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

“Moseley's law works because the K and L electrons of copper really have hydrogen-like energies with a charge of 28.”

They do not: the model puts copper's K level at 10.67 keV and its L at 2.67, against measured bindings of 8.98 and about 0.95, because the other electrons screen too. The difference comes out right for Kα — 8.000 against 8.048 keV — because the extra screening shifts both ends almost equally; for Kβ it gives 9.48 against 8.905.

Tested in The X-ray line that counts the protons · the atomic spectra reading path

“Franck and Hertz measured the energy needed to ionise mercury.”

That is what they first believed. Mercury's ionisation energy is 10.44 eV; the 4.9 eV they found is the energy to lift an electron to the first excited state, and the atoms give it back as 253.7 nm ultraviolet light, which they saw only above 4.9 V. Bohr's model, published the year before, explained it.

Tested in The electrons that pay in fixed amounts · the atomic structure reading path

“To reach a farther circular orbit a spacecraft should point its engine at the target.”

Read on the velocity circles, the cheapest transfer is two tangential burns, each a straight jump between two circles along the direction of motion: from 29.78 to 32.73 km/s leaving the Earth's orbit, and from 21.47 to 24.12 km/s arriving at an orbit at 1.524 AU.

Tested in The velocity that always lies on a circle · the orbit stability reading path

“The Earth's flattening tells how fast it spins, by Newton's calculation for a uniform fluid.”

Newton's uniform-Earth calculation gives a flattening of 1/230; the measured one is 1/298, and Jupiter and Saturn are likewise less flattened than uniform bodies at their spins. The difference measures how strongly each planet's mass is concentrated towards its centre.

Tested in The spinning fluid that becomes a cigar · the Self-gravity reading path

“A strong enough direct current in the coil should launch the ring too.”

The push grows as the square of the frequency at low frequency — a hundredth of R/L gives 1 × 10⁻⁴ of the limit — and a steady current, frequency zero, gives none once it has settled. Only the moment of switching on, while the current is still rising, gives a single kick.

Tested in The ring that jumps because its current is late · the induction reading path

“A water dropper's voltage grows without limit until something sparks.”

Even without a spark it stops: a drop carrying the charge its ring induces must climb its own can's potential, and the drops are turned back when that costs more than their fall supplies. For 3 mm drops falling 30 cm the ceiling is 15.8 kV, at √(mgh/c).

Tested in The water drops that charge themselves to a spark · the potential reading path

“Raising the laser photon's energy always raises the gamma-ray energy a given electron beam can reach, so the hardest laser available is the best.”

The share of the electron's energy the photon takes rises as x/(1 + x), but past x = 2 + 2√2 = 4.83 the hardest scattered photons have enough energy, with the laser photons still arriving, to make electron–positron pairs, and are lost. Designs stop just below that line.

Tested in The photon a fast electron throws back · the length contraction reading path

“Since a plane wave cannot give an electron net energy, laser light cannot accelerate electrons at all.”

The theorem holds for an infinite plane wave in empty space. A focus breaks it: an electron pushed out sideways by the intensity gradient leaves before the pulse has finished with it, keeps its energy, and leaves at tan θ = √(2/(γ − 1)) — 45° for 1 MeV. Plasmas, gratings and magnets break it in other ways, and every laser accelerator uses one of them.

Tested in The light that moves an electron and pays it nothing · the relativistic dynamics reading path

“Momentum is mass times velocity, so a body moving straight can have momentum only along its motion.”

A body under stress carries momentum in its energy currents. The moving lever's momentum has a component FLv cos 2α/c² across its motion — 3 × 10⁻¹⁰ kg m/s for a kilonewton on a one-metre arm at the Earth's orbital speed — although every part of it moves in a straight line.

Tested in The lever that is twisted and does not turn · the Mass-energy reading path

“Any electron beam passing near a grating will make useful light from it.”

The field at wavelength λ reaches only βγλ/4π from the beam, and the emission falls as e^(−2h/hᵢₙₜ). For green light a 300 keV electron's field reaches 49 nanometres; at a millimetre a 15 MeV beam has 2.4 mm. That is why the effect is used at terahertz frequencies with relativistic beams.

Tested in The rainbow a passing electron leaves on a grating · the radiating charge reading path

“At wavelengths shorter than the Debye length the ion-acoustic wave goes on as sound with a higher pitch.”

Once kλD approaches one the electrons stop shielding the ions' bunching — they cancel 99 per cent of it at kλD = 0.1 and 10 per cent at 3 — and the frequency stops rising in proportion, levelling towards the ion plasma frequency. The wave has become the ions' own oscillation.

Tested in The sound a plasma carries on its electrons' heat · the plasma oscillation reading path

“Once the current exceeds the critical value, the wire becomes an ordinary normal metal.”

It goes into an intermediate state of normal and superconducting regions. London's model puts its resistance at half the normal value the moment the critical current is passed, rising to 0.93 of it at twice the critical current.

Tested in The current a superconductor stops with its own field · the superconductivity reading path

“A magnetron makes microwaves by sending electrons from cathode to anode.”

A magnetron oscillates only in the window below Hull's cut-off and above the Buneman–Hartree line — 2.7 to 5.7 kV at 170 mT for an oven-sized tube — where electrons cannot reach the anode directly but drift round the cathode at E/B fast enough to bunch on the cavities' travelling wave.

Tested in The field at which no electron reaches the anode · the magnetism reading path

“A thin air gap offers almost no resistance, since air is a thousand times less viscous than water.”

The inverse cube more than makes up for it. A silicon plate 0.6 mm across on a 2 kHz spring is critically damped by the air beneath it at a gap of 24 micrometres and overdamped tenfold at 10, which is why micro-machined accelerometers have holes etched through their plates.

Tested in The wet plates that hold on when pulled · the viscosity reading path

“A block resting on a spillway crest is safe against a flood as long as its weight is enough to resist the water pushing on its face.”

Water reaching underneath adds uplift on the base, which can exceed the face pressure's effect. A solid block 0.7 times as long as it is high would tip on its face pressure alone at 1.85H; flooding a chamber beneath it through an inlet at 1.4H makes it tip at 1.4H. Fusegates on dam spillways are designed to do exactly that.

Tested in The floodgate that opens at a chosen depth · the hydrostatics reading path

“Pulling a bicycle's lower pedal backwards turns the cranks forwards and so drives the bicycle forwards.”

The bicycle is a spool with an effective hub radius of the crank length times the rear teeth over the front, 0.16 to 0.82 of the wheel's radius in real gears. That is below one, so the bicycle rolls backwards, towards the pull, and its cranks turn backwards with it.

Tested in The spool that rolls towards the hand pulling it · the Free-body reading path

“The top of a falling ladder hits the floor no harder than a stone dropped from the same height, since the ladder's energy is shared among its parts.”

The top is the far end of a rod turning about its foot, and late in the fall it moves faster than a free-falling stone: released at 15°, a 4 m ladder's top strikes the floor at 10.3 m/s against the stone's 8.7 m/s.

Tested in The ladder that lets go of the wall · the energy reading path

“Astigmatism in an eye can be corrected by finding the right spherical lens.”

A sphere moves both line foci together; it can only choose where between them the retina sits. For 1.5 dioptres of astigmatism and a 4 mm pupil the best sphere leaves 10.3 arcminutes of blur, ten times the eye's resolution. A cylinder of −1.5 dioptres moves one focus onto the other and removes it.

Tested in The focus that is two lines · the imaging reading path

“Spikes are a cosmetic flaw that costs nothing; the light in them would otherwise be lost anyway.”

By Babinet's principle the spikes carry the light the vanes block, spread along their length: the share in the spikes grows in proportion to the vanes' width, while the star's peak falls only as the square of the open area. Along a spike the light falls as the inverse square of the angle against the inverse cube between spikes — 24 times brighter at 20 λ/D — which is where faint companions are lost.

Tested in The spikes a bright star wears · the diffraction reading path

“Quantum amplifier noise matters only at microwave frequencies and millikelvin temperatures.”

Expressed as a temperature, half a quantum is ħω/2k: 0.12 K at 5 GHz, but 4,655 K at the 1550 nm wavelength of optical fibre. Every fibre amplifier adds noise as though it were that hot, and its 3-decibel minimum noise figure is the quantum limit.

Tested in The noise every amplifier must add · the uncertainty reading path

“A resistance standard must be made from a material whose purity and dimensions are controlled precisely.”

The quantised Hall resistance depends on neither: it is h/νe² in silicon, gallium arsenide and graphene alike, agreeing between them to parts in ten thousand million, because it counts edge channels rather than measuring a material. Since 2019 it defines the ohm.

