True as far as it goes, and silent on what decides — part 5 of 5

“A Fresnel lens is optically equivalent to the lens it was cut from.”

For rays, yes. For waves, each 1 mm step removes 890.9 wavelengths of path, and the fractional parts differ from ring to ring, so the rings no longer arrive in step: the focal peak of a cylindrical Fresnel lens falls to 16 per cent of the solid lens's. Only steps of whole wavelengths, λ/(n − 1) = 1.12 μm in acrylic at 550 nm, restore it.

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

“A laser cavity must be stable, so that light stays in it.”

Only if the gain is small. A confocal unstable resonator with magnification 1.5 lets out 56 per cent of the light on each round trip past the edge of a small mirror, filling a large high-gain medium with light that leaves after a few passes; high-power lasers are built that way.

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

“An electron struck head-on by a heavy particle leaves at twice the particle's speed.”

In the heavy particle's frame the electron bounces back at the speed it arrived with; back in the laboratory the two speeds combine relativistically, giving 2v/(1 + v²/c²) — 0.800c, not c, for a particle at 0.5c. Its Lorentz factor is 2γ² − 1, which is why its energy grows without limit while its speed does not.

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

“Frame dragging turns everything near a spinning body in the same direction as its spin.”

It has the pattern of a dipole field: along the spin over the poles, at twice the rate, and against it over the equator. Round a polar orbit the rate swings between 163 and −81 mas/yr and averages 41 in the sense of the spin.

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

“The frozen share is the heat capacity times the supercooling divided by the latent heat.”

With the heat capacity water has at 0 °C, that gives 45.6 per cent at −38 °C; the true share is 51 per cent, because supercooled water's heat capacity rises steeply below −20 °C. The answer is an integral of the measured heat capacity, and does not depend on the path the enthalpy is imagined to take.

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

“Insulation on an electrical cable is there only to stop it touching things, and costs it some cooling.”

On a wire thinner than the critical radius the sheath improves the cooling. A 2 mm copper wire carrying 20 A in still air runs at 60 °C bare and 44 °C with a 1 mm PVC sheath; the current that keeps it at 70 °C rises from 22.7 A to 31.0 A.

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

“Stochastic resonance is a resonance at a particular frequency, like a tuned circuit's.”

It is a resonance in the noise, not the frequency: the response peaks when the mean time between noise-driven hops matches half the push's period. A slower push peaks at less noise, a faster one at more; and a bare threshold with no dynamics at all shows the effect, passing a signal best at a noise close to the threshold itself.

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

“The P-wave shadow zone is a region no earthquake wave reaches.”

No direct P wave reaches it, but rays that just enter the solid inner core are bent back out early and land as close as 115°, inside the shadow; those weak arrivals are how Inge Lehmann found the inner core in 1936. Waves diffracted round the core also leak into it, as rays cannot.

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

“A heavy block hitting a light one simply knocks it out of the way.”

Against a wall the light block comes back and hits the heavy one again, and again, each time taking a little of its momentum. A block a hundred times heavier is stopped at 5 per cent of its starting distance from the wall and pushed back out by 31 collisions; its velocity falls as a cosine of the collision count.

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

“A disk settling on a table approaches lying flat gradually, getting ever closer without quite arriving.”

Under either of the two loss mechanisms proposed, the tilt reaches zero at a definite time. With air drag under the rim the cube of the tilt falls linearly; with rolling resistance its 3/2 power does; one second before the end the tilt is 2.15° or 0.46° from a start of 10° a hundred seconds earlier.

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

“Moseley's law, Kα = (3/4) Ry (Z − 1)², gives the X-ray energy of every element.”

To about a per cent up to zinc — 1.006 times the law for copper — but the measured energy climbs above it for heavy elements, to 1.108 for gold and 1.120 for lead, because the innermost electrons there move at a sizeable fraction of light speed and relativity deepens their binding.

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

“The spacing between the dips of a Franck–Hertz curve is the atom's excitation energy.”

Only if each collision happens the instant the electron has enough energy. With a finite free path every dip moves out in proportion and the spacing becomes E1/(1 − λ): 4.95 V and 5.06 V for free paths of 2 and 5 per cent of the gap, against 4.886 V. Real tubes also show spacings growing with order.

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

“The velocity circle is a curiosity of elliptical orbits.”

It holds for every orbit under an inverse square, closed or open: a circle centred on the origin for a circular orbit, one through the origin for a parabola, and an arc for a hyperbola, cut off at the two directions it has at infinity, where it still moves at 1.12 GM/h for e = 1.5. Add a small inverse-cube term and it stops being a circle at all.

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

“Spin a fluid faster and it flattens further.”

Only up to a point. The spin a Maclaurin spheroid needs peaks at Ω²/πGρ = 0.449 at e = 0.930 and then falls: flatter spheroids need slower spin, because their angular momentum grows through their size rather than their rate. A uniform fluid cannot be held together by its gravity at any faster spin.

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

“A better conductor is always pushed harder.”

Per unit weight, aluminium beats copper by a factor of two despite copper's lower resistance, because copper is three times as dense. And the gain from lower resistance stops at the inductive limit: the average push is x²/(1 + x²) of the perfect conductor's, x = ωL/R, so it barely grows once x passes a few.

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

“The dropper's polarity is fixed by how it is built.”

The device is symmetric; which can goes positive is decided by the stray imbalance it starts from. Any small difference in potential between the cans grows, any small common potential dies away, so every start is swept onto the line of equal and opposite potentials — one way or the other. A residue left after a spark keeps the polarity; drained completely, the next sign is a fresh choice.

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

“A laser-Compton source makes a gamma-ray beam with a broad spectrum, like bremsstrahlung.”

The spectrum is broad, but energy and angle are locked together: a photon at angle θ carries the edge energy divided by about 1 + γ²θ². A collimator passing γθ < 0.10 selects a band 1 per cent wide at the top, so an aperture makes the beam monochromatic.

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

“An electron in a light wave simply oscillates back and forth across the beam along the electric field.”

Only while a₀ is small. The magnetic force pushes it along the beam at twice the light's frequency, so it drifts forward at a₀²/(4 + a₀²) of c — a fifth of c at a₀ = 1 — and in the frame moving with the drift it traces a figure eight.

