length — and a coupler cut twice as long as intended transfers nothing at all rather than transferring more.

Tested in Two tails that swap everything · the guided waves reading path

“A neutrally buoyant body is always in unstable equilibrium, so it cannot be left at a depth.”

That holds for a compressible body in a uniform fluid, which is the case computed for a uniform fluid. A rigid body in a column whose density rises with depth is pushed back whichever way it is displaced — the slope of the net force is positive — so its neutral depth is stable. The two cases are drawn together here and their slopes have opposite signs.

Tested in The body that displaces two things · the buoyancy reading path

“A freely falling frame is inertial, so the equivalence principle holds in a falling laboratory.”

It holds at a point and to a stated precision over a stated size. Comparing the tidal acceleration across a region with the local gravity gives a patch of δr/2 — the tolerance times the distance to the centre — with Newton's constant, the mass and the speed of light all cancelling out. On the Earth's surface that is 3.2 millimetres at a part in a thousand million, and a hundredth of a millimetre at a neutron star.

Tested in How big now is · the simultaneity reading path

“An absorption edge is a step, and above it the coefficient falls smoothly.”

It falls smoothly only for a free atom. For copper metal the same curve carries a modulation reaching eleven per cent of the absorption, extending seven hundred electronvolts above the edge, and the modulation vanishes if the copper is vaporised. What survives the vaporisation is the atom; what is lost is the neighbours, and the ripple is the difference.

Tested in The ripple that counts the neighbours · the attenuation reading path

“A semiconductor does not absorb photons below its band gap.”

It absorbs them exponentially weakly rather than not at all. A hundred millielectronvolts below its gap, gallium arsenide still has an absorption coefficient of two thousandths of an inverse centimetre and amorphous silicon has five hundred — a ratio of a quarter of a million between two materials neither of which has a single state in its gap in the description that produces the number.

Tested in Below the gap, where there is nothing to absorb · the attenuation reading path

“A sound wave carries no net flow, because the fluid returns to where it started each cycle.”

It returns to where it started to first order in the amplitude and not to second. What survives the cycle average is the product of two oscillating quantities that are not in phase, and in a standing wave it organises into four closed cells per wavelength drifting at a millimetre a second under a very loud tone — small beside the acoustic velocity of a metre a second, and not zero, and unidirectional.

Tested in The drift a sound leaves behind · the viscosity reading path

“A lubricating film survives because the oil is dragged in fast enough.”

Speed is necessary and nowhere near sufficient. The Dowson–Higginson calculation with the pressure dependence removed and everything else unchanged gives 5.6 nanometres at two metres a second against 906 with it — a factor of 163, and the smaller number is far below the roughness of any real surface. What holds the surfaces apart is a liquid that becomes a solid on the way in.

Tested in The oil that is a glass for a quarter of a millisecond · the viscosity reading path

“A bound like that must come from string theory.”

A kinetic-theory estimate reaches the same order. For a dilute gas the ratio is proportional to the mean free path divided by the particles' de Broglie wavelength, so it falls under compression until that ratio reaches one — and a mean free path shorter than a wavelength is not a meaningful quantity. The calculation hits the floor at exactly the density at which it stops applying.

Tested in Whether a fluid can be made arbitrarily thin · the viscosity reading path

“How loudly each mode rings is predicted by the theory of the remnant.”

Only the frequencies and decay times are. The amplitudes depend entirely on how the remnant was made — on the mass ratio, the spins and the geometry of the collision — and are computed from numerical simulations of the merger rather than from the spectrum. That separation is what makes the test possible: the part that is predicted is the part that is tested.

Tested in A few cycles that are only mass and spin · the gravitational waves reading path

“A gravitational wave leaves the matter it passes through exactly as it found it.”

It leaves a permanent change in the separations. The strain does not return to zero after the wave has gone; it settles at an offset of order a fifth of the peak amplitude, and the masses are then at rest at new distances rather than oscillating about a new centre. Nothing in the linearised theory of a passing wave produces that, which is why the effect was not found until 1991.

Tested in The ring that does not come back · the gravitational waves reading path

“H is B divided by the permeability of the material.”

That holds only where the magnetisation is produced by the field, which a permanent magnet's is not. Inside a uniformly magnetised sphere B is two thirds of μ₀M along the magnetisation and H is minus a third of M — opposite in direction, so no positive permeability relates them. Dividing B by μ inside a permanent magnet returns a field pointing the wrong way.

Tested in The field that points against the magnet it is in · the ampere law reading path

“A magnetic field cannot be written as the gradient of a potential.”

It can, everywhere there is no current. The obstruction is not local — the circulation round every small loop off the wire is exactly zero, which is the condition for a gradient — and the potential that results reproduces the field of a circular loop on its axis to the last digit. What fails is single-valuedness, and only for paths that encircle a current.