Tested in The resistance counted in whole numbers · the bands reading path

“Because the transition depends on the atoms' quantum wavelengths overlapping, heavier atoms in the same trap must be cooled further to condense.”

In a harmonic trap the transition temperature, 0.94 ħω N^⅓/k, contains the trap's frequency and the number of atoms and not the mass. Rubidium and sodium in traps of the same frequency condense at the same temperature; the heavier atoms simply sit in a smaller cloud, at a higher density.

Tested in The condensate a trap makes · the superfluidity reading path

“The more layers of reflective film packed into an insulating blanket, the better it insulates.”

Pressed together, the layers touch, and the conduction through their contacts grows with the packing faster than the radiation falls. In the model drawn, a 2 cm blanket between 293 K and 77 K passes least, 346 mW/m², at about 14 layers per centimetre; at 80 per centimetre the contacts carry more than the radiation ever did.

Tested in The thin foils that stop radiant heat · the blackbody reading path

“A store that has been safe for a year at one size will stay safe if the same material is piled in the same way next summer.”

The critical size falls as exp(E/2RTₐ). For a material with an activation energy of 80 kJ/mol that is just safe 1 m thick at 40 °C, the safe thickness is 2.7 m at 20 °C and 42 cm at 60 °C — each 20 °C warmer roughly halves it.

Tested in The heap that sets itself alight · the diffusion reading path

“The still points of a hanging chain's higher modes divide it into equal parts, as a string's do.”

They crowd towards the free end, where the tension and wave speed are least: the single node of the second mode is at 0.190 of the length above the bottom, not at half, because a wave climbing the chain at √(gy) is slowest and shortest near the free end.

Tested in The chain that swings in Bessel's notes · the standing waves reading path

“A singing tube can be silenced by turning the heat down just below the point at which it started singing.”

The onset is subcritical. In the model drawn, raising the heat-transfer strength K starts the note only past the linear threshold of 0.629, and already loud; lowering it, the note persists down to K = 0.55. Between the two, silence and song are both stable, and which one the tube is in depends on its history.

Tested in The tube that sings when heated low down · the resonance reading path

“A shift of ten kilometres a second is easy to measure, since a good spectrograph resolves a few.”

One galaxy's velocity can be measured that well; the cluster's members scatter about their mean by a thousand. The uncertainty of the mean falls only as σ/√N, so reaching 3 km/s with σ = 1000 km/s takes about 110,000 galaxies, and no single cluster has more than a few hundred.

Tested in The light that climbs out of a cluster of galaxies · the gravitational redshift reading path

“Atoms cooled below the recoil limit have reached a lower temperature.”

They have reached no temperature. The distribution never stops changing: the share within a tenth of a photon's momentum goes from 55 to 87 to 96 per cent between 100, 1,000 and 10,000 scattering times, and its width is set by how long the light has been on, not by a balance between heating and cooling.

Tested in The atoms the light stops touching · the radiation pressure reading path

“Fast electrons in a magnetic field radiate at their circling frequency, so a hotter population radiates more strongly.”

Spontaneous emission does, weakly. Amplification depends on the sign of the slope of the velocity distribution at the resonant speed: a Maxwellian, however hot, has a negative slope everywhere and absorbs at every frequency. Only a population with more fast electrons than slow ones at some speed — a shell or a horseshoe — can amplify.

Tested in The radio that only heavier electrons can make · the radiating charge reading path

“A blocked island is simply an open circuit, like a gap in a wire.”

Its blockade can be switched by a gate that carries no current: at a gate charge of e/2 the states with zero and one extra electron cost the same and the island conducts at any bias, and the conductance oscillates with period e in gate charge. That periodic switching is what makes the island an electrometer sensitive to a small fraction of an electron's charge.

Tested in The island that takes electrons one at a time · the conductors reading path

“Any defect where the anisotropy is reduced lowers the coercivity to the anisotropy of the defect.”

Only a wide one. Exchange stiffness forbids the magnetisation from turning inside a layer much thinner than √(A/K) alone: a completely soft layer a fifth of the wall parameter wide leaves the nucleation field at 0.99 of the anisotropy field. The defect matters in proportion to how much of the first unstable mode it can hold.

Tested in The magnet that reverses where its crystal is weakest · the magnetisation reading path

“The quiet side of a Halbach row is shielded, as by a sheet of iron.”

There is no shield, and nothing on the quiet side absorbs flux. Each half of the pattern alone — the blocks magnetised up and down, or those magnetised sideways — makes equal fields on both sides, 0.275 of μ₀M₀ at a twentieth of a wavelength; the two fields are shifted a quarter wavelength one way above the row and the other way below, so they add to 0.549 above and cancel below.

Tested in The magnets that put their whole field on one side · the magnetism reading path

“Two identical nuclei scatter like two distinguishable ones, since the classical calculation is exact for the Coulomb force.”

The classical count for identical nuclei adds the rates for the struck and the striking nucleus reaching the detector. Quantum mechanics adds the amplitudes first, and the cross term oscillates in angle: for spinless nuclei the rate at 90° is exactly twice the classical sum at every energy, and between 30° and 90° there are about 7 fringes at η = 16.

Tested in The scattering quantum mechanics returns unchanged · the correspondence reading path

“The extra flicker of thermal light, its photon bunching, is the evidence that light is made of particles.”

The extra flicker is the wave term: classical waves with random phases produce exactly n̄² per mode, and Hanbury Brown and Twiss used it on starlight with no photons in their theory. The particle term can be mimicked by a detector that absorbs energy in quanta. The evidence no wave can mimic is the opposite effect, antibunching, a variance below n̄.

Tested in The flicker that is half wave and half particle · the photon reading path

“An accelerating observer has no consistent notion of simultaneity at all.”

Radar time gives one: an event the observer can signal and hear back from is assigned the midpoint of the round trip. For the travelling twin its surfaces of equal time never cross, so no event is dated twice and none is skipped, and for the inertial legs it reproduces the rest frame exactly — the home clock reads 0.8 of the twin's time until a signal home can no longer return before the turnaround, at the twin's 2.5 years.

Tested in The now a travelling twin can measure · the simultaneity reading path

“A stronger laser always makes the electron lose a larger share of its energy, in proportion to the classical radiated power.”

Only while the field in the electron's frame is small compared with the field that pulls pairs from the vacuum. At χ = 1 the classical formula overestimates the radiated power by a factor of 5.5; a 2 GeV electron at a₀ = 100 keeps 7 per cent of its energy classically and 23 per cent with the quantum correction.

Tested in The light that takes back an electron's energy · the relativistic dynamics reading path

“If the bottom of the spring stays still, the spring as a whole cannot be falling freely.”

The centre of mass falls at exactly g from the instant of release, because gravity is the only external force; the top makes up for the bottom, falling the whole metre in 0.261 s where a dropped stone would need 0.452 s.

Tested in The spring whose bottom does not know it has been dropped · the momentum reading path

“A contact force on a body that has not hit anything must stay finite.”

Inside the paradox's region a slightly compliant floor's force spikes as the rod jams — 20, 56 and 163 times the rod's weight for stiffnesses of 10⁴, 10⁵ and 10⁶ mg/l — while the impulse stays at 0.239 m√(gl). For a rigid floor that becomes a finite impulse delivered in no time: an impact with no collision.

Tested in The rod that friction cannot decide about · the friction reading path

“Near an absorption line the refractive index cannot be measured, because the light is absorbed there.”

Inside the line it is absorbed; beside it the vapour is transparent and its index still carries the line's strength, falling off as 1/(λ − λ₀). The hooks of 10¹⁴ sodium atoms per cubic centimetre in 10 cm sit 100 picometres either side of a line a few picometres wide, where nothing is absorbed.

Tested in The hooks that weigh an absorption line · the dispersion reading path

“Every prism that reflects light twice by total reflection blurs the image in the same way.”

Only when the two faces' planes of incidence differ. A ray square to a roof's edge meets both faces in the same plane, and the two halves leave identical; that is the geometry of a Porro prism, whose faces are met square to their edges and which needs no phase correction.

Tested in The roof that splits a binocular's image · the total internal reflection reading path

“A person stuck in quicksand should be pulled out hard and fast.”

The resistance is the suction of water drawn through packed sand into the space the foot leaves, which by Darcy's law is proportional to the speed: from compacted 0.1 mm sand, 2.5 kN at a centimetre a second, four times a person's weight. The speed at which the suction equals the body's weight is 2.71 mm/s in clean sand and a thousand times slower when clay fills the pores.

Tested in The sand that lets a body in and will not let it out · the rheology reading path

“A flowing layer of sand moves like a thick liquid, with a parabolic velocity profile and a speed growing as the square of its depth.”