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

“The paradox is resolved because torque is not frame-independent, so the unbalanced torque in the moving frame is simply an artefact.”

The torque is real in that frame, and it has to go somewhere. The forces do work at the rate F·v; the energy flows through the arms and carries momentum −(FLv/c²) cos 2α across the motion. Angular momentum about the moving pivot changes at τ − v × G, and v × G equals the torque at every orientation.

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

“Smith–Purcell light is diffraction of the electron's field by a grating, so its colours follow the grating equation for light.”

The colours follow the grating equation with the charge's speed in place of an incoming light wave: 1/β replaces the sine of an incidence angle. That is why the wavelength straight back is d(1/β + 1), longer than any light the same grating could diffract back on itself.

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

“A plasma carries sound waves whenever it is quasineutral and the wavelength is long.”

Only when its electrons are much hotter than its ions. The kinetic dispersion relation gives a damping of 0.45 of the amplitude per radian for equal temperatures in hydrogen — gone within a cycle — falling to 0.031 at Tₑ/Tᵢ = 10 and to the electrons' floor of 0.0126 by 100.

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

“A superconductor's current capacity is a current density times its cross-section, like any conductor's.”

For a type I superconductor the current flows in a surface layer a penetration depth thick and the limit is Ic = 2πaHc, proportional to the radius. A 1 cm lead wire carries ten times what a 1 mm wire carries, at a tenth of the density.

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

“To find the cut-off field one needs the potential between the electrodes, which depends on the space charge of the electrons themselves.”

The electron's canonical angular momentum is fixed by the field alone, and its energy at the anode by the anode voltage alone, so Vc = (eB²b²/8m)(1 − a²/b²)² whatever the potential does in between. Integrated paths for a vacuum field, a uniform field and a space-charge-limited profile reach the anode at exactly the same field.

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

“Wet plates stick together because of surface tension.”

A meniscus at the rim adds a little, but the large force when they are pulled apart quickly is the viscous suction of liquid drawn into the gap, 3πμR⁴v/2h³ — 29.5 N for water between 10 cm discs 10 micrometres apart opening at a micrometre a second. At faster pulls it is capped only where the water cavitates, at ½πR²pₐₜₘ, 398 N.

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

“Rising water on the upstream side of a hinged gate pushes it harder and harder towards opening.”

For a gate hinged above its centre of pressure the rising water first presses it harder against its sill — the moment about a hinge at 0.4H is most negative at a depth of 0.80H — and then loosens it as the centre of pressure climbs, reversing at 1.33H.

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

“The thread's pull makes the spool roll; friction only resists.”

The floor's friction is what turns the spool: rolling towards the hand, the friction on the spool points away from the hand and its moment about the axle spins the spool forwards. Without enough of it — μ below about 0.3 for a pull of half the spool's weight — the spool slides instead of rolling.

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

“Where the ladder leaves the wall depends on its length and how heavy it is.”

Neither. Energy and the circle its centre must follow fix the ladder's turning rate at every angle, and the wall's push vanishes at cos θ = ⅔ cos θ₀ for any length or mass of uniform ladder — always at two-thirds of the starting height of the top.

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

“The best focus of an astigmatic lens lies halfway between its two line foci.”

Halfway in dioptres, not in distance. The circle of least confusion lies at the harmonic mean, 2f₁f₂/(f₁ + f₂), because a beam's width is proportional to its distance from focus measured as power — 111.1 mm for foci at 100 and 125 mm, not 112.5.

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

“A telescope with four vanes makes four spikes, so the number of spikes is the number of vanes.”

Each vane makes two spikes, pointing opposite ways. A cross of four vanes is two straight bars, whose spikes coincide in pairs, giving four; three vanes at 120° are not opposite each other and give six. A hexagonal pupil's three pairs of edges give six more, which is why the Webb telescope shows eight.

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

“Quantum mechanics forbids noiseless amplification.”

It forbids amplifying both quadratures at once without noise. A phase-sensitive amplifier that stretches one quadrature by √G and squeezes the other by 1/√G keeps the commutator with no extra mode and can be noiseless for the quadrature it amplifies; it is the same device that makes squeezed light.

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

“The plateaus appear because each Landau level holds exactly eB/h electrons per unit area, so the filling counts in whole numbers.”

That only fixes where integer filling occurs, at single values of the field. The plateaus are wide because disorder broadens each level and traps the states in its tails; while the Fermi level crosses trapped states, the current-carrying states do not change and neither does the Hall resistance.

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

“The condensate fraction of an ideal Bose gas is 1 − (T/Tc)^(3/2).”

For a gas in a box. In a harmonic trap the number of states grows as the square of the energy rather than its square root, the excited states fill as T³, and the fraction is 1 − (T/Tc)³ — which ten million atoms follow to within a per cent, and a thousand atoms only roughly, their transition rounded and lowered to about 0.94 Tc.

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

“Radiation shields work by being cold, or by blocking the line of sight between the hot and cold walls.”

They work by being poor emitters. Each shield floats at the temperature at which it radiates away what it receives, and N identical shields divide the flux by N + 1. A single aluminised foil between black walls passes 6.2 W/m², less than thirty black shields, 11.0 W/m², because a shiny shield's resistance is 54 times a black one's.

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

“A heap is safe if its temperature has stopped rising.”

Just above critical the temperature rises, levels off at about one unit of RTₐ²/E — some ten kelvin — and lingers: at ψ = 0.37, 0.6 per cent above the critical 1/e, for 55 cooling times, before running away in a few. A steady-looking rise of ten degrees is exactly what a heap past the point of no return shows.

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

“A hanging chain swinging as a whole behaves as a rigid rod pivoted at its top, at √(3g/2L).”

Nearly: it swings at 1.2024√(g/L) against the rod's 1.2247, because a chain bends and its lower part lags. A weight on its end lowers the frequency towards a simple pendulum's √(g/L): 1.0564 with a weight equal to the chain's own mass.

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

“The tube sings because the hot gauze's heat makes the air expand and push on the sound.”

Expansion alone is a quarter cycle out of step: heat transfer that follows the air's velocity instantly does nothing. It is the lag of the gauze's heat release behind the air's motion — a tenth of a period or so — that puts part of the heat in phase with the pressure; the drive is largest at a quarter-period lag and reverses beyond half a period.

