Simply false — part 3 of 5

“A wave reflecting from a boundary leaves at the angle it arrived at, measured from the boundary.”

That is true of every wave whose speed is a scalar, and false of an internal wave. Its frequency fixes the angle its energy makes with the horizontal, because gravity supplies the restoring force and gravity is vertical, so a reflection must preserve that angle and not the angle to the wall. The figures draw incident and reflected beams at the same angle to the horizontal off a slope, which for a mirror would be impossible, and the wavelength changes by sin(θ+α)/sin(θ−α) rather than staying fixed.

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

“The inertial period is twelve hours.”

It is π over Ω sin φ, so it is twelve hours at the pole and longer everywhere else: 13.8 hours at 60°, 16.9 at 45°, 23.9 at 30°, and 69 at ten degrees of latitude. It is exactly half a pendulum day, checked at five latitudes against an independently computed Foucault period, and it diverges at the equator where f vanishes.

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

“Water drains out of a bath in opposite directions in the two hemispheres.”

The Rossby number of a draining bath is about four thousand, computed from its own speed, size and latitude. The Coriolis acceleration is four thousand times smaller than the accelerations the flow already has, so it is beaten by the shape of the basin, the residual swirl from filling, and the asymmetry of the plug. The experiment can be made to work — it has been, in a symmetric tank left still for a day — and the conditions required are exactly the conditions a bath does not meet.

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

“Snell's law with a complex index gives the refraction angle into an absorbing medium.”

It gives a complex number, and a complex angle is not the direction of anything. What the boundary condition actually fixes is the tangential wavenumber, which stays real; the normal component is complex, and its real and imaginary parts point along different directions. Surfaces of constant amplitude lie parallel to the interface and surfaces of constant phase are tilted, so there are two angles rather than one — and the construction reduces to Snell's law exactly, to twelve decimal places, when the absorption is set to zero.

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

“A flat slab cannot focus, because it has no curvature.”

A flat slab of index −1 focuses twice: once inside at the source's own distance from the front face, and once beyond it at the remaining thickness. Both foci are read off the traced rays here and are identical for every ray angle drawn, so there is no aberration of any kind — because the refraction is exact rather than paraxial. What the slab has instead of a focal length is a working distance that moves with the source.

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

“The continuous beta spectrum shows that energy is not exactly conserved in nuclear decay.”

That was the live hypothesis for four years and it was wrong. A continuum is what three bodies in the final state produce, because the energy can be divided in a continuum of ways; a line is what two produce. The shape is a phase-space count with no free parameters beyond a normalisation, and it fits — so the observation is evidence for an unseen third body rather than against a conservation law.

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

“In secular equilibrium the members of a decay chain are present in equal amounts.”

Their activities are equal and their amounts are not. Equal activity means λN is the same for every member, so N is inversely proportional to λ — the abundance ratio is the half-life ratio. In the uranium series that spans twelve orders of magnitude, from a whole gram of ²³⁸U to 2.3 × 10⁻¹² grams of ²²²Rn, and each ratio here is computed from half-lives alone and agrees with tabulated abundances to two per cent.

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

“A half-life is a fixed property of a nuclide and nothing outside the nucleus can change it.”

Electron capture is a nuclear process whose rate is proportional to the electron density at the nucleus, and that density is a chemical quantity. Beryllium-7 decays 0.83 per cent faster as the metal than as the fluoride, and its rate changes with pressure and with what cage it is put inside. Those are small numbers and they are not zero, and the mechanism is not in dispute.

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

“Conservation of charge is a separate experimental law, additional to the field equations.”

Taking the divergence of the Ampère–Maxwell equation gives ∂ρ/∂t + ∇·J = 0 with no assumption at all, so the field equations already contain it. Written covariantly it is the vanishing four-divergence of the four-current — a single scalar equation, the same in every frame — and if it failed the four-current would not be a four-vector, so the transformation the previous rung established would have nothing to act on.

Tested in Charge and current are one thing · the field transformation reading path

“Writing Maxwell's equations covariantly is a notational convenience.”

It changes the count. Four vector equations become two tensor ones: the inhomogeneous pair collapses into a single equation with a free index and the homogeneous pair into a cyclic identity on F, which is the statement that F is a curl of the four-potential. The second pair stops being a law at all and becomes an identity, which is a change in what kind of statement they are rather than in how they are written.

Tested in Six numbers, one object · the field transformation reading path

“The Kauzmann paradox is that a supercooled liquid's entropy falls below its crystal's.”

No measured entropy does that. The paradox is that the *extrapolation* of the equilibrium liquid's entropy crosses zero excess at a finite temperature — 152 K for the substance drawn — and the liquid always falls out of equilibrium first. What is paradoxical is that the intervention appears to be needed, not that the crossing is observed.

Tested in The entropy that depends on how fast it was cooled · the third law reading path

“A sample at a microkelvin has essentially zero entropy.”

Its lattice does. Its nuclear spins carry R ln(2I+1) — 11.5 joules per kelvin per mole for copper's spin-3/2 nuclei — down to the temperature at which their own dipolar interactions order them, which for copper is about 58 nanokelvin. A sample at a microkelvin is three orders of magnitude above that, and its nuclear spin system is as disordered as it was at room temperature.

Tested in A law about spectra, not about heat · the third law reading path

“Acoustic streaming is a nonlinear effect that only matters at high amplitude.”

It is nonlinear in the sense that it comes from a second-order term, and its speed is linear in the intensity while the oscillation itself goes as the square root. So streaming grows faster than the motion that drives it: doubling the drive doubles the flow and increases the oscillation by forty per cent, and at ordinary diagnostic intensities the flow in water is millimetres per second.

Tested in Where the loudness goes · the attenuation reading path

“Absorption falls smoothly with photon energy, because a harder photon interacts less.”

It falls steeply and it is not smooth. At the binding energy of an inner shell the coefficient jumps upward — by a factor of 4.4 for iodine at 33.17 keV and 4.0 for lead at 88, computed here from tabulated values on either side. A material becoming more opaque as the photon gets harder is a discontinuity in a material property as a function of frequency, and nothing in the classical account of absorption produces one.

Tested in The steps in an absorption curve · the attenuation reading path

“Light absorbed by a gas can only heat it.”

Each absorption from the beam an atom is moving towards takes a photon's momentum off its motion, and the light it scatters leaves bluer than it arrived by the Doppler shift that motion gave it — so the kinetic energy goes out with the light. An ensemble followed photon by photon here falls from 1.10 millikelvin to 240 microkelvin in a few hundred microseconds, after about 1,200 scatterings per atom.

Tested in The friction made of light · the radiation pressure reading path

“Nothing that scatters light can cool an atom below the Doppler limit ħΓ/2kB.”

That limit is exact for an atom with one ground state and says nothing about one with several. In crossed-polarised beams a spin-½ atom is cooled by the light shift rather than by the Doppler shift, and the stochastic model here settles caesium at 6.96 µK in wells a hundred recoil energies deep, against a Doppler limit of 125 µK. The first measurement of a sodium molasses found 43 µK against a limit of 240.

Tested in The limit that belonged to a simpler atom · the radiation pressure reading path

“Throwing uphill and downhill are the same problem with the range reduced or increased a little.”

The reach along a line at elevation α is (v²/g)/(1 + sin α), which is not symmetric about level ground and diverges as the line turns straight down. A 45° uphill line gets 0.586 of the level range and a 45° downhill line 3.41 times it — nearly six times as far — from the same arm at the same speed.

Tested in One curve answers every slope · the projectile reading path

“Thrown from a moving platform, the best angle is still 45° — the platform just adds distance.”

The platform shifts the whole circle of available launches, and the tangency moves. At a platform speed of half the throwing speed the best aim is 53.6° relative to the platform, the ball actually leaves at 36.4° to the ground, and it goes 1.76 times as far as a standing throw. Searching every angle and solving 2cos²θ + u cos θ − 1 = 0 give the same answer.

Tested in The best throw is a tangency · the projectile reading path

“Water vapour only condenses into liquid when the air is saturated.”

That is true over a flat surface. A concave meniscus lowers the vapour pressure it is in equilibrium with by exp(−2γVₘ/rRT), so a pore narrow enough is full of liquid in unsaturated air: at 50 per cent humidity every pore below 1.51 nm radius holds condensed water, and at 90 per cent every pore below 9.96 nm.

Tested in The pore that fills from dry air · the capillarity reading path

“The contact angle a liquid makes on a solid is fixed once the solid, the liquid and the surrounding medium are chosen.”

A voltage across a thin insulator under the drop lowers the effective solid–liquid energy by ε₀εᵣV²/2d without changing any of the three. A 2 µL water drop on a 1 µm fluoropolymer coating goes from 115° to 50° between 0 and 95 V, and its footprint from 1.53 mm to 2.72 mm across, at constant volume — which the figure checks.

Tested in The angle a voltage can set · the capillarity reading path

“A light-emitting diode that turns every electron into a photon emits all of that light.”

Only the light inside the escape cone crosses a flat face. Generated in every direction inside gallium arsenide, 1.97 per cent of it lies within that cone and 1.31 per cent survives the reflection at the face; for gallium nitride the figures are 4.35 and 3.41 per cent. The rest is reflected back inside, and every gain in extraction has come from changing where that light goes next.

Tested in The cone light has to find to get out · the etendue reading path

“A fibre has a polarisation delay that can be measured once and compensated with a fixed device.”

The delay is a random variable of the fibre's condition, not a constant of its construction. One simulated 100 km fibre built from a thousand randomly oriented sections has a delay that wanders between 0.6 and 9.8 ps across a 4.4 THz band, and the same fibre watched over days samples a Maxwell distribution as temperature and handling reshuffle its sections.

Tested in The delay that is a random variable · the dispersion reading path

“Gravity gets weaker as one goes down into the Earth.”

