Simply false — part 4 of 5

“The smaller the pinhole, the sharper the picture.”

Only down to a size set by the wavelength and the camera's depth. For a camera 100 mm deep in 550 nm light, the diameter holding half the light of a point's image is least, 0.21 mm, for a hole 0.34 mm across; a hole half that size makes a blur more than twice as wide, all of it diffraction.

Tested in The hole that makes the sharpest picture · the imaging reading path

“A full moon is about twice as bright as a half moon, because twice as much of the lit surface faces the Earth.”

A perfectly diffusing sphere would be 3.1 times brighter at full than at half. The Moon is about ten times brighter. In the drawn soil model, hiding the shadows between grains alone doubles the ratio, and the shadows of larger rocks and craters and the grains' own backward scattering supply the rest.

Tested in The full moon that is too bright · the scattering reading path

“Electrons in a metal repel each other so strongly that they must collide constantly.”

An electron can collide only into empty states, and at zero temperature an electron ε above the Fermi level has a share of collisions proportional to ε², which vanishes at the Fermi level. In copper, one at the Fermi level at room temperature travels about 5 µm before another electron deflects it; one 1 eV above travels about 35 nm.

Tested in The collisions the exclusion principle forbids · the exclusion reading path

“A satellite measuring infrared radiance from the Earth measures the temperature of the surface.”

Only in the transparent parts of the spectrum. Across a model CO₂ band the brightness temperature falls from 287 K in the wings, which see the ground, to 224 K on the shoulders, which see the tropopause, and rises to 255 K at the band centre, which sees the stratosphere.

Tested in The spectrum that is a thermometer at every height · the blackbody reading path

“The repulsion between the particles of a beam grows weaker as the beam goes faster.”

The electric repulsion grows as γ: 2.29 times the Coulomb force at β = 0.9, 7.1 at 0.99. What grows faster is the magnetic attraction of the parallel currents, γβ², so the net sideways force is γ(1 − β²) = 1/γ, and the sideways acceleration, against a transverse inertia of γm, is 1/γ² of Coulomb's.

Tested in The beam that stops pushing itself apart · the field transformation reading path

“An electron riding an accelerating wave at nearly the speed of light keeps gaining energy for as long as the wave lasts.”

Once it is faster than the wave it slips forward through it into the decelerating half. In a wave of 0.3 E₀ with phase-velocity factor γₚ = 20 it peaks at γ = 519 after about 740 plasma wavelengths and then loses energy; at γₚ = 40, γ = 1999 after about 2,500.

Tested in The wave an electron rides to a gigaelectronvolt · the relativistic dynamics reading path

“Large molecules react with each other faster than small ones, because they present bigger targets.”

For two equal spheres the diffusion-limited rate is 4π(D(A) + D(B))(R(A) + R(B)), and with D = kT/6πηR the radius cancels: 8kT/3η, 7.4 × 10⁹ M⁻¹ s⁻¹ in water at 25 °C for molecules of 0.1 nm or 100 nm alike.

Tested in The speed limit that does not care how big the molecules are · the diffusion reading path

“Reducing an oscillator's damping reduces its thermal motion.”

The mean-square thermal displacement is kT/k at every damping; the drawn spectra for Q = 3, 30 and 300 each enclose it exactly, and simulated oscillators at Q = 3 and 300 average 1.00 and 0.92 of it. Damping changes only how the motion is spread over frequency.

Tested in The noise a high Q moves out of the way · the equipartition reading path

“Compressing hydrogen always drives its ionisation equilibrium towards neutral atoms.”

Only while its atoms have room for their ground state. With the binding lowered by the neighbouring charges, the ionised fraction at 10,000 K falls to 0.41 per cent near 10²⁷ nuclei per cubic metre and then rises to one by 6 × 10²⁸ m⁻³, where the ion sphere shrinks to three Bohr radii and the ground state disappears.

Tested in The pressure that ionises · the chemical potential reading path

“A seismic survey can measure the thickness of any layer it can detect, from the time between the echoes of its top and base.”

Below about a quarter of a wavelength the time between the combined echo's peak and trough stops following the thickness and settles at about 0.34 of a pulse period, the spacing within the pulse's own derivative. The layer is still detectable — a tenth of a wavelength thick it returns 1.07 times one face's echo — but its timing carries no information about its thickness.

Tested in The layer too thin to see · the impedance reading path

“A perfectly regular train of waves, once made, keeps its shape if nothing disturbs it.”

On deep water, and for light in a fibre with the corresponding dispersion, every modulation longer than a certain length grows. For the nonlinear Schrödinger equation the unstable wavenumbers run from zero to twice the amplitude, and the fastest grows at the amplitude squared: a one per cent ripple becomes a crest 1 + √2 times the train's height by t ≈ 5.6.

Tested in The even wave train that cannot stay even · the wave packets reading path

“A weak periodic push can only slightly perturb a clock that keeps its own time.”

It can take the clock's frequency over entirely. In Adler's equation, every natural frequency within ±K of the drive's runs at exactly the drive's frequency, with a fixed phase lag — 0.93 rad at a detuning of 0.8 K.

Tested in The clock that is pulled into step · the resonance reading path

“A neutron star is made of neutrons because gravity crushes electrons into protons.”

Gravity supplies the density and nothing else. What converts the matter is the balance μₙ = μₚ + μₑ between three Fermi seas. At nuclear density, with free nucleons, it settles at 0.49 per cent protons. The electrons must sit 56 MeV deep while the neutrons fill to only 57.9 MeV above a proton at rest, so a proton made there costs more than a neutron.

Tested in The protons a star cannot afford · the chemical potential reading path

“The ideal-gas law holds for any sufficiently dilute gas, at any temperature.”

It forces α = 1/T, so the expansion coefficient grows without limit as the gas is cooled. By a Maxwell relation α is minus the pressure derivative of the entropy per unit volume, which the third law sends to zero. A Fermi gas at the same density agrees with the ideal law when hot. At a tenth of its Fermi temperature αT is 0.048, and at a hundredth 0.002.

Tested in The two laws absolute zero will not allow · the third law reading path

“Longer chains, being heavier, reach the semidilute regime at higher concentrations.”

The reverse. A coil's volume grows faster than its mass, as the 1.76 power of its length, so the overlap concentration falls as chains lengthen, by a factor of 5.8 for each tenfold increase. Polystyrene of 10⁶ g/mol overlaps at 4.2 g/L, and of 10⁵ at 24 g/L.

Tested in The pressure that stops counting molecules · the osmosis reading path

“A resist line that is strong enough to stand dry will survive being rinsed and dried.”

Drying exposes it to the suction of the meniscus in the gaps beside it. With a modulus of 2.5 GPa and equal lines and spaces, lines dried from water collapse beyond an aspect ratio of 4.1 at 100 nm width, 3.1 at 30 nm and 2.3 at 10 nm.

Tested in The walls a rinse pulls together · the capillarity reading path

“Below the friction limit, two surfaces in contact are stuck and do not move relative to each other.”

Between elastic bodies any sideways load at all makes a ring at the edge of the contact slip. The stuck disc has radius (1 − Q/μP)^(1/3) of the contact: at a tenth of the sliding load 7 per cent of the area is already slipping, and at half of it 37 per cent. A 20 mm steel ball pressed with 100 N creeps 2.1 μm before it slides.

Tested in The contact that slips before it slides · the friction reading path

“A bound molecule's atoms spend their time near the bottom of the potential well that binds them.”

Only for deeply bound levels. For a state bound by almost nothing the wavefunction's exponential tail reaches far beyond the force's range. On a square-well model the helium dimer has 94 per cent of its probability outside a 5 Å well and a mean separation of 44 Å; the measured value is 52 ± 4 Å.

Tested in The molecule that lives outside its own well · the harmonic approximation reading path

“Any dispersion can be undone by the opposite dispersion, returning a pulse to its original length.”

Only the second-order term cancels at one spacing. A 20 fs pulse stretched by 100 mm of silica to 501 fs comes back to 28.8 fs at the best grating spacing, not to 20. The third-order terms of glass and gratings add, and near the optimum that residue sets the length: 30.3 fs two millimetres either way.

Tested in The dispersion made of angles · the dispersion reading path

“A reflected wave can only weaken the field at a surface, since some of the light goes back.”

At the critical angle the incident and totally reflected waves add in phase, and the intensity at the surface is 4.00 times the incident for s polarisation and 5.21 for p, with glass at 1.518 and water at 1.33. Nothing is created: all of the light is still reflected.

Tested in The mirror that lights a tenth of a micrometre · the total internal reflection reading path

“The barrier at a metal–semiconductor contact is set by the metal's work function, so choosing the metal chooses the barrier.”

That is Schottky and Mott's rule, with a slope of one: 0.23 to 1.60 eV from aluminium to platinum on n-silicon. With 10¹⁴ interface states per cm² per eV across a 0.5 nm gap, Bardeen's slope is 0.10, and the same metals give 0.71 to 0.84 eV. Measured slopes for silicon and gallium arsenide lie between about 0.05 and 0.3.

Tested in The barrier the metal cannot choose · the bands reading path

“Where incoming waves lose height over a bar field, the energy must have been dissipated by friction or breaking.”

In the shallow-water equations used here, with nothing that dissipates, the amplitude of a resonant wave falls steadily through an 8-bar patch and only 27 per cent of the energy emerges behind. The rest is sent back: reflected and transmitted energy add to exactly the incident, and a partial standing wave forms in front.

Tested in The sandbars that reflect the sea · the periodic media reading path

“Losses in metal conductors always grow with frequency, because the skin depth shrinks.”