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

“Galaxies in a cluster that look blueshifted relative to its central galaxy are moving towards the observer.”

Some are, but averaged over thousands of clusters the motions cancel and a blueshift remains: the members' light leaves a shallower part of the well than the central galaxy's. For a cluster of 10¹⁵ solar masses the prediction is −13 km/s at half a megaparsec and −22 km/s at two.

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

“A sub-recoil scheme that works along one axis works in all three.”

In three dimensions the target near zero is a tiny ball of momentum space that random landings rarely hit. A filter whose rate grows as p² gathers only 2.1 per cent of the atoms below 0.1 ħk, a steady value it never passes; one growing as p⁴ holds what it catches long enough to reach 22 per cent and keep rising.

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

“Radio from a planet's aurora comes out at the cyclotron frequency of the field where it is made.”

Just below it, by roughly the electrons' energy over mc² — 1.7 per cent for 10 keV electrons, 14 per cent for 100 keV — and because the field falls as the cube of distance along a polar field line, each frequency maps a height: on the Earth, 300 kHz is emitted about 0.8 Earth radii up and 50 kHz about 2.2.

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

“The charge on a conductor is proportional to its voltage, with the capacitance as the constant of proportionality.”

On average, for anything large or warm. A metal island joined to its leads only by tunnel junctions holds a whole number of electrons, and with no gate charge its current stays at zero until the bias reaches e/CΣ — no electron can step on until the bias pays the charging energy. At a fiftieth of the charging energy in temperature the current-voltage curve is flat across that gap.

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

“Coercivity is a property of a magnetic material, set by its composition.”

It is set mainly by the outer few nanometres of each grain and by the grain's shape. With μ₀Hₖ = 7.66 T and μ₀Mₛ = 1.61 T, a neodymium magnet reaching 1.5 T needs a defect factor α of 0.41 if its grains feel their full demagnetising field and 0.20 if they feel none — which is why raising the anisotropy only at the grain boundaries, with a little dysprosium diffused in from the surface, buys most of the coercivity of alloying the whole grain.

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

“No arrangement of permanent magnets can produce a field larger than the remanence of the material.”

A Halbach cylinder, whose magnetisation turns twice as it goes once round, has a uniform field in its bore of Bᵣ ln(rₒᵤₜ/rᵢₙ), with nothing outside: 1.54 T at a radius ratio of 3 and 3.22 T at 10 for 1.4-tesla magnets. What stops it is the coercivity of the segments nearest the bore, which feel a reverse field larger than they can hold.

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

“The classical answer returns when the interference between the two identical nuclei dies away at high energy or for heavy particles.”

The interference never dies away; its fringes become finer, their number growing in proportion to η, the closest approach in wavelengths. A detector ±2° wide at 60° reads 0.93 of the classical value at η = 100 and within 0.6 per cent of it by η = 3000 — the classical answer returns because the fringes become too fine to resolve.

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

“Light behaves as waves in some experiments and as particles in others, so each picture applies to its own kind of measurement.”

One measurement — the energy fluctuation of a single volume of black-body radiation — needs both at once: n̄ + n̄² per mode, the two equal at hν = kT ln 2. Wien's law, the particle picture, gives 0.009 of the right value at hν = kT/10; the Rayleigh–Jeans law, the wave picture, gives 5.9 times the right value at hν = 5kT.

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

“The stay-at-home twin ages more because the turnaround makes the traveller's view of home's clock run fast for an instant.”

By radar the catching-up is spread over years, not an instant: at 0.6c home's clock runs at 0.8 of the twin's rate before 2.5 years of the twin's time and after 5.5, and at 2.0 times it in between. At 0.9c the stretch is longer and the peak rate 4.36. The total under the curve is home's whole elapsed time, 10 years against the twin's 8.

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

“An electron in a plane-wave laser pulse leaves with the energy it entered with, since a plane wave cannot give it net energy.”

True without radiation reaction, and the electron then leaves exactly as it came. With it, the light-front momentum γ(1 + β) is no longer conserved but falls as 1/(1 + (2/3)rₑk₀γ₀(1 + β₀)∫a²dφ), and the energy goes into gamma rays the electron radiates while it is shaken.

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

“The released spring's elastic energy is turned into the kinetic energy of its fall.”

Less than half of it is. Each coil meets the descending block in an inelastic collision, and by the end of the collapse 46 per cent of the energy released has gone into those collisions as heat and sound, leaving 54 per cent as motion.

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

“Painlevé's paradox needs implausibly high friction and so never matters.”

It needs μ above 2√(1 + κ)/κ with κ = ml²/I: 4/3 for a uniform rod, but only 0.60 for a body with its mass concentrated near its centre and a long light arm to the contact. Bodies that turn easily about their centre reach the region at ordinary coefficients.

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

“To get an oscillator strength from a spectrum, the shape of the line must be known.”

Not from the hooks. Their separation stays within 1.8 per cent of 2√(A/K) for Lorentzian or Doppler lines up to 10 picometres wide, because in the far wings every line's dispersion falls as A/(λ − λ₀), and A contains only the number of atoms, the oscillator strength and the path.

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

“A phase-correcting coating must hold the roof's polarisation behaviour exactly to an ideal mirror's.”

The loss grows slowly at first: a coating whose s–p phase departs from a mirror's by 30° on each face still restores the peak to 0.998. Bare glass departs by 139° and leaves 0.70, so a coating need only come far closer to a mirror than glass does, across the whole visible spectrum.

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

“Quicksand is a liquid, which is why things sink into it.”

At rest it is a soft solid: a clay network gives it a yield stress, here 30 Pa, below which its apparent viscosity is a million pascal-seconds. It becomes a liquid only when a disturbance breaks the network, and once the grains have settled — in 2 to 27 minutes for grains of 0.2 to 0.05 mm — it is packed more densely than before it was disturbed.

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

“The friction of a granular flow depends on its shear rate.”

It depends on the shear rate only through the inertial number, which also contains the pressure and the grain size. The same shear rate gives a high friction under light pressure and a low one under heavy pressure, which is why every depth of a steady layer, though sheared at a different rate, has the same friction, tan θ.

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

“γ is 5/3 for monatomic gases, 7/5 for diatomic ones and 4/3 for polyatomic ones.”