It gets stronger through the whole mantle. Gauss's law applied to the reference model's density gives 9.82 m/s² at the surface and 10.69 m/s² at the core–mantle boundary, 2,891 km down — 8.8 per cent more — and gravity only starts falling once the core itself is being entered.

Tested in The pull that grows on the way down · the Gauss's law reading path

“A steady voltage across a junction between two superconductors drives a steady current through it.”

It drives an alternating supercurrent, because a voltage makes the phase difference advance at 2eV/ħ and the supercurrent is the sine of that phase. In a shunted junction biased at 1.3 times its critical current the voltage is a train of pulses between 0.3 and 2.3 in units of IcR, each one phase slip, with a mean of 0.831 — √(i² − 1) exactly, as the integration checks.

Tested in The voltage that is a frequency · the superconductivity reading path

“A wave packet that has spread out has lost its initial shape for good.”

In an oscillator whose levels carry a small quadratic term, a coherent state of nine quanta collapses in about nine periods, reassembles whole on the opposite side at 120 periods and exactly as it began at 240, and at an eighth and a quarter of that time it is four and two whole copies of itself. A classical ensemble started from the same distribution collapses the same way and never regroups.

Tested in The return a classical cloud never makes · the correspondence reading path

“Any quantum state of a particle can be pictured as a cloud of classical possibilities in position and momentum, blurred by the uncertainty principle.”

The phase-space function whose shadows are the true position and momentum distributions is the Wigner function, and for a superposition of two packets it runs from 0.289 down to −0.289, with its interference stripes taller than the packets themselves at 0.159. No cloud has a negative density, so no cloud reproduces the state; the momentum density's zeros are exactly where the negative stripes cancel the positive ones.

Tested in The probability that goes below zero · the correspondence reading path

“At full power a car's speed rises steadily, like a body under a constant force.”

Constant power means the kinetic energy rises steadily, so the speed rises as the square root of the time: without resistance the second 50 km/h takes three times as long as the first would at the same power. The integrated run is a straight line for 0.86 s while the grip limits it and a square root afterwards, matching the closed form to better than a millimetre per second.

Tested in The speed at which grip hands over to power · the energy reading path

“The causal structure of spacetime sets the length of a second.”

A uniform stretch of spacetime about any event keeps every cone and every causal relation, and multiplies every squared interval by the square of the stretch: by 4 for a stretch of 2, checked on 500 pairs. Nothing about what can influence what distinguishes the two, so the unit of time has to come from a clock.

Tested in What the light cones alone can decide · the spacetime diagram reading path

“A drawing of the whole of an infinite spacetime has to distort its light cones.”

Applying arctan separately to u = t − x and v = t + x maps the infinite plane into a diamond, keeps every light line at 45° to a part in a billion, and changes the causal relation of none of 4,000 sampled pairs of events. What it gives up is distance: the same half-unit of proper time looks 0.426, 0.094 and 0.010 times its size at the origin when it starts at t = 1, 3 and 10.

Tested in The five places infinity turns out to be · the spacetime diagram reading path

“Burning the fuel hotter is what makes an ideal Otto engine more efficient.”

Its ideal efficiency depends only on the compression ratio. Cut into twelve thin Carnot strips, every strip of the cycle has the same efficiency, 58.48% at a ratio of 9, because the curves bounding it are one exponential scaled by the compression's temperature ratio; adding more heat stretches the loop along the entropy axis without changing any strip's efficiency.

Tested in The temperature an engine really takes its heat at · the heat engines reading path

“An engine running between a kilogram of water at 90 °C and a kilogram at 10 °C can convert 22% of the heat it takes.”

22% is the Carnot efficiency at the start, and it falls as the hot water cools and the cold water warms. The best possible engine draws 177.8 kJ from the hot water and delivers 20.8 kJ of work, 11.70% of the heat — exactly 1 − √(283.15/363.15) — before the two meet and it stops.

Tested in The work left in two buckets of water · the heat engines reading path

“A state inside a band gap needs a defect or an impurity to hold it.”

A perfectly regular chain with alternating couplings v and w holds two states at zero energy, in the middle of its gap, whenever v < w. For a chain of 20 cells at v/w = 0.5 both lie within a millionth of the coupling of zero; at v/w = 1.5 the same chain has none, and its nearest level is 0.527 from zero. Nothing was broken — only where the chain ends.

Tested in The end that knows how the middle was cut · the periodic media reading path

“A structure that repeats in only one direction cannot reflect light arriving from every direction; that needs a three-dimensional photonic crystal.”

A quarter-wave stack of indices 4.6 and 1.6 forbids both polarisations at every angle of incidence from air between 0.848 and 1.321 of its design frequency, a window 43.6% wide, and ten pairs reflect more than 99.99% at every angle from normal to 89° in the middle of it. What a three-dimensional crystal adds is a gap for light arriving from inside a dense medium.

Tested in The mirror that works from every direction · the periodic media reading path

“Gibbs's entropy, which never gives a negative temperature, shows that negative temperatures are an artefact of a bad definition.”

Two systems of two-level units, 1,000 and 200 of them, with 600 and 145 excited, have equal Gibbs temperatures. In contact the energy still moves: the pair has a few hundred times more arrangements with 21 quanta passed from the smaller system to the larger, so that is where the energy goes. It moves from the system with the lower Boltzmann inverse temperature, −0.964 against −0.405, and stops where the Boltzmann temperatures agree, to 0.006.

Tested in The count that decides which entropy is right · the third law reading path

“A charge accelerating steadily loses energy to radiation reaction at the rate it radiates.”

The reaction force is proportional to the rate of change of the acceleration and is exactly zero while the acceleration is steady, while the radiated power is the full Larmor rate. Through a push that rises, holds and stops, the reaction force takes nothing during the steady part and 0.775 units while the push stops, against 0.750 radiated over the whole push.

Tested in The bill that arrives when the pushing stops · the radiating charge reading path

“A difference in how the Earth and the Moon fall towards the Sun would be too small to see in the Moon's orbit.”

The Sun pulls at 5.93 mm/s², the two bodies' self-energies differ by 4.3 × 10⁻¹⁰, and the push that difference would produce turns once a synodic month, close enough to the orbital frequency that the orbit magnifies it about twenty-two times. The result is a displacement of 13.1 metres per unit η, and the Earth–Moon distance has been ranged to centimetres and then millimetres.

Tested in The binding energy that has to fall too · the equivalence principle reading path

“An electrode driven by a symmetric alternating voltage through a capacitor sits at zero average voltage.”

In a plasma it charges negative until one cycle's electron charge matches one cycle's ion charge. With an argon plasma at 3 eV, a 100 V drive leaves the electrode at a mean of −106 V, and integrating the charging cycle by cycle gives the same number as the charge balance, which involves a Bessel function and nothing else about the circuit.

Tested in The bias no battery supplies · the plasma oscillation reading path

“Charge spreads evenly over a conducting surface, because like charges repel equally in every direction.”

It collects where a walker arriving from far away is most likely to touch first. On a thin charged strip the density follows 1/π√(1 − x²), and walkers from outside land in the outermost bins 6.4 times as often as in the same width at the centre; near a square's corner the density rises as distance to the −⅓ and near a knife edge as −½.

Tested in The potential is where the wanderers stop · the potential reading path

“A polymer chain stretches in any fast enough flow.”

In simple shear at λγ̇ = 1 a dumbbell's mean square stretch settles at 1.67 times its resting value, because the flow's rotation turns each chain away from the stretching direction as fast as it is stretched. In uniaxial extension at the same rate nothing rotates it, and by eight relaxation times the stretch is nearly two thousand times its resting value and still growing.

Tested in The stretch a chain cannot outrun · the rheology reading path

“Diluting a charged polymer frees its counterions, since each is then surrounded by more space to explore.”

Only while the charges along the chain are farther apart than the Bjerrum length. For DNA, a charge every 0.17 nm against a Bjerrum length of 0.71, the Poisson–Boltzmann solution keeps 76.2 per cent of the counterions inside the inflection of the charge profile in cells of 10², 10⁴ and 10⁶ rod radii alike. Diluting a millionfold does not release them.

Tested in The counterions that never leave the chain · the osmosis reading path

“A critical latitude is a sharp line past which nothing happens.”

The instability is a tongue of finite width, so a subharmonic detuned slightly by the inertial frequency still grows. The fastest available growth stops at 30.0° with a modulation strength of 0.1 and at 31.5° with 0.3, and a wave integrated at 29.5° grows at 0.145 e-folds a period, nearly as fast as a resonant one.

Tested in The latitude past which a tide cannot split · the stratification reading path

“A chaotic system is irregular all the time.”

Two hundred-thousandths below the setting where its three-cycle is born, the logistic map repeats itself to within 0.004 every third step for calms averaging 173 iterates, between bursts that cross the whole interval. Nothing random is added: the same start gives the same sequence of calms and bursts every time.

Tested in The calm that is the ghost of a cycle · the chaos reading path

“Light passing through a scattering medium falls off exponentially with thickness.”

Only the light that crosses without being scattered does. Thirty mean free paths of cloud droplets keep 9.4 × 10⁻¹⁴ of a beam unscattered, and let through 29 per cent of the light by diffusion. Beyond a few transport lengths the diffuse transmission falls as one over the thickness — 0.180, 0.060 through 8 and 30 mean free paths of an evenly scattering medium — which is Ohm's law, not an exponential.

Tested in The cloud light has to walk through · the scattering reading path

“The frequency of light cannot change in a linear medium unless something moves.”

A change of the medium everywhere at once keeps the wavenumber and therefore changes the frequency. Switched from index 1 to 1.5, the grid's period goes from 30.04 to 45.07 — a ratio of 1.500 — in a linear medium in which nothing moves and nothing is emitted.

Tested in The reflection that needs no surface · the refraction reading path

“Measuring a particle's polarisation from its decay requires knowing the direction the particle was travelling.”