Not for the TE₀₁ mode of a round pipe. In a 60 mm copper pipe its loss is 0.51 dB/km at 60 GHz and 0.20 dB/km at 110 GHz, falling as the −3/2 power of frequency far above cutoff. TE₁₁ and TM₀₁ in the same pipe rise as the square root; TE₁₁ is at 21 dB/km at 60 GHz.

Tested in The loss that falls as the frequency rises · the guided waves reading path

“Retardation adds a correction whose size depends on the details of the atom's spectrum.”

It adds exactly one power of distance, for two atoms (6 → 7) and for an atom and a mirror (3 → 4). The far-field law depends only on the static polarisability, because only fluctuations slower than a light crossing survive at long range.

Tested in The attraction that weakens when light is too slow · the retardation reading path

“A zero of the electric field is an equilibrium where a charge could be held.”

Every zero is a saddle. The field grows linearly away from it at rates that add to zero, because it has no divergence there. Midway between two equal charges those rates are −4, 2 and 2 in units of the charge over the half-separation cubed, pushing a charge back along the axis and out across it.

Tested in The zeros a handful of charges can make · the the field concept reading path

“Angle and angular momentum obey ΔφΔL ≥ ħ/2, like position and momentum.”

A state of definite angular momentum has ΔL = 0 and Δφ = π/√3, a product of zero. For von Mises states the product sits just above ħ/2 when they are tightly bunched, crosses below it at κ = 1.58, and falls towards zero as the bunching relaxes.

Tested in The angle the uncertainty principle cannot be written for · the uncertainty reading path

“A measurement of a spin-½ component can only ever read +1 or −1, so no averaging of measurements can give a larger number.”

An average over post-selected weak measurements can. With a coupling of 0.1 pointer-widths and a post-selection kept in 0.99 per cent of runs, the kept pointers are centred at 8.0 times the coupling, approaching the weak value 10. Either spin value alone would move the pointer by at most one coupling.

Tested in The value a gentle measurement returns · the measurement reading path

“Hawking radiation is too faint to be seen from any horizon.”

From astrophysical black holes, yes: a solar-mass hole is at 6.2 × 10⁻⁸ K and the one at the galaxy's centre at 1.5 × 10⁻¹⁴ K, both far colder than the microwave background. An analogue horizon's temperature is ħκ/2πk. A flowing atomic condensate has been measured at about 3.5 × 10⁻¹⁰ K, a hot object in a system cooled to nanokelvin.

Tested in The river that sound cannot swim up · the Velocity-addition reading path

“The Lorentz–FitzGerald contraction was an arbitrary hypothesis invented to rescue the ether.”

It was motivated by electromagnetism. Heaviside's 1888 field of a moving charge has equipotentials flattened along the motion by exactly 1/γ: 0.8 at 0.6c, 0.44 at 0.9c. Charges moving together along the line of motion push each other with 1/γ² of the force at rest. A body held together by such forces must contract to stay in balance.

Tested in The contraction the forces work out for themselves · the length contraction reading path

“The diffusion equation is an exact law of heat flow, limited only by the accuracy of the diffusivity.”

It has an infinite signal speed. Released at a point, heat is put at once at every distance: after one relaxation time 48 per cent lies beyond the reach of a signal at the medium's speed, and at a tenth of it 82 per cent. A thermometer 4vτ away reads 0.0052 by t = τ, before any signal could arrive.

Tested in The heat that arrives before it could · the relativistic thermodynamics reading path

“The Planck scale is so far beyond reach that no observation can test a prediction made there.”

Some predictions accumulate over distance. If light slowed in proportion to its energy at the Planck energy, a 31 GeV photon from z = 0.903 would arrive 1.16 s late. The 31 GeV photon from GRB 090510 arrived within 0.83 s of the burst's start, and Fermi's analysis set the scale above 1.2 times the Planck energy.

Tested in The photons that raced for seven billion years · the planck scale reading path

“A mountain on a star small enough to be undetectable could be as large as a terrestrial one.”

A neutron star's gravity limits it far more strictly. Its crust is estimated to hold an ellipticity of at most about 10⁻⁵, roughly 12 cm on a 12 km star. The slow spin-down of millisecond pulsars caps theirs at tens to hundreds of micrometres: J0437−4715's at about 240 μm.

Tested in The mountain a spinning star is allowed · the gravitational waves reading path

“Near the Earth, general relativity only changes the rate of clocks; distances are Euclidean.”

The radius a tape would measure from the Earth's centre to its surface exceeds the circumference divided by 2π by GM/3c² = 1.48 mm for a uniform Earth and 2.00 mm for one with a dense core. Outside the Sun, the radial distance from its surface to the Earth's orbit is 7.9 km longer than the difference of the two radii. Both come from the spatial part of the same metric that slows the clocks, and at these speeds it is the same size as the time part.

Tested in The radius a mass adds that no circumference shows · the gravitational redshift reading path

“Newton's third law holds for gravity automatically, since action and reaction are equal by definition.”

It holds only if every material's active mass is the same multiple of its passive mass. If aluminium pulled a thousandth harder per unit of passive mass than iron, a dumbbell of the two, 0.5 kg each and 10 cm apart, left alone would push itself 830 m in a year. The law is an empirical claim about gravity, and it has been tested.

Tested in The pull that has to equal the pull back · the equivalence principle reading path

“A star made of stiff enough matter could be as heavy as it likes; the only limit is how hard the matter can push.”

In general relativity the pressure is itself a source of gravity and of inertia, so raising it raises the load it must carry. For a uniform-density star the central pressure needed becomes infinite when the radius reaches 9/8 of the Schwarzschild radius, whatever the matter is. With any matter whose sound speed stays below light's, no neutron star can exceed about 3.2 solar masses.

Tested in The pressure that weighs down what it holds up · the Self-gravity reading path

“A mirror returns circularly polarised light as it arrived.”

It reverses the handedness of every photon at normal incidence: a perfect conductor reverses all of it at every angle, and glass reverses all of it straight on, returns an even split at Brewster's angle (56.3° for n = 1.5) and keeps 60 per cent at 60°. Only a surface whose electric and magnetic responses are equal keeps the handedness at every angle.

Tested in The handedness a mirror cannot keep · the maxwell equations reading path

“The magnetic forces between two current elements are equal and opposite, as Newton's third law requires.”

By the Biot–Savart law they are not. An element pointing straight at a second element exerts no force on it, while the second, turned at 90°, pushes the first with the full unit of force: a net push of 1.00 in units of μ₀I₁I₂/4πr². Ampère's own law of 1826 makes every pair equal and opposite. Both laws give identical forces between complete circuits.

Tested in The force between pieces of a current · the ampere law reading path

“Measuring a film's resistivity needs a sample of known, regular shape, such as a long strip.”

Four contacts on the edge of a film of any shape without holes suffice. For a square, a disc with unevenly spaced contacts and an irregular lamina of the same material, the readings are 0.221 and 0.221, 0.089 and 0.450, and 0.209 and 0.232 ohms, and exp(−πR₁/Rₛ) + exp(−πR₂/Rₛ) = 1 returns 1.000, 0.999 and 0.999 ohms per square from all three.

Tested in The sheet resistance that forgets the shape · the conductors reading path

“A paste can be made to flow as slowly as desired by pushing it gently enough.”

A thixotropic paste has a critical stress, 1.89 Pa in this model, below which no steady flow exists. Held at 1.8 Pa it creeps to a halt, its viscosity growing without limit; at 2.0 Pa it runs steadily at about 1 Pa s. A gap sheared at a mean rate below the critical 0.63 per second divides into a flowing band and a band at rest.

Tested in The paste that forgets it was stirred · the rheology reading path

“Blood flowing through a capillary has the same viscosity as blood in a large artery.”

In glass tubes blood at 45 per cent red cells is 3.2 times as viscous as plasma in wide tubes, and about 1.25 times in tubes 7 μm across. Below about 5 μm the apparent viscosity rises again steeply, to 5.5 times plasma's at 3 μm, where red cells about 8 μm across have to fold to pass.

Tested in The viscosity that belongs to the tube · the rheology reading path

“A falling domino can only topple a neighbour of the same size or smaller, since energy cannot be created.”

No energy is created. Each domino stores energy when it is stood on its end, and the push only has to lift it over a small hill: for a standard domino 0.156 as thick as tall, the hill is 0.006 of its weight times its height and the drop before it strikes a neighbour half its height away is 0.028, 4.6 times as much. A barrier grows as the fourth power of size, so the next domino can be up to 1.47 times larger.

Tested in The push that pays only for the hill · the energy reading path

“A gyroscope keeps pointing in a fixed direction relative to the ground, which is how a gyrocompass points north.”

A free gyroscope keeps its direction relative to the stars, so relative to the ground it appears to turn once a day. A gyrocompass is a gyroscope constrained to stay nearly level, and the constraint turns the Earth's rotation into a torque that swings the axis towards the meridian. Released 20° east of north at latitude 50°, it swings through north with an 84.4-minute period and, damped, settles within 0.03° after eight hours.

Tested in The compass that finds the axis the Earth turns about · the rotation reading path

“A refractive index is always greater than one, since nothing travels faster than light in vacuum.”

For X-rays every material's index is less than one: 1 − 5.3 × 10⁻⁶ for beryllium, 1 − 7.5 × 10⁻⁶ for silicon and 1 − 5 × 10⁻⁵ for gold at 8 keV. What exceeds c is the phase velocity, which carries no signal; the energy and any information travel more slowly than c, as the requirement that a response follow its cause guarantees.

Tested in The index that falls below one · the refraction reading path

“With the right lenses and mirrors, the beams of several identical lasers can be merged into one beam as bright as all of them together.”

Passive optics conserve étendue, and N identical single-mode beams occupy N modes however they are arranged. Their combined radiance can be no greater than one beam's. Only beams that differ in some other label — polarisation, wavelength — or whose phases are locked into one mode can be made brighter together than apart.