Tested in A potential that does not come back to itself · the ampere law reading path

“The floor does work on a jumper, which is where the kinetic energy comes from.”

The floor's force acts on a patch of skin that is momentarily at rest, so it acts through zero displacement and does zero work. The kinetic energy comes from chemical energy inside the muscles. Integrating the floor's force over the centre of mass's rise does give the kinetic energy — 293 joules for the push drawn here — and that integral is not work and does not measure any transfer.

Tested in The floor that does no work · the energy reading path

“Random collisions with moving walls cannot heat anything, because the gains and losses cancel.”

The gains and losses cancel at first order in the wall's speed and not at second. The energy then grows in proportion to the number of collisions rather than as its square, which is the difference between the mechanism Fermi proposed for cosmic rays in 1949 and the one that works — a factor of sixteen hundred over two thousand collisions in the case drawn here.

Tested in The wall that moves while the ball is in flight · the energy reading path

“Diffraction rings are caused by imperfections at the edge of the optic.”

They are caused by the edge existing. A perfectly clean, perfectly circular aperture has them, and the calculation that produces them assumes exactly that. What removes them is not a better edge but a transmission that falls smoothly to zero — a Blackman taper puts the first sidelobe 58 decibels down, from the same aperture with the same edge quality.

Tested in The rings that belong to the edge · the diffraction reading path

“Aberration blurs the image.”

It does not widen the core appreciably; it empties it. The computed point-spread function keeps its width and loses its height, and every photon lost from the core reappears in a halo — at a fifth of a wave the peak falls by a factor of 6.8 and the light outside the core rises by 8. Blurring and scattering are different failures and they ruin different measurements.

Tested in How accurate a mirror has to be · the diffraction reading path

“The photon has spin one, so it has three spin states and one of them is hard to observe.”

It has two, and there is no third state of the free electromagnetic field at all. A longitudinal light wave — the field oscillating along the direction of travel — is not a weak solution of Maxwell's equations; it is not a solution, because in empty space the divergence of the electric field is zero and that forces the field perpendicular to the wavevector.

Tested in Two states where the counting says three · the spin reading path

“A Stern–Gerlach experiment on electrons would work with a strong enough magnet.”

The magnet cancels out of the comparison. The spin splitting is proportional to the field gradient and so is the Lorentz blurring produced by the same gradient across the beam's width, so their ratio contains neither the gradient nor the magnet's length — it is the electron's de Broglie wavelength divided by the beam's width, which at a hundred electronvolts and a micrometre of beam is two parts in a hundred thousand.

Tested in The experiment that defines spin and cannot be done on it · the spin reading path

“Moving heat from cold to hot requires work.”

It requires a source of low entropy, and heat at a high temperature is one. A machine with three reservoirs and no work crossing any boundary moves heat up a gradient, at a ceiling of the engine efficiency between the hot reservoir and ambient times the fridge coefficient between ambient and cold — 3.4 for a 450-kelvin flame lifting heat from freezing point into a 300-kelvin room.

Tested in A fridge with no work going into it · the heat engines reading path

“A better thermoelectric needs a material that conducts electricity better.”

Conductivity enters the group linearly and the Seebeck coefficient enters squared, and doping a semiconductor to raise the first lowers the second. The product has a maximum at about 10^20 carriers a cubic centimetre and falls on both sides, so raising the conductivity past that point makes the material worse. Every thermoelectric in use is a heavily doped semiconductor for exactly that reason.

Tested in An engine with one number in it · the heat engines reading path

“A moving blackbody is no longer a blackbody.”

It is one in every direction. The Planck form is exactly invariant under a Doppler shift, with the temperature multiplied by the shift — so a body at 100 kelvin moving at half light speed is a perfect 173-kelvin blackbody looking one way and a perfect 58-kelvin one looking the other. What is not a blackbody is the sum over directions, which is a sum of Planck spectra at different temperatures and is therefore not a Planck spectrum.

Tested in The body that has no temperature when it moves · the relativistic thermodynamics reading path

“If entropy is invariant and volume contracts, the entropy density is unchanged.”

The entropy density rises by exactly the Lorentz factor, because the same entropy occupies a smaller volume. The energy density rises by the factor squared, because the energy rises as well as the volume falling — and the ratio between those two powers is the reason the boosted radiation is not a blackbody at any temperature whatever.

Tested in The count that no observer can disagree about · the relativistic thermodynamics reading path

“A preferred frame would contradict relativity.”