Those are the values for translation and rotation alone. Carbon dioxide's two bending vibrations are partly awake at room temperature, giving γ = 1.289, as if it had 6.9 degrees of freedom; sulphur hexafluoride's soft vibrations bring it to 1.10, or about 20; and nitrogen's γ falls from 1.400 at 300 K to 1.304 at 2000 K as its stretch wakes.

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

“Knock happens because the compressed fuel–air mixture is hotter than its autoignition temperature.”

At the end of compression the charge is already above its autoignition temperature at any practical ratio, but its ignition delay is several to hundreds of milliseconds. Knock is a race: the burning gas squeezes the end gas further, shortening its delay to about a millisecond — 0.56 ms at a ratio of 10 on 90-octane fuel — comparable to the 2.2 ms the flame takes to cross the chamber.

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

“A folded velocity in a radar image is simply an error to be discarded.”

It is the true velocity shifted by a whole number of windows, and it can be recovered: two pulse rates in the ratio 4:3 fold differently, and the pair of readings identifies the true speed uniquely across ±75 m/s, three times the faster rate's window — or by continuity from slower winds nearby, which is how a tornado's folded couplet is read.

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

“A chirp radar measures a target's range from its compressed echo's delay.”

Only if the target is stationary. Along a linear sweep a frequency shift is indistinguishable from a time shift, so a Doppler shift of 0.05 B moves the spike by 2.5 units of 1/B and lowers it to 0.950 of its height; a moving target is misplaced in range in proportion to its speed unless the radar sweeps both up and down.

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

“A beam carrying a tiny fraction of a plasma's electrons can only disturb it by a correspondingly tiny amount.”

The fastest growth of a cold beam through cold plasma is 0.687 ωₚ times the cube root of its density over the plasma's: 0.136 ωₚ for one electron in a hundred, 0.0068 ωₚ for one in a million — still a growth of e in 23 plasma periods.

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

“The direction of a self-made magnetic field is fixed by how the machine is built.”

The equations are unchanged when the current changes sign, so above the critical speed there are two steady states, with equal speed, torque and current, differing only in the field's direction. Runs from seeds of +10⁻⁶ and −10⁻⁶ reach mirror images. The machine's build fixes the speed it needs; the stray field it starts from fixes the direction.

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

“Charging a capacitor from a supply always wastes as much energy as it stores.”

Only if the supply is switched straight to its final voltage. Raised in N equal steps, each allowed to settle, the loss is CV²/2N of the energy stored — 12.5 per cent in eight steps, 1.6 per cent in sixty-four — because the loss is set by the voltage across the resistance while the charge moves.

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

“To hold off breakdown in a coaxial gap, make the inner conductor as thick as possible so its surface field is small.”

Thickening the conductor lowers its surface field but narrows the gap. In a tube of 10 cm radius the onset voltage rises from 8 kV for a 0.1 mm wire to a maximum of 128 kV at an inner radius of 3.41 cm, a ratio of 0.341, and falls beyond it; past that ratio the first ionisation becomes a spark at once.

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

“Relativity is a small correction to atomic binding.”

For the single electron of a hydrogen-like ion it is not small in heavy elements: Schrödinger gives 115.2 keV for uranium and Dirac 132.3, within half a keV of the measured value; at Z = 120, 196 against 264 keV.

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

“An intense laser ionises atoms by tunnelling.”

Only when the field changes slowly compared with the tunnelling time, so that γ = √(Iₚ/2Uₚ) is less than one. For hydrogen at 800 nm that needs more than 1.1 × 10¹⁴ W/cm²; at 400 nm more than 4.6 × 10¹⁴; at 3.2 µm only 7.1 × 10¹². Below those intensities the atom absorbs many photons at once instead.

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

“With enough blocks a single stack can lean out as far as is wanted.”

True, because ½Hₙ diverges, but the cost is exponential: one block length needs 4 blocks, two need 31, three 227, four 1674 and five 12367, each extra length about e² = 7.4 times the blocks of the last.

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

“The pressure at the bottom of a container of water is atmospheric pressure plus the weight of the water above, per unit area.”

Plus the weight of the water above and whatever is pressing on the water's surface. In a sealed bottle the air pocket sits below atmospheric by exactly the weight of the water standing above the tube's tip — 2.0 kPa at a 30 cm fill, falling to 73 Pa as the level reaches the tip — so the outlet feels 906 Pa above atmospheric at every level.

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

“A clock's accuracy is the escapement's business.”

The escapement's error is tan φ/2Q of every period, because the push only has to replace what the pendulum loses. A push a fifth of the swing after the bottom costs 8.8 seconds a day with Q = 1,000 and 0.088 with Q = 100,000; the pendulum's quality hides the escapement's faults.

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

“Pseudorapidity is just rapidity measured with a detector.”

It equals rapidity only for massless particles. At fixed transverse momentum dy/dη = p/E, so a distribution flat in rapidity dips at η = 0 to the particles' transverse speed: for 0.5 GeV/c, 0.96 for pions, 0.71 for kaons and 0.47 for protons.

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

“The periodic wobble of an eccentric satellite's clock is a gravitational effect, because the satellite's height changes.”

Half of it is the satellite's speed. The gravitational slowing swings between −1.37 and −1.88 parts in 10¹⁰ round the orbit and the speed's between −0.58 and −1.09; both are deepest at closest approach, and integrated they contribute exactly equal halves, √(GMa) e sin E/c² each.

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

“The thickness of an ice sheet at the coast is set by how much snow falls on it and how fast it flows.”

At the grounding line it is set by buoyancy alone: the ice is exactly thick enough to touch the bed, the density of seawater over that of ice times the water depth — 1.11 times it in the benchmark, 803 m where the bed is 723 m below the sea. The snowfall and the flow then decide where the line sits, not how thick the ice is there.

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

“The thicker the twisted layer, the better it turns the light.”

Transmission between crossed polarisers is 1 − sin²((π/2)√(1 + u²))/(1 + u²), which is exactly one at u = √3, √15 and √35 and dips between; a thicker layer guides better on average, its leakage falling as 1/(1 + u²), but at its dips it is worse than a thinner cell at its maximum.

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

“Measuring a temperature optically needs a calibration against a contact thermometer.”