Tested in The beams that add only if they differ · the etendue reading path

“A hydrogen atom cannot hold a second electron, since a neutral atom has no charge left to attract one.”

It holds one by 0.754 eV. A neutral atom's electron cloud is polarised by an approaching electron, and a second electron in a diffuse orbit nearly four times further out than the first is bound by that polarisation. The one-parameter calculation that gets helium within 2 per cent says the ion is unbound by 0.74 eV; letting one electron sit in and one out binds it by 0.36 eV.

Tested in The ion that exists because its electrons keep apart · the atomic structure reading path

“The Δ⁺⁺, three up quarks with aligned spins in the lowest state, violates the exclusion principle.”

It would if the three quarks were identical in every respect. The state is symmetric in space, spin and flavour, so something else must be antisymmetric. Giving each quark a three-valued label — colour — and combining the three values antisymmetrically makes the whole state antisymmetric and allowed. Three values is the least number that works for three quarks.

Tested in The label three identical quarks forced · the exclusion reading path

“What an accelerating observer sees must depend on the frequency of the waves it is looking at.”

Every original frequency fades along the same curve, merely shifted in the observer's time: a wave ten times higher in frequency passes each seen frequency 2.3 units of c/a later. The shape of the fading, and so its spectrum, is identical, and the temperature depends on the acceleration alone.

Tested in The heat hidden in a Doppler shift · the accelerated frames reading path

“A box of light weighs more than E/c², because the light's pressure adds to its gravity.”

The light's energy and pressure each contribute E/c², but the walls that hold the light are under tension, and tension is negative pressure: it contributes −E/c². In any body in equilibrium the stresses sum to zero, so the box weighs exactly E/c² more than when empty, as a balance would show.

Tested in The light that pulls twice as hard as its mass · the Mass-energy reading path

“Any random walk, however irregular its timing, spreads as the square root of time at long times.”

Only if the waiting times between steps have a finite mean. With waits falling as t^(−1−α) and α < 1 the mean is infinite, and 1,500 simulated walkers spread with mean-square displacement growing as t^0.69 for α = 0.75 and t^0.49 for α = 0.5 over the last decade — not in proportion to time, and not for any choice of diffusion coefficient.

Tested in The diffusion that slows the longer it is watched · the diffusion reading path

“Ice cannot exist above 0 °C.”

Ice VII, the form stable above 2.2 GPa, is solid at 139 °C at 3 GPa and at 442 °C at 20.6 GPa, according to the international standard equations for water's melting curve. It has been found trapped in diamonds that formed deep in the Earth's mantle.

Tested in The melting line that turns round · the phase change reading path

“Beer's law gives the attenuation of any layer from the average amount of absorber in it.”

It gives the attenuation of a uniform layer. For any other the transmission averaged over the layer exceeds e^(−average optical depth), by Jensen's inequality: at an average optical depth of 2 an even layer transmits 13.5 per cent, the same absorber in patches over half the area 50.9 per cent, and over a tenth, 90 per cent.

Tested in The pigment that absorbs less for being packed · the attenuation reading path

“Sound passes easily from air into water, since water carries sound well.”

Water carries sound well once it is in it. Crossing the boundary is another matter: the characteristic impedances of air and water differ by a factor of 3,600, and at normal incidence only 0.11 per cent of the power crosses, −29.5 dB. The rest is reflected.

Tested in The lever that lets air speak to water · the impedance reading path

“A black hole is a larger target for light than for anything travelling slowly.”

The reverse. Light is captured inside an impact parameter of √27 GM/c², a disc 6.75 times the horizon's cross-section. A particle arriving at speed v is captured inside roughly 4GM/(cv) — at a tenth of light speed a disc 404 times the horizon's, at the 220 km/s of the Galaxy's dark matter 7.4 million times. The slower the traveller, the larger the target, without limit.

Tested in The target that grows as the traveller slows · the horizons reading path

“Radiating energy away can only damp a vibration.”

Only if the vibration carries positive energy and angular momentum in the frame where the radiation is measured. A pattern that runs backward through a spinning star's material but forward across the sky carries negative angular momentum; the gravitational waves it sends out carry positive angular momentum away, and the pattern grows. The ℓ = m = 2 r-mode of a 1.4-solar-mass neutron star at 700 Hz grows this way in about 160 seconds.

Tested in The wobble that radiating makes larger · the gravitational waves reading path

“A superconductor spun in zero field stays field-free, since it expels magnetic fields.”

It expels the field its carriers feel, and in the rotating frame they feel the Coriolis force as a field of 2mω/e. To cancel it the superconductor generates a real field of the same size along the spin axis — 1.137 × 10⁻¹¹ tesla per radian per second, 5.7 nanotesla in a rotor at 80 Hz.

Tested in The field a spinning superconductor makes of itself · the superconductivity reading path

“Light already has a mass inside a plasma or a superconductor, so a mass for the free photon would change nothing new.”

In a plasma or superconductor the mass-like behaviour comes from charges responding to the field, and once their charge is counted Gauss's law holds exactly. A mass in empty space would have no charges to count: the flux through a surface would fall short of the enclosed charge, with nothing to account for the difference.

Tested in The mass that would make Gauss's law leak · the Gauss's law reading path

“Light can be diffracted by matter, but matter cannot be diffracted by light, which has no structure to scatter from.”

A standing light wave is a periodic potential with period half a wavelength, and a particle crossing it is diffracted exactly as light is by a grating. Sodium atoms were diffracted this way in 1986 and electrons in 2001, and atom interferometers now use such gratings routinely as their beam splitters and mirrors.

Tested in The grating made of light · the matter waves reading path

“A surface that emits poorly at some wavelength will exchange little heat at that wavelength with anything.”

Silicon carbide reflects almost everything between 10.3 and 12.6 µm and so barely glows there — its emissivity at 11 µm is about 5 per cent. Yet across a 10 nm gap nearly all of its heat is carried at 10.55 µm, inside that band, by a surface wave that exists only because its permittivity there is negative.

Tested in The heat that crosses a gap too narrow for light · the blackbody reading path

“A ball set down on a spinning turntable is flung outward and off the edge.”

A solid ball rolling without slipping moves, as seen from the room, on a circle traversed at 2/7 of the table's rate, whatever it was doing when set down, and returns to its starting point every 3.5 turns of the table. Set down at rest relative to the room, it stays put, spinning in place. Nothing flings it outward, because the only horizontal force on it is friction, and friction here turns its velocity rather than increasing it.

Tested in The ball a turntable will not throw off · the circular motion reading path

“A ball placed on a turntable so that it moves with the surface simply rides round with it.”

Riding round would need a steady inward force from friction, and a steady friction force at the contact would also steadily change the ball's spin, which the rolling condition does not allow. The ball instead circles, seen from the room, on a circle three and a half times the radius it started at — 8.75 cm for a start 2.5 cm from the axis.

Tested in The ball a turntable will not throw off · the circular motion reading path

“A spring with a mass on it has one natural frequency.”

It has one for every standing wave that fits along its coils. With a spring a tenth of the load's mass the next two are 10.2 and 20.3 times the bounce; for a spring alone they are the odd harmonics, 3 and 5 times. When one of an engine's harmonics reaches the valve spring's second mode the coils surge and the valve stops following its cam.

Tested in The third of itself a spring carries · the harmonic approximation reading path

“A ball thrown straight up comes down where it was thrown from, since it and the ground both move with the Earth.”

It lands west. While it is above the ground its eastward speed is the ground's speed at the height of the hand, and the ground beneath it is moving at the same speed only at that height; above it, the ball lags behind the sky it is passing through. Thrown up to 100 m at latitude 45°, it lands 62 mm west.

Tested in The ball thrown up that lands to the west · the projectile reading path

“Light from a moving source could travel at c plus the source's speed, and nothing astronomical would show it.”

A star in a 10-day binary orbit at 100 km/s would then send light that overtakes its own earlier light at any distance beyond 4 parsecs; spectroscopic binaries far beyond that would show velocity curves distorted into apparent eccentric orbits, or be seen in two or three places at once. Algol, 28 parsecs away, is eleven times past that distance, and its velocity curve is a clean sine wave.

Tested in The binary star that would be seen twice · the Velocity-addition reading path

“The speed of sound in any matter can approach the speed of light as the matter is heated or compressed without limit.”

Not in a gas of free particles. An ideal relativistic gas has cₛ² = kT(1 + k/cᵥ)/(e + kT), which rises from 5kT/3mc² to exactly one-third of c² and never exceeds it: 0.312 at kT = mc². Exceeding one-third requires forces between the particles, of a kind that stiffens matter under compression.

Tested in The sound speed a neutron star needs · the relativistic thermodynamics reading path

“A particle moving close to the speed of light in a magnetic field has its spin turned much more slowly than its motion, because of time dilation.”

Both turn more slowly — the momentum at eB/γm, the spin at eB/m·(1/γ + a) — and the difference between them, a·eB/m, is the same at every speed. For a muon in 1.45 tesla it is 229 thousand turns a second whether the muon crawls or moves at γ = 29.3.

Tested in The spin that runs ahead of its own motion · the field transformation reading path

“Mixing two fluids of the same density gives a fluid of that density.”

Only if density depends linearly on what is mixed. Seawater's density curves with temperature, while mixing averages temperature and salt in a straight line, so an even mixture of two waters on one density surface lands on the dense side: 0.18 kg/m³ denser for parents 4 and 16 °C apart. The mixture sinks away from both.

Tested in The mixture heavier than either water · the stratification reading path

“A floating square beam always lies flat, one face up, since that keeps its weight lowest.”

