True where it was derived, used where it was not — part 4 of 5

“The front of a light pulse is its wavefront, perpendicular to the direction it travels.”

Angular dispersion separates the two. Behind a grating at 800 nm with 30° incidence the intensity front is tilted from the wavefronts by 25.6° for 600 lines per millimetre, 47.2° for 1,200 and 76.7° for 1,800.

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

“Totally reflected light never enters the second medium, so it cannot be used to illuminate anything there.”

It leaves a field there that carries no energy away but excites anything it touches. In water beyond a 1.518 coverslip, 488 nm light at 65° makes a field 3.07 times the incident intensity at the surface, falling by e in 110 nm, so a cell's contact region fluoresces while its interior stays dark.

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

“An ohmic contact is made by choosing a metal whose work function removes the barrier.”

Pinning prevents that. The barrier stays and is made thin instead. Behind a 0.8 eV barrier the depletion region is 301 nm wide at 10¹⁶ donors per cm³, 30 nm at 10¹⁸ and 3.0 nm at 10²⁰. Tunnelling takes over once the characteristic tunnelling energy passes kT, near 6 × 10¹⁸ per cm³.

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

“A sandy seabed only affects waves by making them break or by slowing them in shallow water.”

A patch of gentle sinusoidal bars reflects waves without breaking them. In 4 m of water with bars 50 m apart and 0.8 m in amplitude, 4 bars reflect 56 per cent of the amplitude of a 100 m wave and 8 bars 86 per cent, while waves well away from twice the bar spacing pass almost untouched.

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

“Waveguides are only for short runs inside equipment, because coaxial cable is better over distance.”

At high frequency the reverse holds. A 9.5 mm air coaxial cable loses 72 dB/km at 1 GHz and 557 dB/km at 60 GHz, rising as the square root of frequency. A 60 mm pipe in TE₀₁ loses 0.51 dB/km at 60 GHz, about 1,100 times less.

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

“Because light is so fast, retardation matters only over astronomical distances.”

It matters over any distance comparable with the wavelength the source works at. Here that is an atomic transition, so the crossover is c/ω₀: 9 nm for helium, 94 nm for sodium and 136 nm for caesium. A sodium atom 1 μm from a mirror feels an attraction 11 times weaker than the unretarded inverse cube would give.

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

“Counting the zeros of a field is a routine calculation.”

Only in two dimensions. There, n equal line charges make exactly n − 1 zeros, the roots of P′(z). In three dimensions a search of forty random arrangements found at most 4, 3 and 6 zeros for three, four and five charges, against Maxwell's conjectured maxima of 4, 9 and 16. No proof of the maximum exists.

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

“The phase of a light wave is always well defined and can be measured to any precision if the light is bright enough, so its phase uncertainty is merely technical.”

Phase and photon number form the same circle-and-integers pair. A coherent state with 10 photons has a circular phase spread of 0.026; with 2, 0.195; with 0.5, 0.620. The vacuum, with none, has its phase spread evenly round the circle.

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

“Weak-value amplification makes a small effect easier to detect, so it beats ordinary measurement.”

It concentrates information and does not create it. The Fisher information in the kept runs is A²/(1 + A²) of the most any run can carry: 0.50 at a weak value of 1, 0.990 at 10, 0.9999 at 100. That is still no more than all runs could carry, and the kept fraction is exactly 1/(1 + A²). Its advantage is against technical noise, not quantum limits.

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

“Because light obeys relativistic velocity addition, light in a moving medium cannot be brought to a standstill against the flow the way sound can.”

The relativistic law changes how the upstream speed falls — at half the wave speed it is 0.696 of it rather than 0.5 — but not where it vanishes. Upstream light in water stops exactly when the water moves at c/n, as sound stops when the flow reaches the speed of sound.

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

“Length contraction is only a disagreement between observers about simultaneity; nothing physically happens to a moving object.”

Both accounts give the same number. An electron orbiting a nucleus gently accelerated to 0.6c, computed from Maxwell's field of a moving charge and the relativistic law of motion alone, ends in an orbit 0.8003 as wide along the motion as across, against 1/γ = 0.8000. The same calculation stretches its period by 1.2504 against γ = 1.2500.

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

“Because diffusion is acausal, every calculation of heat flow in ordinary matter is suspect.”

Relativity requires only τ ≥ D/c². For heat carried by particles of speed u colliding every mean free path, the real relaxation time exceeds D/c² by 3c²/u²: 1.1 × 10¹² for air molecules and 1.1 × 10⁵ for electrons in copper. The bound bites only for carriers near c, such as photons in the Sun (3) and a quark–gluon plasma (3.1).

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

“A minimum length at the Planck scale can be added to physics without touching relativity.”

A length is contracted and an energy is shifted by a boost. The photon measured at 31 GeV on Earth would be measured at the Planck energy by an observer approaching it at rapidity 40.5. An energy-dependent speed picks out a frame, unless relativity is deformed to keep the Planck energy the same for everyone.

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

“Not detecting a signal from a pulsar says nothing about the pulsar.”

It bounds the pulsar's shape. Searches have pushed the Crab's strain below about a hundredth of its spin-down limit, which limits its ellipticity to 7.7 × 10⁻⁶, a mountain of about 9 cm. Gravitational waves then carry less than a ten-thousandth of the energy the Crab loses.

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

“The radius of a planet is its circumference divided by 2π.”

That is the only radius a surface survey can find, and it is what general relativity's coordinate r means. The distance from the centre, measured with rulers, is larger, because the space inside a mass is curved like the cap of a sphere. For a neutron star of compactness 0.34 the difference is 6.8 per cent of the radius, 0.83 km in 12 km.

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

“The offset between the Moon's centre of mass and its centre of figure is too small to matter.”

About 1.9 km in 1,737 km is small, but it never averages away: the Moon keeps one face to the Earth, so any self-force from a difference between its aluminium-rich crust and its iron-bearing interior would push always along the Earth–Moon line. Four decades of laser ranging see no such push, and that bounds the difference at a few parts in 10¹⁴.

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

“Compressing a star makes it able to hold more mass, as it does in Newton's theory.”

In Newton's gravity the mass of a cold neutron star rises with its central density all the way to 5.8 solar masses. In general relativity it peaks at 0.71 solar masses at a central density of 3.6 × 10¹⁸ kg/m³ and then falls. Stars past the peak are unstable: squeezed slightly, they gain more pull than push.

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

“Matching a material's impedance to free space only removes the reflection at normal incidence.”

It does more: with ε = μ the material treats s and p light identically at every angle of incidence, so circular light keeps its handedness on reflection and on transmission however steeply it arrives, though it still reflects obliquely — 5.8 per cent at 60° for ε = μ = 1.5. A small particle with equal electric and magnetic responses scatters nothing backwards at all.

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

“Experiments can decide between Ampère's force law and the Biot–Savart law.”

Only experiments on part of a circuit could, and a steady current cannot flow in part of a circuit. The difference between the laws, summed round any closed source, is exactly zero: a unit circle of current pushes an element outside it with 1.2025 units by either law, though the contributions of its individual pieces differ by factors of two, and two coaxial loops attract identically by both to four figures.

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

“Van der Pauw's method works for any sample, since the shape drops out.”

Only for a sample with no holes. A square with a circular hole a fifth of its side across returns 1.215 ohms per square for a film of 1.000, and one with a hole 0.4 of its side returns 1.80. The contacts on the outer edge cannot tell a hole is there, and the relation assumes there is none.

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

“A yield stress measured in a laboratory is a fixed property of the material.”