Along the line of flight it needs only the parent's energy. For taus from Z decays, the fraction of the tau's energy carried by the pion is spread as 1 + P(2x − 1), and 100,000 decays drawn with P = −0.14 read back −0.150 ± 0.0055. The curve moves by 0.085 per cent at ten thousand times the energy, so the neutrino hiding the tau's direction does not matter.

Tested in The slope a spin leaves in a spectrum · the relativistic dynamics reading path

“Determining a particle's spin requires measuring the spins of the particles it decays into.”

For a resonance decaying to two spinless particles, the spin is written along its band: at fixed pair mass the other mass squared is a straight line in the decay angle, and 30,000 decays each through spins 0, 1 and 2 go dark at zero, one and two places, following the squared Legendre polynomials at χ² of 18, 17 and 13 over 20 bins.

Tested in The plane in which three bodies are flat · the relativistic dynamics reading path

“A sail that reflects nearly all the light stays cool.”

In a 100 GW beam, reflecting all but one part in 100,000 still leaves a megawatt absorbed over a few square metres. Radiating it from both faces, a 4 m sail settles at 915 K with an emissivity of 1 and 1936 K with an emissivity of 0.05; the temperature falls only as the fourth root of the absorption, so a thousand times better mirror is only 5.6 times cooler.

Tested in The rocket that leaves its fuel at home · the Mass-energy reading path

“Energy levels shift in straight lines as a magnetic field is increased.”

Only levels that nothing else shares a quantum number with. Of sodium's six 3p states, the two with mⱼ = ±3/2 are straight lines; the other four bend, and each level's moment changes from its Landé value, such as 2/3, to its strong-field value, such as 1, over about two decades of field. States with equal mⱼ never cross — the mⱼ = −½ pair comes no closer than √2ζ, 486.0 GHz, at 12.3 T.

Tested in The field an atom calls strong · the atomic spectra reading path

“Faster computation needs more energy per operation.”

It needs more energy per operation per second. Margolus and Levitin's theorem caps the rate of orthogonal steps at 2⟨E − E₀⟩/πħ, 6.04 × 10³³ per second for each joule above the ground state — 5.43 × 10⁵⁰ a second for a kilogram of rest energy — and says nothing about what each step must dissipate, which can be made as small as desired by going slowly.

Tested in The fastest a state can stop being itself · the uncertainty reading path

“A diffusion flux stops when the concentration gradient of that species is zero, and not before.”

Nitrogen's flux in the two-bulb experiment is 309 times what its own starting gradient of 0.00207 would drive, and 6.6 hours in it passes through zero with a gradient of 0.1468 still in place. A species' flux depends on every species' gradient, so it can flow with no gradient of its own and stop with a large one.

Tested in The gas that flows towards more of itself · the diffusion reading path

“The sharpest resonance of a structure is the one its transmission spectrum shows most clearly.”

At a bound state in the continuum it is the one the spectrum does not show at all. Far from the bound state the slower mode's feature has a width of 0.029; nearer, 0.0099 and 0.0018; at the bound state it is gone, because a mode that cannot leak into the channel cannot be excited from it either, while its quality factor is infinite.

Tested in The resonance that refuses to leak · the resonance reading path

“With enough matching elements any load can be matched to any source over any band.”

Only a load that stores no energy. A resistance shunted by a capacitance with RCω = 2 is held across the band to 0.707, 0.392, 0.320, 0.289, 0.275 and 0.270 by optimised networks of none to five elements, and can never go below exp(−π/RCω) = 0.208 however many are added. A plain resistance, by contrast, is matched perfectly at every frequency by an ideal transformer alone.

Tested in The mismatch no network can remove · the impedance reading path

“The Fresnel approximation is accurate once the observer is in the near field of a wide aperture.”

Its accuracy depends on how much light goes out at wide angles, not on the regime's name. At a Fresnel number of one, a slit two wavelengths wide differs from the exact result by 17.5 per cent rms and one sixteen wide by 2.4 per cent, falling roughly as the inverse of the width — the rate at which a sharp edge's wide-angle share of the power falls.

Tested in The fan of plane waves inside every beam · the huygens reading path

“A ferromagnet is a set of little magnets that have lined themselves up with each other.”

The magnetic energy of two neighbouring moments in iron is 0.20 K in temperature units, computed here from the moment and the lattice spacing. Iron orders at 1,043 K, five thousand times higher, so an arrangement held together by that interaction would be destroyed by any temperature at which it could be handled. Four other ferromagnets are computed in the same figure and the smallest ratio is 112.

Tested in What holds a magnet together is not magnetism · the magnetisation reading path

“A permanent magnet keeps its magnetisation indefinitely if nothing is done to it.”

It keeps it for a time that is an exponential of a barrier, and every barrier is finite. The lifetime figure here reads the barrier needed for each of four lifetimes off its own curve: one second needs 20.7 thermal energies, ten years needs 40.3. A magnet is stable in the same sense a supercooled liquid is — long-lived, not permanent — and the creep figure shows a sample drifting steadily through eight decades of time with nothing applied to it.

Tested in Nothing keeps a magnetisation for ever · the magnetisation reading path

“A structure with no load on it has no forces in it.”

Only if it is determinate. Two steel bars and an aluminium bar between rigid crossheads, heated uniformly by a hundred kelvin with nothing applied, carry 77.7 kN of compression and 38.8 kN of tension each — computed here, and summing to zero at every temperature to twelve figures, because there is nothing outside the assembly for them to balance against. Remove one bar and the forces vanish.

Tested in The load nobody applied · the Free-body reading path

“A prestressed beam is at its most stressed when it is fully loaded.”

The top fibre is in tension at 1.7 MPa with the prestress applied and nothing whatever opposing it, because a force below the centroid bends the beam upwards. The beam's own weight is what rescues it, bringing the top back to 0.3 MPa of compression the moment it is lifted. The condition that decides how much eccentricity is permitted is the empty one, not the loaded one, which is the upper bound in the tendon-zone figure.

Tested in A state no load could reach · the Free-body reading path

“A structure fails when its most heavily loaded member reaches its yield stress.”

It fails when enough members have yielded to leave it a mechanism, which for the propped cantilever computed here is twelve and a half per cent later, and for the four-legged table is anywhere from a third later to four times later depending on a manufacturing error. Between first yield and collapse the structure is redistributing: the yielded member holds a constant force and the others take the increment.

Tested in The one number the tolerances cannot touch · the Free-body reading path

“In a graded fibre the rays near the axis arrive first, because they have the shortest path.”

They have the shortest path and the lowest speed, because the index is highest on the axis, and in a parabolic profile the two cancel. Four rays launched between three and a hundred and twenty-five milliradians are traced here and cross the axis within 0.36 per cent of a pitch of each other; the pitch itself, 1.11 millimetres, contains no launch angle at all.

Tested in The same cone, and a different arrival · the etendue reading path

“A nonlinear coupling between two modes and a third lets energy pass between them.”

It lets energy pass and come back. Two oscillators coupled with strength V and mismatched in frequency by Δ exchange a fraction 1/(1 + (Δ/2V)²) of their energy and no more, periodically — 5.9 per cent at a mismatch of eight couplings, integrated here and checked against the closed form. Accumulation needs the mismatch to be zero, and on a mass chain it never is.

Tested in The condition three modes never meet · the equipartition reading path

“A larger resistor delivers more noise power.”

It delivers a larger voltage into a proportionally larger impedance and the same watts. The available power is kTΔf — 4.14 × 10⁻²¹ W per hertz at room temperature, −174 dBm in a hertz — with no resistance in it at all, identical at every point on every curve of the third figure. A 50 Ω input and a 10 GΩ electrometer differ by four decades in noise voltage and not at all in noise power.

Tested in Half a kT in a piece of wire · the equipartition reading path

“A mass estimated from velocities is a model-dependent number and could be wrong by a large factor.”

The mass goes as the square of the velocity dispersion, so a dispersion wrong by thirty per cent moves the mass by less than a factor of two. Coma's virial mass is 10¹⁵ solar masses against 2 × 10¹³ in stars, and no plausible error in the dispersion, the radius or the geometric factor closes a ratio of fifty. What the estimate does depend on is whether the cluster has settled, which is the third figure and is checkable separately.

Tested in Weighing what cannot be put on a scale · the equipartition reading path

“The estimate is too large because the cutoff was put absurdly high, and a reasonable cutoff fixes it.”

The density goes as the fourth power of the cutoff, so thirty decades of cutoff buys a hundred and twenty of density and a hundred and twenty is what has to be found. The cutoff that would give the measured answer is 8 millielectronvolts — a wavelength of a tenth of a millimetre — which corresponds to no known physics, and which is a strange enough number to be a clue rather than a resolution.

Tested in The estimate that misses by a hundred and twenty · the planck scale reading path

“Tabletop gravity experiments are gradually closing in on ever shorter distances.”

The departure from an inverse square is an exponential of the separation divided by a range, so a measurement at a given distance excludes ranges above it almost completely and ranges an order of magnitude below it not at all — 4.0 and 1.3 × 10⁻⁴ respectively, computed here at the fifty-two micrometres reached. The programme is a series of definite exclusions rather than a tightening limit, and pushing to ten micrometres would test ten micrometres and little shorter.

Tested in The scale that may not be where it looks · the planck scale reading path

“Entanglement swapping sends something from the middle to the ends.”

The four outcomes of the middle measurement leave the outer pair in four different Bell states, all maximally entangled and all equally likely, and their correlations average to exactly zero at every analyser angle — checked here at seven hundred and twenty angles to twelve figures. Until two classical bits arrive saying which occurred, the outer parties' records are indistinguishable from noise. The entanglement appears at once; nothing usable does.

Tested in A link between two that never met · the entanglement reading path

“Ground states are typical states of their Hilbert space.”

They are extraordinarily atypical. A randomly chosen state's block entanglement is the block's length times ln 2 — a volume law — while a gapped ground state's saturates, varying by less than a twentieth of a per cent from a block of eight sites to one of forty, and a gapless one grows only as the logarithm with a coefficient measured here as 0.333 against the third conformal field theory gives. At forty sites the random state's entanglement is more than eight times either.