Its shear rate grows as the square root of the pressure above, not in proportion to depth, so the profile is Bagnold's, u ∝ h^(3/2) − (h − z)^(3/2), and the surface speed grows as h^(3/2): on a 25° slope, 2.57 cm/s at three grains deep and 75.2 cm/s at thirty, where a liquid's speed would have grown a hundredfold.

Tested in The sand whose friction depends on how fast it flows · the granular matter reading path

“A gas compressed and released is adiabatic, because the compression is quick.”

Only if it is quick compared with the time heat takes to reach the walls. The gas spring's effective exponent runs from 1, isothermal, when ωτ ≪ 1, to γ when ωτ ≫ 1, and between the two heat flowing back and forth each cycle damps the swing, most strongly near ωτ = 1; at ωτ = 3 the swing falls to 7 per cent in ten periods.

Tested in The ball that weighs how a gas stores heat · the equipartition reading path

“A higher-octane fuel contains more energy and makes an engine more powerful.”

Octane measures how long the unburned charge can wait before igniting by itself, not its energy: the ignition delay grows by a factor of 1.43 from 90 to 100 octane. Its only value is permission to compress further — here from a ratio of 10.8 to 11.9, raising the ideal efficiency of a fuel–air cycle from 53.3 to 54.8 per cent.

Tested in The squeeze a petrol engine is not allowed · the heat engines reading path

“A radar can measure faster winds simply by pulsing faster.”

Pulsing faster widens the velocity window, λ·PRF/4, and shrinks the unambiguous range, c/2·PRF, in exact proportion: their product is cλ/8, 3.75 million m²/s for an S-band radar. Pulsing to see 200 km limits speeds to ±19 m/s; pulsing to see ±50 m/s limits the range to 75 km.

Tested in The speeds a pulsed radar folds into one · the doppler reading path

“Compressing a chirp is free: the spike is as clean as a short pulse would have been.”

The matched filter's spike has sidelobes, the first at −13.6 dB, in which a weak target beside a strong one can be lost. Tapering the filter with a Hamming weighting pulls them below −37.5 dB, but widens the spike 1.49 times and costs some signal-to-noise ratio.

Tested in The long pulse that arrives as a spike · the wave packets reading path

“Without collisions the beams' drift energy, once given to the wave, comes back to them when the wave oscillates.”

In a 20,000-electron simulation the field takes 37% of the drift energy by ωₚt = 26 and then rocks as trapped electrons swing in it, but between ωₚt = 45 and 60 the field holds 8%, the electrons' disordered motion about their beams' mean 91%, and the drift 1%. The energy is not lost; it is filamented into phase space, where no field can collect it again.

Tested in The two beams that unravel each other · the plasma oscillation reading path

“A dynamo whose field reverses at irregular intervals must be driven by something irregular.”

Rikitake's two coupled discs, driven by a perfectly constant torque with no noise, reverse 1,187 times in 12,000 circuit times, at intervals averaging 10.1 with a standard deviation of 8.9; an interval that has already lasted 5, 10 or 15 goes on for a further 10.2, 9.3 or 10.0 on average, as though each reversal were a random event.

Tested in The dynamo that reverses with nothing pushing it · the flux freezing reading path

“A lower resistance wastes less energy, so a better conductor makes charge-sharing more efficient.”

The resistance sets the time, RC/2, and the peak power, V₀²/R, and their product is fixed. For two 1 nF capacitors at 10 V, 0.01 Ω burns 10 kilowatts for 5 picoseconds and a megohm a ten-thousandth of a watt for half a millisecond; both lose 25 nanojoules, half the start.

Tested in The half a wire always takes · the field energy reading path

“A power line at a higher voltage needs a thicker single conductor to stay below corona.”

It uses a bundle. For a line 15 m up with 16 mm conductors, a single conductor reaches the fair-weather onset of 31.6 kV/cm at about 470 kV, a pair at 680, and bundles of three and four stay below it to 800 kV, because a ring of 45 cm-spaced subconductors looks from outside like one conductor of 21 cm radius.

Tested in The glow a thin wire wears before it sparks · the dielectrics reading path

“A supercritical atom needs a single nucleus with more than about 170 protons.”

The threshold depends on how the charge is spread, not only on its total. Two uranium nuclei, 184 protons between them, pull the lowest level into the negative-energy sea while they are within about 35 fm of each other, which they are, for about 10⁻²¹ s, in a collision near the Coulomb barrier.

Tested in The nucleus that would make positrons from nothing · the atomic spectra reading path

“An electron freed by a laser just flies off with the energy it absorbed.”

Freed by tunnelling, it is driven back to its ion by the reversing field. One that leaves 18° after the field's peak returns 234° of the field's cycle later with 3.17 times the quiver energy — up to 19 eV at 10¹⁴ W/cm² and 800 nm — and if it recombines it emits a photon of that energy plus the binding energy, a high harmonic of the laser.

Tested in The atom a light wave opens like a wall · the tunnelling reading path

“Leaving a safety margin at every block will eventually stop a stack from growing, since the margins add up while the offsets shrink.”

A margin ε kept at every interface carries down to every block beneath it, and the stack reaches Hₙ(½ − ε) exactly: it is the ideal stack scaled down, still unbounded. With ε = 0.05 a thousand blocks reach 3.37 lengths instead of 3.74.

Tested in The stack that leans past its own table · the Free-body reading path

“A Mariotte bottle holds its outlet at exactly the height of the tube's tip.”

At the tip less the overpressure a bubble needs to leave the tube, 2σ/r: 7.4 mm of water for a 2 mm tube and 29.7 mm for a 0.5 mm one, costing 3.8 and 16.2 per cent of the flow from a 10 cm head.

Tested in The bottle that pours at one rate however full it is · the hydrostatics reading path

“The best escapement always pushes exactly at the bottom of the swing.”

Only if nothing else depends on the drive. With Q = 1,000 at a 2° swing, a push at the bottom leaves the circular error exposed: 5.3 seconds a day across a 20 per cent change in drive. A push placed after the bottom, where the slopes of the two errors cancel, changes by 1.29; one placed before, by 11.9.

Tested in The kick that does not change the beat · the pendulum reading path

“Most of a collision's products fly out sideways, since that is where the detector is largest.”

A flat spray in pseudorapidity, the shape collisions tend to produce, is 11.5 times denser per degree at 5° to the beam than at 90°, and 132 times denser per steradian; as many particles fall between 1° and 2.7° as between 90° and 40°.

Tested in The angle a collider counts in logarithms · the Velocity-addition reading path

“Testing the gravitational slowing of clocks needs a clock flown specially for the purpose.”

Two navigation satellites stranded on an eccentric orbit, carrying the hydrogen masers they had for navigation, confirmed it to about 2.5 parts in 10⁵ from a thousand days of their own wobble — about six times better than Gravity Probe A's dedicated rocket flight of 1976.

Tested in The satellites launched into the wrong orbit · the time dilation reading path

“An ice sheet's response to its climate is reversible: restore the conditions and the ice returns to where it was.”

Between A = 5.0×10⁻²⁶ and 2.1×10⁻²⁵ Pa⁻³ s⁻¹ the benchmark sheet has two stable grounding lines, one on each side of the inland-deepening stretch. A sheet that has retreated across it at 3×10⁻²⁵ does not return when A is set back to 1×10⁻²⁵; it stays on the inner branch until A falls below 5.0×10⁻²⁶.

Tested in The grounding line with nowhere to stop · the buoyancy reading path

“Rotating light's polarisation needs an optically active substance, whose molecules are themselves handed.”

A twisted layer of birefringent molecules with no handedness of their own rotates it by 90° through a few micrometres — Mauguin guiding — by a different mechanism from the circular birefringence of a sugar solution, which rotates by tens of degrees per decimetre.

Tested in The twist that carries light round a corner · the polarisation reading path

“At absolute zero a vibration neither gives nor takes energy, so the two Raman lines both vanish.”

Only the anti-Stokes line vanishes. The rates go as n and n + 1, and the +1 is the vibration's zero-point motion: a 4 GHz resonator at 50 mK holds 0.022 quanta and its gain-to-loss ratio is 0.022, but light can still always lose a quantum to it.

Tested in The light that comes back bluer when it is warm · the scattering reading path

“Once a system's entropy has reached its maximum, nothing in it remembers the state it started from.”

The coarse-grained entropy, computed from the count of colours, forgets; the arrangement does not. Run the ring backwards after 60 ticks and at tick 120 it is exactly all white again, while Boltzmann's equation, which kept only the count, predicts only further mixing.

Tested in The ring that forgets and then remembers · the entropy reading path

“The pattern of the rings is fixed by the chemistry of the two salts.”