For a thixotropic material it depends on how long the sample stood before it was measured, because the structure resisting flow keeps growing at rest. Two laboratories testing the same paste after different waiting times report different yield stresses, and both are correct.

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

“A lower apparent viscosity in narrow tubes means the red cells there are fewer or thinner, so the blood is different.”

The blood leaving the tube has the same red-cell fraction as the blood entering it. The cells are fewer inside the tube at any instant — 0.72 of the entering fraction at about 11 μm — because they travel in the fast core and cross the tube sooner, and the viscosity is lower because a cell-free sleeve of plasma a few micrometres thick lubricates the wall. Neither changes the blood.

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

“In a chain of growing dominoes, the energy of the first push is amplified along the chain.”

Nothing is amplified. Each domino releases energy that was stored in it when the chain was set up, and uses a small part of it to tip the next over its hill. A chain of fourteen wooden dominoes growing by 1.5 a step, starting at 5 mm, ends with one 0.97 m tall releasing 13 J — 6.6 × 10⁹ times the first push, all of it supplied by whoever stood the dominoes up.

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

“A gyrocompass works anywhere on the Earth.”

Its north-seeking torque goes as the cosine of the latitude. A compass tuned at 50° has 0.40 of its design torque at 75° and 0.14 at 85°, its swing grows 1.6 and 2.7 times slower, and bearing friction leaves a proportionately larger error. At the pole the Earth's axis is vertical and there is no horizontal direction for the compass to find.

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

“Total reflection happens only when light inside a denser medium meets a boundary with a less dense one.”

It happens wherever light meets a medium of lower index, and for X-rays every solid has a lower index than vacuum. X-rays arriving from outside are totally reflected at grazing angles below √(2δ): 10.0 milliradians (0.57°) for gold and 3.9 (0.22°) for silicon at 8 keV.

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

“A diffraction grating that combines lasers of different colours into one beam beats the conservation of étendue.”

It conserves it. Each wavelength occupies its own mode, and a grating fed at 0.12° per nanometre of wavelength sends every one out in the same direction as a single spatial beam holding many colours. The étendue per colour is unchanged; what has changed is that the beams were never copies of one another.

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

“In helium each electron screens the nucleus from the other by one full unit of charge.”

It screens by 5/16. With both electrons in one orbital shaped for a charge ζ, the energy ζ² − 4ζ + 5ζ/8 is least at ζ = 27/16, giving −77.5 eV against a measured −79.0 eV. Screening by a whole charge would treat each electron as always inside the other's orbit, when it is outside about as often as inside.

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

“The exclusion principle is a rule about electrons in atoms.”

It applies to every particle of half-integer spin. The same requirement that keeps two electrons out of one state forced quarks to carry a three-valued label that none of them shows individually, and the label turned out to be the charge of the strong force.

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

“A black hole's temperature and an accelerating observer's temperature are two different effects that happen to share a formula.”

They share a mechanism. Light from something falling into a black hole of one solar mass fades, seen from far away, with an e-folding time of 4GM/c³ = 19.7 μs, and ħ/2πk times that rate is Hawking's 6.2 × 10⁻⁸ K. Both horizons produce an exponential redshift, and an exponential redshift's spectrum is thermal.

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

“Pressure is a small correction to gravity that only matters inside neutron stars.”

In ordinary matter it is: a part in a million at the Sun's centre. But for anything whose pressure is comparable with its energy density it is not small, and for a substance with p < −ρc²/3 it reverses the sign of gravity. A vacuum energy, with p = −ρc², repels with twice its energy, which is why a universe dominated by one accelerates.

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

“A diffusion coefficient measured by tracking a molecule is a property of the molecule and its medium.”

For heavy-tailed trapping it is also a property of the measurement: the time-averaged displacement at a fixed lag falls as the record lengthens, as T^(α − 1) — measured slopes −0.30 and −0.51 for α = 0.75 and 0.5. A longer record is more likely to contain a very long trap, and the apparent diffusion slows.

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

“Water's backwards melting slope shows that water is thermodynamically anomalous.”

Clapeyron's relation is obeyed exactly by every phase of water. The slope is negative for ice Ih because ice Ih, with its open hydrogen-bonded lattice, is less dense than the liquid; the four other ices that meet the liquid are denser than it, and their slopes are positive, from about 30 to 60 K per GPa. The anomaly is a structural fact about one crystal.

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

“The absorption of a suspension is the absorption of the pigment it contains, whatever form the pigment is in.”

Only if the pigment is spread finely enough that no grain shades itself. Packed into spheres whose own optical thickness is 1, a pigment absorbs 0.71 of what it would dissolved; at 10, 0.15. In cells of the same internal concentration, a 100 μm cell gets 0.32 of the dissolved value at the blue band of a chlorophyll-like pigment.

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

“An ideal middle ear would multiply the pressure by 60, the square root of the air–water impedance ratio.”

That is the figure for sound entering open water. The inner ear is a fluid-filled coil with a lower input impedance, and the lever that matches it is smaller. Besides, a transformer's match is broad: in the plane-wave picture a gain within a factor of two of the ideal still passes about two-thirds of the power.

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

“Capture by a black hole is Newtonian gravitational focusing with a very small target.”

A Newtonian point mass captures nothing at all: any angular momentum gives a barrier L²/r² that wins at small distances. Relativity adds a term falling as 1/r³ that beats the barrier inside a few gravitational radii, and that is what captures. The result looks like focusing onto a ball of radius 8 GM/c² — four horizon radii — but the ball is not there.

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

“The London field of a spinning superconductor measures the material's properties.”

It measures only the carriers' ratio of mass to charge, which for electron pairs, 2m over 2e, is the same as for single electrons. The penetration depth, the transition temperature and the metal decide where the current flows and how deep; they do not change the field inside. That universality is what made it usable as a mass balance.

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

“The more sensitive the laboratory test of Coulomb's law, the better it bounds the photon's mass.”

A photon mass m introduces a length ħ/mc, and a test of size R sees it only as (R·mc/ħ)²/6. Sensitivity helps linearly; size helps quadratically. A metre-scale shell sensitive to one part in 10¹⁶ bounds the mass near 10⁻¹⁴ eV, while the solar wind's magnetic field, measured crudely but over 40 astronomical units, bounds it at 10⁻¹⁸ eV.

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

“Any laser strong enough to diffract atoms will diffract electrons.”

Atoms near a resonance respond to light enormously more than a free electron does, and a slow atom spends far longer in the beam. For a one-radian grating a quarter of a millimetre wide, sodium atoms 1 GHz from resonance need about 3 × 10³ W/m²; 380 eV electrons in infrared light need about 3 × 10¹² W/m², a billion times more.

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

“No surface can radiate more heat than a black body at the same temperature.”

True of radiation that travels into the far field, which is all Planck's law counts. Surfaces closer together than the thermal wavelength, about 17 µm at room temperature, also exchange evanescent waves that tunnel across the gap. Two silicon-carbide half-spaces 10 nm apart exchange 9,300 W/m²K against the black-body 6.1; at 1 nm, 930,000.

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

“A third of the spring's mass is the exact correction.”

It is exact only in the limit of a light spring, where every coil moves in proportion to its distance from the fixed end. A heavy spring bows into part of a sine wave, more of it moves, and the effective share rises: 4/π², about 0.405, for a spring vibrating with nothing on it, where the third predicts a frequency 10.3 per cent too high.

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

“A ball thrown up and one dropped from the same height are mirror images, so their deflections should cancel or match.”

The deflection is proportional to the area under the height–time curve, measured from the starting height. A drop from h sweeps h·t/3 below its start; a throw to h sweeps (4/3)h·t above its start over its longer flight. The throw lands four times as far west as the drop lands east — 62.1 against 15.5 mm from 100 m at 45°.