Only if it is lighter than 0.211 or heavier than 0.789 of water's density. Between about 0.28 and 0.72 it floats on a corner with a diagonal vertical, and in the ranges between it floats tilted — 26.6° at a density of 0.25. What a floating body minimises is not the height of its centre of gravity but that height relative to its centre of buoyancy.

Tested in The log that floats on its corner · the buoyancy reading path

“The more monochromatic the light, the better an interferometer can locate a reflecting surface.”

A monochromatic source makes fringes at every path difference, so it cannot say where equal paths are, only how far from one fringe to the next. A broad spectrum makes fringes only where the paths match: a 50 nm band at 840 nm confines them to an envelope 6.2 µm wide, which locates each reflector in depth to that precision.

Tested in The depth a broad spectrum can see · the coherence reading path

“Wet things are darker because the water absorbs light.”

A millimetre of water absorbs a negligible fraction of visible light. The darkening comes from the path the light takes: a water film returns 47 per cent of the light scattered back up by the surface beneath it, for another chance of absorption, and water in the pores lets light go deeper before it escapes. A mid-grey surface under a water film falls from 0.5 to 0.32 reflectance with no absorption in the water at all.

Tested in The sand that darkens when it is wet · the scattering reading path

“The hotter a surface, the faster it boils water off it.”

Only up to the critical heat flux, about 1.1 MW/m² at some twenty degrees above boiling for water at one atmosphere. Beyond that, vapour blankets the surface, and the flux falls as the surface gets hotter — to about 19 kW/m² at the Leidenfrost point, a surface three times hotter passing fifty-eight times less heat. A drop on a pan at the Leidenfrost point outlasts one on a much cooler pan.

Tested in The boiling curve that turns back · the phase change reading path

“A system further from equilibrium must take longer to reach it.”

Only if relaxation has a single timescale. With a fast and a slow mode, the late approach is set by how much of the slow mode the starting state contains. In the three-state model here a start at T = 0.901 is five times further from the bath's equilibrium than one at T = 0.50, carries almost none of the slow mode, and overtakes it at a time of about 7 units, staying ahead by orders of magnitude.

Tested in The hotter start that cools first · the entropy reading path

“Scattering between a source and a receiver destroys any possibility of focusing back onto the source.”

It destroys an ordinary lens's focus, which relies on a known, simple medium. A time-reversal mirror needs only the wave that arrived: in the model here, a mirror 10 wavelengths wide and 40 away focuses to 4.8 wavelengths in empty space, and to 1.5 wavelengths through a slab of 200 lossless scatterers, because the scatterers redirect waves into the mirror from a wider range of angles than it subtends.

Tested in The mirror that sends a wave back to where it began · the huygens reading path

“Light launched into one of a row of coupled waveguides spreads into a hump centred on the guide it started in.”

It spreads as |Jₙ(2Cz)|²: two beams moving outward at 2C guides per unit length, with most of the power near their fronts. After seven coupling lengths the starting guide holds 2.9 per cent of the light and the brightest guides, twelve away on either side, 8.1 per cent each.

Tested in The light that spreads in two beams · the periodic media reading path

“Warming the plates can only add to the attraction, since it adds fluctuations.”

For ideal mirrors it does add. For plates that let the static magnetic fluctuation through, as an ordinary metal does, removing that one term makes the pressure fall below its zero-temperature value: to 0.66 of it at 3.2 μm at 300 K, before the classical part takes over further out.

Tested in The force that forgets Planck's constant · the retardation reading path

“How the plasma is distributed across the column changes how much current is needed to confine it.”

Solving J × B = ∇p radius by radius for flat-topped, parabolic, peaked and hollow profiles with the same particles and temperature gives 0.800 MA for every one. The profile sets where the current flows and how strong the field is inside, not the total.

Tested in The current that squeezes what carries it · the ampere law reading path

“The light a gas scatters is the light each atom scatters, times the number of atoms.”

Only for a gas whose occupations are small. In an ideal Fermi gas the scattering per atom is the classical value times S(q), the share of atoms whose recoil lands in an empty state. With the Fermi momentum equal to the photon's and the gas at 0.05 of the Fermi temperature that is 0.78 of the classical total; with it twice the photon's, 0.47.

Tested in The light a full sea will not scatter · the exclusion reading path

“Gravity is far too weak to quantise anything in a laboratory; quantum states of a body in a gravitational field are a matter of black holes and the early universe.”

A neutron on a horizontal mirror has quantised states in the Earth's field with a natural height (ħ²/2m²g)^(1/3) = 5.87 μm and energies of 1.41, 2.46, 3.32 and 4.08 peV. They were seen in 2002 as steps in the transmission of a slit between a mirror and an absorber, and since 2011 as resonances at 255 Hz and above.

Tested in The bounce that comes in fixed heights · the matter waves reading path

“A softer, more yielding material always rolls with more resistance.”

A material that deforms without loss returns all its energy and gives a symmetric pressure. On the computed bed the pressure is symmetric at 1 mm/s, where the bed relaxes fully, and at 1000 m/s, where it has no time to relax and behaves as a stiffer elastic solid. Resistance comes from the lag between load and response, not from softness.

Tested in The push a millimetre ahead of the axle · the friction reading path

“Everything falls at g in a vacuum, so a drop falling through mist with no air resistance falls at g.”

A body that gathers what it meets at rest has to share its momentum with it. A drop sweeping a still mist settles to exactly g/7, from any starting size and in any density of mist: integrated from 0.2 mm, 0.05 mm and effectively zero, all three are at 0.143 g after 8 s.

Tested in The raindrop that falls at a seventh of g · the momentum reading path

“Length contraction is too small to matter outside thought experiments.”

For a 6 GeV electron γ is 11,742, and a row of magnets alternating every 3 cm is, in the electron's frame, a row alternating every 2.55 μm. The electron radiates at that period, and its light reaches the laboratory Doppler-shortened by a further 2γ, as 1.6 Å X-rays: the contraction is one of the two factors of γ in every undulator's wavelength.

Tested in The magnet period that comes out as an X-ray · the length contraction reading path

“Only an accelerating charge radiates.”

In a medium a charge in uniform motion radiates if it outruns light there. Frank and Tamm's formula gives 214 visible photons per centimetre of water for a singly charged particle near the speed of light, from an electron that neither turns nor slows appreciably. The radiating is done by the medium's electrons, polarised by the passing field.

Tested in Why the glow of a fast charge is blue · the radiating charge reading path

“A radio station is heard most easily close to its transmitter and fades with distance.”

For a sky wave above the critical frequency there is a skip zone round the transmitter that no single hop reaches: at 14 MHz with an 8 MHz layer it extends to about 900 km with the layer treated as a mirror, and over 1,100 km with rays traced through it. Inside it the station is heard only by its ground wave, which dies within tens of kilometres.

Tested in The frequency the sky returns only at a slant · the plasma oscillation reading path

“The coloured bands seen in plastic between polarising filters are a property of the plastic, like a dye.”

They are contours of stress. Between crossed circular polarisers the transmitted light is sin²(πΓ/λ), with the retardation Γ = Ct(σ₁ − σ₂) proportional to the local difference of principal stresses; unloaded, the same plastic is dark everywhere. A 50 mm polycarbonate disc squeezed by 500 N shows order 3.61 at its centre, rising without limit at the loads.

Tested in The stress that polarised light can see · the polarisation reading path

“A bigger telescope shows a faint nebula brighter.”

No lossless instrument can raise the surface brightness of an extended source above what the naked eye sees. Through a 200 mm telescope it stays at the naked-eye value up to 28.6×, where the exit pupil equals a 7 mm eye pupil, and falls as the square of the exit pupil beyond: to 0.33 at 50×, 0.082 at 100×. The bigger telescope makes the nebula larger at the same brightness.

Tested in The nebula no telescope can brighten · the etendue reading path

“Every lens makes the edge of the image darker by the same law.”

The law depends on how the lens maps angles onto the image. At 60° off axis an ordinary rectilinear lens delivers 0.063 of the central light, a stereographic fisheye 0.281, an equidistant 0.413 and an equal-area fisheye 0.500, from the same light through the aperture spread over different areas of image.

Tested in The four cosines at the edge of a photograph · the imaging reading path

“A free wave packet must spread, and its peak must move in a straight line.”

Berry and Balazs's packet |Ai(x − t²/4)|² is an exact free solution that never spreads and whose main lobe accelerates along x = t²/4: from −1.02 at t = 0 to 2.98 at t = 4 and 5.23 at t = 5, in units with ħ = m = 1.

Tested in The packet that bends with nothing pushing it · the wave packets reading path

“A Doppler shift needs the source or the observer to move towards or away from the other.”

Light with helical wavefronts exp(iℓφ) scattered from a surface that spins in place is shifted by ℓΩ: 100 Hz for ℓ = 1 at 100 revolutions per second, 2,000 Hz for ℓ = 20. Circularly polarised light through a half-wave plate spinning at Ω is shifted by exactly 2Ω. Nothing approaches or recedes.

Tested in The Doppler shift with nothing coming closer · the doppler reading path

“A bottle's note is a standing wave in the bottle, like a flute's, so its pitch depends on the bottle's length.”

The note's wavelength, about 3 m at 109 Hz for a 750 ml bottle, is ten times the bottle. It is the plug of air in the neck bouncing on the air in the body, f = (c/2π)√(A/VL′), and a sphere, a cube, a tall cylinder and a flat box of the same volume with the same neck all sing 109 Hz while their own standing-wave modes lie between 572 and 2,016 Hz.

Tested in The note a bottle sings whatever its shape · the resonance reading path

“The same attraction assembles microscopic particles floating at an interface.”

The meniscus a floater makes by its weight shrinks as the cube of its radius, so the binding energy falls as about the sixth power. For spheres of half water's density it is 3.6 × 10¹⁰ kT at 1 mm and equals kT near 12 μm; smaller particles at interfaces clump by charges and by contact-line undulations, not by their weight.