The anti-Stokes to Stokes ratio is exp(−ħΩ/kT) times a known frequency factor, with no other property of the material in it: 0.146 for silica's 440 cm⁻¹ band at room temperature, and the formula inverts to the temperature exactly. A one per cent error in the ratio is 1.42 K at 300 K for silica, 0.47 K for diamond.

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

“Boltzmann's equation describes how a gas actually evolves.”

It describes how almost every large system evolves, for times short compared with recurrence. A single ring fluctuates about the law by about 0.26 at 100 sites, 0.07 at 1,000 and 0.013 at 20,000, and every ring returns to its start after 2N ticks, which no solution of Boltzmann's equation can do.

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

“Liesegang bands are spaced by the diffusion length, so they grow further apart as the square root of time.”

Their positions follow xₙ = 2.88√tₙ — the time law — but their spacing follows a separate law: each position is a fixed multiple of the one before, xₙ₊₁/xₙ → 1.092 or 1.192 for nucleation thresholds of 0.35 and 0.5, a geometric progression.

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

“Salt dissolves above 75 per cent humidity and crystallises again below it.”

Going down, the drop must supersaturate before a crystal can nucleate, and it stays liquid to about 45 per cent. On a coast where the humidity swings between 52 and 92 per cent every day, a salt crust dissolves the first night and stays liquid thereafter, though the air is below 75 per cent for hours every afternoon.

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

“A moiré pattern is just an optical curiosity of overlapping fabrics and screens.”

It magnifies mismatch by the ratio of the bands to the lines, P/p = 1/ε for a strain ε: a 1,200-line-per-millimetre grating on steel strained by 10⁻³ shows a fringe every 0.83 mm, each a contour of equal displacement, and the same magnification aligns microchip layers to nanometres.

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

“The wavelength of the ripples in a shaken dish is set by the size of the dish.”

In a dish many wavelengths across it is set by the liquid's dispersion relation: the wave whose own frequency is half the drive's. Water 1 cm deep shaken at 10 Hz makes 55.5 mm waves, at 50 Hz 9.9 mm, at 200 Hz 3.63 mm.

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

“A spin echo cancels the effect of an uneven magnetic field.”

Of an uneven field that each spin sees unchanged. A molecule that diffuses through a 5 mT/m gradient sees different fields before and after the pulse, and the echo keeps only exp(−γ²G²D·TE³/12): 0.71 at 100 ms and 0.06 at 200 ms for room-temperature water, as a 3,000-walker simulation reproduces.

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

“Squeezing reduces the noise of a measurement.”

It moves the noise. The disk of variance 25 for 100 atoms becomes an ellipse of variance 1.22 in one direction and 535 in the other, and twisted too far the ellipse bends round the sphere and the state is worse than independent atoms: the gain peaks at 12.0 dB for 100 atoms at a twist of about one unit of 1/√N and is below zero past about 2.3.

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

“Radiation pressure on a cloud of atoms pushes it from outside, so a laser trap's beams can only squeeze it.”

The beams squeeze it through the shadow each atom casts, and the light each atom rescatters pushes its neighbours apart. Both fall as the inverse square of the separation; with rescattered light absorbed 1.3 times as readily as the laser, the net is a repulsion 1.6 × 10²⁴ times the atoms' gravity — the force between charges of 1.0 × 10⁻⁴ e.

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

“The hydrogen spectrum is a property of the proton and the electron.”

It is a property of an inverse-distance potential with a wall or a centre the electron cannot pass. The electron above helium shows the same 1/n² series scaled by Λ², 0.657 meV instead of 13.6 eV, and its transitions at 125.9 and 149.4 GHz stand in the hydrogenic ratio 0.843; the sharp-surface model gives 119.2 and 141.3 GHz, ratio 0.844.

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

“An eruption is driven by the steam produced by the heat flowing in from below.”

It is driven by heat already stored. Throwing out the top of the column lowers the pressure on everything beneath, superheating it: 5 m removed superheats the water from 5 to 12 m deep by up to 11.8 °C, and at the surface 10 °C of superheat flashes into 39 times the water's own volume of steam.

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

“An outlet wider than about five grain diameters cannot jam.”

Under the law fitted with a critical size, R* = 4.94, it cannot; under a law with no threshold that agrees with it to within a factor of 2.7 everywhere from 3 to 4.4 diameters, it jams on average once an hour at 5.01 diameters and once a year at 6.04 for 1 mm beads. The two differ only beyond about a million grains between jams, where measurements stop.

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

“The pitch-drop experiment drips once every eight or nine years.”

It did until 1988: seven intervals averaging 8.3 years. Since the building was air-conditioned the next two took 12.4 and 13.4 years. With an activation energy of 200 kJ/mol each degree cooler lengthens the wait by 32 per cent, so a fall of 1.6 °C in the average temperature accounts for the change.

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

“A water clock's vessel should be wider at the top to make up for the faster outflow when it is full.”

Wider at the top in one particular way: the surface area must grow as the square root of the depth, so the radius as its fourth root. A straight-sided cone matched to that shape at the rim and near the floor still runs from 24.0 to 30.3 mm/h and errs by up to 41 minutes on an even scale; the quartic bowl lowers its level at 27.0 mm/h at every depth.

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

“A falling column is simply falling, so its parts exert no forces on one another.”

Only a column falling freely with no pivot is unstressed. About a pivot it turns at α = 1.5·g·sinθ/L, and only the point at two-thirds of the height has exactly the sideways acceleration its own weight would give it; above, the material must be dragged along, and below, held back, which is the bending.

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

“The taper of a whip is what makes its tip go supersonic.”

In a whip that loses no energy, the end's speed is √(2E/m), m the mass of the tip, whatever the taper: uniform, linearly tapered and faster-tapered whips with the same 0.2 g cracker all reach 387 m/s. The taper decides how gradually the speed builds — sonic 1.6 cm from the tip in a tapered whip, only in the last millimetre in a uniform one — not how fast the end goes.

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

“The swing's plane precesses steadily, as a Foucault pendulum's does.”

It turns only while the energy is in the bounce, when the swing is too small to see, and stands still while the swing holds the energy: a staircase of 43.6° steps, not a slope. The step depends on the twist, 9.1° at an angular momentum of 0.12 × 10⁻⁵ and 67.3° at 1.0 × 10⁻⁵.