The microwave background has one, it is measured to four significant figures, and nothing is contradicted. The laws are identical in every frame; the radiation's state is not, and a state may pick out a frame exactly as a body of water does. What relativity forbids is a frame picked out by the laws, which is a different claim.

Tested in The bath that pushes back · the relativistic thermodynamics reading path

“The level curves of the action for a swarm thrown from one point are centred on the point the swarm has fallen to.”

They are centred the same distance ABOVE the launch point. In a uniform field the arriving velocity of every particle is its displacement from that raised point divided by the elapsed time, so every trajectory crosses the curves at right angles about a centre nothing ever occupies.

Tested in The action that knows where every path ends · the least action reading path

“Where a target is does not decide how best to aim at it; only how far away it is does.”

A hoop and a vertical board centred on the same point give the same thrower best aims 11.5° apart, on opposite sides of the least-speed launch: 58.5° for the hoop, whose opening grows as the ball comes down more steeply, and 47.0° for the board, whose size does not depend on how the ball arrives and which a flatter, faster throw hits with less error of height.

Tested in The throw most likely to go in · the projectile reading path

“An eccentric orbit that dives closer to the Earth runs its clock slower overall than a circular one of the same period.”

Every orbit of the same period loses exactly the same time, 7.47 µs here for eccentricities of 0, 0.2 and 0.4, checked by quadrature along each against the closed form. The time-average of 1/r over any Kepler orbit is 1/a, so the depth term and the speed term each come out the same whatever the shape.

Tested in The orbit that ages less than a throw · the time dilation reading path

“Two parts of a system in thermal equilibrium are always at the same temperature.”

In a static gravitational field the local temperature times the local clock rate is what equalises. Two cavities of radiation sharing a conserved energy, with the lower clock running at 0.8 of the upper one's rate, reach their entropy maximum — found by search — when the upper cavity is at 0.800 of the lower one's temperature. Where the local temperatures are equal the entropy is 1.82 per cent below its peak.

Tested in The column that is hotter at the bottom · the relativistic thermodynamics reading path

“A reaction with a positive standard Gibbs energy does not proceed.”

It proceeds until the mixture's Gibbs energy is least, and that minimum is always strictly inside. For A ⇌ B at 298 K with ΔG° = +10 kJ/mol the Gibbs energy of the mixture first falls, to −43 J/mol at 1.74 per cent B, found by search and matching exp(−ΔG°/RT). The mixing term's slope is infinite at the pure end, so no finite ΔG° can hold a reaction at nothing converted.

Tested in The reaction that cannot go all the way · the chemical potential reading path

“No light source can put out more light power than the electrical power it consumes.”

Each photon leaves with about the gap energy plus a little, 0.619 eV for a 0.58 eV gap at 408 K, while each electron is given only qV. An ideal diode at 0.15 V puts out 4.1 times its electrical input, the difference drawn from the crystal's heat. A diode at 408 K was reported in 2012 to emit 2.3 times its electrical input at picowatt power.

Tested in The glow that carries a voltage · the chemical potential reading path

“Photons have to be measured as distinguishable for their interference to disappear.”

It is enough that they could be told apart. Photons whose polarisations differ by 60° part 37.5 per cent of the time at zero delay rather than never, although no polariser is anywhere in the apparatus; the two ways of parting end in different final states and stop adding as amplitudes.

Tested in The outcomes identical photons refuse · the photon reading path

“More squeezing always gives a quieter measurement.”

Loss replaces the lost light's noise with vacuum noise, and the measured noise is the surviving fraction of the squeezed noise plus the lost fraction of ordinary noise. With 80 per cent of the light surviving, no amount of generated squeezing can show more than 7.0 dB; and a reference angle that wanders by 3° turns 12 dB of squeezing into 9.7.

Tested in The noise pushed below the floor · the photon reading path

“Only gravity changes the phase when a neutron interferometer is turned.”

The Earth's rotation turns the laboratory under the neutrons while they are in flight, and the Sagnac phase for a 10.1 cm² interferometer at latitude 42.3° is 1.73 rad, 3 per cent of the 57.4 rad gravitational phase. Unlike the gravitational phase it does not depend on the neutrons' wavelength, which is how the two are separated.

Tested in The fall that leaves the mass in the phase · the matter waves reading path

“Measuring gravity precisely enough outside a planet reveals how its mass is arranged inside.”

A uniform ball, a body with a dense core under a light mantle and a hollow shell of the same mass and radius have the same field everywhere outside — checked at 1.7 radii by adding up the pull of every mass element — while their moment-of-inertia factors are 0.400, 0.330 and 0.551. No precision of exterior measurement separates them.

Tested in The field outside that cannot find the core · the Gauss's law reading path

“Any large enough disturbance of either sign breaks into solitons.”