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

“Measuring the speed of light from a moving source in a laboratory is the direct test, and astronomy only adds reach.”

The laboratory measures the speed at the source; the binary measures its accumulated effect over thousands of light years, which multiplies any fraction k of the source's speed by the ratio of distance to critical distance — 16,000 for Her X-1, 120,000 for SMC X-1. That leverage is what takes the bound on k from 10⁻⁴ in the laboratory to 2 × 10⁻⁹.

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

“In the muon's rest frame the laboratory magnetic field is γ times stronger, so its spin should precess γ times faster.”

The rest-frame field is γ times stronger, and the rest-frame clock runs γ times slower, so the Larmor precession clocked in the laboratory is (g/2)eB/m at every speed. The spin's lead over the momentum then needs one more term: the Thomas precession, (1 − 1/γ)eB/m backwards, which exists because the rest frame is being turned.

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

“A lake's winter ice forms because cold water rises.”

Cold water rises only below 4 °C, and it does so because fresh water's density curve has a peak there. Salt moves the peak down four times faster than it lowers the freezing point; above 24.6 g/kg the peak lies below freezing and the lake-like behaviour vanishes.

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

“A plank floats flat because it is flat.”

It floats flat because it is wide enough. A rectangular timber lies stably broad face up only if its width exceeds √(6s(1 − s)) times its thickness; at half density that is 1.22. A square fails the test between densities 0.211 and 0.789, and an iceberg, at 0.89 of seawater's density, capsizes if it is narrower than about 0.77 of its height.

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

“Only a source's spectral width matters for OCT, not the shape of its spectrum.”

The depth response is the Fourier transform of the spectrum, so every edge and ripple in the spectrum reappears as a ghost in depth. A flat-topped spectrum of the same 50 nm width as a Gaussian gives a central peak a third wider and side lobes reaching 22 per cent — faint copies of each reflector at depths where there is nothing.

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

“Since dark things absorb more, getting them wet makes little difference to how dark they look.”

The reverse: the film's extra return trips cost a dark surface most. A surface of dry reflectance 0.9 keeps 86 per cent of it when wet, one of 0.5 keeps 64 per cent, and one of 0.2 only 54 per cent — and a coloured surface becomes more saturated, since the colours it absorbs are absorbed again on every return.

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

“Boiling works the same way in orbit, only without convection.”

Gravity is what lifts bubbles off a heated surface and lets liquid back in. The critical heat flux goes as the fourth root of gravity — 707 kW/m² on the Moon, 111 kW/m² at a ten-thousandth of Earth's gravity — and in true weightlessness a bubble grows where it forms until it covers the surface, so boiling reaches its crisis at a small fraction of the flux safe on the ground.

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

“A hot system reaching equilibrium first would violate the second law.”

The second law requires the distance from equilibrium, measured as relative entropy, to fall for each system; it says nothing about how two different systems' distances compare. Both curves fall monotonically. One simply falls along a faster route.

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

“Playing a wave backwards makes it return to a perfect point at its source.”

The reversed wave converges and then diverges again, since nothing at the source absorbs it, and together they make a spot about half a wavelength across — 0.60 wavelengths at half amplitude for a closed mirror. The evanescent near field of the source, which holds the finer detail, never reached the mirror and cannot be sent back.

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

“Diffraction always makes a beam wider; only a lens can make it narrower again.”

At a phase step of π the band's curvature is reversed and a beam spreading with ordinary diffraction contracts instead. Flipping the sign of every other guide's field halfway along moves the beam there, and a beam that spread for four coupling lengths contracts for four more back to its starting width of 0.75 guides.

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

“The thermal correction to the Casimir force is a small detail of the metal's optics that a better material model will settle.”

It turns on one fluctuation at zero frequency, the static magnetic field, which a metal of finite conductivity lets through and a perfect conductor or superconductor excludes. That term alone is half the force at large separation and 15 per cent of it at 1 μm in the ideal-mirror model, and measurements have favoured opposite answers.

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

“A narrower plasma column needs a stronger current to hold it together, since its pressure is higher.”

Bennett's relation, μ₀I²/8π = Nk(Tₑ + Tᵢ), contains the number of particles per metre and the temperature and not the radius. Squeezing a column at fixed temperature raises its pressure and its field together, in the same proportion. A column of 10¹⁹ ions per metre at 10 keV needs 0.80 MA at any radius.

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

“Since the exclusion principle concerns identical particles, it cannot affect how atoms interact with light, which is a different particle.”

The photon is not what is excluded. The atom must end in a state of motion, and that final state is what the other atoms already occupy. The scattering rate per atom is suppressed by the occupation of the recoil state, exactly as an electron's absorption is suppressed in a semiconductor whose conduction band is already filled.

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

“Since everything falls the same way, the mass of a falling body cannot affect anything about how it falls.”

Its trajectory does not depend on the mass; its quantum states do. The natural height is (ħ²/2mm′g)^(1/3), with m the inertial mass from the kinetic energy and m′ the gravitational mass from the potential; the two do not cancel, and the energies depend on the mass as m^(1/3).

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

“Rolling resistance is friction between the wheel and the ground, and a smoother surface reduces it.”

A wheel rolling freely has no sliding to resist. The ground's push acts a distance d ahead of the axle and its torque about the contact needs a force F = W·d/R to balance. In the computed bed d is 6.4 mm on a 300 mm wheel at the worst speed, a coefficient of 0.021, with no slip anywhere; typical tyres imply 1 to 6 mm.

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

“The g/7 result explains why raindrops fall slowly.”

It belongs to a drop with no air resistance. With drag, a 0.25 mm drop reaches its terminal speed of 3.3 m/s and a 1 mm drop 6.6 m/s within about a second, and in 4 s of gathering at 0.5 g/m³ their radii grow by under 1 per cent. Real drops fall at their terminal speed; the sweeping only enlarges them, by 0.125 mm per kilometre of cloud.

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

“The X-rays come from the electron moving at nearly the speed of light, which makes its radiation very energetic; the frame is irrelevant.”

The wavelength can be found in the laboratory alone, but only by counting how much the light emitted in each period gets ahead of the electron — λᵤ(1/β̄ − 1) — and that slip is the contraction and Doppler factors in other words. For γ = 3 and K = 1.2 the computed slip is 0.102 of a period against 0.096 from the large-γ formula.

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

“Cherenkov radiation is how fast particles lose their energy in matter.”

It is a trace, not a brake. A fast particle in water loses about 2 MeV per centimetre to ionisation and a few hundred electronvolts to Cherenkov light — about a four-thousandth. Its value is that the light's angle measures the speed, which with a momentum gives the mass.

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

“The time an echo takes to return from the ionosphere gives the height of the layer that reflected it.”

It gives a virtual height, half the delay times c, which is larger because the wave slows as it nears its turning point. For a Chapman layer peaking at 300 km, a 7.5 MHz echo appears to come from 357 km while the wave turns at 268 km, and the virtual height diverges as the frequency approaches the critical frequency.

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

“A thicker photoelastic model gives more fringes and so a more sensitive measurement.”

In a two-dimensional model the stress falls as 1/t while the light path grows as t, so they cancel: the centre's fringe order is 8CP/πDλ, independent of thickness. Sensitivity comes from the material's coefficient — 78 Brewsters for polycarbonate against 2.8 for optical glass.

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

“Lower magnification is always better for faint extended objects, because it concentrates the light.”