Tested in The cereal that gathers at the edge of the bowl · the capillarity reading path

“A balloon rises until the air is too thin to support its gas.”

A balloon floats where the air's density makes its lift equal its weight, set by the envelope's volume and the mass it carries. A zero-pressure balloon of 1.1 million m³ carrying 3 tonnes floats at 41.3 km with 3,500 or 5,000 cubic metres of helium at launch alike; the extra gas only vents. A sealed 1,500 m³ balloon weighing 95 kg floats where the air is 0.063 kg/m³, 22.0 km.

Tested in The balloon that floats at a density · the buoyancy reading path

“Ice on a lake thickens at a steady rate while the cold lasts.”

Its growth rate falls as the ice thickens, because the latent heat released at the underside must be conducted up through all the ice above. At 10 K below freezing Stefan's law gives 11 cm after a day, 29 cm after a week and 61 cm after a month: h = √(2kΔT t/ρL), the square root of time.

Tested in The ice that grows more slowly the thicker it gets · the diffusion reading path

“Good conductors of heat are metals, because heat is carried by free electrons.”

In an insulator heat is carried by phonons, and diamond, with none of the free electrons, conducts about 2,200 W/m K at room temperature against copper's 400. Kinetic theory of the phonon gas, κ = Cvℓ/3, with diamond's sound speed and Debye temperature, rises through 10⁴ W/m K near 100 K.

Tested in The gas of sound that carries a diamond's heat · the kinetic theory reading path

“Salting the water makes pasta cook faster because salt water boils at a much higher temperature.”

Ten grams of salt in a litre is 0.34 mol of ions per kilogram, which raises the boiling point by 0.513 × 0.34 = 0.18 K. The effect is real and a thousand times too small to matter in a kitchen.

Tested in The degree a spoonful of solute buys · the chemical potential reading path

“A charge moving at constant velocity can radiate only by outrunning light in the medium, as Cherenkov radiation does.”

It also radiates wherever the medium changes, at any speed. Crossing from vacuum into polypropylene, a singly charged particle with γ = 4,000 radiates αγħωₚ/3 ≈ 203 eV, in a hollow cone at angle about 1/γ, without accelerating and far below any Cherenkov threshold for X-rays, where every material's index is below one.

Tested in The light a charge makes by changing medium · the radiating charge reading path

“Antimatter's fall is easy to measure: drop a positron or an antiproton and watch.”

An antiproton weighs 1.6 × 10⁻²⁶ N, and an electric field of 1.02 × 10⁻⁷ V/m balances it. Stray fields from surface patches on any metal are millivolts per metre, ten thousand times its weight, and its own image charge in a wall a centimetre away pulls with 35 times its weight. A direct measurement needed neutral antihydrogen, held and released in a magnetic trap.

Tested in The antiatom that had to be weighed anyway · the equivalence principle reading path

“An electrostatic lens can be made converging or diverging by the sign of the voltage on its electrode, as a glass lens can by its shape.”

A lens with the same potential on both sides converges whichever way its centre is charged. Traced through three coaxial tubes, the focus lies 4.0 tube radii beyond the centre with the middle tube at four times the outer potential and 6.0 radii beyond with it at three tenths; neither sign makes a diverging lens, because in each gap the electron is slower, and lingers longer, where the push is towards the axis.

Tested in The voltage that bends an electron like glass · the potential reading path

“Tuning two coils to resonance makes their magnetic coupling stronger.”

The coupling coefficient is set by geometry alone — 0.099 for two loops one radius apart, 0.0028 at five radii — and no capacitor changes it. What resonance changes is how many cycles the energy survives: the link's best efficiency depends only on U = k√(Q₁Q₂), so k = 0.003 with Q = 1,000 gives U = 3 and 54 per cent.

Tested in The coupling a resonance rescues · the induction reading path

“When the ocean warms and the sea rises, the pressure on the sea floor rises too, because there is more water above it.”

Thermal expansion adds height and no mass. A warmed column stands taller and is less dense in exact proportion, so the weight per square metre — the pressure at the bottom — is unchanged. A pressure gauge on the sea floor sees melting ice arrive and is blind to the sea's thermal swelling.

Tested in The rise a joule of heat buys · the hydrostatics reading path

“A red cell in dilute salt water bursts because its membrane is weak and gives way under the osmotic pressure.”

Before it bursts the membrane carries almost no tension: the cell takes in water by changing shape, from a 94 fL disc towards the 146 fL sphere its 134 μm² of membrane can enclose. Only at the sphere does the membrane have to stretch, and it tears at about 2 per cent stretch, holding 6 kPa against an osmotic pressure of about 700 kPa. The burst is set by the shape running out.

Tested in The cell that bursts by geometry · the osmosis reading path

“An atmosphere that cools with height more slowly than the dry adiabatic rate cannot convect.”

It cannot convect dry air. At 6.5 K/km it is conditionally unstable at every temperature above about −2 °C at 850 hPa: saturated air cools at 3.8 to 6 K/km there. A parcel lifted from 30 °C with a 23 °C dew point saturates at 890 m and is warmer than its surroundings from 2.1 to 13.4 km, with 2,543 J/kg of available energy.

Tested in The cloud that cools more slowly than the air · the stratification reading path

“A flea scaled up to the size of a person would jump over a skyscraper, since it jumps a hundred times its own length.”

Jump height does not scale with size at all. Muscle work and the work of lifting a body both grow with mass, so the rise is the same at every size: about 0.61 m with a tenth of the body as jumping muscle doing 60 J/kg. The flea's 18 cm is a hundred body lengths only because it is small; a person-sized flea would rise about the same half-metre as a person.

Tested in The jump that does not get higher with size · the energy reading path

“A centrifuge separates isotopes because the heavier ones are thrown outward by a stronger force.”

Every molecule is thrown outward and every one is held by the gas's pressure; at equilibrium the gas obeys Boltzmann's law in the centrifugal potential, n ∝ exp(mω²r²/2kT). Two isotopes differ only through m in that exponent, so their ratio changes from axis to wall by exp(Δm v²/2kT): 1.24 for uranium hexafluoride at a 600 m/s wall.

Tested in The spin that sorts by a difference of mass · the circular motion reading path

“A poor reflector such as water or soil reflects only a small fraction of a radio wave, so the ground hardly affects a signal travelling across it.”

At grazing incidence every surface reflects with a coefficient of −1. Over flat ground the reflected wave cancels the direct one more and more completely with distance: from a 30 m mast at 900 MHz to a receiver 1.5 m up, the power falls as d⁻⁴ beyond 540 m, 10 dB below free space at 5.4 km and 30 dB below at 54 km.

Tested in The surface that cancels whatever skims it · the refraction reading path

“Tilting a lens gives a photograph more depth of field.”

It gives the same depth of field in a different direction. The near and far limits of sharpness swing about the same hinge as the plane of focus, so the sharp zone becomes a wedge with its edge at the hinge — for a lens of focal length 50 tilted 3° at f/8, from 1,264 to 1,444 units along the axis — thin near the hinge and thicker far from it. It is pointed along the subject instead of across it.

Tested in The plane of focus a tilted lens tilts · the imaging reading path

“A magnetic resonance scanner forms its image with radio waves, so its resolution should be limited to about their wavelength.”

Nothing is focused and nothing is imaged by a wave. A gradient writes each position into the precession frequency, 425.8 Hz per mm at 10 mT/m, and the image is the Fourier transform of the signal: its pixel is 1/(γ̄GT), 0.62 mm for a 3.76 ms readout, where the 63.87 MHz wave is 4.7 m long in air.

Tested in The image made of frequencies · the spin reading path

“Radiocarbon dating stops at about 50,000 years because after nine half-lives there is too little carbon-14 left to measure.”

An accelerator counts carbon-14 atoms at a fraction of a per cent of the modern level easily. What stops the method is contamination: a sample holding a fraction f of modern carbon reads at most τ ln(1/f) — 24.8 thousand years for 5 per cent, 38.1 thousand for 1 per cent, 57 thousand for 0.1 per cent — whatever its true age.

Tested in The clock read by counting atoms · the decay reading path

“The most useful place for a detector is straight down the axis of a neutrino beam, where the flux is highest.”

On the axis the neutrino takes a fixed 0.427 of each pion's energy, so the spectrum copies the pions' and is broad — 1.5 GeV wide at half maximum in the computed beam. At 2.5° off the axis the same pions give a peak 0.10 GeV wide near 0.68 GeV, with almost nothing above it: fewer neutrinos, far more of them at the energy an oscillation experiment over 295 km needs.

Tested in The neutrino energy an angle fixes · the relativistic dynamics reading path

“The ideal gas law pV = NkT is a low-speed approximation that relativity corrects at high temperature.”

Pressure is n⟨p·v⟩/3, and for any relation between energy and momentum Boltzmann's weighting gives ⟨p·v⟩ = 3kT exactly. Averaged over the Maxwell–Jüttner distribution it equals 3kT to four decimal places from kT = 0.003 mc² to 300 mc². It is the energy, not the pressure, that relativity changes.

Tested in The gas law relativity leaves exact · the relativistic thermodynamics reading path

“A nucleus emits a gamma ray with exactly the energy of the transition between its two levels.”

It emits the transition energy less the recoil, E²/2Mc², because the photon carries momentum and the energy that leaves is fixed by the invariant mass of the decay, (M² − M′²)c²/2M with M and M′ the masses before and after. For iron-57's 14.4 keV line the deficit is 1.96 meV, 420,000 times the line's natural width; an identical absorber needs as much again in excess, so the two miss by 3.9 meV.

Tested in The gamma ray its own kind cannot absorb · the Mass-energy reading path

“Water in a cloud freezes as soon as the cloud is colder than 0 °C.”