The same reaction gives coarse or fine rings depending on how supersaturated the product must become before it nucleates, which gels, concentrations and impurities change: as the threshold rises from 0.3 to 0.55 the spacing coefficient rises from 0.069 to 0.243 and the number of bands falls from 45 to 19.

Tested in The rings a reaction lays down in a gel · the diffusion reading path

“Raoult's law, counting the dissolved ions, gives the humidity at which a salt dissolves.”

Only for dilute solutions. A saturated salt solution is far from ideal: at 6.144 mol/kg Raoult's law gives a water activity of 0.819, the measured value is 0.753, and the difference — water bound in the ions' hydration shells — moves the threshold by seven percentage points.

Tested in The salt that dissolves in the air's own water · the chemical potential reading path

“Stacking two identical crystal sheets gives a material with the same lattice, only thicker.”

Twisted by an angle θ, two lattices of spacing a make a moiré lattice of period a/(2 sin(θ/2)) — 12.8 nm for graphene at 1.1°, about eleven thousand atoms per repeat — and at that angle electrons moving through the moiré slow almost to rest and the bilayer becomes a superconductor.

Tested in The pattern two gratings make that neither contains · the superposition reading path

“Any shaking, however gentle, will eventually build up waves on a liquid, since the forcing is resonant.”

Damping lifts the tip of the instability tongue: the shaking must exceed 4γω/(k tanh kh), 0.80 m/s² (0.081 g) for water at 50 Hz, and the threshold climbs steeply with frequency, to 2.8 g at 400 Hz for water and 84 g for a 20 cSt oil.

Tested in The ripples that answer at half the shaking · the standing waves reading path

“Diffusion-weighted images show where water is.”

They show where water is hindered from moving. Each tissue's signal is multiplied by exp(−bD), so at b = 1000 s/mm² free water keeps 0.050 and normal brain 0.449, while brain whose cells swelled in the first hours of a stroke, at half the diffusion coefficient, keeps 0.670 and appears bright.

Tested in The echo that gives back everything but the wandering · the spin reading path

“A squeezed state of ten thousand atoms will make a clock 26 dB better.”

Only if the detector resolves the squeezed noise. A detector that miscounts by 3 atoms leaves 22.1 dB of the 26.2 available at 10,000 atoms, and one that miscounts by 10 leaves 13.5 dB; with 100 atoms the 10-atom detector is 6.4 dB worse than independent atoms read perfectly.

Tested in The atoms that measure better together · the measurement reading path

“The density limit of a magneto-optical trap is a fixed property of the atom.”

It is 3κc/[(σᵣ − σₗ)σₗ I], set by the trap's spring constant and by how much more strongly rescattered light is absorbed than the laser: 2.5 × 10¹⁰ per cm³ at a ratio of 1.3, 7.5 × 10¹⁰ at 1.1, without limit as the ratio approaches 1. It also stops holding once the cloud dims a beam by two-thirds, at about 8.8 × 10⁷ atoms here.

Tested in The light that makes a cold cloud repel itself · the radiation pressure reading path

“A field applied to an atom shifts its lines by a small amount proportional to the field.”

Only while the field is weak compared with what binds the atom. The image atom's binding is so weak that 100 V/cm moves its 1→2 line from 119.2 to about 178 GHz, half as much again, and near 860 V/cm the field rather than the image begins to decide where the electron sits.

Tested in The atom whose nucleus is a reflection · the potential reading path

“Any boiling in a hot spring's tube sets off an eruption.”

Only boiling that throws out enough water. Removing 1 cm releases 0.59 litres of steam per square metre, 0.06 of the room it made, and the column settles back; past about 14 cm the steam takes more room than the water it displaced and the column runs away.

Tested in The column that boils from the middle · the phase change reading path

“A hopper that has run for a long time without jamming is due to jam.”

The sizes of avalanches between jams are exponentially distributed — 4,000 at each of three outlets fall on one line e^(−s/⟨s⟩) — so the chance of the next grain completing an arch is the same however many have passed. A long run is evidence about the outlet, not about the next second.

Tested in The outlet that jams by chance · the granular matter reading path

“Pitch is a solid, since it shatters when hit with a hammer.”

It shatters because a hammer blow lasting a millisecond is far shorter than its relaxation time, η/G = 0.23 s. Held for an hour, it has flowed some 16,000 times further than it sprang; over the eight years between drops the elastic part is about one part in 10⁹ of the motion.

Tested in The drop that takes a decade · the viscosity reading path

“The quartic vessel is the shape any water clock needs for an even scale.”

It is the shape for an outflow governed by Torricelli's law. If viscosity governs the outlet, the flow is proportional to the depth and the even-falling shape is r ∝ h^(1/2): through a 0.3 mm tube 30 mm long viscosity takes 96% of the head — but such a tube would take about 13 days to pass 25 litres.

Tested in The vessel whose level falls evenly · the hydrostatics reading path

“A tall brick chimney will only crack once it has leaned far over.”

Brick carries compression and almost no tension, and the bending stress grows with the square of the height while the compression that has to be overcome grows only with the height. A chimney forty times as tall as its radius first develops tension a third of the way up at a tilt of 12.1°; one eighty times as tall, at 6.3°.

Tested in The chimney that breaks a third of the way up · the Free-body reading path

“Since speed goes as the square root of energy, a whip can lose most of its energy and still crack.”

Only with a margin. A 0.2 g cracker on a 15 J throw has a ceiling of 387 m/s, and losing more than 11% of the moving energy per metre of whip — about a fifth over its 2 m — brings the tip in below 343 m/s. Sound needs a cracker under 2E/c², 0.25 g at 15 J.

Tested in The crack at the end of a whip · the momentum reading path

“The size of each step is set by how much sideways twist the spring was given.”

By the twist relative to the energy of the oscillation. Two springs bounced to 0.08 and 0.048, their twists scaled so the ratio of angular momentum to energy is the same, turn by steps agreeing within 1.3° over the range from 9° to 67°, though their twists differ by a factor of 0.36.

Tested in The swing that turns a step at a time · the chaos reading path

“Defocusing a microscope slightly is an equivalent way to see a transparent specimen.”

It makes the specimen visible, 0.96 to 1.06 at 8 μm of defocus here, but only where its phase changes sharply — edges and nucleus — as bright and dark fringes that do not follow its thickness, and whose sign flips through focus. The phase plate gives brightness that follows the phase itself, in focus.

Tested in The specimen that is invisible until its light is delayed · the imaging reading path

“A beam that heals itself behind an obstacle is unaffected by it.”

Only beyond the obstacle's shadow. With the centre of the axicon blocked by a 3 mm disc the axis is dark to about 17 cm, the distance b/α at which rays from the disc's edge arrive; at 22 cm the core is back at 88% of the unblocked beam's.

Tested in The beam that does not spread and the price it pays · the diffraction reading path

“Visibility measures how much light the air absorbs.”

It measures how much the air scatters, almost all of it by particles: clean air alone allows 337 km, particles scattering 0.1 per km bring it to 35 km and 1 per km to 3.9 km. A reported visibility is a measurement of the air's extinction read with the eye's threshold.

Tested in How far the air lets a mountain be seen · the scattering reading path

“The harder a source's spectrum, the hotter the gas it heats, without limit.”

Thomson's rule says so — 1.9 × 10⁹ K for a flat spectrum cut off at 10 MeV — but near mc² the Klein–Nishina cross-section falls and a photon hands over only part of its energy, and the computed Compton temperature levels off at 7.2 × 10⁷ K.

Tested in The temperature a gas takes from the light around it · the relativistic thermodynamics reading path

“Since the transverse mass is unaffected by motion along the beam, it measures the mass exactly.”

It is unaffected by a boost along the beam, and by one across it only at second order — but its edge is blurred by the W's natural width, 2.1 GeV, so that 9.8% of decays lie above 80.37 GeV, and its position depends on how well the neutrino's sideways momentum is reconstructed. The measurements differ by 75 MeV, about one part in a thousand.

Tested in The mass weighed without one of its pieces · the Mass-energy reading path

“An acoustic resonance measured to a part in a million gives the speed of sound to a part in a million.”

Only after corrections far larger than that: the thermal boundary layer at the wall, where the gas's compressions are isothermal rather than adiabatic, lowers the first mode of a 62 mm sphere by 215 parts per million at 100 kPa, and must be computed to better than one per cent of itself.

Tested in The note that fixed the kelvin · the equipartition reading path

“Since the panel's effective mass can become many times its weight, a light panel can be made to insulate like a heavy wall across the board.”