Below D/7 mm the exit pupil is wider than the dark-adapted eye's pupil and the extra light falls on the iris: a 200 mm telescope at 15× wastes 71 per cent of what it collects. The image is no brighter than at 29×, only smaller.

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

“Dark corners in a wide-angle photograph are a defect of the lens that a better lens would remove.”

An ideal rectilinear lens, or a pinhole, illuminates the image of a uniform scene as cos⁴θ of the centre: one cosine from the aperture seen at a slant, two from the longer path, one from the slanted arrival. A perfect 14 mm lens on a 36 × 24 mm frame loses 3.52 stops at the corner, delivering 9 per cent of the central light.

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

“An accelerating packet in free space violates Newton's first law, or Ehrenfest's theorem.”

The bright lobe accelerates; the packet does not. For the finite packet Ai(x)e^(0.04x) the lobe goes from −1.02 to 23.96 by t = 10 while the intensity-weighted centroid stays at −6.25 to within 0.004. The dim oscillating tail carries the balancing motion.

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

“Turning a receiving antenna about the line of sight to a satellite cannot change the measured range, since nothing has moved closer.”

For a circularly polarised signal each turn of the antenna adds or removes one cycle of carrier phase — 19.0 cm of apparent range at GPS L1 — the spin rotational shift integrated over the turn. Precise positioning corrects for the wind-up.

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

“The low throb of a car with one window open is the engine or the road.”

It is the cabin resonating as a Helmholtz resonator, its air the spring and the air in the window opening the mass, driven by the turbulence of the flow past the opening. With 3 m³ of air and an opening equivalent to a circle 36 cm across, the formula gives about 18 Hz.

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

“Floating objects attract each other by surface tension the way magnets do, whatever they are.”

Only if their menisci have the same sign. Two floaters that both raise the surface, or both depress it, attract; a raised one and a depressed one repel. A light piece of cereal drifts to a wall the milk wets, which raises the surface, and a heavy floating needle drifts away from it.

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

“Since every balloon floats where it is neutrally buoyant, all are equally stable there.”

Stability depends on compressibility relative to the air. A full open balloon or a sealed one is less compressible than the air and is pushed back if displaced — a sealed one bobs with a period of 3.2 minutes at 20 km — but an open balloon that has cooled and gone slack is exactly as compressible as the air, with no restoring force at all, and sinks at night until ballast is dropped.

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

“Freezing is a diffusion problem, so the frozen depth spreads like heat, as the square root of time divided by the diffusivity.”

The square root survives, but the rate is set by the latent heat, not by the ice's heat capacity: h = √(2kΔT t/ρL). The exact Neumann solution, which adds the heat stored in the cooling ice, differs by 1 per cent for lake ice at 10 K and by 16 per cent only at a Stefan number of 1.5, where the latent heat is small.

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

“Thermal conductivity is a property of a material.”

At low temperature it is a property of the specimen. Below the peak the phonons cross the crystal without colliding and the walls set their mean free path, so the conductivity is proportional to the sample's size: at 5 K the model crystal conducts 3.0, 30 and 301 W/m K for samples 0.1, 1 and 10 mm across.

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

“Freezing-point depression depends on what is dissolved, so salt is a better antifreeze than sugar because of its chemistry.”

In dilute solution it counts particles per kilogram, nothing else: a gram of sucrose in 100 g of water lowers the freezing point 0.054 K, a gram of salt 0.636 K, mostly because salt's formula unit is six times lighter and splits into two ions. Chemistry enters only as the solution concentrates; half-and-half ethylene glycol freezes near −37 °C against an ideal −26 °C.

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

“Thicker foils always give more transition radiation, since there is more material to radiate.”

The radiation comes from the two surfaces, with opposite signs, and the foil adds nothing of its own. A 5 μm polypropylene foil gives about a third of what a 15 μm foil does above 4 keV, because the two surfaces' wavelets have not slipped out of phase within it; a 50 μm foil gives less than the 15 μm one, oscillating about the sum of two independent surfaces.

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

“If antimatter fell upwards, the effect would show up only in experiments made with antimatter.”

Every nucleus carries part of its electrostatic energy in virtual electron–positron pairs, and that energy is 0.5 parts in a thousand of beryllium's mass and 4.1 of platinum's. If positrons fell up, elements would fall at rates differing in proportion — and torsion balances and the MICROSCOPE satellite find titanium and platinum falling alike to about a part in 10¹⁵.

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

“Light bends towards the normal entering glass because it speeds up there, as a particle would.”

That was Newton's corpuscular account, and it is exactly right for electrons crossing into a higher potential, which really are faster there: √V sin θ is conserved. Light is slower in glass, as Foucault measured in 1850, and bends for the opposite reason — its wavefronts, not its momentum, are what the boundary constrains.

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

“A higher quality factor lets a link reach as far as needed.”

Coupling falls as the cube of the distance, so each tenfold gain in Q extends the range by only about the cube root of ten: identical loops pass 50 per cent efficiency out to 2.1 radii with Q = 100 and to 5.0 radii with Q = 1,000.

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

“Heat that reaches the cold deep ocean contributes little to sea-level rise, because cold water barely expands.”

Cold water barely expands at the surface. At 2 °C its expansion coefficient is 0.87 × 10⁻⁴ per kelvin at the surface and 1.60 × 10⁻⁴ at 40 MPa, so a joule stored at 4 km raises the sea nearly twice as far as one stored in the same water at the top; across a subtropical column the 2–6 km layer yields 1.22 mm per watt-year against 1.35 mm for the top 2 km.

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

“Since osmotic pressure is set by the counting of particles, a cell's response to dilution is the same as a perfect osmometer's.”

Only the water-holding part responds. With about 40 per cent of a red cell's volume in haemoglobin and other solids, the volume is the inactive part plus the water-holding part scaled by c₀/c, and at 1.5 per cent salt a 90 fL cell shrinks to 68 fL, not to the 54 fL a perfect osmometer would reach.

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

“Air descending a mountain warms only back to the temperature it had before it climbed.”

If it rained on the way up, it does not. Air at 20 °C with a 14 °C dew point crossing a 3 km ridge cools to 1.0 °C at the crest, rains out 4.2 of its 10.0 g/kg, and descends at the dry rate, arriving at 30.2 °C — 10 K warmer — with a dew point of 5.9 °C. The heat is the latent heat of the rain left on the far slope.

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

“With a strong enough catapult, the smallest animals could jump highest of all.”

Below a few milligrams air drag, which grows with area while weight grows with volume, takes over. A flea leaving at 1.9 m/s rises 15.8 cm instead of the 18.4 it would reach in vacuum, and at 5 m/s it would reach 65 cm instead of 127.

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

“A bigger rotor spins its gas harder and so separates better.”

The separation depends on the wall speed, not the radius, and so does the stress in the rotor: a thin hoop carries σ = ρv² whatever its size. The highest speed a material allows, √(σ/ρ), is about 423 m/s for an aluminium alloy and 500 m/s for maraging steel, and Dirac's bound on separative power grows as its fourth power.

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

“A radar mounted above the sea should see a low-flying target best at the horizon, where nothing is in the way.”

The sea's reversed reflection puts the first null of the radar's pattern exactly at the horizon. For an antenna 20 m up at 3 GHz the lobes are 2.5 mrad apart, and a target 5 m above the water 20 km away sits where the one-way field is 0.62 of free space — weaker by the square for an echo.

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

“A sensor must be tilted as much as the subject is to keep a tilted subject sharp.”

The image plane leans by tan θ′ = m tan θ, where m is the magnification. At a twentieth, a road leaning 80° from square needs the image plane to lean only 15.8°; only at life size do the two lean equally, and at twice life size an object plane at 30° needs the image plane at 49.1°.