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

“Phase contrast makes a cell's brightness proportional to its thickness.”

For thin cells, as 2φ: 0.210 brighter at 0.1 rad. The relation bends as the phase grows, 1.20 at 0.5 rad, and a plate wide enough to pass useful light also shifts the slowly varying part of a large object's own diffracted light, so the middle of a 50 μm region fades to 0.86 or 0.98 of the background and only its edge shows.

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

“An axicon focuses light, like a lens, only to a line instead of a point.”

It spreads it. A lens of the same 10 mm aperture puts 188,000 times the incident intensity at its focus and is 520 times dimmer 3 cm beyond; the axicon's axis rises steadily to at most 3,476 times the incident, because each point on it is fed by a ring of the aperture whose circumference grows with distance.

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

“Distant mountains look blue because the air absorbs red light.”

Air absorbs almost no visible light. A dark ridge looks blue because the air in front of it scatters sunlight towards the eye, most strongly in blue: at 20 km it already glows with 41% of the sky's blue and 11% of its red, 4.5 times bluer than the sky for a near ridge in clean air. Far ridges turn pale, the glow saturating at the sky's own colour.

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

“Photons scattering off electrons always lose energy to them, by Compton recoil.”

Only photons harder than 4kT. Off electrons at 10⁷ K, a photon of 1 keV gains on average 4.8 eV per scattering from the electrons' motion, and one of 10 keV loses 128 eV; the crossover is at 3.45 keV, and soft photons cool a gas while hard ones heat it.

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

“Half the W's mass can be read off the peak of the electron's momentum across the beam, where the decay puts it.”

The peak is there for a W at rest across the beam, at M/2 = 40.19 GeV, because sin θ is flat near 90°. But it moves with the W's own sideways motion at first order: 6 GeV of it shifts the electron's falling edge by 2.44 GeV of mass equivalent, while the transverse-mass edge moves by 0.01.

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

“The speed of sound in a gas gives its ratio of heat capacities, γ, which has to be measured.”

For a gas whose molecules can turn or vibrate, yes: nitrogen's γ is 1.3997 at 300 K and 1.294 at 3000 K as its vibration wakes. For a monatomic gas γ = 5/3 exactly at any temperature drawn, because its atoms can only move three ways, so its speed of sound has no unknown in it but k.

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

“Adding tuned resonators to a panel improves its sound insulation.”

In a band near their tuning. Just above it the panel's dynamic mass passes through zero, at f₀√(1 + μ) = 367 Hz, and its loss falls to 5.2 dB, far below even the bare panel's 15.1: the resonators make the panel worse there than without them.

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

“Critical slowing down always warns before a combustor starts to oscillate.”

Where the onset is subcritical, a loud state coexists with the quiet one before the quiet one loses stability, and noise can carry the mode over the barrier between them. With noise strength 0.005 the jump comes after a mean of 136 natural times at a growth rate of −0.24, where the quiet state's variance is a modest 0.042 and its decay time still four natural times.

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

“Thunder stops being heard beyond about twenty kilometres because the air absorbs it.”

Air absorbs thunder's low notes at a fraction of a decibel per kilometre. What silences it is refraction: in air cooling 6.5 K per kilometre every ray curves upwards with a radius of about 91 km, and a listener 2 m up is in shadow beyond 14.0 km of sound from 1 km up the channel and 23.7 km of sound from 3 km up.

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

“A restoring force makes a stable equilibrium.”

Only if it arrives on time. The light in the cavity takes 31.8 μs to follow the mirror, so the restoring push lags the motion, and a lagging restoring force feeds energy in: a free mirror held by 100 W on the blue side swings at 625 Hz with an amplitude growing by e every 4.0 ms.

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

“Lensing changes only the brightness of what is behind the lens.”

For a wave near w ≈ 1 the lens multiplies each frequency by a different amount: across a detector's band a 30-solar-mass lens raises a chirp's amplitude by 3 per cent at 10 Hz and by up to 93 per cent near 500 Hz, a distortion the unlensed signal does not have.

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

“A ratio of nineteen orders of magnitude between two masses requires a parameter tuned to nineteen decimal places.”

Only if the mass depends on the parameter as a power. The strong scale depends on the coupling at the Planck mass exponentially: a change of one unit in 1/αₛ = 52.5, under two per cent, changes it by a factor of 2.01. The Higgs boson's mass, which depends on the Planck scale quadratically, is the case that does need tuning.

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

“Once the optical properties of a metal are measured, the Casimir force between plates of it is determined.”

Gold's measured permittivity fixes the Lifshitz sum at every Matsubara frequency but zero, where it is never measured. The Drude and plasma extrapolations to zero frequency agree to 18 per cent at the first Matsubara frequency and less beyond, yet the pressures they predict differ by 16 per cent at 1 μm and 41 per cent at 3 μm, entirely through the static TE term.

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

“Field lines show where charged particles go.”

Only for particles whose motion is dominated by drag, which move at the field divided by the drag and so have no velocity of their own. With a drag β = 10 in the field's own units a released charge strays from its line by 0.027 charge-separations, and the departure falls as β to the power −1.98 — but in a vacuum, with no drag, the same charge never reaches the end of the line.

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

“The size of the normal regions in the intermediate state is set by the critical field.”

The critical field sets how much is normal, not how finely it is divided. The period comes from the balance between wall energy, falling as 1/a, and the energy of the field fanning out of the surface, rising as a: for a 1 mm lead plate half normal, with a wall length of 0.1 μm, it is 54 μm, and it scales as the square root of the thickness.

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

“Relativity only adds a factor γ to the mass in Newton's parabola.”

That is right at the launch, where the exact path is a parabola with inertia γ₀m — 7.09 times flatter than Newton's for a launch at 0.99c. It fails further out, where the path becomes a hyperbolic cosine, y = (ε₀/qE)(cosh(qEx/p₀c) − 1), and a 1 MeV electron in 1 MV/m is deflected 38.9 cm in a metre against 37.4 cm from the γm parabola.

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

“A charge in crossed fields drifts at E/B and gyrates on an orbit of fixed size.”