Only in narrow bands, widening with the mass put into the resonators: beating a plain panel of the same weight by 10 dB takes 280–311 Hz with half the mass resonating, 267–323 Hz with 150%, and none with 10%. Each tuning brings its own pass band, and between tunings the panel is no better than its weight.

Tested in The wall made heavy only at one note · the impedance reading path

“Any resonator's noise grows as its damping falls, since only damping holds the noise down.”

In thermal equilibrium the kicks come from the damping itself, and equipartition fixes the variance at kT over the stiffness whatever the damping is. The growth needs a noise source independent of the mechanism eroding the margin — turbulence in a combustor, whose kicks stay the same while the flame's feedback cancels the acoustic damping.

Tested in The noise that warns a tube is about to sing · the resonance reading path

“Wind carries sound downwind the way a river carries a boat, so a following wind only makes it arrive sooner.”

What matters is the wind's change with height. A wind strengthening by 4 m/s per kilometre adds its shear to the temperature gradient: upwind the effective speed falls 7.8 m/s per kilometre and the shadow comes in at 9.8 km; downwind it rises 0.21 m/s per kilometre, the rays curve down, and there is no shadow at all.

Tested in The thunder that passes over the listener's head · the huygens reading path

“Light pushing on a mirror can only heat its motion, since every photon delivers a random kick.”

With the laser on the red side of the cavity's resonance the same delay removes energy: a mirror on a 2 kHz mount is softened to 1,905 Hz and its ringing dies by e every 4.0 ms. This is how mechanical oscillators have been cooled to their quantum ground state.

Tested in The mirror held on a spring of light · the radiation pressure reading path

“Whether a mass acts as a lens depends on how large it is compared with the wavelength.”

It depends on the Schwarzschild radius, which is millions of times smaller than the region the bent rays pass through. A lens with w well below 1 — one lighter than about 81 solar masses for a wave of 100 Hz — leaves a gravitational wave nearly untouched although, half-way to a source a few billion light-years off, its Einstein radius is more than ten thousand times the distance from the Earth to the Sun.

Tested in The lens a long wave walks round · the light deflection reading path

“Gravity is weak because Newton's constant is small.”

G has units and so no size of its own; it sets the Planck mass, 1.22 × 10¹⁹ GeV. What is small is the proton: gravity between two protons is the square of the proton's mass in Planck masses, 5.9 × 10⁻³⁹, and the proton is light because the strong coupling, 1/52.5 at the Planck mass, needs nineteen decades of energy to become strong.

Tested in The nineteen decades a slow coupling buys · the planck scale reading path

“The force measurements have settled which description of a metal is correct.”

They disagree with each other. Micromechanical oscillators between 0.16 and 0.75 μm, where the descriptions differ by 3.1 to 11.9 per cent, agree with the plasma description; a torsion pendulum between 0.7 and 7 μm, where they differ by 11 to 49 per cent but surface potentials had to be subtracted, agreed with the Drude description.

Tested in The static field a metal may or may not reflect · the retardation reading path

“Since a charged particle cannot follow an electric field line without friction, it cannot follow a magnetic one either.”

A magnetic force is always perpendicular to the velocity and does no work, so a charge gyrates round a magnetic field line and slides along it. In a dipole field a charge launched at 30° to the line at the equator follows it, within 1.7 hundredths of the line's scale, to latitude 33.2° and back, its speed unchanged.

Tested in The path a charge takes is not its field line · the the field concept reading path

“The domains of a superconductor in the intermediate state and the domains of a ferromagnet are unrelated, since one expels a field and the other makes one.”

Both are the same compromise between a wall energy per unit area and the energy of a field leaking from the surface, and both give a period proportional to the square root of the sample's thickness times a microscopic wall length — 5.43√(δd) for stripes half normal in a superconducting plate.

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

“A charge accelerated by a uniform field for long enough travels as far sideways as it likes.”

Its sideways distance is its proper time multiplied by p₀/m, and the proper time of a charge accelerated by a constant force grows only as the logarithm of the laboratory's time. Launched at 0.6c, it is 2.60 units of mc²/qE across at t = 20 mc/qE, where Newton's law says 12.

Tested in The throw a uniform field bends into a chain · the relativistic dynamics reading path

“Whether a charge stays trapped in crossed fields depends on which frame it is watched from.”

It depends only on the sign of E² − c²B², which every observer measures the same. Every frame agrees that the orbit is bounded below the critical ratio and that the charge is carried off above it; at exactly E = cB it moves along x = s³/6, y = s²/2 in its proper time s, for ever.

Tested in The orbit that opens when E reaches cB · the field transformation reading path

“Fringes of high contrast in an entanglement experiment show that the correlations are nonlocal.”

For a Werner state the fringe visibility is p itself. Entanglement needs visibility above 33 per cent and a CHSH violation above 71 per cent; fringes at 60 per cent are plain and the state is entangled, but every CHSH test on it is passed by a list of answers fixed in advance.

Tested in The entangled pair a list of answers can imitate · the entanglement reading path

“Molecules that lose their interference have been scattered out of the beam or broken up.”

They arrive in the same numbers and at the same places on average. Behind the interferometer the mean arrival rate is unchanged at every temperature; only the fringes on it fade, from 46 per cent at 1000 K to 4 per cent at 3000 K.

Tested in The molecule that interferes until it glows · the matter waves reading path

“Building a pendulum's rod from the stiffest, strongest steel makes it the most stable timekeeper.”

Stiffness is irrelevant to the period; expansion decides it. Invar, an iron–nickel alloy that is weaker than good steel, expands a tenth as much and loses 0.052 s a day per kelvin against steel's 0.50, which is why it replaced compensated pendulums in the best clocks after 1900.

Tested in The pendulum that must not grow when it warms · the pendulum reading path

“The thicker the liquid inside a spinning container, the more energy the container loses while the liquid catches up.”

The energy lost is fixed by the moments of inertia alone — the fraction the contents carry, 84 per cent for an egg — exactly as in a perfectly inelastic collision. Viscosity sets only how long the catching up takes: an Ekman spin-up time of 5.1 s for water at 3 revolutions a second, 0.51 s for a liquid a hundred times as viscous.

Tested in The egg that keeps turning after it is stopped · the rotation reading path

“Since gravity holds a liquid surface flat, a surface in weak gravity is much rougher.”

Gravity damps only the ripples longer than the capillary length and enters the roughness through a logarithm: from a hundred times the Earth's gravity to a ten-thousandth of it, water's roughness changes from 0.363 to 0.439 nm, by 13 per cent for a ten-thousandfold change.

Tested in The surface whose thickness depends on the view · the surface tension reading path

“A building ventilated by its own warmth ventilates itself equally well in all seasons.”

The flow grows as the square root of the temperature difference: a two-storey house with 500 cm² of openings low and high exchanges 0.83 of its air an hour at −20 °C outside, 0.30 at 15 °C and none at 21 °C. In summer, with the inside cooler, the stack runs backwards.

Tested in The building that breathes through its doors · the hydrostatics reading path

“A phase measurement weighs a cell absolutely, since the phase is measured to a few milliradians.”

The noise is small — 0.09 pg for a 1000 μm² cell at 10 mrad a pixel — but the conversion uses one assumed refractive increment, 0.18 mL/g. Proteins are near it; a cell made of nucleic acid would be understated by 6 per cent and one made of carbohydrate by 19, so the absolute accuracy is a few per cent, set by composition.

Tested in The cell weighed by the light it delays · the imaging reading path

“A diffractive lens's colour error is like a glass lens's, only larger, and so can be corrected the same way.”

It runs the opposite way: from 450 to 650 nm a BK7 singlet's focus moves 2.1 per cent with blue focusing short, a kinoform's 38 per cent with red focusing short. That reversal is what lets a weak kinoform correct a glass lens; a flat lens on its own still lacks the delay, and smears a 10 fs pulse into a 166 fs plateau.

Tested in The delay a flat lens has no room for · the refraction reading path

“A petrol engine's efficiency is set by its compression ratio, so knock caps its efficiency.”

It is set mostly by how far the burned gas expands. In the ideal cycle, compressing by 10 and expanding by 13 gives 59.0 per cent, within 0.3 percentage points of an Otto engine compressing and expanding by 13 — which would knock — and letting the gas expand all the way back to the intake pressure, 26.8 times, gives 63.6.

Tested in The engine that expands further than it compresses · the heat engines reading path

“A regular geyser erupts at a regular interval, which is what makes it predictable.”

Its intervals cluster round two values, not one. In the model the waits form two humps near 56 and 86 minutes and the average of all of them, 76 minutes, falls in the valley between, among the least likely: announcing it misses by 16.0 minutes rms, where the duration of the last eruption predicts the next to 5.5.