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

“The frequency a proton precesses at tells only where it is, once a gradient is applied.”

It also tells what molecule it is in: protons in fat precess 3.5 parts per million slower than those in water, 224 Hz at 1.5 T, and frequency encoding reads that as position, drawing fat 0.52 mm out of place at 10 mT/m and 2.1 mm at 2.5 mT/m. The same shift, read as chemistry, is NMR spectroscopy.

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

“A radiocarbon age quoted as 10,000 years means the sample is 10,000 calendar years old.”

Conventional radiocarbon ages are computed with Libby's original half-life of 5,568 years rather than the measured 5,730, and assume the atmosphere's carbon-14 level never changed. Both are corrected by calibration against tree rings and other dated records, which shift ages by up to several thousand years.

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

“Neutrinos from fast pions come out in all directions, since the decay is isotropic.”

Isotropic in the pion's frame. In the laboratory every neutrino thrown into the forward half of that frame lands within 1/γ of the pion's direction — 1.91° for γ = 30 — and even one thrown back at 154° comes out at 8.3°.

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

“Heating a gas of fast particles makes them move faster.”

Their speeds are capped at c and crowd against it: the most probable speed is 0.93c at kT = 0.5 mc², 0.98c at mc² and 0.996c at 3 mc². Further heat goes into momentum and energy, which grow without limit, and hardly at all into speed.

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

“A nucleus in a crystal recoils just as a free one does, only less, because its neighbours slow it.”

Either the crystal takes the recoil as a vibration, in quanta, or it takes it as a whole, with essentially infinite mass and no energy lost. The recoil-free fraction is 0.93 for iron-57 in iron at absolute zero and 0.82 at room temperature; for Mössbauer's 129 keV iridium line it is 0.14 at zero and 0.082 at 88 K.

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

“Seeding a cloud adds water to it and so makes more rain.”

Seeding adds ice nuclei, not water. A supercooled cloud with too few ice crystals cannot rain because its water is spread over a hundred thousand droplets per litre too small to fall; one crystal per litre taking all 0.3 g/m³ of liquid grows to 0.85 mm and falls. Too many nuclei backfire: at ten thousand per litre each crystal reaches only 0.04 mm.

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

“Sound absorption in air grows with frequency, so the right fix for distance is simply more power.”

Below the collision limit, more power does buy distance. Near it no amount does: at ωτ = 0.23 a wave loses a factor e of its amplitude in a single wavelength whatever its amplitude, and above it the gas has no continuum response to carry a wave. At 100 km that limit is near 90 Hz.

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

“Refraction surveys reveal every layer beneath the surface, since every boundary refracts.”

A layer slower than the one above it bends waves towards the vertical and never reaches a critical angle, so it sends back no head wave. With 1 km at 3 km/s over 1 km at 2 km/s over rock at 5 km/s, the first arrivals show two segments, and a two-layer reading puts the rock at 2.72 km instead of 2 km.

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

“A more powerful transmitter lets a high-frequency scanner see deeper without limit.”

Depth is bought with dynamic range, the ratio of the strongest echo to the noise, at 2α₀f decibels per centimetre: 10 dB/cm at 10 MHz. Each extra centimetre needs ten more decibels of range, and the transmitted power is capped by tissue heating and cavitation, so in practice dynamic ranges of 50–80 dB set the depth.

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

“The more compact a star, the more its light can be reddened, without limit, up to the horizon.”

Only for something not standing still. Buchdahl showed that a static body whose density does not increase outward must be larger than nine-eighths of its Schwarzschild radius, where the surface redshift is exactly z = 2. A neutron star reaches about 0.24.

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

“The harmonics of cyclotron emission sit at exact integer multiples of the fundamental.”

Exactly so only in a weak field. The nth Landau level lies at mc²(√(1 + 2nb) − 1), b being the field over the critical 4.41 × 10⁹ T, so at a tenth of that field the second level is 1.92 times the first and the third 2.78 times; at the critical field, 1.69 and 2.25.

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

“Air breaks down at three megavolts per metre.”

That is the field for a gap of about a centimetre at one atmosphere. A metre gap between flat electrodes breaks down at about 2.5 MV/m, a millimetre at about 4.5, a hundred micrometres at about 9, and a gap of ten micrometres needs about 30 MV/m. The field needed depends on how many ionising collisions the gap has room for.

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

“Earnshaw's theorem says a magnet can never float stably above other magnets, so the Levitron must work by some hidden feedback.”

The theorem applies to a magnet of fixed orientation, whose energy follows one component of the field and has no minimum. A fast-spinning top's axis follows the local field direction, its energy is μ|B| + mgz, and |B| can have a minimum in empty space. Above a ring 10 cm across the sideways energy of a vertical magnet falls by 113 μJ at 15 mm, while that of a field-following one rises by 46 μJ.

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

“A perfect railgun with no resistance would convert all its electrical energy into the projectile's motion.”

At constant current a lengthening circuit stores as much magnetic energy as it does work, so half the supplied energy is left in the field behind the armature — 31.3 MJ beside the projectile's 31.3 — and is released at the muzzle unless the circuit is made to return it.

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

“Pressing a floating needle a little deeper only makes the water push back harder, as it does for a floating boat.”

Only up to a point. The support rises with depth until the meniscus at the contact line is nearly vertical — for the 1.36 mm needle, 160 mN per metre with its centre 2.78 mm down — and then falls, so a needle loaded past density 11.2 or pushed past that depth does not settle lower; it goes through.

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

“Sound travels through sand at roughly the speed it has in quartz, since sand is quartz.”

Quartz carries compressional waves at about 6,000 m/s. A dry quartz packing carries them at a few hundred, rising as the sixth root of the pressure on it — 316–362 m/s at 1 kPa, six centimetres down, and 681–780 m/s at 100 kPa, six metres down — because its stiffness is that of its contacts, not of the mineral.

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

“A vibrating ring responds to any rotation, however slow, so its sensitivity is limited only by noise.”

Only if its two vibration orientations have exactly the same frequency. A split of 0.2 per cent between them pins the pattern to the flaw for rotations below Δω/2nK, 1.25 thousandths of the vibration frequency in the case drawn, and below that the pattern rocks and goes nowhere.

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

“A more massive, more robust resonator makes a more sensitive balance.”

The smallest detectable mass is twice the resonator's effective mass times the smallest fractional frequency change that can be resolved, so it falls in proportion to the resonator's own mass. A tenth-of-a-gram quartz crystal at a part in a hundred million notices two nanograms; a carbon nanotube of 10⁻²² kg at a part in a hundred thousand notices about one atomic mass unit.

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

“Index matching works best with the finest powder, since small particles scatter least.”

For a fixed index mismatch a grain's phase shift, and so its scattering, grows with its size, so coarse grains turn opaque closer to the crossing and make the narrower filter: 52 nm wide for 5 μm grains, 26 nm for 20 μm, in a 5 mm layer.

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

“Whether a surface is glossy or matt is a property of the surface.”

It is a property of the surface's roughness measured in wavelengths times the sine of the grazing angle. Office paper with 2 μm of roughness is matt in light at steep angles and mostly specular below about 1.3°; a sea with 0.5 m of height variation is rough to 3 cm radar above 0.3° and smooth to 1 m radio waves below about 9°.

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

“Every narrow feature in a Doppler-free spectrum is a transition of the atom.”

Midway between two transitions sharing a lower level there is a feature that is no transition at all: atoms moving at ±λΔν/2, 104 m/s for rubidium's F′ = 2 and 3, are pumped on one transition while probed on the other. These crossovers sit at −345.1, −211.8 and −133.3 MHz, between the lines at −423.6, −266.65 and 0 MHz.