Pure droplets freeze on their own only near −38 °C; above that they need a rare particle to nucleate the ice, and most clouds between 0 and −20 °C are largely liquid. That supercooled liquid has a higher chemical potential than ice, 217 J/mol higher at −10 °C, which is why ice that does appear grows at its expense.

Tested in The ice that grows by stealing from the droplets · the chemical potential reading path

“Sound cannot travel in the upper atmosphere because the air is too thin to be loud.”

A thin gas carries sound at the same speed as a dense one at the same temperature; what it cannot carry is high frequencies. The loss per wavelength is 4.26ωτ with τ = η/p, and since η does not depend on pressure, the highest frequency surviving a wavelength falls in proportion to p: 212 MHz at the ground, 3 kHz at 80 km, 92 Hz at 100 km.

Tested in The note too high for thin air · the kinetic theory reading path

“The first wave to reach a seismometer from an earthquake is the one that travelled the shortest path.”

It is the one that took the least time. Beyond the crossover distance — 185 km for a 35 km crust at 6 km/s over mantle at 8 km/s — the head wave that dives to the mantle, runs along it and climbs back arrives first, beating the direct wave by 4.8 s at 300 km although its path is 26 km longer.

Tested in The wave that arrives first by going the long way · the huygens reading path

“A higher-frequency ultrasound scanner makes a better image.”

It makes a smaller one. Axial resolution, Nc/2f, and the deepest echo a 60 dB scanner recovers, D/2α₀f, both fall as 1/f in soft tissue: 0.51 mm and 20 cm at 3 MHz, 0.10 mm and 4 cm at 15 MHz. The number of resolvable depths along a line, D/(α₀Nc), is about 390 at every frequency.

Tested in The depth an ultrasound image pays for its sharpness · the attenuation reading path

“Adams's 1925 measurement of the redshift of Sirius B confirmed general relativity.”

It agreed with Eddington's forecast of about 20 km/s, but the forecast rested on a radius of 19,600 km inferred from an assumed surface temperature of 8,000 K. The star is about 25,000 K and under 6,000 km across, and its redshift, measured with its light separated from Sirius A's, is 80.4 ± 4.8 km/s. The agreement was between two wrong numbers, the measured one almost certainly contaminated by the companion's scattered light.

Tested in The redshift that weighs a dead star · the gravitational redshift reading path

“The energy of a cyclotron line gives the surface magnetic field of a neutron star directly, at 11.6 keV per 10⁸ tesla.”

The line is seen redshifted by the star's gravity, about z = 0.24 for a 1.4 solar-mass, 12 km star, and it forms where the scattering is, which may be above the surface in a dipole field falling as the cube of the distance. A 38 keV line read naively gives 3.3 × 10⁸ T; with the redshift and the relativistic level spacing, 4.2 × 10⁸ T at the surface, and a layer 1 km up would show the same field as 31 keV.

Tested in The circling that comes in quanta · the radiating charge reading path

“Lowering the pressure of a gas always makes it easier to break down.”

Only on the high-pressure side of Paschen's minimum. A 1 mm gap in air holds about 4.5 kV at sea level and about 300 V at roughly a hundredth of an atmosphere, but at a thousandth of that it holds more again, and in a good vacuum the gas cannot break down at all; what fails then is the metal.

Tested in The spark that needs room to start · the dielectrics reading path

“The faster the top spins, the more stable it floats.”

Spin holds the axis against the torque that would flip it, so a slow top falls. But the axis follows the field direction by precessing about it at μB divided by the spin angular momentum, and a top spun so fast that this precession is slower than its sideways swing in the trap keeps its axis fixed in space and is a fixed magnet again. For the model top the window runs from about 12 to about 92 revolutions a second.

Tested in The window a spinning magnet floats in · the magnetism reading path

“A railgun has no recoil, because the magnetic force acts only on the projectile.”

The current turns the corner at the breech and the breech sits in the field of the rails; integrated from Biot–Savart it is pushed backward by the same 3.13 MN that drives the armature forward. The rails are pushed apart as well, by 18.9 MN on every metre, six times the forward push.

Tested in The push at the end of a pair of rails · the ampere law reading path

“A floating needle is held up by surface tension instead of by buoyancy, so Archimedes' principle does not apply to it.”

Keller's theorem: the vertical pull of surface tension along the contact lines equals the weight of the water displaced by the dent in the surface. Counting that water, the needle displaces exactly its own weight — 79 per cent of it in the dents and 21 per cent under the needle, for a 1.36 mm steel needle.

Tested in The water the dent displaces · the buoyancy reading path

“Sound from a source on a beach travels outward through the sand in straight lines, weakening with distance.”

Its speed rises with depth, so every ray bends back towards the surface. Where the speed grows as a power of depth every ray is a scaled copy of every other, and each returns to the surface 5.89 times its deepest point away from the source.

Tested in The sound sand carries faster the deeper it goes · the granular matter reading path

“A standing vibration in a ring is fixed to the material, so when the ring turns the pattern turns with it.”

The Coriolis force couples the two equal-frequency orientations of the vibration, and the pattern slips back against the material by $2/(n^2+1)$ of the rotation. For the lowest mode a quarter turn of the ring moves the pattern 54°, three-fifths of the way.

Tested in The vibration that turns part of the way · the rotation reading path

“Without gravity there is no way to weigh anything.”

Weighing by vibration uses inertia, not weight. A chair on springs of 900 N/m swings with a period that changes by 11 ms per kilogram of occupant near 75 kg, and timed to a millisecond it gives the mass to about 0.1 kg — which is how the crews of Skylab were weighed in 1973.

Tested in The weight a frequency can find · the harmonic approximation reading path

“A mixture of two clear, colourless materials cannot be coloured.”

A layer of glass powder in a liquid matched at one wavelength transmits that wavelength and scatters the rest out of the beam. A 5 mm layer of 10 μm grains passes a band 36 nm wide centred where the dispersion curves cross, and looks coloured in transmission though neither ingredient absorbs.

Tested in The colour at which glass disappears · the dispersion reading path

“A mirror smooth to a few atoms reflects X-rays as well as it reflects light.”

An 8 keV X-ray's wavelength is 0.15 nm. A silicon mirror reflects it almost totally below its critical angle of 0.22° and above it falls as the fourth power of the angle, faster still if rough: at 0.6° from 1.4 × 10⁻³ for a perfect surface to 2.1 × 10⁻⁶ with 3 nm of roughness.

Tested in The rough surface that is a mirror at a glance · the refraction reading path

“To remove Doppler broadening a gas must be cooled until its atoms barely move.”

A pump beam sent against the probe empties the atoms it is resonant with, and the two beams share atoms only if those atoms are at rest along the beams. The probe's absorption then shows dips about 8.6 MHz wide at each transition in a cell at 300 K, in which atoms move at 169 m/s along the beam.

Tested in The lines a warm vapour hides · the atomic spectra reading path

“Superfluid helium climbs walls because, having no viscosity, it is drawn up them by a special force.”

The film is held by ordinary van der Waals attraction to the wall and is as thick above the λ point as below it — 30 nm at 1 cm, falling as the cube root of the height. What changes below the λ point is that the film can flow along the wall without friction.

Tested in The film whose thinnest point sets the flow · the superfluidity reading path

“Electrons can be drawn out of a metal only by giving them enough energy to climb over its work function, by heat or by light.”

A field of a few gigavolts per metre tilts the surface step into a triangle 1–2 nm wide at the Fermi level, and electrons tunnel through it from a cold metal: tungsten emits 5 × 10⁵ A/m² at 4 GV/m with no heating at all.

Tested in The electrons a field pulls from cold metal · the tunnelling reading path

“As a particle approaches the speed of light, it becomes impossible to give it more energy.”

Energy goes in without limit; only the speed stops responding. An electron needs 0.66 MeV for 0.9c, 10.9 MeV for 0.999c and 361 MeV for six nines, each further nine costing about √10 times more, and colliders have given electrons 104.5 GeV and protons 6.8 TeV.

Tested in The speed that stops while the energy keeps coming · the relativistic dynamics reading path

“A charged particle circling in a fixed magnetic field always goes round at the same frequency, whatever its speed.”

Only while it is slow. The frequency is eB/2πγm, so a proton in a 1.45 T field goes round 22.1 million times a second when slow and 2.1 per cent less often at 20 MeV, enough to fall out of step with a fixed-frequency cyclotron within a few hundred turns.

Tested in The turns a muon makes at any energy · the time dilation reading path

“Frost heave happens because water expands by nine per cent when it freezes.”

Stephen Taber froze clay saturated with benzene and with nitrobenzene, both of which contract on freezing, and the clay heaved. Heave in the field comes from lenses of clear ice that grow by drawing liquid up from below; a metre of silt with forty per cent porosity could gain at most about 3.6 cm from expansion in place, and susceptible ground lifts several times that.

Tested in The frost that lifts by drinking · the chemical potential reading path

“Water evaporates slowly because freeing each molecule from the liquid takes a lot of energy.”

The energy is already accounted for in the vapour pressure, and at that vapour pressure molecules leave water at 20 °C at 2.54 kg per square metre per second. A still puddle loses about 4 × 10⁻⁵ — some sixty thousand times less — because the vapour it makes cannot get away and almost all of it returns.

Tested in The puddle that could dry in a second · the kinetic theory reading path

“Where light meets a mirror, its brightest layer lies against the mirror surface.”

A metal reflects light inverted, so the electric field of the incoming and reflected waves cancels at the surface at every instant; the first bright sheet lies a quarter-wavelength out, 92 nm in an emulsion of index 1.5. Wiener's film, sloping up through the pattern, was clear where it touched the mirror.