A trough of depth 4 holds no levels when turned upside down, because it is a barrier, and releases no solitons: its deepest point shrinks from −4 to −1.46 by t = 0.6 as it spreads into a train of ripples. A positive hump of any height, however small, holds at least one level and releases at least one soliton.

Tested in The solitons a hump already contains · the wave packets reading path

“A coupler cut to exactly the right length for 1550 nm works across a telecommunications band.”

The evanescent tails that set the coupling reach further at longer wavelengths, so the beat length changes with colour. The coupler cut to 37.6 µm transfers above 95 per cent over only 94 nm and falls to 3 per cent at 1800 nm; tapered couplers of 300 and 800 µm stay above 95 per cent over 313 and 438 nm.

Tested in The coupler that does not care about the colour · the guided waves reading path

“An optical time crystal is only an engineering matter of modulating strongly enough.”

The gap for light of wavelength 1550 nm sits at a modulation frequency of 3.9 × 10¹⁴ hertz, twice the light's own, and the index has to swing by a sizeable fraction of itself at that rate. Microwave versions at 2 × 10¹⁰ hertz have been built with switched circuits; no optical material is known to respond fast and strongly enough.

Tested in The crystal made of moments · the refraction reading path

“Light in a disordered medium is always concentrated near the surface it enters.”

At a resonance of one random stack of 120 layers, where it transmits 78 per cent, the energy density inside rises to over a thousand times that of the transmitted light at 62 per cent of the way through and falls off exponentially on both sides — a mode trapped by the disorder, far from either surface.

Tested in The walk that interference can stop · the scattering reading path

“A ship that is inexplicably slowed in calm water must be fouled, overloaded or caught in a current.”

Over a 1.5 m layer of water 2 per cent lighter than the water beneath, the fastest wave on the interface travels at 0.52 m/s, 1.02 knots. A hull moving just below that speed drags a train of interfacial waves 53 m long behind it, and the drag they carry away peaks at 0.96 knots — invisible at the surface, which moves by a fiftieth of the interface's displacement.

Tested in The wave that holds a ship back · the stratification reading path

“Turbulence scrambles a plasma's magnetic field, so the state it settles into depends on the details of how it was driven.”

The settled state depends on two numbers, the flux and the helicity. A field of 1,000 twisted modes decaying under resistivity has lost half its energy by the time it has lost 3.3 per cent of its helicity; every spectrum drawn here then falls onto the same floor, energy equal to the smallest wavenumber times the helicity, which is a single field shape. Pinches in different machines, started differently, land on one curve of reversal parameter against pinch parameter.

Tested in The twist that outlives the turbulence · the flux freezing reading path

“Clocks on the Earth run at a steady rate through the year, since nothing about their surroundings changes.”

Against a clock far from the Sun they do not. The orbit's eccentricity of 0.0167 takes the Earth deeper into the solar potential at perihelion, and every terrestrial clock runs slower in early January than in early July by 3.3 × 10⁻¹⁰ — about half as much as the Earth's own gravity slows a clock on its surface. It is invisible to any comparison made on the Earth, which is what the equivalence principle requires.

Tested in The clocks that must all slow together · the gravitational redshift reading path

“Two surfaces with the same charge always repel in an electrolyte, since every contribution to the force is a repulsion or a screened one.”

Mean-field theory says so and proves it: the Poisson–Boltzmann pressure between two like-charged planes with their counterions equals the counterion density at the midplane, 1.71 in natural units at one Gouy–Chapman length apart and 0.29 at four, never negative. In the strong-coupling limit the same system has a pressure of 2/d − 1, which is negative beyond two Gouy–Chapman lengths, and calcium silicate hydrate sheets, clay platelets and DNA in spermine are held together by exactly that attraction.

Tested in The like charges that pull together · the osmosis reading path

“A current in a plasma is the charge of the particles times the velocity of their guiding centres.”

Not across a pressure gradient. In a uniform field every guiding centre of a Maxwellian plasma is at rest, yet the ions at any point move on average at vₜ²/ΩL along the gradient's perpendicular — 0.33 thermal speeds for a density that falls by e every three gyroradii — because more circles are centred on the dense side. Counted particle by particle across a plasma edge, that flow reproduces the fluid current p′/B to within six per cent of its peak.

Tested in The current no particle carries · the magnetism reading path

“An electrical circuit made of metal wires is a classical object whose energy can take any value.”

A superconducting island joined through a junction has discrete levels. At a Josephson energy fifty times the charging energy its lowest transitions are 18.94, 17.79 and 16.50 charging energies, computed by diagonalising the circuit's Hamiltonian, and each can be driven separately with microwaves because no two are equal.

Tested in The circuit that forgets its charge · the superconductivity reading path