Tested in The geyser that remembers one eruption · the phase change reading path

“A wave that travels at a well-defined speed delivers a pulse at that speed.”

A pulse travels at the group velocity, and for a bending wave that is twice the phase velocity: at 0.05 MHz·mm in aluminium the crests move at 0.69 km/s and the energy at 1.33. A five-cycle burst sent 300 mm along a 1 mm plate arrives as a compact packet in its stretching wave after 55 μs and as a smeared one in its bending wave, centred near 130 μs.

Tested in The plate that carries two kinds of sound · the guided waves reading path

“Earthquake waves that travel along the surface sample only the surface.”

A Love wave's period chooses its depth. In a 35 km crust an 8-second wave dies away within 7 km below the base and travels at 3.56 km/s; an 80-second wave reaches 276 km into the mantle before falling by a factor e and travels at 4.41. The longest periods arrive first because they run through the fastest rock.

Tested in The shaking a uniform Earth could not carry · the wave motion reading path

“Maxwell's equations in matter, with D = εε₀E and B = μμ₀H, apply in whatever frame the calculation is done.”

Those relations hold only in the material's rest frame. For a material moving at v through a magnetic field they become D = εε₀E + ε₀((εμ − 1)/μ) v × B in the laboratory, and using the rest-frame form there misses the whole voltage.

Tested in The voltage a moving magnet adds to an insulator · the maxwell equations reading path

“Nothing can recede faster than light, so the Hubble law must bend over near c.”

Distance measured along the cloud's own instant grows at a rate equal to the distance over τ, passing c at a distance cτ and three times c at 3cτ, while every particle's velocity relative to the centre, tanh of the same number, stays below c. The two are different quantities, and both hold at once.

Tested in The universe one explosion makes in empty space · the spacetime diagram reading path

“A particle's frequency must rise with its energy, γν₀, but its internal clock slows to ν₀/γ, so de Broglie's wave contradicts time dilation.”

Both hold for the same readings sampled along two lines: watched at a fixed place the pattern beats at 1.25 times the rest frequency for 0.6c, and followed along one clock it ticks at 0.75 times. The clock rides its own pattern and stays in phase with it, while the crests overtake it at c²/v.

Tested in The wave a row of moving clocks leaves on an instant · the simultaneity reading path

“A nucleus that is heavier than a neighbouring isobar will decay to it.”

Only to an isobar one proton away. Germanium-76 is 2.039 MeV heavier than selenium-76, two protons on, but the odd–odd arsenic-76 between them is 923 keV heavier still, so the only route is double β decay, with a half-life of 1.9 × 10²¹ years.

Tested in The parabola that decides which way a nucleus decays · the Mass-energy reading path

“Cooling a resonant bar a hundredfold makes it a hundred times more sensitive.”

The burst strain needed to match the bar's thermal energy falls only as the square root of the temperature: 1.6 × 10⁻¹⁶ at 300 K, 1.8 × 10⁻¹⁷ at 4.2 K and 2.8 × 10⁻¹⁸ at 0.1 K for a 2.3-tonne aluminium bar. The cryogenic bars reached a few times 10⁻¹⁹ only by amplifiers whose effective noise was near a millikelvin.

Tested in The bar a wave of gravity barely touches · the gravitational waves reading path

“A plasma is too hot for anything in it to form an ordered solid.”

Order depends on the ratio of interaction energy to thermal energy, not on the temperature of the electrons. Grains of 5 μm carrying 25,000 charges a third of a millimetre apart, jostled by gas at room temperature, have an effective coupling of about 4 × 10⁴, some four hundred times beyond where screened charged particles freeze.

Tested in The dust a plasma freezes into a crystal · the plasma oscillation reading path

“Stars, being made of ionised plasma, are electrically neutral.”

Gravity pulls a proton 1,836 times harder than an electron, so a star in equilibrium holds the field (mₚ − mₑ)g/2e that keeps them together, and by Gauss's law carries a net positive charge of 77 C per solar mass, independent of its radius: about 4 × 10⁻³⁷ of an electron's charge per nucleon.

Tested in The imbalance of charge that would outweigh gravity · the Gauss's law reading path

“Spin is a rotation of the electron, so the electron's spin angular momentum must behave like that of a spinning ball.”

The Einstein–de Haas effect shows spin angular momentum is real angular momentum — it is handed to the lattice and turns the rod — but its ratio to magnetic moment is twice that of any rotating charged body, g′ = 2 against 1. Spin is angular momentum without a rotation.

Tested in The rod that turns when its magnetism flips · the spin reading path

“A state with a large product of position and momentum spreads is far from the limit the uncertainty principle sets.”

A particle split between two packets 8 units apart, each of width 0.5 (ħ = 1), has σₓσₚ = 4.03, eight times Heisenberg's minimum, yet its entropic sum is only 2 ln 2 − 1 = 0.39 nats above the entropic bound: it is two sharp places, not one blurred one.

Tested in The sharpness a variance cannot see · the uncertainty reading path

“Quantum localisation of a kicked rotor means the rotor's momentum is held by some restoring force or damping.”

Nothing in the quantum equation of motion damps or restores; the evolution is unitary and conserves information. The growth stops by interference among the momentum states, and adding a few per cent of noise to the kick strength — destroying the phases without adding energy — restores the growth: 280 after 400 kicks with ±5 per cent, 1,098 with ±20 per cent.

Tested in The heating a quantum rotor refuses · the correspondence reading path

“Oil and gas leaking from the deep seabed rise to the surface, where any spill can be seen and skimmed.”

In the stratified deep ocean a blowout's plume entrains so much dense deep water that it can be trapped hundreds of metres above the source. The 2010 Gulf of Mexico blowout left a persistent plume of dissolved hydrocarbons spreading at about 1,100 m depth, some 400 m above the 1,500 m wellhead, close to what the theory gives for its buoyancy flux.

Tested in The plume that stops below the surface · the stratification reading path

“A balloon rises when the burner is lit and stops when it is turned off.”

The air inside, nearly three tonnes of it, takes minutes to cool through the fabric and the balloon's mass answers slowly. A 5-second, 3 MW burn starting at 10 s warms the air by 5.6 °C, the balloon begins to rise only at 17 s and is still climbing four minutes later, 400 m higher.

Tested in The balloon a cold morning lifts · the buoyancy reading path

“Since the Caldirola–Kanai Lagrangian gives the damped equation of motion, it is a correct description of the damped oscillator in every respect.”

Its canonical momentum is e^(γt)mẋ, so the phase-space area it conserves is not the area in the physical variables, which shrinks as e^(−γt). Quantised, [x, mẋ] = iħe^(−γt), and a thermal oscillator's uncertainty product falls below ħ/2 at t ≈ 8 for γ = 0.2 — a violation no real damped oscillator shows.

Tested in The friction an action can hold only by changing its clock · the least action reading path

“A loop must be circular for a car to go round it at steady effort.”

Nothing in the physics prefers a circle. A clothoid loop 20 m high — Euler-spiral sides with a circular arc of 8.4 m radius over the top — lets a car with half a g at the top enter at 22.7 m/s instead of 23.2 and peaks at 4.3 g partway up instead of 6.5 g at the entry.

Tested in The loop that is not a circle · the circular motion reading path

“Beating a laser against a delayed copy of itself measures its linewidth directly.”

Only when the delay exceeds the coherence time, so the two copies are independent; then the beat is a Lorentzian of twice the laser's width. With a delay shorter than 1/πΔν most of the beat collapses into a narrow spike and the rest into fringes spaced by the inverse delay, and the width read off it is the wrong number.

Tested in The width a perfect laser still has · the coherence reading path

“A metal mesh with holes much smaller than a wavelength blocks only one polarisation, like a wire grid.”

A square mesh has walls in both directions, so either field finds conductors along it, and the mesh stops both — a microwave oven's door mesh lets through about a tenth of a per cent of its 12.2 cm waves in the thin-sheet model while passing most visible light, because at wavelengths far below the period it is simply a window.

Tested in The grid that stops the light along its wires · the polarisation reading path

“An entropic attraction is the same everywhere a particle goes.”

A wall is a sphere of infinite radius, so it returns more of a particle's exclusion shell than another particle does: for the same bath the contact depth is 3.2 kT beside a sphere, 6.2 kT beside a wall and about 12 kT in a right-angled corner. Particles in such a bath are pushed onto walls and into corners and edges.

Tested in The attraction that is the small ones' room · the entropy reading path

“With enough circulation a dilution refrigerator can be run as close to absolute zero as desired.”