Tested in The photograph that keeps the spectrum · the superposition reading path

“The Earth's lowest resonance is light going once round the planet, about 7.5 hertz.”

For perfectly conducting walls the lowest tone of a spherical cavity is √2 higher than a ring's, (c/2πR)√2 = 10.6 Hz, because a wave on a sphere is not a wave on a loop. The measured 7.8 Hz lies close to 7.5 only because the leaky ionosphere slows the wave to about three-quarters of the speed of light; two corrections almost cancel.

Tested in The tone the whole Earth rings at · the guided waves reading path

“A fast enough flow of liquid iron can keep up any magnetic field, including the simple symmetric dipole the Earth appears to have.”

Not a field that is exactly symmetric about an axis. Its meridional field lines must circle a ring on which the meridional field vanishes, and in a steady state the only thing that can drive current round the axis there is the flow crossing that field. In the slowest dipole mode of a sphere the ring is at 0.873 of the radius and the current required there is 89 per cent of its largest value; the speed a flow would need to supply it diverges at the ring, whatever the flow.

Tested in The field no symmetric flow can keep · the flux freezing reading path

“A heavy enough clock can time a distance as precisely as wanted, since its quantum spread falls as its mass grows.”

Its spread falls as the inverse square root of the mass, but a clock must be larger than its own Schwarzschild radius to read anything, and that size grows with the mass. For a one-metre distance the sum is least at a 5.4-tonne clock, and any heavier clock does worse.

Tested in The clock that cannot be both light and good · the planck scale reading path

“A graviton with a tiny enough mass would be indistinguishable from a massless one everywhere outside cosmology.”

In the linear theory the opposite holds. A massive spin-2 field always carries a scalar state that pulls on matter and not on light, so with the planets' orbits matched, light bending comes out at three-quarters of general relativity's value for every mass, however small. Cassini measured 1.00001 ± 0.00001.

Tested in The four shapes a wave of gravity leaves out · the gravitational waves reading path

“Inside a cage of wires the static field leaking through the gaps dies away exponentially, so a few spacings in, the cage is as good as a solid box.”

Only the ripple dies exponentially — at e^(−2πd/s), measured as 11.9 per radius against 2π/s = 12.0 for a twelve-wire cage. The uniform part of the field is not periodic and does not decay at all: a cage of twelve joined wires, each 2 per cent of the spacing in radius, lets 25.8 per cent of an applied field through to its centre.

Tested in The field a ring of wires lets through · the potential reading path

“Because a superconductor has no resistance, it dissipates nothing in any field.”

In a steady field it dissipates nothing. In an alternating field the pinned vortices are dragged in and out each cycle, and the heat per cycle is the area of the hysteresis loop: 2μ₀H³/3H* per unit volume below full penetration, 2μ₀H*H − 4μ₀H*²/3 above it.

Tested in The field a superconductor keeps after all · the superconductivity reading path

“In a coarsening froth the small bubbles shrink and the large ones grow, because a small bubble has the higher pressure.”

That is the rule for isolated bubbles. In a two-dimensional froth a cell's area changes at (πκγ/3)(n − 6), set by its number of sides alone: computed at three sizes four times apart, a pentagon loses area at −1 and an octagon gains at +2 in every case. A small seven-sided cell grows while a large five-sided neighbour vanishes.

Tested in The bubble that grows by counting its sides · the surface tension reading path

“Near the critical point the surface tension falls in proportion to the distance from it, as Eötvös's rule says.”

Eötvös's straight line holds over most of the range and fails near the end. Water's tension goes as τ^1.256, τ = 1 − T/Tc; the van der Waals theory, which treats each molecule as feeling the average of its neighbours, gives τ^1.50, and the measured exponent is the one shared by every fluid and every uniaxial magnet.

Tested in The tension that vanishes where the phases meet · the surface tension reading path

“Removing energy from a bound system always makes it cooler and slower.”

For an inverse-square force the time-averaged kinetic energy is minus half the potential (measured: −0.500), so the total energy is minus the kinetic. Taking energy out raises the kinetic energy: a satellite losing energy to drag speeds up, and a star radiating heats up.

Tested in The third law that falls out of a change of scale · the least action reading path

“A projectile's path is a parabola.”

It is an arc of an ellipse with the Earth's centre at one focus. The parabola is the top of that ellipse when the throw is small compared with the Earth: the far focus of a 5 km/s throw's ellipse lies 896 km below the top of the flight, where a parabola with the same speed and gravity at the top would put its focus at 795 km, and the two converge for slower throws.

Tested in The parabola that is the top of an ellipse · the projectile reading path

“Any bundle of light rays, however it is arranged, can be produced from a point source by a suitable arrangement of lenses and mirrors.”

Only a bundle whose rays are the normals of some surface. A twisted bundle of skew lines, whose slopes summed round a loop grow with the area enclosed — 8.5 waves at the edge of the drawn example — is normal to no surface and cannot be made from a point by any optics.

Tested in The surface every ray is normal to · the fermat reading path

“A small change to the light — an adjustment of one wave's strength or phase — can remove a dark point.”

A dark point's winding is a whole number and cannot change by a small amount, so a single point cannot disappear; it can only meet one of opposite winding and annihilate with it. Raising one of three waves' amplitude from 1 towards 2 brings an opposite pair together, and they vanish only at a = 2, where that wave can no longer be cancelled anywhere.

Tested in The dark lines a wave cannot avoid · the diffraction reading path

“Dense coding beats the one-bit-per-qubit limit and so sends information more cheaply than is otherwise possible.”

Counting the qubit that delivered the shared pair, two bits cost two qubits, the ordinary rate. What changes is timing — the pair can be delivered before the message exists — and security: an eavesdropper holding only the travelling qubit of a maximally entangled pair can extract 1 − H(1/2) = 0 bits.

Tested in The qubit that carries two bits · the entanglement reading path

“Ramsey's method works on a cloud of atoms however their individual frequencies are spread.”

Each atom's fringes sit at its own frequency, so the cloud averages shifted copies and the fringes wash out once the spread approaches the fringe spacing. With a free time of 10τ, a spread of 0.01 cycles per τ cuts the central contrast to 0.77 and one of 0.025 to 0.21.

Tested in The wait that sets the precision · the measurement reading path

“A rod too long for a hole in its own frame cannot fall through it, whatever another observer sees, since whether it passes through is a fact.”

Whether it passes through is a fact, and it does in both frames. A rod of rest length L sliding at 0.8c over a hole of rest length L passes through level in the plate's frame; in its own frame its front starts falling 0.80 L/c before its rear, it tilts by 21.8°, and it threads through a hole contracted to 0.60 L front first.

Tested in The rod that is level for only one observer · the simultaneity reading path

“The equivalence principle predicts the full bending of starlight by the Sun.”

Applied in flat, accelerating frames it captures only the effect of clocks running at different rates, which gives the Newtonian 0.875″. General relativity's 1.751″, confirmed by radio interferometry to parts in ten thousand, needs the curvature of space around the Sun as well, which no flat rocket contains.

Tested in The rocket in which light travels on circles · the accelerated frames reading path

“A moving object, however fast, has a definite direction of travel that every observer agrees on.”

Any real object, yes. A spot moving at u > c along a screen is seen by an observer moving along it at v with speed (u − v)/(1 − uv/c²): infinite at v = c²/u — 0.333c for a 3c spot — and reversed for faster observers. Its successive positions are spacelike-separated events, whose order is not absolute.

Tested in The spot that outruns light and carries nothing · the Velocity-addition reading path

“Water's large latent heat of boiling shows only that its molecules are strongly attracted to each other.”

Strong attraction would raise its boiling point without raising its entropy of boiling. Water's entropy of boiling, 108.9 J/(mol K), is 25 per cent above Trouton's value: its hydrogen bonds hold the liquid in an ordered network, so boiling destroys order as well as separating molecules.

Tested in The entropy every liquid gains on boiling · the phase change reading path

“Particles that stick wherever they land, with no preference for any direction, build a compact, roughly round clump.”

Only if they arrive anywhere on the surface with equal chance, as in an Eden cluster, whose mass grows as r^2.00. If they arrive by random walks, the tips catch them first: a cluster of 3,000 is 220 steps across against 80 for the Eden cluster, and its mass grows as r^1.65, near the 1.71 of large simulations.

Tested in The cluster that grows where the walkers arrive · the diffusion reading path

“Narrowing a harbour's entrance protects it from waves at every frequency.”

It shields the harbour from waves off its resonance and amplifies the resonance itself: in a channel model, the peak amplification at the harbour's head rises from 7.6 with no constriction to 8.6 at a mouth one tenth as wide and 11.5 at three hundredths, because less of the energy that gets in can get back out. Miles and Munk called it the harbour paradox.

Tested in The tide a lake makes by itself · the standing waves reading path

“A thicker, stiffer pane of glass always insulates better at every frequency.”

Stiffness opens a window. Where the bending wave in the pane travels along it as fast as the sound's trace along its surface, stiffness cancels mass and sound passes almost freely; for 4 mm glass this coincidence is near 3 kHz, and just above it the field-incidence loss is 24 dB below the mass law. A thicker pane moves the window lower, into the range where speech carries its consonants.

Tested in The wall that stops sound by its weight · the impedance reading path

“From any direction an observer can see at most half of a sphere.”

Only if light travels in straight lines. Light leaving a neutron star of 1.4 solar masses and 12 km is bent round it on the way out, and the last ray to escape leaves tangentially from a point 121° from the one facing the observer, so 76 per cent of the surface is in view at once. A star of 1.76 Schwarzschild radii would show all of it.

Tested in The star whose far side can be seen · the light deflection reading path

“A black hole swallows only what is aimed almost straight at it, so in a thin gas it grows at a negligible rate.”