Tested in The corner of Hilbert space that is ever visited · the entanglement reading path

“A stronger laser will reach the nonlinear regime soon.”

The strongest focus achieved, 10²² W/cm², corresponds to 2.7 × 10¹⁴ V/m, which is 2 × 10⁻⁴ of the Schwinger field — so the correction is of order 4 × 10⁻⁸. The departure goes as the square of the field and the intensity as the square of the field too, so closing four decades of field means eight decades of intensity. No projected facility is within six.

Tested in The one medium that was supposed to add exactly · the superposition reading path

“The transfer is partial, so a long enough coupler moves all the power across.”

It is complete at the coupling length and complete back at twice it, and goes on alternating for as long as the guides run together. A coupler that divides power in a chosen ratio is therefore made by cutting the interaction region to a length — half the power at half the coupling length — and a coupler cut twice as long as intended transfers nothing at all rather than transferring more.

Tested in Two tails that swap everything · the guided waves reading path

“A neutrally buoyant body is always in unstable equilibrium, so it cannot be left at a depth.”

That holds for a compressible body in a uniform fluid, which is the case computed for a uniform fluid. A rigid body in a column whose density rises with depth is pushed back whichever way it is displaced — the slope of the net force is positive — so its neutral depth is stable. The two cases are drawn together here and their slopes have opposite signs.

Tested in The body that displaces two things · the buoyancy reading path

“A freely falling frame is inertial, so the equivalence principle holds in a falling laboratory.”

It holds at a point and to a stated precision over a stated size. Comparing the tidal acceleration across a region with the local gravity gives a patch of δr/2 — the tolerance times the distance to the centre — with Newton's constant, the mass and the speed of light all cancelling out. On the Earth's surface that is 3.2 millimetres at a part in a thousand million, and a hundredth of a millimetre at a neutron star.

Tested in How big now is · the simultaneity reading path

“An absorption edge is a step, and above it the coefficient falls smoothly.”

It falls smoothly only for a free atom. For copper metal the same curve carries a modulation reaching eleven per cent of the absorption, extending seven hundred electronvolts above the edge, and the modulation vanishes if the copper is vaporised. What survives the vaporisation is the atom; what is lost is the neighbours, and the ripple is the difference.

Tested in The ripple that counts the neighbours · the attenuation reading path

“A semiconductor does not absorb photons below its band gap.”

It absorbs them exponentially weakly rather than not at all. A hundred millielectronvolts below its gap, gallium arsenide still has an absorption coefficient of two thousandths of an inverse centimetre and amorphous silicon has five hundred — a ratio of a quarter of a million between two materials neither of which has a single state in its gap in the description that produces the number.

Tested in Below the gap, where there is nothing to absorb · the attenuation reading path

“A sound wave carries no net flow, because the fluid returns to where it started each cycle.”

It returns to where it started to first order in the amplitude and not to second. What survives the cycle average is the product of two oscillating quantities that are not in phase, and in a standing wave it organises into four closed cells per wavelength drifting at a millimetre a second under a very loud tone — small beside the acoustic velocity of a metre a second, and not zero, and unidirectional.

Tested in The drift a sound leaves behind · the viscosity reading path

“A lubricating film survives because the oil is dragged in fast enough.”

Speed is necessary and nowhere near sufficient. The Dowson–Higginson calculation with the pressure dependence removed and everything else unchanged gives 5.6 nanometres at two metres a second against 906 with it — a factor of 163, and the smaller number is far below the roughness of any real surface. What holds the surfaces apart is a liquid that becomes a solid on the way in.

Tested in The oil that is a glass for a quarter of a millisecond · the viscosity reading path

“A bound like that must come from string theory.”

A kinetic-theory estimate reaches the same order. For a dilute gas the ratio is proportional to the mean free path divided by the particles' de Broglie wavelength, so it falls under compression until that ratio reaches one — and a mean free path shorter than a wavelength is not a meaningful quantity. The calculation hits the floor at exactly the density at which it stops applying.

Tested in Whether a fluid can be made arbitrarily thin · the viscosity reading path

“How loudly each mode rings is predicted by the theory of the remnant.”

Only the frequencies and decay times are. The amplitudes depend entirely on how the remnant was made — on the mass ratio, the spins and the geometry of the collision — and are computed from numerical simulations of the merger rather than from the spectrum. That separation is what makes the test possible: the part that is predicted is the part that is tested.

Tested in A few cycles that are only mass and spin · the gravitational waves reading path

“A gravitational wave leaves the matter it passes through exactly as it found it.”

It leaves a permanent change in the separations. The strain does not return to zero after the wave has gone; it settles at an offset of order a fifth of the peak amplitude, and the masses are then at rest at new distances rather than oscillating about a new centre. Nothing in the linearised theory of a passing wave produces that, which is why the effect was not found until 1991.

Tested in The ring that does not come back · the gravitational waves reading path

“H is B divided by the permeability of the material.”

That holds only where the magnetisation is produced by the field, which a permanent magnet's is not. Inside a uniformly magnetised sphere B is two thirds of μ₀M along the magnetisation and H is minus a third of M — opposite in direction, so no positive permeability relates them. Dividing B by μ inside a permanent magnet returns a field pointing the wrong way.

Tested in The field that points against the magnet it is in · the ampere law reading path

“A magnetic field cannot be written as the gradient of a potential.”

It can, everywhere there is no current. The obstruction is not local — the circulation round every small loop off the wire is exactly zero, which is the condition for a gradient — and the potential that results reproduces the field of a circular loop on its axis to the last digit. What fails is single-valuedness, and only for paths that encircle a current.

Tested in A potential that does not come back to itself · the ampere law reading path

“The floor does work on a jumper, which is where the kinetic energy comes from.”

The floor's force acts on a patch of skin that is momentarily at rest, so it acts through zero displacement and does zero work. The kinetic energy comes from chemical energy inside the muscles. Integrating the floor's force over the centre of mass's rise does give the kinetic energy — 293 joules for the push drawn here — and that integral is not work and does not measure any transfer.

Tested in The floor that does no work · the energy reading path

“Random collisions with moving walls cannot heat anything, because the gains and losses cancel.”

The gains and losses cancel at first order in the wall's speed and not at second. The energy then grows in proportion to the number of collisions rather than as its square, which is the difference between the mechanism Fermi proposed for cosmic rays in 1949 and the one that works — a factor of sixteen hundred over two thousand collisions in the case drawn here.

Tested in The wall that moves while the ball is in flight · the energy reading path

“Diffraction rings are caused by imperfections at the edge of the optic.”

They are caused by the edge existing. A perfectly clean, perfectly circular aperture has them, and the calculation that produces them assumes exactly that. What removes them is not a better edge but a transmission that falls smoothly to zero — a Blackman taper puts the first sidelobe 58 decibels down, from the same aperture with the same edge quality.

Tested in The rings that belong to the edge · the diffraction reading path

“Aberration blurs the image.”

It does not widen the core appreciably; it empties it. The computed point-spread function keeps its width and loses its height, and every photon lost from the core reappears in a halo — at a fifth of a wave the peak falls by a factor of 6.8 and the light outside the core rises by 8. Blurring and scattering are different failures and they ruin different measurements.

Tested in How accurate a mirror has to be · the diffraction reading path

“The photon has spin one, so it has three spin states and one of them is hard to observe.”

It has two, and there is no third state of the free electromagnetic field at all. A longitudinal light wave — the field oscillating along the direction of travel — is not a weak solution of Maxwell's equations; it is not a solution, because in empty space the divergence of the electric field is zero and that forces the field perpendicular to the wavevector.

Tested in Two states where the counting says three · the spin reading path

“A Stern–Gerlach experiment on electrons would work with a strong enough magnet.”

The magnet cancels out of the comparison. The spin splitting is proportional to the field gradient and so is the Lorentz blurring produced by the same gradient across the beam's width, so their ratio contains neither the gradient nor the magnet's length — it is the electron's de Broglie wavelength divided by the beam's width, which at a hundred electronvolts and a micrometre of beam is two parts in a hundred thousand.

Tested in The experiment that defines spin and cannot be done on it · the spin reading path

“Moving heat from cold to hot requires work.”

It requires a source of low entropy, and heat at a high temperature is one. A machine with three reservoirs and no work crossing any boundary moves heat up a gradient, at a ceiling of the engine efficiency between the hot reservoir and ambient times the fridge coefficient between ambient and cold — 3.4 for a 450-kelvin flame lifting heat from freezing point into a 300-kelvin room.

Tested in A fridge with no work going into it · the heat engines reading path

“A better thermoelectric needs a material that conducts electricity better.”

Conductivity enters the group linearly and the Seebeck coefficient enters squared, and doping a semiconductor to raise the first lowers the second. The product has a maximum at about 10^20 carriers a cubic centimetre and falls on both sides, so raising the conductivity past that point makes the material worse. Every thermoelectric in use is a heavily doped semiconductor for exactly that reason.

Tested in An engine with one number in it · the heat engines reading path

“A moving blackbody is no longer a blackbody.”

It is one in every direction. The Planck form is exactly invariant under a Doppler shift, with the temperature multiplied by the shift — so a body at 100 kelvin moving at half light speed is a perfect 173-kelvin blackbody looking one way and a perfect 58-kelvin one looking the other. What is not a blackbody is the sum over directions, which is a sum of Planck spectra at different temperatures and is therefore not a Planck spectrum.

Tested in The body that has no temperature when it moves · the relativistic thermodynamics reading path

“If entropy is invariant and volume contracts, the entropy density is unchanged.”

The entropy density rises by exactly the Lorentz factor, because the same entropy occupies a smaller volume. The energy density rises by the factor squared, because the energy rises as well as the volume falling — and the ratio between those two powers is the reason the boosted radiation is not a blackbody at any temperature whatever.