The drift at E/B holds below E = cB; the orbit's size does not. From rest the loops are 2γ²mE/qB² high, γ the Lorentz factor of the drift: 0.66, 1.88 and 6.13 mc/qB at E/cB = 0.3, 0.6 and 0.85, against Newton's 0.6, 1.2 and 1.7, and without limit as E approaches cB.

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

“A pair that cannot violate a Bell inequality has no entanglement worth anything.”

Teleportation with Werner's state reaches the fidelity (1 + p)/2, beating the best classical scheme's 2/3 from p = 1/3 onwards, while the CHSH inequality is violated only above p = 1/√2: at p = 0.6 the pair teleports at 0.80 and its largest CHSH value is 1.70.

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

“The interference of hot molecules fades because they emit photons, so the number of photons sets the loss.”

The number and the wavelength together. For paths a micrometre apart and molecules at 3000 K, three photons emitted would leave 2.3 per cent visibility if every one revealed the path; because many are longer than twice the separation, 4.5 per cent remains, and between paths 0.1 μm apart 92 per cent of the fringes survive the same glow.

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

“A compensated pendulum is immune to temperature.”

To slow, uniform changes. After a sudden 1 K warming a mercury pendulum loses 0.035 s before its mercury, which warms more slowly than its rod, catches up, and that loss is never recovered; and if the top of the case warms 1 K more per metre than the bottom, the mercury pendulum loses 0.25 s a day.

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

“A stopped raw egg restarts because some of the liquid sloshes against the shell when it is released.”

Because the liquid never stopped. Held for half a second, the shell is still while its contents, spinning down only as viscosity drags on them, keep 72 per cent of their speed; on release they share it with the shell, which restarts at 60 per cent of its former speed. A longer hold leaves less: 22 per cent after two seconds.

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

“The thickness of a liquid's surface is a property of the liquid, like its density.”

It is a property of the liquid and of the measurement. The roughness depends logarithmically on the largest ripple counted, set by the instrument's coherence length or, at most, by the capillary length: an X-ray measurement that sees ripples up to a few micrometres reports about a third of a nanometre for water, and in free fall the same surface would reach 0.43 nm over a metre.

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

“Warm air rises, so a heated building leaks warm air at the top and nothing else happens.”

It leaks out above the neutral plane and draws cold air in below it, with the pressure difference growing linearly from that plane: 69 Pa outward at the roof and 69 Pa inward at the ground of a 100 m building at 21 °C in −10 °C air with even openings. Where the plane sits is set by the openings: three times more leakage low down puts it at 10 per cent of the height.

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

“Measuring a cell's volume measures how much it has grown.”

A cell can double its volume by taking up water with no change in what it is made of. In the model a cell swelling to twice its volume halves its refractive index excess, from 0.0796 to 0.0398, and its integrated phase — its dry mass, 300 pg — stays exactly the same.

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

“A lens focuses all colours together if it gives every ray the right phase at every wavelength.”

That requirement is the same as giving every ray the same group delay to the focus, and the edge ray's extra path, √(R² + f²) − f, must be made up by holding the central light back. For a 10 mm focal-length lens of 5 mm radius that is 3,937 fs, which a glass lens supplies by being 2.27 mm thicker in the middle.

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

“Letting an engine's gas expand further is free extra work.”

It costs power. The work per cycle rises but the piston must sweep a larger volume for it, and the mean effective pressure falls from 11.0 bar for the Otto cycle at a ratio of 10 to 8.8 bar expanding to 13 and 4.4 bar at complete expansion: more work from each charge, less from each litre of engine.

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

“An earthquake that changes a geyser's intervals must have changed its eruptions.”

It need only change the refill. Cutting the model's refill rate by 25 per cent leaves both kinds of eruption as they were and lengthens every wait, the long ones from about 86 to about 104 minutes, raising the slope of the rule from 10.5 to 14.1 minutes per minute of eruption.

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

“Thin-plate bending theory gives the speed of flexural waves in a plate.”

Only while the wavelength is long against the thickness. Below about 0.1 MHz·mm in aluminium it agrees with the exact Rayleigh–Lamb result to a few per cent; above about 0.3 it runs away, predicting a speed that grows without limit, while the true wave levels off at the 2.92 km/s of a Rayleigh wave on each face.

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

“The strongest surface-wave shaking arrives at the speed of the strongest waves.”

It arrives at the slowest group velocity, not at any phase speed. For a 35 km crust the group velocity dips to 3.39 km/s near 19 seconds, below the crust's own shear speed, and waves over a band of periods arrive together there: 3,000 km away the largest motion comes near 14.8 minutes, three minutes after the first 80-second waves.

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

“The polarisation of a moving insulator comes from the magnetic force on its charges, so it depends only on its permittivity.”

That was Lorentz's account, and it gives (ε − 1)/ε of a metal's voltage at any permeability. Relativity adds the electric dipole that a moving magnetisation carries, and gives (εμ − 1)/εμ: 0.833 against 0.944 for ε = 6, μ = 3, a voltage 13 per cent larger. The two agree only for a non-magnetic material.

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

“Cosmological redshift is a stretching of space and not a Doppler shift.”

In the cloud the redshift of light from a particle of rapidity η is 1 + z = e^η, which is exactly the relativistic Doppler factor for that relative motion and exactly the ratio of the receiver's clock to the emitter's at emission. In empty space the two descriptions are the same number; what separates them in the real universe is gravity, which the empty cloud does not have.

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

“The de Broglie wavelength h/p is an independent postulate of quantum mechanics.”

Given a clock that ticks at its rest energy divided by Planck's constant, h/p follows from the Lorentz transformation alone: the readings of a moving row of such clocks along a laboratory instant repeat every c²/γvν₀ = h/γmv, 0.123 nm for a 100 eV electron. What quantum mechanics adds is that the clock exists and that the wave interferes.

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

“Each radioactive nucleus has one way to decay, towards stability.”

Potassium-40 is heavier than both of its neighbours, by 1.311 MeV than calcium-40 and 1.504 MeV than argon-40, and decays both ways: 89 per cent by β⁻ decay, nearly all the rest by electron capture. An odd–odd nucleus sits on the upper of two parabolas and can fall to either side.

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

“Weber's bars were simply not sensitive enough to detect gravitational waves.”