Its cooling is 84ṅT² watts, so against a heat leak Q it holds T = √(Q/84ṅ): 3.5 mK for 100 nW at 100 µmol/s, and halving the temperature costs four times the flow. Below a few millikelvin the boundary resistance between helium and metal, growing as T⁻³, makes getting heat out of the sample the limit rather than the cooling.

Tested in The mixture that stays mixed at absolute zero · the third law reading path

“The farther from the pyramid, the clearer the chirp.”

From far away every step is seen at nearly the same low elevation, the reflections arrive almost evenly spaced, and the echo flattens to a near-steady tone: 669 Hz falling only to 569 Hz from 100 m, against 923 Hz to 470 Hz from 5 m. The sweep is widest close to the stair.

Tested in The chirp a staircase sings back · the periodic media reading path

“Everyone in a room hears a rotating loudspeaker's vibrato in the same way.”

Each wall returns an image of the horn that sees its motion from another direction, so the direct sound and each reflection swing in pitch with different timing. Their sum swells and collapses several times a turn, to as little as 6% of its peak in the case computed, in a pattern that changes with where the listener stands.

Tested in The loudspeaker that goes round · the doppler reading path

“Averaging the shift over the whole solar disc removes the Sun's motions and leaves the gravitational part.”

Averaging removes rotation, which is odd about the centre. It does not remove convection, which is even: rising gas is brighter than sinking gas everywhere on the disc, so the brightness-weighted average is a blueshift at every point, largest at disc centre and falling as the cosine of the viewing angle towards the limb.

Tested in The redshift the Sun's boiling hides · the gravitational redshift reading path

“Because gravity takes time to travel from Jupiter, a passing ray is deflected towards where Jupiter was some fifty minutes before it is observed, and measuring that delay measures the speed of gravity.”

The deflection does point away from Jupiter's position about fifty minutes earlier — but that is the position at which the ray passed the planet, and the delay is the time for light to cross from Jupiter to the Earth. A field from a uniformly moving mass points at its extrapolated present position, as a charge's does; the experiment confirmed the geometry of light, and the speed of gravity enters only at second order in Jupiter's speed.

Tested in The bend a moving planet gives · the light deflection reading path

“A flat plate carries its charge evenly over its faces, because nothing on a flat surface is curved.”

A charged disc carries its charge as 1/√(1 − ρ²), 2.29 times the central density at nine-tenths of the radius and without limit at the rim. Half the charge lies in the outer 13% of the radius. The flat faces have zero curvature everywhere; the density is set by the edge, where the shape ends.

Tested in The charge that follows no local rule · the conductors reading path

“A transformer gets louder in proportion to how hard its core is driven.”

The strain rises slowly at low flux and steeply near saturation. In the model, the core-face level is 46 dB at 1.2 T, 63 dB at 1.7 T and 75 dB at 1.9 T: the last two tenths of a tesla add 12 dB, nearly as much as the half tesla below them.

Tested in The hum at twice the mains · the magnetisation reading path

“Aluminium cannot be heated by an induction hob, because it is not magnetic.”

What matters is the resistance the metal presents, which depends on its thickness as well as its material. Thick aluminium presents 5.1 × 10⁻⁵ Ω per square; 20 µm foil, thinner than its 0.52 mm skin depth, presents 1.3 × 10⁻³ Ω — 26 times as much and more than thick steel — and heats violently.

Tested in The pan a better conductor will not heat · the induction reading path

“Fermi pressure is a force between the atoms, a repulsion strong enough to resist the trap.”

The atoms in the comparison are ideal: they do not interact at all. Each fermion still carries three-quarters of the Fermi energy on average at zero temperature, because the 10⁵ atoms must be stacked one to a state up to (6N)^(1/3) ħω. The pressure is the energy of that stacking, and it would be the same for non-interacting particles.

Tested in The cloud that stops shrinking · the exclusion reading path

“More field always means more current.”

In a band with a scattering time τ the drift velocity is the band's maximum velocity times Ωτ/(1 + (Ωτ)²): proportional to the field while Ωτ is small, peaking at half the maximum band velocity at Ωτ = 1 and falling beyond, because more electrons are on the return half of their swing when they scatter. That negative differential resistance is what superlattice oscillators use.

Tested in The push that makes an electron swing · the bands reading path

“The best moment in an acceleration cycle is the one at which a spread of momenta produces no spread in arrival times, because the bunch then holds together by itself.”

At transition nothing restores a proton's arrival time, so the bunch's longitudinal oscillation stops — its frequency falls to zero — while the energy keeps rising; the motion cannot follow adiabatically, and the bunch length and momentum spread swing to extremes. The stable phase must also jump from the rising to the falling slope of the accelerating wave.

Tested in The energy above which faster arrives later · the relativistic dynamics reading path

“In a neutron capture all the lost mass leaves as the gamma ray's energy.”

The gamma ray carries momentum E/c, and the atom recoils with the same momentum: for ²⁹Si the recoil keeps 1.33 keV, one part in about 6,400 of the 8.47 MeV released — some 400 times the comparison's final precision. The test is of Δmc² against the gamma-ray energy plus that recoil, and the recoil is computed from relativity.

Tested in The test of mass and energy that needs a ruler · the Mass-energy reading path

“Walking droplets behave like quantum particles, and so show that quantum mechanics is a hidden classical wave.”

A walker is guided by a Faraday wave of 4.75 mm wavelength at 80 Hz that its own bounces lay down, and some of its statistics resemble quantum ones. But the wave lives in ordinary space and the drop can be watched without disturbance; reported single-drop interference through two slits did not survive careful repetition, and no local wave of this kind can reproduce the correlations of entangled pairs.

Tested in The drop that will not merge · the surface tension reading path

“The pressure in a plant cell is small, because the cell wall is thin.”

The pressure is set by the solute, not by the wall: 0.3 osmol/L of sap against pure water gives Π = cRT = 0.74 MPa, three times a car tyre's. The thin wall carries it in tension — 30 MPa in a half-micrometre wall around a 20 µm cell — and the thinness is why the stress is so high, not why the pressure is low.

Tested in The stem that stands by pressure · the osmosis reading path

“If the motion of a coin is deterministic, a skilled tosser can always produce the outcome they want.”

Near the origin of the launch map the bands are wide, and slow tosses with little spin can indeed be controlled. At a typical 2.4 m/s and 240 rad/s the bands are 6.4 rad/s wide, and a scatter of only one per cent in spin and speed already brings the chance of the starting face from certainty to 0.36, and two per cent to 0.49.

Tested in The toss that is random without chaos · the chaos reading path

“An object dropped in a spinning station falls straight down to the point beneath it, only more slowly or quickly depending on the spin.”

Released at rest relative to the floor, it moves at the speed of the place it was let go, which is slower than the floor's, and flies straight until the floor catches it — landing 106 cm behind the point beneath it from 2 m in a 10 m station, and taking 0.76 s against the Earth's 0.64 s.

Tested in The throw a spinning station bends · the projectile reading path

“Because water is transparent, a thin layer of it on glass should change little about the light inside.”

Transparency is not the question; contact is. At 45° a dry glass–air face reflects all the light and a wet glass–water face reflects 1.2% of it. Across four bounces, a wetted tenth of each footprint removes 34% of the signal, and a quarter removes 68%.

Tested in The windscreen that feels the rain · the total internal reflection reading path

“Vision blurs under water because the eye is slightly out of focus, as a short-sighted eye is.”

Under water the model eye has 18 dioptres instead of 61 and would bring distant light to a focus 73 mm behind its principal plane, three times its length. With a 3 mm pupil and the eye relaxed, each point of the scene is spread into a disc 7.3° across — a defocus of 43 dioptres, not a fraction of one.

Tested in The eye that loses its lens in water · the imaging reading path

“Atoms in a solid can diffuse only if there is a force driving them, such as a concentration gradient pushing them along.”

No atom in the lattice simulation is pushed: each move is a vacancy swapping with a randomly chosen neighbour, always with A and a third of the time with B. The asymmetry appears anyway — the vacancies gather on A's side, 56 against 38 — because a vacancy beside fast neighbours moves more often. The gradient is a statistical consequence, not a force.

Tested in The markers that move because atoms are unequal · the diffusion reading path

“Cooling Newcomen's cylinder less would have saved steam without cost.”

Leaving the cylinder warmer leaves more vapour in it: at 40 °C the atmosphere pushes the piston with 0.93 atm and the walls waste 7.0 cylinder-fulls of steam; at 70 °C the waste halves but the push falls to 0.69 atm. Every setting traded work for steam.

Tested in The cylinder that must not be cooled · the heat engines reading path

“A heat pipe works in any orientation, since capillary action does not depend on gravity.”