Tested in The count that no observer can disagree about · the relativistic thermodynamics reading path

“A preferred frame would contradict relativity.”

The microwave background has one, it is measured to four significant figures, and nothing is contradicted. The laws are identical in every frame; the radiation's state is not, and a state may pick out a frame exactly as a body of water does. What relativity forbids is a frame picked out by the laws, which is a different claim.

Tested in The bath that pushes back · the relativistic thermodynamics reading path

“The level curves of the action for a swarm thrown from one point are centred on the point the swarm has fallen to.”

They are centred the same distance ABOVE the launch point. In a uniform field the arriving velocity of every particle is its displacement from that raised point divided by the elapsed time, so every trajectory crosses the curves at right angles about a centre nothing ever occupies.

Tested in The action that knows where every path ends · the least action reading path

“Where a target is does not decide how best to aim at it; only how far away it is does.”

A hoop and a vertical board centred on the same point give the same thrower best aims 11.5° apart, on opposite sides of the least-speed launch: 58.5° for the hoop, whose opening grows as the ball comes down more steeply, and 47.0° for the board, whose size does not depend on how the ball arrives and which a flatter, faster throw hits with less error of height.

Tested in The throw most likely to go in · the projectile reading path

“An eccentric orbit that dives closer to the Earth runs its clock slower overall than a circular one of the same period.”

Every orbit of the same period loses exactly the same time, 7.47 µs here for eccentricities of 0, 0.2 and 0.4, checked by quadrature along each against the closed form. The time-average of 1/r over any Kepler orbit is 1/a, so the depth term and the speed term each come out the same whatever the shape.

Tested in The orbit that ages less than a throw · the time dilation reading path

“Two parts of a system in thermal equilibrium are always at the same temperature.”

In a static gravitational field the local temperature times the local clock rate is what equalises. Two cavities of radiation sharing a conserved energy, with the lower clock running at 0.8 of the upper one's rate, reach their entropy maximum — found by search — when the upper cavity is at 0.800 of the lower one's temperature. Where the local temperatures are equal the entropy is 1.82 per cent below its peak.

Tested in The column that is hotter at the bottom · the relativistic thermodynamics reading path

“A reaction with a positive standard Gibbs energy does not proceed.”

It proceeds until the mixture's Gibbs energy is least, and that minimum is always strictly inside. For A ⇌ B at 298 K with ΔG° = +10 kJ/mol the Gibbs energy of the mixture first falls, to −43 J/mol at 1.74 per cent B, found by search and matching exp(−ΔG°/RT). The mixing term's slope is infinite at the pure end, so no finite ΔG° can hold a reaction at nothing converted.

Tested in The reaction that cannot go all the way · the chemical potential reading path

“No light source can put out more light power than the electrical power it consumes.”

Each photon leaves with about the gap energy plus a little, 0.619 eV for a 0.58 eV gap at 408 K, while each electron is given only qV. An ideal diode at 0.15 V puts out 4.1 times its electrical input, the difference drawn from the crystal's heat. A diode at 408 K was reported in 2012 to emit 2.3 times its electrical input at picowatt power.

Tested in The glow that carries a voltage · the chemical potential reading path

“Photons have to be measured as distinguishable for their interference to disappear.”

It is enough that they could be told apart. Photons whose polarisations differ by 60° part 37.5 per cent of the time at zero delay rather than never, although no polariser is anywhere in the apparatus; the two ways of parting end in different final states and stop adding as amplitudes.

Tested in The outcomes identical photons refuse · the photon reading path

“More squeezing always gives a quieter measurement.”

Loss replaces the lost light's noise with vacuum noise, and the measured noise is the surviving fraction of the squeezed noise plus the lost fraction of ordinary noise. With 80 per cent of the light surviving, no amount of generated squeezing can show more than 7.0 dB; and a reference angle that wanders by 3° turns 12 dB of squeezing into 9.7.

Tested in The noise pushed below the floor · the photon reading path

“Only gravity changes the phase when a neutron interferometer is turned.”

The Earth's rotation turns the laboratory under the neutrons while they are in flight, and the Sagnac phase for a 10.1 cm² interferometer at latitude 42.3° is 1.73 rad, 3 per cent of the 57.4 rad gravitational phase. Unlike the gravitational phase it does not depend on the neutrons' wavelength, which is how the two are separated.

Tested in The fall that leaves the mass in the phase · the matter waves reading path

“Measuring gravity precisely enough outside a planet reveals how its mass is arranged inside.”

A uniform ball, a body with a dense core under a light mantle and a hollow shell of the same mass and radius have the same field everywhere outside — checked at 1.7 radii by adding up the pull of every mass element — while their moment-of-inertia factors are 0.400, 0.330 and 0.551. No precision of exterior measurement separates them.

Tested in The field outside that cannot find the core · the Gauss's law reading path

“Any large enough disturbance of either sign breaks into solitons.”

A trough of depth 4 holds no levels when turned upside down, because it is a barrier, and releases no solitons: its deepest point shrinks from −4 to −1.46 by t = 0.6 as it spreads into a train of ripples. A positive hump of any height, however small, holds at least one level and releases at least one soliton.

Tested in The solitons a hump already contains · the wave packets reading path

“A coupler cut to exactly the right length for 1550 nm works across a telecommunications band.”

The evanescent tails that set the coupling reach further at longer wavelengths, so the beat length changes with colour. The coupler cut to 37.6 µm transfers above 95 per cent over only 94 nm and falls to 3 per cent at 1800 nm; tapered couplers of 300 and 800 µm stay above 95 per cent over 313 and 438 nm.

Tested in The coupler that does not care about the colour · the guided waves reading path

“An optical time crystal is only an engineering matter of modulating strongly enough.”

The gap for light of wavelength 1550 nm sits at a modulation frequency of 3.9 × 10¹⁴ hertz, twice the light's own, and the index has to swing by a sizeable fraction of itself at that rate. Microwave versions at 2 × 10¹⁰ hertz have been built with switched circuits; no optical material is known to respond fast and strongly enough.

Tested in The crystal made of moments · the refraction reading path

“Light in a disordered medium is always concentrated near the surface it enters.”

At a resonance of one random stack of 120 layers, where it transmits 78 per cent, the energy density inside rises to over a thousand times that of the transmitted light at 62 per cent of the way through and falls off exponentially on both sides — a mode trapped by the disorder, far from either surface.

Tested in The walk that interference can stop · the scattering reading path

“A ship that is inexplicably slowed in calm water must be fouled, overloaded or caught in a current.”

Over a 1.5 m layer of water 2 per cent lighter than the water beneath, the fastest wave on the interface travels at 0.52 m/s, 1.02 knots. A hull moving just below that speed drags a train of interfacial waves 53 m long behind it, and the drag they carry away peaks at 0.96 knots — invisible at the surface, which moves by a fiftieth of the interface's displacement.

Tested in The wave that holds a ship back · the stratification reading path

“Turbulence scrambles a plasma's magnetic field, so the state it settles into depends on the details of how it was driven.”

The settled state depends on two numbers, the flux and the helicity. A field of 1,000 twisted modes decaying under resistivity has lost half its energy by the time it has lost 3.3 per cent of its helicity; every spectrum drawn here then falls onto the same floor, energy equal to the smallest wavenumber times the helicity, which is a single field shape. Pinches in different machines, started differently, land on one curve of reversal parameter against pinch parameter.

Tested in The twist that outlives the turbulence · the flux freezing reading path

“Clocks on the Earth run at a steady rate through the year, since nothing about their surroundings changes.”

Against a clock far from the Sun they do not. The orbit's eccentricity of 0.0167 takes the Earth deeper into the solar potential at perihelion, and every terrestrial clock runs slower in early January than in early July by 3.3 × 10⁻¹⁰ — about half as much as the Earth's own gravity slows a clock on its surface. It is invisible to any comparison made on the Earth, which is what the equivalence principle requires.

Tested in The clocks that must all slow together · the gravitational redshift reading path

“Two surfaces with the same charge always repel in an electrolyte, since every contribution to the force is a repulsion or a screened one.”

Mean-field theory says so and proves it: the Poisson–Boltzmann pressure between two like-charged planes with their counterions equals the counterion density at the midplane, 1.71 in natural units at one Gouy–Chapman length apart and 0.29 at four, never negative. In the strong-coupling limit the same system has a pressure of 2/d − 1, which is negative beyond two Gouy–Chapman lengths, and calcium silicate hydrate sheets, clay platelets and DNA in spermine are held together by exactly that attraction.

Tested in The like charges that pull together · the osmosis reading path

“A current in a plasma is the charge of the particles times the velocity of their guiding centres.”

Not across a pressure gradient. In a uniform field every guiding centre of a Maxwellian plasma is at rest, yet the ions at any point move on average at vₜ²/ΩL along the gradient's perpendicular — 0.33 thermal speeds for a density that falls by e every three gyroradii — because more circles are centred on the dense side. Counted particle by particle across a plasma edge, that flow reproduces the fluid current p′/B to within six per cent of its peak.

Tested in The current no particle carries · the magnetism reading path

“An electrical circuit made of metal wires is a classical object whose energy can take any value.”

A superconducting island joined through a junction has discrete levels. At a Josephson energy fifty times the charging energy its lowest transitions are 18.94, 17.79 and 16.50 charging energies, computed by diagonalising the circuit's Hamiltonian, and each can be driven separately with microwaves because no two are equal.

Tested in The circuit that forgets its charge · the superconductivity reading path

“Light is bent twice as much as Newton predicts because a photon is not an ordinary particle.”

Every body is bent by more than Newton predicts, by the factor 1 + v²/c², and light is the case v = c. The figures integrate exact geodesics at six speeds and every one lies on that curve to 0.22 per cent at two thousand masses; nothing special happens at the speed of light except that the second term has grown to equal the first.