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

“A film's flow is limited by how thin it is everywhere along its path.”

The same volume passes every point, so the film runs fastest where it is thinnest, at the rim, its highest point. Only there does it reach the critical speed, about 27 cm/s; just above the inner liquid it moves at 22 cm/s and near the outer bath at 8 cm/s.

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

“The work function of a surface can only be found by measuring the energy of the electrons that come out.”

On a Fowler–Nordheim plot, the logarithm of current over field squared against one over the field, field emission is a straight line whose slope is −bφ^(3/2); reading the slope back gives the work function, 4.50 eV for the case drawn, with no knowledge of the emitting area.

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

“The faster of two beams in a collider is the one with the higher speed.”

At LEP's 104.5 GeV an electron's speed fell short of c by 1.2 × 10⁻¹¹; at the LHC's 6.8 TeV a proton's by about 10⁻⁸. Both are c for every practical purpose, and what distinguishes beams is their energy and momentum, which keep growing.

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

“A pion lives 26 nanoseconds, so a beam of pions cannot travel more than about eight metres.”

That is cτ, the distance at the speed of light without time dilation. The decay length is γβcτ, (p/mc)·7.8 m: at 10 GeV/c, 560 m, so 84 per cent of the pions cross a 100 m beamline, and neutrino beams are made from pions decaying along tunnels hundreds of metres long.

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

“Growing ice pushes with the Clapeyron slope of the melting line, about 13.5 MPa per kelvin.”

That slope holds when ice and water are squeezed equally, as in a sealed vessel. In soil the liquid stays near atmospheric pressure and flows to the ice, and then the pressure ice can sustain is ρᵢLΔT/Tₘ, 1.12 MPa per kelvin — twelve times smaller, but available without any expansion and without any container.

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

“A light bulb is filled with gas to stop the filament burning.”

A vacuum already keeps oxygen out. The gas is there to make evaporating tungsten atoms diffuse instead of flying straight to the glass, so many return to the filament. In a simple model a millimetre of argon slows the loss about seventy-fold, letting a filament run at 2730 K for the life it would have at 2450 K in vacuum.

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

“A thicker emulsion just makes a brighter photograph.”

Thickness sets the purity of the colour, not mainly its brightness. A 550 nm record 2 μm thick holds 11 developed sheets and replays a peak 46 nm wide; 5 μm holds 27 and replays 18 nm; 15 μm holds 82 and replays 6 nm — the resolving power of a grating, set by the number of its lines.

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

“The Schumann resonances are a steady hum the planet gives off.”

They are a statistical feature of incessant noise. Each stroke's ring decays by e in about 0.16 s, little more than a cycle, and strokes arrive at fifty a second with random strengths; the tones exist only as peaks in spectra averaged over minutes, and a station 90° from the storms misses the odd ones entirely.

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

“The Earth's field is a tilted dipole, so its symmetry is merely tipped over, and Cowling's theorem does not touch it.”

A tilted dipole is not symmetric about the axis the core turns round, and the theorem is about the field's own symmetry, not the frame it is described in. The geomagnetic dipole carries 2.7 per cent of its power in the part that is not symmetric about the rotation axis, and the next two degrees 77 and 81 per cent. The field is nearly symmetric, never exactly, which is all the theorem requires.

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

“The Planck length is the limit on how precisely any distance can be known.”

It is the limit for a distance of about the Planck length. Timing a longer distance with light against a clock gives a floor that grows with the distance, 1.89 (ℓₚ² l)^(1/3): 1.2 × 10⁻²³ m for a metre, 6.4 × 10⁻²⁰ m for the distance to the Sun, about 10⁻¹⁴ m for the observable universe.

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

“A single interferometer measures a wave's polarisation content.”

It measures one combination of the six. A wave from overhead registers only in its tensor part; the breathing and longitudinal patterns are equal and opposite everywhere, so no interferometer can separate them. GW170814 could favour pure tensor over pure vector by 200 to one and over pure scalar by 1,000 to one only because Virgo's arms point differently from the two LIGO detectors'.

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

“A thicker wire shields in proportion to its thickness.”

A wire's ability to take up charge at a given potential depends on the logarithm of the spacing over its radius, so each tenfold thickening removes about the same fraction of the leak: 45.8 per cent at a thousandth of the spacing, 31.6 at a hundredth, 7.3 at a tenth.

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

“A superconductor always expels magnetic field, so it cannot be magnetised and cannot trap a field.”

A type-I superconductor expels it; a type-II superconductor above its lower critical field lets it in as quantised vortices, and if those are pinned it keeps them. Cooled in any field of at least H* and then left in zero field, a pinned slab keeps an average of H*/2, with H* = Jc a at the middle; a 24 mm disc of gadolinium barium copper oxide has held 17.6 T.

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

“The hexagonal cells of a froth are stable because a hexagon is the most efficient shape.”

A hexagon is the only polygon whose 120° corners can be joined by straight films, so it is the only cell whose sides are flat, whose pressure matches its neighbours' and whose area does not change. Its stability is a matter of counting corners — six corners each turning π/3 make up a full turn — not of efficiency.

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

“Surface tension is a property of a liquid, like its density or viscosity.”

It is the free energy of the boundary between two coexisting phases, and it exists only while there are two. Water's falls from 72.7 mN/m at 20 °C to 14.4 at 300 °C and 0.4 at 370 °C, and is exactly zero at the critical point, 373.946 °C, where liquid and vapour become one fluid.

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

“A pendulum keeps time at any amplitude because its restoring force is proportional to displacement.”

A spring's restoring force is, and a spring's period does not depend on its amplitude — the scaling exponent 1 − k/2 is zero for k = 2. A pendulum's force is a sine, not a single power, so it has no scaling law; it keeps time only while its swing is small enough to be a spring.

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

“In a vacuum the angle that throws furthest is always 45°.”

On a flat Earth with uniform gravity. On a round one the ground curves away under a long throw and a flatter launch wins: 44.8° at 1 km/s, 37.8° at 5 km/s, 25.0° at 7 km/s, and zero at the circular-orbit speed, 7.91 km/s, where the best throw is horizontal.

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

“Once a lens has aberrations its rays no longer meet at a point, so there is no wavefront left to speak of, only a blur of rays.”

The rays no longer meet, but they are still the normals of a family of surfaces — the surfaces of equal optical path from the source. Traced through one curved glass surface with strong spherical aberration, the rays cross those surfaces at 90° to within 0.005°. The aberration is the departure of that surface from a sphere: 205 waves at the 2.8 mm edge.

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

“A beam with a dark centre is a beam with something blocking its middle.”

A beam whose phase winds ℓ times round its axis must be dark on the axis, where the phase has no single value; its brightness rises from zero as r^(2ℓ) and peaks at w√(ℓ/2) — 0.71, 1.00 and 1.22 beam radii for ℓ = 1, 2 and 3 — with nothing in the way.

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

“Any entangled pair can be used to send more than one bit per qubit.”

Not a noisy one. For a pair mixed with complete noise in proportion 1 − p, the capacity max(1, 2 − S) exceeds one bit only above p = 0.748, although the mixture is entangled for every p above 1/3. Between the two thresholds the entanglement exists and cannot be spent on this task.

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

“The longer an atom is driven at a trial frequency, the more precisely its own frequency can be found.”

The precision is set by how long the atom evolves, not how long it is driven. Two π/2 pulses separated by a free time give a central fringe 1/2T wide; computed exactly for pulses of length τ and a wait of 10τ it is 0.044 cycles per τ, against 0.067 for a single pulse lasting the same 12τ.