That is true of collisionless particles, captured inside a cross-section of 4(c/v)² horizon areas. Gas collides: streams passing either side meet behind the hole, lose their sideways speed as heat and fall in. Its effective cross-section is λ(c/cₛ)⁴ horizon areas — 2.5 × 10⁸ times the particles' at 10 km/s, a sphere about the size of Saturn's orbit around a hole of one solar mass.

Tested in The gas a hole catches from far away · the horizons reading path

“Once a current is switched on, the field around it is given by the Biot–Savart law.”

Only after the switch-on has been seen from every part of the wire. One metre from a long wire the field is zero for 3.3 ns, then overshoots — 1.46 times the steady value at 1.5 light-metres — and settles from above as the second of Jefimenko's two terms, the retarded rate of change of the current, dies away along the wire.

Tested in The field a changing field does not make · the maxwell equations reading path

“Moving two coils closer together always makes the field between them more uniform.”

Closer than one radius the field peaks at the midpoint and falls on either side; further apart it dips there between two peaks. Only at one radius does its curvature vanish, and the measured departure from uniform then grows as the fourth power of distance rather than the second.

Tested in The two coils one radius apart · the magnetism reading path

“A semiconductor absorbs no light below its band gap, since no state there can be reached.”

The electron and hole the light would create attract, and bind into states below the gap: a hydrogen series at Eg − Ry*/n², with Ry* = 13.6 eV × (μ/m₀)/ε². For GaAs that is 4.8 meV below the gap and 11.6 nm across; for cuprous oxide, near 92 meV, with lines resolved up to n = 25.

Tested in The hydrogen atom inside a crystal · the bands reading path

“An atom's binding energy depends only on the charges, so positronium, with the same charges as hydrogen, is bound by the same 13.6 eV.”

Binding depends on the reduced mass too. Hydrogen's is 0.99946 of the electron's; positronium's, with equal partners, is exactly a half, so its ground state is at −6.80 eV and its orbit is twice hydrogen's size. Muonium, with a partner 207 times heavier, differs from hydrogen by half a per cent.

Tested in The atom with no nucleus · the atomic structure reading path

“The top of a rolling wheel always moves at twice the speed of the axle.”

Only at everyday speeds. The top's speed is the axle's speed composed with the rim's, 2v/(1 + v²/c²): 0.8c for an axle at 0.5c and 0.976c at 0.8c. The correction for a car at motorway speed is about 10⁻¹⁴, which is why it is never noticed.

Tested in The wheel whose top cannot go twice as fast · the Velocity-addition reading path

“A rod in a steady shear flow lines up with the flow and stays there.”

In creeping flow it never stops turning. It spends most of its time near alignment — within 10° of the flow 46 per cent of the time for an aspect ratio of 5 and 82 per cent for 20 — and then flips through half a turn, every π(r + 1/r) units of strain. Only inertia, Brownian motion, contact with other rods or a non-Newtonian liquid can stop it.

Tested in The rod that tumbles for ever in a stream · the viscosity reading path

“A drop of liquid lying in a horizontal tube stays where it is unless something pushes it.”

Only if the tube is the same width at both ends of the drop. In a tapered tube the two menisci have different radii and lower the pressure inside by different amounts — 269 and 252 Pa for a 2 μL water plug in a glass taper — and the 17 Pa difference drives the water towards the narrow end, with nothing outside acting on it.

Tested in The drop that runs to the narrow end · the capillarity reading path

“A system's heat capacity rises with temperature towards a constant, its equipartition value.”

Only if its energy can rise without limit. A system with two levels a gap Δ apart has a heat capacity that peaks at 0.439k when kT = 0.417Δ and then falls as 1/T², because once both levels are nearly equally full extra heat has nowhere to go. Any finite stack of levels does the same.

Tested in The heat capacity that goes up and comes down · the equipartition reading path

“An object that feels cold is drawing more and more heat from the hand the longer it is held.”

The surface temperature jumps to its final value the moment of contact and stays there, and the heat drawn falls as one over the square root of time: from steel at 20 °C, 8,060 W/m² after one second and a tenth of that after a hundred. The cold is greatest at the first touch.

Tested in The metal that feels colder than the wood · the diffusion reading path

“Weak disorder only slightly reduces how far a wave gets through a one-dimensional medium.”

In one dimension any disorder localises every wave, and weak disorder only lengthens the distance. The localisation length scales as the inverse of the reflectance per surface: 4.5 plates for glass of index 2.6, 120 plates for 1.2, nearly 2,000 for 1.05 — long, but finite, and beyond it the transmission still falls exponentially.

Tested in The pile of glass that stops every colour · the periodic media reading path

“Anything moving across the surface of water makes waves, more slowly the more slowly it moves.”

A steady object can only feed waves whose phase speed equals its own, and on clean water no wave is slower than 23.1 cm/s, reached at a wavelength of 1.71 cm. Below that speed there is no wave to feed, and an object leaves no wake at all.

Tested in The speed below which nothing on water leaves a wake · the wave motion reading path

“With fast enough electronics, active cancellation can remove noise at every frequency.”

Any delay τ between hearing the noise and playing its inverse leaves a residual 4 sin²(πfτ). With 50 μs it is 30 dB down at 100 Hz, 10 dB at 1 kHz, and adds noise above 3.3 kHz. Random noise cannot be predicted around the delay, so cancellation is a low-frequency method; a headphone's cup handles the rest.

Tested in The silence that fits in a tenth of a wavelength · the superposition reading path

“A weight pulled straight down on a spring and released can only bounce straight up and down.”

Only if the bounce is not near twice the swing frequency. At exactly 2:1 the slightest tilt grows: energy drains from the bounce into the swing until all of it is swinging, then drains back, over and over. At a ratio of 2.4 the swing never takes more than about 1.5 per cent.

Tested in The spring that swings when it was only bounced · the chaos reading path

“Every damped oscillation dies away exponentially, losing the same fraction of its amplitude each cycle.”

Only if the damping force grows with speed. Dry friction is a constant force, so it removes a fixed amount — twice the friction over the stiffness — every half-swing. A mass released ten dead-band widths out stops after five half-swings, the turning points falling in a straight line, while viscous damping that starts at the same rate is still swinging after twenty-three.

Tested in The swing that dies in a straight line · the friction reading path

“Friction in a speed governor's linkage only makes it less accurate, so the best governor is the one with the least friction.”

A governor with no friction cannot settle. The linearised loop has the characteristic equation s³ + βs² + cs + aK/J = 0, stable only if βc > aK/J; below that damping the engine hunts, its speed swinging ever wider. The governor drawn here needs a damping of at least 0.40 and hunts at 0 and at 0.25.

Tested in The governor that needed friction to work · the circular motion reading path

“Rainbows form because sunlight is totally reflected inside raindrops.”

No ray entering a sphere can be totally reflected inside it. A ray entering at height b meets every internal surface at arcsin(b/n), which is below the critical angle arcsin(1/n) for every ray that gets in. The internal reflection behind the primary rainbow is partial: for a ray entering at 0.85 of the radius, 8.3 per cent of the entering light leaves after one reflection.

Tested in The drop that cannot hold the light it lets in · the total internal reflection reading path

“A light wave passing through a focus keeps the same phase it would have had without the lens.”

It gains half a cycle on a plane wave of the same frequency: arctan(z/zR) runs from −π/2 far before the focus to +π/2 far after. Most of it is gained within a Rayleigh range, 0.25 of a cycle by z = zR. A line focus gains a quarter cycle.

Tested in The half cycle a focus adds · the diffraction reading path

“Frost cannot form while the air is above freezing.”

A surface that sees a clear sky radiates more to it than it receives back, and in still air it settles well below the air temperature: 12.3 °C below for an insulated surface under a clear dry sky in still air at 5 °C. Allowing for the latent heat of condensation and the ground's warmth, ground frost forms with the air at 3 to 5 °C.

Tested in The cold reservoir overhead · the heat engines reading path

“Hot water rises, so a body of water heated by the sun is always warmest at the top.”

Only if nothing else sets the density. Brine at 80 °C and 20 per cent salt weighs 1122 kg/m³ and fresh water at 25 °C weighs 1011; the hot layer is the heavy one and stays at the bottom. Solar ponds hold 70–90 °C at two metres down, and a natural one in Transylvania was found at about 70 °C at 1.3 m in 1901.

Tested in The pond that is hottest at the bottom · the stratification reading path

“Floating ice cannot raise the sea level when it melts, since it is already displacing its own weight of water.”

It displaces its weight of seawater and melts into fresh water, which is less dense. In a sea of 35 g/kg at 0 °C the meltwater occupies 2.75 per cent more than the water the ice displaced, even after mixing; a thousand cubic kilometres of floating ice raises the sea 0.068 mm, a thirty-seventh of the 2.54 mm from the same ice on land.

Tested in The floating ice that still raises the sea · the buoyancy reading path

“The most power from a pressure-retarded osmosis membrane comes from holding the seawater at its full osmotic pressure, where each litre does the most work.”

At the full osmotic pressure no water crosses at all. The flux falls in a straight line with pressure, so power — flux times pressure — is a parabola peaking at half the osmotic pressure: 8.38 W/m² at 14.2 bar for a membrane of 1.5 L/m²h per bar against 28.4 bar of seawater, with each litre doing half the work it could.

Tested in The waterfall at every river mouth · the osmosis reading path

“A beam self-focuses when its intensity is high enough, so focusing a laser more tightly makes self-focusing more likely.”

Narrowing a beam of fixed power raises its intensity and its self-focusing angle, but raises its diffraction angle in exactly the same proportion. The competition is decided by power alone: 4.3 MW in fused silica at 1064 nm, 3.0 GW in air at 800 nm, whatever the beam's size.

Tested in The beam that focuses itself · the fermat reading path