Tested in The half of the bend a slow body never feels · the light deflection reading path

“Mercury's perihelion precession is an effect of special relativity — the planet is moving fast, so its mass changes.”

Special relativity alone, with the planet's inertia growing as its energy does in Newton's attraction, gives one sixth of the observed 43 arcseconds a century. A Lorentz-invariant scalar theory of gravity gives minus one sixth. The figures integrate all three couplings in the weak field and measure +0.99, −1.01 and +6.11 in units of πGM/c²ℓ; the remaining five sixths come from the curvature of space and the second-order shape of the field, which special relativity does not supply.

Tested in The orbit special relativity cannot close · the orbit stability reading path

“A dielectric particle is always pulled towards the strongest field.”

Only if it polarises more than the liquid it displaces, and with conduction included that comparison changes with frequency. The figures compute the complex factor K for three beads: a 200 nm latex bead in water of 0.01 S/m is pulled at low frequency and pushed above 3.3 MHz, a 2 µm bead of the same latex is pushed at every frequency, and a ceramic bead does the opposite of the small latex one.

Tested in The force whose sign a frequency chooses · the dielectrics reading path

“A helium balloon moves forward when a car accelerates because the air in the car sloshes backward and pushes it.”

In a closed cabin accelerating steadily there is no sloshing: the air is at rest relative to the cabin, with a pressure gradient pointing backward-and-down that is exactly what accelerates it. The balloon sits in that gradient. The figures balance the forces on a 30 cm party balloon at 3 m/s² — buoyancy 174 mN along −(g − a), weight 53 mN, string 119 mN — and the string's lean comes out at 17.0°, the tilt of g − a, with the 16.2 mN net forward force the balloon's own mass needs.

Tested in The balloon that leans the wrong way · the buoyancy reading path

“A boat in a lift accelerating upward floats lower, because it is heavier.”

It is heavier, and the water pushes up harder by exactly the same factor, because the pressure at every depth scales with g + a as well. The figures solve the draught of a block of relative density 0.6 by balancing the two forces at every acceleration from free fall to three gravities, and it is 0.60 of the block's height throughout.

Tested in The balloon that leans the wrong way · the buoyancy reading path

“A gas of grains left to itself cools uniformly, like a hot gas losing heat to cold walls.”

It cools uniformly only at first. The simulation of 2,000 discs with e = 0.7 follows Haff's law, T₀/(1 + t/t₀)², for tens of collision times and then leaves it, running a factor of two hotter than the law after 83 collision times, because by then the grains have gathered into clusters with empty space between: grain counts in a 10 × 10 grid end with a variance seventeen times their mean.

Tested in The gas that cools itself into clumps · the granular matter reading path

“A ball that loses a fixed fraction of its speed at each bounce keeps bouncing for ever, just ever lower.”

It bounces infinitely often and stops in a finite time. Each flight is e times as long as the one before, so the flight times are a geometric series: dropped from 1 m with e = 0.8, the first fall takes 0.452 s and the ball has made every one of its bounces by 4.064 s, t₁(1 + e)/(1 − e). The figures step sixty bounces from the rule and add the geometric remainder, and the total agrees with the closed form to a part in a billion.

Tested in The bounces that add up to a stop · the momentum reading path

“A pendulum's plane of swing stays fixed unless the Earth turns under it.”

Only a swing in a perfectly straight line keeps its plane. Any ellipse turns by itself, in the sense the bob goes round, at Airy's rate (3/8)ωab/L². A 1 m pendulum swung 30° with a sideways push making b/a = 0.3 turns its long axis 5.3° every half swing in the integration, with a fixed pivot and no rotation of anything.

Tested in The ellipse a pendulum turns by itself · the pendulum reading path

“No optical system can image more than a single pair of points perfectly.”

No system of surfaces between two uniform media can, apart from a flat mirror. A continuously graded medium can: every ray from each of four sources in Maxwell's fish-eye, traced through the index n₀/(1 + r²/R²), passes through its image point to within 8 × 10⁻⁴ of the radius — the integration's own error — for every source at once.

Tested in The medium that images every point · the fermat reading path

“Since a sphere looks the same from every direction, a glass ball is an aberration-free lens for every direction.”

It is equally good in every direction and perfect in none. A uniform ball of index 1.5 focuses its central rays 1.50 radii from its centre and a ray entering at 0.9 of the radius at 1.10; at index 2 the spread is smaller but never zero. The symmetry removes off-axis aberrations and leaves spherical aberration, which the ball has in full for every direction alike.

Tested in The lens with no axis · the fermat reading path

“The Fermi–Dirac distribution is a special property of electrons, arising from the exclusion principle and from nothing else.”

It is the occupancy of any site that can hold nothing or one, in contact with a reservoir at chemical potential μ: two states, weights 1 and e^((μ−ε)/kT). A surface site covered by a gas molecule obeys it exactly — Langmuir's coverage p/(p + p½), plotted against ln(p/p½), falls on the Fermi function to a part in 10¹² — with no electrons and no quantum mechanics beyond the fact that a site holds one molecule.

Tested in The site that fills like an electron level · the chemical potential reading path

“Doping a semiconductor with donors adds electrons and leaves the holes alone.”

The holes fall in exact proportion as the electrons rise, because generation and recombination hold the product n·p at nᵢ². In silicon at 300 K the figures solve charge neutrality with the product law from 10⁸ to 10¹⁸ donors per cubic centimetre, and √(n·p) stays at 10¹⁰ to a part in a billion; at 10¹⁶ donors the electrons outnumber the holes by 10¹² to one.

Tested in The product doping cannot move · the chemical potential reading path

“Relativity matters only for particles in accelerators and for astronomy; chemistry is non-relativistic.”

The innermost electron of an atom of nuclear charge Z moves at about Zα times the speed of light — 0.58 of it in gold. Dirac's exact ground state for that charge has a mean radius 0.878 of Schrödinger's, drawn from the two closed forms. The contraction propagates to the outer s orbital, closes gold's d-to-s gap from about 3.7 eV, where silver's 3.9 eV lies, to 2.4 eV, and turns a white metal yellow.

Tested in The metal that is yellow because it is heavy · the atomic structure reading path

“A Bose–Einstein condensate is automatically a superfluid, because all its atoms are in one state.”

An ideal Bose gas, with ε = p²/2m, has ε/p starting at zero: an arbitrarily slow body can make an arbitrarily soft excitation, and the gas has no critical velocity at all. It is the interactions that turn the bottom of the spectrum into sound and give ε/p a floor; for a weakly interacting gas the floor is the speed of sound.

Tested in The speed below which nothing can be made · the superfluidity reading path

“When a moving ball hits an identical ball at rest and both move off, they always separate at a right angle.”

Only when the incoming ball is slow compared with light. The figures boost the back-to-back centre-of-momentum pair into the laboratory and find a symmetric opening of 83.6° at a kinetic energy of half the rest energy, 70.5° at twice it and 44.4° at ten times it; the right angle survives only as the limit γ → 1.

Tested in The right angle a fast collision closes · the relativistic dynamics reading path

“An ideal spring obeying Hooke's law is isochronous at any amplitude.”

Only while the mass is slow. With a relativistic mass the period grows with amplitude: 1.01, 1.17, 2.13 and 6.43 times the slow period at amplitudes of 0.2, 1, 3 and 10 times c/ω₀, with a first correction of (3/16)(ω₀a/c)². At large amplitude it approaches 4a/c, the time light takes to cross the swing four times.

Tested in The spring that becomes a light-clock · the relativistic dynamics reading path

“Locking the modes concentrates more energy into the output.”

The average intensity is N in both cases — 20 for twenty modes, locked or random — because cross terms between different frequencies average to zero whatever their phases. Locking moves the same energy into a fraction 1/N of the time, where it is N times the average; it creates none.

Tested in The phases that turn a glow into pulses · the superposition reading path

“An interference pattern made by several laser beams changes shape whenever the beams' phases drift.”

Only when there are more beams than one plus the number of dimensions. Three beams in a plane have two relative phases and the pattern can slide in two directions, so every phase change is a slide: for two arbitrary changes a shift reproduces the original triangular lattice to within 2 per cent of its peak on a 60-by-60 search. The shape is immune to the phases, which is why the first two- and three-dimensional optical lattices for atoms used three and four beams.

Tested in The lattice no phase can bend · the superposition reading path

“A field containing no spatial frequency above some limit cannot vary faster than that limit anywhere.”

(cos x + i·2·sin x)^10 multiplies out to waves e^(inx) with n from −10 to 10 and nothing beyond — checked by projecting it onto n = 12 and 14, which gives zero — yet its local wavenumber at the origin is 20, twice the band limit, and it completes 3.0 cycles within |x| < 0.62 where its fastest component completes 2.0. The band limit constrains the field on average over a period, not at every place.

Tested in The wiggle faster than any wave in it · the huygens reading path

“A charge on a circle radiates at the frequency it goes round.”

Only when it is slow. The field arriving from a charge at 0.2c is close to a sine wave at the orbital frequency; at 0.9c it is a spike 0.7 per cent of a turn wide, and its Fourier transform has its strongest harmonic near the 14th and power down to one per cent only by about the 126th. At 0.99c the strongest is near the 453rd.

Tested in The flash a circling charge sends once a turn · the radiating charge reading path

“Small particles always scatter blue light more strongly than red.”

Only below their resonance. A bound electron's cross-section rises as ω⁴ up to its resonance and then settles, within 10 per cent of the Thomson value, from 9.8 eV to 29 keV for a resonance at 589 nm. Sunlight scattered off the free electrons of the corona keeps the colour of sunlight exactly.

Tested in The cross-section that forgets the colour · the radiating charge reading path

“A theory of gravity that differs from general relativity will show up first in the fastest, most violent systems.”

Not for a dipole. Its power relative to the quadrupole goes as (5/96)Δα²(c/v)², falling with slope −2 in orbital speed, so it is largest in slow orbits. For Δα² = 10⁻⁵ a dipole would equal the quadrupole at v/c = 7.2 × 10⁻⁴ and be negligible in a merging pair at 20 Hz, where v/c = 0.15.