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

“'The rod is level' is a statement about the rod and must be true in every frame if it is true in one.”

It is a statement about the positions of all the rod's points at one instant, and which events make up one instant depends on the frame. A rod dropped level at speed u while sliding at speed v is tilted by atan(γuv/c²) in its own frame — 21.8° for v = 0.8c, u = 0.3c.

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

“Light bends inside an accelerating rocket because spacetime there is curved, as near a mass.”

Spacetime in the rocket is flat; every ray is a straight line in an inertial frame. The bending comes from describing it with the rocket's clocks, which run at a rate proportional to height above the horizon, so light's speed timed by a reference clock is c·x/x₀ and the rays bend towards where it is slow, as into glass.

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

“Nothing can move faster than light, so a spot of light swept across a distant screen is limited to c.”

The limit is on things that carry energy, matter or a signal. A spot is a succession of separately lit points, each lit by light sent from the source, and a beam sweeping a screen at distance D moves its spot at ωD at the nearest point — three times light's speed for ωD = 3c, and without limit nearer its birth point.

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

“The heat needed to boil a liquid is a property of its chemistry and has to be measured for each one.”

For ordinary liquids it is mostly fixed by the boiling point. Those boiling between 250 and 700 K need 87.2 ± 2.8 J/(mol K) times their boiling temperature — benzene 87.0, chloroform 87.5, hexane 84.4 — so a boiling point alone predicts the latent heat to a few per cent.

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

“The branches of a frost pattern, an electrodeposit or a lightning scar are shaped by the chemistry of the material.”

The branching is set by the growth rule: wherever growth is fed by something that diffuses or is pushed by a potential, the supply concentrates on protrusions, as charge does on a sharp point. Among 3,000 probe walkers, the outermost ring of a grown cluster — 5 per cent of its particles — catches 19 per cent of arrivals, and the inner three rings, 79 per cent of its particles, catch 24 per cent.

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

“A lake's water rises and falls only with the tide or the weather, the way the sea's does.”

Every lake also rings at its own periods, as a standing wave of the whole basin set going by wind or pressure and left to slop: Lake Geneva at 73.5 minutes, Loch Ness at 31.5, Lake Erie at 14.4 hours. Merian's formula 2L/√(gh), from a lake's length and mean depth alone, gives 61.7 minutes, 33.6 minutes and 15.8 hours.

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

“A wall stops sound by absorbing it.”

A wall that absorbed nothing would still stop most sound: the sound's pressure must shake the wall's mass to pass, and the shaken wall reradiates only a fraction. For a wall with only mass the loss is 10 log(1 + (ωm/2ρc)²): 37.6 dB at 500 Hz for 20 kg/m². The energy that does not pass is reflected back into the room it came from.

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

“A star with two opposite hot spots must pulse as it turns, since each spot is alternately in front and behind.”

Bending hides less of each spot's swing. In Beloborodov's approximation the two spots' brightnesses add to 2rₛ/R whatever the rotation, wherever both are in view; on the exact paths the sum varies by 60 per cent at 5 rₛ, 34 per cent at 3 rₛ and only 7 per cent at 2 rₛ.

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

“A growing black hole's mass increases exponentially, like anything whose growth rate is proportional to its size.”

Bondi's rate is proportional to the square of the mass, and a rate that grows as M² runs to infinity in a finite time, M₀/Ṁ₀. Exponential growth is what the radiation from the infalling gas imposes once it reaches the Eddington rate; before that, small holes grow slower than exponentially and large ones would grow faster.

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

“A changing magnetic field creates an electric field, and a changing electric field creates a magnetic field; that mutual creation is what keeps a light wave going.”

Faraday's law and the displacement term relate the fields at one place and time; they do not say which made which. Solved, Maxwell's equations give each field as an integral over charges, currents and their rates of change on the past light cone, with no field anywhere in the source. Round a wire switched on, ∂B/∂t and ∂E/∂r computed separately from those integrals agree to a few parts in a hundred thousand, because both come from the same current.

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

“A region of zero magnetic field is a region of no magnetic force.”

Two coils with opposite currents make a field that is exactly zero at the midpoint and grows linearly in every direction from it. An atom whose energy rises with field strength is pushed towards the zero from all sides; that is the magnetic trap of cold-atom physics, and its gradient is most uniform with the coils √3 radii apart.

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

“An exciton is like a hydrogen atom, so it holds together at room temperature as a hydrogen atom does.”

Its binding is scaled down by μ/ε²: 4.2 meV in GaAs against the 25.9 meV of kT at 300 K, so GaAs's excitons are torn apart by lattice vibrations unless cooled. Only crystals with heavy carriers and weak screening, such as ZnO at 60 meV, keep them at room temperature.

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

“Ortho-positronium's long life is fixed, so its lifetime in a material says nothing about the material.”

In a solid it sits in the gaps between molecules, and wherever its wavefunction overlaps the wall a positron can annihilate with one of the wall's electrons in two photons. Its lifetime falls to 2.1 ns in a pore of 0.3 nm radius and 7.8 ns at 0.6 nm, which makes it a ruler for the empty space in polymers and porous films.

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

“Each turn of a wheel moves it forward by its circumference, so a fast wheel, its circumference contracted, covers less ground per turn.”

It covers more. The contact point is at rest on the ground, so the ground receives the rim's own length, which for a turning rim is γ times the circumference measured from the axle. At 0.8c the axle advances γ·2π = 10.47 radii per turn, not 2π.

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

“Stirring always mixes: a stirred suspension cannot be unstirred.”

In creeping flow it can. With no inertia the motion depends only on the instantaneous shear, so reversing the shear reverses every rod's path, and a population sheared forward and then back by the same strain returns to its starting orientations — to a part in a million in the integration drawn here.

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

“Since a non-wetting liquid is pushed out of a narrow tube, it is pushed out of a tapered tube from both ends equally.”

It is pushed out more strongly from the narrow end, so it moves towards the wide one. Mercury in a glass taper, at a contact angle of about 140°, runs to the mouth with the same mechanism that sends water to the tip.

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

“Since equipartition gives a quarter or a half of k for every degree of freedom, a spin in a magnetic field contributes at least that at high temperature.”

Equipartition applies to energies quadratic in an unbounded coordinate. A spin's energy in a field takes two values and is bounded, so at high temperature its heat capacity goes to zero; it contributes entropy, k ln 2, but no capacity to hold more heat.

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

“Anything at 90 °C burns the skin, so a sauna bench at that temperature must be as dangerous as a pipe.”

Pine at 90 °C meets skin at 33 °C at about 45 °C, at the edge of what skin tolerates for minutes; steel at 90 °C meets it at 85 °C and burns. The difference is the effusivities, which differ by a factor of forty-five, not the temperatures.

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

“Random interference averages to nothing, so a pile of plates with random spacings transmits what the sum of intensities predicts.”

The intensity sum is right for light too mixed in colour to interfere: Stokes's T = (1 − R)/(1 + (2N − 1)R), 0.107 for 100 plates of glass. For one colour the phases are random but fixed, the multiple reflections interfere, and ⟨ln T⟩ falls by 0.084 per plate: about 10⁻³ through 100 plates and 10⁻¹⁴ through 400.

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

“Surface tension matters only for small things, so it cannot set a speed for things on water.”

It sets the threshold for everything. The minimum wave speed is (4gγ/ρ)^¼, which with gravity and density gives water's 23.1 cm/s; soap lowers it to 18.5, liquid helium's tiny tension to 9.6, and Titan's weak gravity takes a methane lake's to 12.