Capillarity supplies a fixed pressure, 2σ/r — 2,718 Pa for 50 µm pores at 50 °C, enough to hold a 28 cm column of water — and gravity takes its share when the hot end is above the cold. The model pipe carries 44 W level and nothing at all when its 300 mm length is stood hot end up.

Tested in The pipe that carries heat as vapour · the phase change reading path

“Any background noise will do, wherever it comes from.”

The noise must arrive from both ends of the line between the stations. With sources only on one side, only one of the two pulses survives; with a single coast 40° off the line, the pulse's envelope peaks at 19.2 s instead of 20.0 and the wave speed read from it is 4% too fast.

Tested in The echo hidden in random noise · the guided waves reading path

“The fork's frequency of 32,768 Hz is a natural property of quartz.”

It is a design choice, the smallest power of two above human hearing, so that fifteen binary halvings give one pulse a second and the fork's hum is inaudible. The frequency is set by the tine's shape: a quartz tine 0.3 mm thick rings at 32,768 Hz when 2.8 mm long and would have to be 24 mm long to ring at 440 Hz.

Tested in The fork inside a watch · the resonance reading path

“A fusion reactor has to reproduce the conditions at the centre of the Sun.”

The Sun's core releases 276 W per cubic metre, less than a resting human body. It succeeds because its energy takes about a hundred thousand years to leave. A tokamak at 14 keV and a density of 10²⁰ per cubic metre releases about 1.7 MW per cubic metre, from a different and vastly faster reaction, at ten times the temperature.

Tested in The product a fusion plasma has to beat · the plasma oscillation reading path

“More twist is always better, since it shorts out the drift more effectively.”

A column whose lines make more than one full turn along its length can lower its energy by bending into a helix, and does so in microseconds. For a torus that limit is a safety factor of 1, and experiments find disruption near 2 at the edge, which sets the largest current a machine can carry.

Tested in The twist a bottle of plasma has to have · the flux freezing reading path

“A magnet can be treated as a pair of north and south magnetic charges, since that gives the correct field.”

It gives the correct field outside the magnet only. Inside the source it gives the opposite mean field, half the size. Applied to the proton it predicts hydrogen's ground-state levels reversed and at half the separation. The measured 1420.4 MHz, with the parallel-spin state above, matches the current-loop value.

Tested in Two dipoles that agree everywhere but inside · the the field concept reading path

“Xenon is used in ion engines because a heavier ion gives more thrust per unit area.”

At the space-charge limit the thrust per area is (8/9)ε₀(V/d)², with the ion's mass cancelled: hydrogen, argon and xenon give the same number. Xenon's advantage is in power: it needs about 19 kW per newton at 1 kV where hydrogen needs about 219 kW.

Tested in The current a vacuum will not carry · the Gauss's law reading path

“The mirror affects the atom's decay by acting on its electron with a force.”

At the distances drawn, a mirror exerts no meaningful force on the atom's electron and shifts its levels negligibly. The emitted wave interferes with its own reflection, which changes how much power leaves. Integrating the far-field pattern over all directions gives the decay rate to better than half a per cent.

Tested in The lifetime that depends on where the mirror is · the decay reading path

“A strong enough light can excite most of the atoms in a gas, so a laser needs only a bright lamp.”

For two levels the upper population is Bρ/(A + 2Bρ), which approaches a half and never reaches it, since the same light that lifts an atom brings it down at the same rate. An inversion needs at least three levels, with pumping and lasing on different transitions.

Tested in The third process equilibrium demands · the blackbody reading path

“The 67.5 MeV peak in cosmic gamma rays is the photon energy of a pion at rest, so most of the pions must be nearly at rest.”

The peak does not depend on the pions' speeds at all. Every box has its edges multiplying to (m/2)², so each one contains 67.5 MeV. Summing pion spectra falling as the powers 2, 2.7 and 4 of energy gives peaks at the same place, with the ratio of the spectrum at a point and at its mirror image equal to 1 to within half a per cent.

Tested in The flat spectrum a fast pion must paint · the relativistic dynamics reading path

“The threshold for a reaction is the total rest energy of what it makes.”

That is the threshold only when the target is very heavy. In general it is ((M + Σm)² − M²)/2M for a stationary target of mass M. For an electron pair on an electron it is 2.04 MeV, twice the rest energy created, and for a muon pair on an electron it is 43.9 GeV, 208 times the 211 MeV it costs on a nucleus.

Tested in The partner that sets the price of a pair · the Mass-energy reading path

“Damp gravel should hold a wall even better than damp sand, since its grains are heavier and have more contact area.”

Bridge force goes as grain size and a grain's weight as its cube. The ratio 6γ/ρgd² is 185 at 0.3 mm and falls to one at about 4 mm. A pebble is held by a small fraction of its weight, so a wet gravel bank behaves like a dry one.

Tested in The sandcastle a spoonful of water holds · the granular matter reading path

“Dissolved gas makes water lighter, so a lake rich in carbon dioxide must be unstable.”

Dissolved carbon dioxide makes water heavier, by about 2.8 kg/m³ from top to bottom here, which is what holds the deep water down. Out of solution the same gas does the opposite: a lifted parcel at 20 m is 66 per cent bubbles. One composition has two opposite effects on density, separated by a pressure.

Tested in The lake that holds its gas until it does not · the stratification reading path

“A bicycle ridden faster is always more stable.”

Above 6.02 m/s the benchmark machine is unstable, by the mode called capsize: a very slow fall into a spiral. It is a small positive growth rate that a rider corrects without noticing, so a fast bicycle feels stable, but the linear equations show it is not self-stable at those speeds.

Tested in The bicycle that steers itself upright · the harmonic approximation reading path

“Small weights could not hold the angular momentum of a large stage.”

Angular momentum goes as distance squared at a given rotation rate, and the cord has only to be long enough. A stage with 1,600 times the weights' mR² needs only forty radii of cord. All of its rotational energy moves into the weights by the time the cord is fully out.

Tested in The weights that stop a spinning satellite · the energy reading path

“Any flexible cord should make a fountain, given enough drop.”

With α = 0, the arch height h₂α/(1 − α) is zero. A thin string picked up from a pile needs no reaction: each piece is pulled straight up by the tension, and all of its momentum comes from the pull. The fountain requires a stiff element that can pivot on the pile.

Tested in The chain that climbs out of the jar · the momentum reading path

“Glories appear because cloud drops are too small for rays to be valid, so they are a small-drop effect.”

A 10 µm drop at 550 nm has a size parameter over 100, where most of what ray optics says is good. In this calculation the back-scatter efficiency stays well above the reflection line out to a size parameter of 300. The ray picture does not fail for being coarse: it lacks a path that exists.

Tested in The rings no ray can reach · the scattering reading path

“A compound eye and a lens eye of the same size should resolve about as well, since both are limited by diffraction.”

A lens eye's whole pupil acts as one aperture, so its blur is λ/D and falls as the inverse of its size. A compound eye's aperture is one facet, so its blur goes only as the inverse square root of its radius. One arcminute takes a 1.7 mm pupil, and takes a 12 m compound eye.

Tested in The eye a fly cannot afford · the diffraction reading path

“Mirrors as good as 99.99 per cent would stop the measurement, because almost no light could enter or leave the cavity.”

Only a tiny fraction enters, and only a tiny fraction leaks out per bounce, and that suffices since a detector can count the small leak. What is measured is the rate of decay, and any level above the noise gives the slope. Very little light is lost to the measurement, as the amount lost to the gas is what changes the slope.

Tested in The absorption measured with a stopwatch · the dispersion reading path

“Kelvin's cooling calculation has no value now that the Earth's age is known to be 4.5 thousand million years.”

Applied to the plate rather than the planet, it fits the data. A slab cooling from 1,300 °C at a ridge gives q = kT/√(πκt), about 96 mW/m² at 25 million years, in agreement with measured heat flow out to about 80 million years, and the same law predicts the ocean's depth against age.

Tested in The age a cooling Earth gives · the diffusion reading path

“An oven fills with a uniform field, as a room fills with air.”

A box 29 × 29 × 19 cm has only 71 modes below 2.45 GHz and about seven distinct patterns in a magnetron's 100 MHz band. Each is a standing wave with nodes, and in the middle plane the sum still has hot patches and near-dark nodes spaced several centimetres apart.

Tested in The oven that cooks in spots · the standing waves reading path

“A wave grows all the way in as the water shallows, since the energy has to go somewhere.”

Conservation of energy flux gives a height that first falls, to 0.91 of its offshore value near 25 m for a 10 s swell, because the group speed first rises as the bottom is felt. Only in much shallower water does it rise above the offshore height, towards Green's law of depth to the minus a quarter.

Tested in The waves that always arrive square to the beach · the wave motion reading path