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