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

“Since sound waves add, playing the inverse of a noise through a loudspeaker silences a room.”

It silences the point where the noise was measured. In a room's diffuse sound, a speaker 50 cm from its microphone makes a zone 10 dB quieter about 12 cm across at 500 Hz — a fifth of a wavelength — and beyond that the speaker adds noise, up to 6 dB louder near itself.

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

“A system governed by a few exact equations with no randomness behaves predictably.”

At small energy the swinging spring does: two springs started a hundred-millionth of a radian apart stay within about 10⁻⁷. At an energy comparable to its weight times its length the same equations put it in a chaotic sea, and the same tiny difference grows until the two springs are doing unrelated things within about 150 units of time, some twenty-five swings.

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

“A quality factor describes how fast any oscillator loses its energy.”

A quality factor assumes a constant fractional loss per cycle. For dry friction the fractional loss per cycle grows as the swings shrink, from a few per cent for a large swing to all of it at the last, so no single Q describes it; a decay record whose logarithm curves rather than falls straight is the sign.

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

“Making a governor more sensitive always makes it regulate better.”

More sensitivity reduces the steady error and raises the friction needed for stability in direct proportion: the minimum damping is aK/Jc, linear in the gain K. A governor made twice as sensitive without twice the damping, or a heavier flywheel, crosses the boundary and hunts — the problem engineers met in the 1860s.

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

“Whether light is trapped in a transparent body depends on its refractive index.”

It depends on its shape. A flat sheet of glass traps 75 per cent of light emitted inside it, a glass cube 23 per cent, and a cube of index below 1.22 none at all; a sphere traps none of the light that enters it from outside, whatever its index.

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

“Gaining phase on a plane wave means part of the beam travelled faster than light.”

The beam is a spread of directions, and the oblique parts advance along the axis with a wavenumber less than k, by ⟨k⊥²⟩/2k = 1/(kw²) on average. A smaller axial wavenumber means fewer crests per metre along the axis, so the crests fall progressively ahead of the plane wave's; the energy travels at or below c throughout.

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

“Since a heat engine needs a cold reservoir, an engine on Earth can only reject its heat to the air, the ground or water.”

The clear sky is a cold reservoir too, reached by radiation. A radiator facing it can shed about 100 W/m² at the air temperature, and a reversible engine between the air and such a radiator could deliver at best about 2 W/m², at a radiator temperature near 4 °C for air at 15 °C.

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

“The thicker the insulating gradient, the more heat a solar pond keeps.”

Thicker insulation also absorbs more of the light on its way down. For brine 60 K above the surface, the share of a 250 W/m² sun kept in the hot layer peaks at 22 per cent with a 1.9 m gradient, and changes by only a few per cent between one and three metres.

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

“An ice sheet raises the sea by the whole volume of its ice when it melts.”

Only the ice above flotation counts. On a bed 500 m below sea level the first 561 m of ice is standing in for seawater it would displace anyway; melting it adds only the 2.7 per cent freshwater excess. Each metre above that adds 0.916 m³ of water per square metre.

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

“A more permeable membrane always gives more power.”

The faster water crosses, the more it dilutes the seawater against the membrane face and the more salt leaks back into the support. In the model drawn here the best power rises to 8.5 W/m² at 5 L/m²h per bar and then falls; an ideal membrane would give six times that.

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

“Self-focusing can make a beam's width shrink to zero.”

The width law says it would, which is how it proves the equation must break down. In air the collapse is stopped by the plasma the light itself makes, which lowers the index again; the beam settles into a filament about a tenth of a millimetre wide that can run for hundreds of metres.

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

“Any lens disperses light the same way: blue focuses closer than red.”

That is true of refraction. A kinoform's focal length goes as one over the wavelength, so blue focuses further away, with an Abbe number of −3.45 against glass's +64. A glass lens taking 5.1 per cent of its power from a kinoform surface brings 486 nm and 656 nm to the same focus.

Tested in The lens with its glass taken out · the refraction reading path

“Mirrors with more curvature always focus the light more tightly and hold it better.”

Bring two mirrors of radius R from the confocal spacing L = R towards L = 2R and the beam shrinks to a point at the centre but grows without bound on the mirrors; past 2R no beam is held at all. On the mirrors, the smallest beam a 1 m-radius pair holds at 633 nm is 0.449 mm, at the confocal spacing.

Tested in The two mirrors that hold a beam · the imaging reading path

“Since an electron is two thousand times lighter than a proton, an electron beam cannot damage the nuclei it passes.”

The relativistic kick an electron can give a nucleus grows as E(E + 2mc²). A 104 keV electron can give a carbon nucleus 21 eV, enough to knock it out of graphene, 11 keV below the slow-rule threshold; microscopes imaging graphene run at 60–80 kV for that reason.

Tested in The kick a fast particle can give an electron · the relativistic dynamics reading path

“A rocket with enough thrust can hover at rest anywhere outside a black hole's horizon.”

Not inside the ergosphere, which on the equator extends to 2GM/c². There the dragging of space is faster than any object can move against it, and every observer is carried round. At spin 0.9 the dragging at the horizon is the hole's rotation rate, 0.313 c³/GM.

Tested in The gyroscope a spinning Earth turns · the field transformation reading path

“A sodium acetate hand warmer releases its heat because the chemical is reacting.”

Nothing reacts. The liquid is supercooled sodium acetate trihydrate, and crystallising it releases latent heat that warms the pack to its melting point, 58 °C; from 20 °C, 43 per cent crystallises at once, and the rest crystallises slowly at 58 °C as the pack loses heat to a hand.

Tested in The supercooled water that only partly freezes · the phase change reading path

“Any object can be kept warm by wrapping it in enough insulation.”

A sphere smaller than k/h cannot be: however thick the coat, its loss tends to k/(h·r) times the bare value, because conduction from a ball into an endless insulating medium carries 4πkrΔT. A 4.8 mm ball under PVC in still air ends at 4 times its bare loss.

Tested in The insulation that makes a wire lose more heat · the diffusion reading path

“The best noise for a bistable system is the one that makes its hops most regular.”

The two measures differ. The response amplitude peaks at a noise of 0.083, but the residence times show their sharpest clustering at half a period at a lower noise, 0.05; at 0.083 hops are already frequent enough to blur the peaks at half, one and a half and two and a half periods.

Tested in The noise that helps a weak signal through · the resonance reading path

“The shadow zones are caused by the core absorbing the waves.”

The P shadow is refraction: the speed drops from 13.7 to 8.1 km/s at the core's surface, so rays entering it bend sharply inward and land far beyond the shadow, not in it. The S shadow is not absorption either: a liquid has no stiffness against shearing, so a shear wave has nothing to travel in.

Tested in The shadow that shows the core is liquid · the wave motion reading path

“With heavy enough blocks this experiment could compute π to as many digits as wanted.”

Each extra digit needs a mass ratio a hundred times larger, and the light block's speed near the wall grows as the square root of the ratio: twenty digits would need it moving 10¹⁹ times faster than the heavy block, which relativity forbids long before. The collisions must also be perfectly elastic, which no real collision is.

Tested in The collisions that count the digits of π · the momentum reading path

“The way a spinning disk speeds up at the end shows that air resistance is what stops it.”

Both mechanisms make the rate run to infinity; they differ in how fast. Air under the rim gives a rate rising as the time remaining to the −1/6 power, rolling resistance as the −1/3. Experiments in a vacuum chamber showed little change, so the air cannot be the main loss.

Tested in The disk that spins faster as it stops · the rotation reading path