Tested in The radiation a difference in falling would make · the equivalence principle reading path

“Light slows down in glass, so its momentum there is less than in vacuum, and that is the end of it.”

A free block crossed by a pulse ends 0.500 EL/Mc² forward, which momentum conservation gives only if the light inside carries E/nc — but a charge moving through water at 0.99c emits at 41.2°, and that angle follows only if a photon in water carries nħω/c. Both measurements are correct; they measure different momenta.

Tested in The momentum light carries into glass · the field energy reading path

“A charge's field energy U gives it an electromagnetic mass of U/c², and that is the mass the field contributes to its inertia.”

The field's momentum for a slowly moving charged shell is (4/3)(U/c²)v, found by integrating ε₀E × B over space. At 0.5c the mass read off the momentum is still 1.2308 times the mass read off the energy, and only as the speed tends to light's does the ratio tend to 1.

Tested in The mass a charge's field gets wrong by a third · the field energy reading path

“A quadrupole is two dipoles back to back, so describing it takes twice a dipole's three numbers.”

The second moment Σq xᵢxⱼ has nine components, six once it is symmetric, and five that reach the far field: the kernel of the Laplacian on quadratic polynomials, computed by elimination, has dimension 5 — and 7 for the octupole, 9 for the hexadecapole. Two of the five are the shape and three the orientation.

Tested in The shape that takes five numbers · the the field concept reading path

“Compressibility is too small to matter to anything as large as the ocean.”

It is small in each layer and adds up over the column as the square of its depth. A column at the ocean's mean depth of 3.68 km would stand 31.0 m higher if its water did not squeeze, 81.4 m over a 6 km abyssal plain and 266 m over the Challenger Deep.

Tested in The sea that stands lower because water gives · the hydrostatics reading path

“Water in a capillary tube rises smoothly to Jurin's height and stops there.”

Only in narrow tubes. Integrating the pull, the weight and the Poiseuille drag with the column's inertia, water in a 1 mm tube overshoots to 1.33 of its Jurin height and in a 2 mm tube to 1.46, then rings about it; the linearised motion is underdamped for any tube wider than 0.48 mm.

Tested in The pore that lifts highest fills slowest · the capillarity reading path

“The Bohr–Sommerfeld rule, action equals a whole number of Planck constants, is the classical limit of quantum mechanics and is exact for simple systems.”

For the simplest systems it is wrong by a fixed fraction of h. The computed levels of a harmonic well enclose 0.500, 1.500, 2.500 and 3.500 h, and those of a box 1.000, 2.000, 3.000 and 4.000 h; for a particle bouncing on a floor, action nh puts the ground state at zero energy.

Tested in The quarter cycle a turning point costs · the least action reading path

“A bat's sweet spot is at its centre of mass, where the most wood is behind the ball.”

The point that sends no jolt to the hands is the centre of percussion for a pivot at the grip, and for a 0.84 m wooden bat held 15 cm from the knob it lies 17.1 cm from the end of the barrel, at 0.797 of the length against a centre of mass at 0.690. For a uniform rod held at one end it is at 2/3 of the length, not 1/2.

Tested in The point that takes the blow without a jolt · the pendulum reading path

“Beyond the critical angle all the light is reflected, whatever is on the other side of the glass.”

Only if nothing on the far side can take the energy the evanescent wave carries. With 50 nm of gold between BK7 and water, p-polarised light at 633 nm is reflected at 0.9 or more on either side of 71.52° and at 0.014 there; s-polarised light is reflected throughout.

Tested in The angle at which total reflection stops being total · the total internal reflection reading path

“A bare glass corner cube reflects totally for any direction within its field of view.”

Tilting the incoming light steepens one face towards its critical angle. For fused silica, every reflection is total from −16° to beyond +45° of tilt in the plane of one edge; for BK7 from −20°; for glass of index 1.8 from −40°. Beyond those, light leaks out through the back.

Tested in The corner that sends light home · the total internal reflection reading path

“A spectral line from a transition that takes eleven million years is too weak to observe.”

The rate per atom is 2.87 × 10⁻¹⁵ s⁻¹, and a cloud of 10²⁰ hydrogen atoms per square centimetre with a velocity spread of 10 km/s still shines at a brightness temperature of about 5.3 K, whatever its temperature — a signal radio telescopes measure routinely.

Tested in The line that takes eleven million years · the atomic spectra reading path

“Atoms are always around a tenth of a nanometre across.”

Only in their ground states. The circular orbit of level n has a diameter of 2n²a₀: 11 nm at n = 10, 1.1 µm at n = 100 — twice the wavelength of green light — and 9.5 µm at n = 300, larger than a bacterium.

Tested in The atom the size of a bacterium · the atomic spectra reading path

“The random thermal motions of atoms in a clock average away, so a clock's temperature cannot shift its rate through relativity.”

The first-order Doppler shifts do average away: over 20,000 caesium atoms sampled at 300 K they average to 3.1 × 10⁻⁹ against a spread of 4.6 × 10⁻⁷. The time dilation has one sign and averages to (3/2)kT/mc² = 3.1 × 10⁻¹³, which a clock at 10⁻¹⁸ must bring below by cooling strontium under 0.63 mK.

Tested in The clock that runs slow because it is warm · the time dilation reading path

“The field of a moving charge can be made into light by moving fast enough.”

Its invariant E² − c²B² is positive in every frame, so it can never become light's zero. But integrated over a passage it falls as 1/γ², slope −2.00, to 6.7 × 10⁻⁷ of E² + c²B² at γ = 1000, and the longitudinal share of the field to 3.3 × 10⁻⁷: a pulse indistinguishable from light by any measurement of it.

Tested in The charge that passes as a flash of light · the field transformation reading path

“A voltage across a boundary needs a source of charge — a battery, a pump or a generator — to maintain it.”

A concentration ratio maintains one at equilibrium with nothing flowing: (kT/e) ln(c₁/c₂), 59.16 mV per decade at 25 °C and 61.54 at 37 °C. Across a boundary with potassium 28 times more concentrated on one side, the electrical step of −89.1 mV cancels the chemical step exactly, and the electrochemical potential is flat.

Tested in The voltage that is a chemical potential · the chemical potential reading path

“A clock can be made as regular as one likes with no minimum heat cost, provided it is run slowly and carefully enough.”

For any process held in a steady state, the squared relative uncertainty of what it counts times the entropy it produces is at least 2 in units of k, and the product does not fall with time because both factors scale with it. Counting to a hundred with a spread of ±10 costs at least 200 k of entropy — 200 kT of heat — however it is done.

Tested in The precision a clock pays for in heat · the entropy reading path

“A thin enough absorber can be made broadband by choosing the right material.”

For any metal-backed layer, the integral over all wavelengths of ln(1/|Γ|) is at most 2π² times the thickness times the static permeability. A 7.5 mm non-magnetic absorber has 147.9 mm of absorption to spend in total, and a band held below −20 dB costs ln 10 of it per unit of wavelength, whatever the material.

Tested in The absorber whose bandwidth is its thickness · the impedance reading path

“A wall oscillating at twice a string's fundamental frequency can only pump the fundamental, as a swing is pumped at twice its frequency.”

A string's modes are evenly spaced, so twice the fundamental is also the gap between every mode and the one two above it. The energy spreads upwards: its mean mode number goes 1.0, 3.3, 13.2 and 47.0 after 0, 10, 20 and 30 round trips, all in odd modes.

Tested in The string whose end will not keep still · the standing waves reading path

“A column of water whose density increases with depth everywhere is stable, and nothing in it will overturn.”

Only if heat and salt diffuse at the same rate. With heat diffusing a hundred times faster, a column with warm salty water above cold fresh water and a density ratio of 2 grows vertical fingers 5.8 cm across at 2.33 e-folds an hour, and any density ratio between 1 and 100 is unstable in the same way.

Tested in The salt that sinks through a stable sea · the stratification reading path

“At low enough speed, every body's drag is proportional to its speed, as Stokes's law says it is for a sphere.”

Not for a long cylinder. Its drag per unit length divided by viscosity and speed is 2.92 at a Reynolds number of 0.1, 1.12 at 10⁻⁴ and 0.69 at 10⁻⁷, from Lamb's formula 4π/(½ − γ − ln(Re/8)): it falls without limit as the flow slows, as one over the logarithm of the Reynolds number.

Tested in The drag a distant wall decides · the viscosity reading path

“Shaking a divided box of particles always shares them equally between the compartments, as the second law requires of a gas.”

Only if the collisions are elastic. With inelastic grains and a flux F(n) = n²e^(−Bn²) through the slot, equal shares are stable only for B below 4; at B = 6 the fuller compartment holds 0.929 of the grains, and 400 grains simulated one hop at a time end with 0.917 on one side.

Tested in The sand that chooses a side · the granular matter reading path

“A wave can travel in either direction along the line it lives on; which way it goes depends only on how it was started.”

Not a Rossby wave. Columns pushed north turn clockwise and columns pushed south anticlockwise, and the flow their spins induce peaks a quarter wavelength west of each crest, so the crest is rebuilt to the west whichever way the meander was first pushed. At 45° a meander 3,000 km long moves 638 km west in two days.

Tested in The wave that can only travel west · the circular motion reading path

“A chaotic system cannot be simulated at all, because every rounding error grows until the computed trajectory is meaningless.”

The computed trajectory is meaningless as a prediction from its starting point and faithful as a picture of the system. A doubling-map computation with an error of up to 0.001 at every step parts from the exact orbit from its own start within eight steps, yet an exact orbit built backwards stays within 8.0 × 10⁻⁴ of it for all 400 steps.

Tested in The wrong orbit beside a right one · the chaos reading path

“A molecule's vibrational spectrum measures only the stiffness of its bond, not its strength.”

Only its lowest line does. The gaps between successive levels fall from the harmonic value towards zero at the dissociation limit, and their sum is the energy to break the bond from its lowest level: for a Morse well with 19 levels, a straight line through the first five gaps encloses 0.9473 of the depth, which is exactly the true value.

Tested in The spectrum that weighs a bond · the harmonic approximation reading path

“Adding a strong beam to a weak one on the same detector can only bury the weak one.”

It lifts it. The detector squares the summed field, and the cross term oscillates with amplitude 2√(Pₗₒ Pₛ): with an oscillator a thousand times stronger the weak signal's contribution to the current swings by ±63 times its own direct-detection current.

Tested in The faint light a bright one makes measurable · the coherence reading path

“Stopping a lens down reduces all its aberrations, colour included.”

It reduces axial colour, which for a 50 mm N-BK7 singlet with its stop 20 mm in front falls as about 775/N µm — 387 µm at f/2, 35 at f/22. Lateral colour at 10° off axis is 55.0 µm at every f-number, because it is carried by the ray through the centre of the stop, which every aperture passes.

Tested in The colour fringe no aperture can close · the imaging reading path

“A diamond sparkles because it is cut with many facets that act like mirrors, and any clear stone cut the same way would do the same.”

The facets reflect only by total internal reflection, which depends on the index. Cut to Tolkowsky's proportions and lit from within 40° of vertical, diamond (n = 2.417) returns at most 88 per cent of the light through its top, cubic zirconia 87, sapphire 48 and glass 21 — in a two-dimensional ray trace through up to eight reflections.

Tested in The window a diamond is cut inside · the total internal reflection reading path

“A gas's particles are fixed in number, so heating it only makes them move faster.”

Hot enough, the gas makes particles. In oxygen at a billion kelvin, where kT is a sixth of an electron's rest energy, equilibrium holds 1.37 positrons per original electron at 10³ g/cm³, and at ten billion kelvin the pairs outnumber the original electrons at every density from 10³ to 10⁶ g/cm³.

Tested in The heat that turns into matter · the relativistic thermodynamics reading path

“The twin who ages less is always the one who accelerated.”

In a universe whose space is a circle ten light-years round, a twin moving at 0.6c goes once round and meets the twin at rest after 16.67 years by the stay-at-home's clock without ever accelerating, and has aged 13.33 years — less by the factor 1/γ = 0.8.

Tested in The twin who never turns round · the time dilation reading path

“A gas ionises when its temperature approaches the binding energy of its electrons, 158,000 K for hydrogen.”

At equilibrium, hydrogen is half ionised at 13,980 K at 10²³ nuclei per cubic metre, 7,235 K at 10¹⁸, 4,519 K at 10¹² and 3,270 K at 10⁶, where the binding energy is between 11 and 48 times kT.

Tested in The electron that leaves because there is room · the chemical potential reading path

“A solid is always more ordered than the liquid it freezes from, so freezing always releases heat.”

Not helium-3 below 0.315 K. There the solid's nuclear spins are disordered while the liquid is a degenerate Fermi liquid, and the melting curve's slope implies the solid holds up to 0.646 R more entropy than the liquid, near 7 mK. Freezing it absorbs heat: compressing liquid starting at 20 mK until 15 per cent has frozen cools it to 1.7 mK.

Tested in The melting curve that has to arrive flat · the third law reading path

“The more energy an excited state has to release, the faster it decays, so a nuclear gamma decay is always practically instantaneous.”

Energy matters far less than angular momentum. At 1 MeV in a nucleus of mass 100, Weisskopf's estimates give 10⁻¹⁵ s for a photon carrying one unit and 17.3 hours for one carrying five; tantalum-180m, 77 keV above its ground state and needing to shed eight units, has never been seen to decay and outlives 10¹⁷ years.

Tested in The spin a photon has to carry away · the decay reading path

“Quantum tunnelling is a property of microscopic particles; a coordinate describing a large piece of metal behaves classically.”

The phase difference across a Josephson junction, a variable shared by every electron pair in two superconducting electrodes, escapes from its well by tunnelling below a crossover temperature ħωₚ/2πk — about 36 mK for a 10 µA, 5 pF junction at 0.99 of its critical current — at about 180,000 escapes a second, whatever the temperature.

Tested in The circuit that goes through a wall · the tunnelling reading path

“A body can rest stably only at a minimum of its potential energy, so the Trojan asteroids must sit in valleys of the effective potential.”

L4 and L5 are maxima of the effective potential in the turning frame, yet for mass ratios below 0.0385 a body displaced from them circles round them: with Jupiter's ratio, 0.000954, bodies started 10° and 25° past L4 at rest swing between 51° and 70°, and between 42° and 85°, from the planet.

Tested in The hilltop that holds the Trojans · the orbit stability reading path

“A small moon's volcanism must be driven by heat left over from its formation or from radioactivity, as the Earth's is.”

Io, forty-three times smaller in volume than the Earth, puts out about 1.0 × 10¹⁴ W, twice the Earth's 4.7 × 10¹³. The tidal formula (21/2)(k₂/Q)GM²R⁵ne²/a⁶ with k₂/Q = 0.015 and Io's eccentricity of 0.0041 gives 9.3 × 10¹³ W: tides account for all of it.

Tested in The moon kept molten by its neighbours · the Self-gravity reading path

“The inverse-square law is a property of gravity and of electric charge, found by measurement.”

It is a property of three-dimensional space. Gauss's law spreads a source's field over the surface around it, whose size grows as r^(d−1) in d dimensions; in two dimensions the same law gives 1/r, in four 1/r³, in five 1/r⁴, with nothing about the force changed.

Tested in The number of dimensions a planet can orbit in · the Gauss's law reading path

“The fair-weather electric field is set by the local weather, so its daily cycle follows the local sun like temperature does.”

Over clean ocean air its daily cycle follows Universal Time. In the model, ships at 120° W, 0° and 120° E see the same field curve peak at 11.4, 19.4 and 3.4 hours local time — one instant, 19.4 UT — because each reads the potential of one ionosphere, which follows the sum of the world's thunderstorms.

Tested in The field that keeps Greenwich time · the potential reading path

“To make a magnetic field that rotates, something has to rotate — a magnet, a coil or a commutator switching the current round.”

Three fixed coils 120° apart carrying currents a third of a cycle apart give a field of constant strength, 1.5 times one coil's peak, pointing at the angle ωt at every instant. Each coil's field only grows, shrinks and reverses along its own axis; the rotation exists only in the sum.

Tested in The field that turns with nothing turning · the superposition reading path

“A lensing galaxy that shows four images of a quasar has made four images.”

A lens with a finite central density maps the sky onto the source plane continuously, and such a map gives an odd number of images for every source position. For a core of 0.01 Einstein radii and a shear of 0.15 the fifth image lies at the centre, magnified about 10⁻⁴, some 1.4 × 10⁵ times fainter than the other four together.

Tested in The fifth image a galaxy hides · the light deflection reading path

“A radio-frequency discharge between two identical electrodes cannot develop a direct self-bias.”

Only if the drive is a pure sinusoid. Driven by a fundamental and its second harmonic of equal amplitude φ₀, it settles at −0.4375 φ₀ with the harmonics in phase and +0.4375 φ₀ with them 90° apart, because the capacitive sheaths clamp the discharge to the midpoint of the waveform's two unequal extremes.

Tested in The bias a waveform can choose · the plasma oscillation reading path

“Two identical neutral particles in a uniform field cannot attract each other.”

Each becomes a dipole along the field, and two aligned dipoles attract when the line joining them lies within 54.7° of the field and repel outside it. At contact along the field the pull is twice the side-by-side push at the same distance, and the sideways part of the force turns any pair towards alignment.

Tested in The liquid a field turns solid · the dielectrics reading path

“Eddy-current drag grows with speed, so a faster magnet is braked harder.”

Over a thin conducting sheet the drag rises in proportion to the speed only while it is slow compared with w = 2/μ₀σd. It peaks at 1.27 w at 0.30 of a perfect image's repulsion and falls as w/v beyond, while lift grows towards the full repulsion.

Tested in The drag that falls as the magnet speeds up · the induction reading path

“Cornflour in water thickens because the water is squeezed out from between the grains, as in wet sand underfoot.”

Dilation plays a part at a free surface, but the thickening persists in closed geometries with no free surface. What changes is the contact: above an onset stress the lubricating films between grains fail and contacts become frictional, which lowers the jamming fraction from about 0.64 to about 0.58 and sends the viscosity of a suspension near 0.57 up forty-fold over a few decades of stress.

Tested in The suspension that seizes when pushed hard · the rheology reading path

“Water at the bottom of a deep trench is warmer than the water above it because heat leaks up from the Earth.”

Geothermal heat flow warms the bottom water by far less than a hundredth of a degree. In the column drawn, whose potential temperature is a uniform 1.00 °C below 4 km, a thermometer reads a minimum of 1.38 °C near 3.9 km and 2.35 °C at 10.9 km: the difference is adiabatic compression of the same water.

Tested in The water that warms on the way down · the hydrostatics reading path

“A coefficient of restitution below one means some of the material deformed inelastically.”

A perfectly elastic ball on a perfectly elastic plate rebounds at 0.80 of its arrival speed when Zener's parameter is 0.2 and 0.33 when it is 1, because the impact launches bending waves that carry energy away through the plate. A 20 mm steel ball at 1 m/s on a 4 mm steel plate comes back at 0.19 of its speed.

Tested in The bounce an elastic plate cannot give back · the momentum reading path

“Euler's calculation of the Earth's free wobble was wrong.”

It is right for a rigid body: with the Earth's dynamical ellipticity of 0.00327 it gives 304 days. The observed 433 days needs the body to yield by k/kₛ = 0.30 of what a fluid Earth would, where kₛ = 0.94 follows from the flattening; the difference is a measurement of the Earth's elasticity, not an error.

Tested in The wobble that is slow because the Earth gives · the rotation reading path