They were not, but the decisive argument against his reported events was not sensitivity. A burst depositing one room-temperature kT in his bar, from the centre of the Galaxy and emitted in every direction, would have cost its source about 80 solar masses of rest energy, and at one event a day the Galaxy would have been radiating some 3 × 10⁴ suns a year.

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

“The grains in a laboratory dusty plasma are held up by the plasma's electric field.”

Only in the sheath, where the field is strong, and only if they are small enough: the charge grows as the radius and the weight as its cube, so a field of 10⁴ V/m balances the weight of a 1.5 g/cm³ grain of 11 μm and no larger. Identical grains therefore settle in one thin layer at the height where the field matches their size.

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

“Lyttleton and Bondi's charged universe was ruled out because a net charge would be noticed on the Earth.”

An Earth-sized body with any imbalance would neutralise itself from its surroundings; at 10⁻²¹ per nucleon its surface field would be 7 × 10⁷ V/m. What ruled the idea out was the neutrality of individual molecules, measured by letting gas out of a tank: below about 10⁻²⁰ within a year, against the 2 × 10⁻¹⁸ the expansion needed.

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

“Einstein and de Haas confirmed that magnetism is caused by electrons circulating in atoms.”

They found the effect and reported the ratio expected for circulating charge, g′ = 1. Later measurements, and Barnett's inverse experiment of spinning a rod to magnetise it, gave g′ close to 2 — 1.92 for iron — which no orbit can give; the magnetism of iron is about 95 per cent electron spin.

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

“Heisenberg's inequality, σₓσₚ ≥ ħ/2, is the strongest statement of the uncertainty principle for position and momentum.”

The entropic relation Hₓ + Hₚ ≥ ln(πeħ), proved in 1975, implies Heisenberg's — no distribution of a given variance has more entropy than a Gaussian — and is strictly stronger. Only a Gaussian meets both bounds.

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

“Quantum mechanics always slows down classical heating.”

At the resonant values of the effective Planck constant, ƛ = 4π times a rational, the free evolution between kicks does nothing to some states and the kicks add coherently: at ƛ = 4π the energy grows as n²K²/4, 63 times the classical energy after 60 kicks.

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

“The more powerful a source, the higher its plume goes.”

Only as the quarter power of the buoyancy flux: a tenfold stronger source rises 1.78 times higher. In a troposphere stratified at N = 0.01 s⁻¹ a 1 GW fire sends its plume 1.7 km up and a 100 GW one 5.4 km; reaching the stratosphere at 11 km takes about 2 TW.

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

“A hot-air balloon can climb as high as its burners can keep its air hot.”

The outside air thins faster than its cooling improves the temperature ratio, so lift per cubic metre falls by about a tenth per kilometre. A 2,800 m³ balloon carrying 550 kg plus its envelope needs 0.241 kg per cubic metre and reaches it at 6.0 km with the envelope at 120 °C; above that, no burner helps.

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

“Friction simply removes energy from a system.”

It moves energy into an environment, and the environment pushes back with fluctuations. An oscillator damped by a reservoir of other oscillators at zero temperature relaxes to its ground state with uncertainty product exactly ħ/2; it is the fluctuations, not the damping, that a Lagrangian for the system alone cannot supply.

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

“The danger of an early looping coaster was the high force at the bottom of the loop.”

Also how suddenly it came. A straight joined to a circle changes curvature from zero to 1/R at a point, so the seat force steps from 1 g to 6.5 g in the time the car's wheelbase takes to pass: the jerk is in principle infinite. On a clothoid entry the force ramps up at a finite rate, here about 5 g per second.

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

“The linewidth of a semiconductor laser is set by the Schawlow–Townes formula.”

Measured diode linewidths in 1981 were about thirty to fifty times the formula's prediction. The gain in a semiconductor is coupled to its refractive index, so every amplitude kick is converted into an extra phase shift as the gain recovers, multiplying the width by 1 + α² — seventeen for α = 4.

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

“Any grid of wires closer together than a wavelength makes a good polariser.”

The leak along the wires is roughly four times the square of (d/λ)·ln(d/2πa): a tenth of the power at a spacing of a tenth of a wavelength for thin wires, a thousandth at a hundredth. A good polariser needs the spacing far below the wavelength, thick wires, and wires tall enough that each gap is a waveguide below its cutoff.

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

“Adding more of a small inert additive just dilutes a suspension of large particles.”

The small particles' pressure pushes the large ones together with a well of depth φ(3/2q + 1) kT at contact: 3.2 kT for an additive a tenth of the large radius at 20% volume fraction. The large particles' second virial coefficient turns negative and falls to the condensation threshold near −1.5 times its hard-sphere value at about 26% for that additive, and the large particles then clump and separate out.

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

“A refrigerator that cools by pumping on a liquid works down to whatever temperature its pump can reach.”

Evaporative cooling is proportional to the vapour pressure, which falls as e^(−L/T): a pumped helium-3 bath delivering 10⁻⁴ W at 0.3 K delivers about 10⁻²⁴ W at 50 mK. The dilution refrigerator's dilute phase keeps its helium-3 density fixed at 6.6%, so its cooling falls only as T², and at 50 mK a 100 µmol/s machine gives 2 × 10⁻⁵ W.

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

“A periodic structure reflects the wavelength whose half equals its period.”

It reflects the wavelength whose half equals the period's projection on the direction of the wave, t cos α + h sin α for a stair seen at elevation α. A long structure seen from close by spans many elevations and so reflects a sweep of wavelengths, from c/2t for steps seen level to c/2√(t² + h²) for steps seen along the slope.

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

“A Doppler vibrato moves a note's frequency up and down by v/c and does nothing else to it.”

A periodically swinging frequency is a comb of lines spaced by the rotation rate, with Bessel-function powers whose spread grows in proportion to the note. A 250 Hz tone through the turning horn keeps 78% of its power at the original frequency; a 4 kHz tone, with a modulation index of 11, spreads its power over about ±80 Hz and keeps almost none there.

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

“A line's Doppler shift is the position of its deepest point.”

For a line made of two Doppler-shifted streams the deepest point and the centre of gravity are different numbers — 615 and 232 m/s to the blue in the two-stream model — and the midpoints at successive depths lean progressively redward. Which point of an asymmetric line is its shift is a definition, and the definition changes the answer by hundreds of metres a second.

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

“A precise enough measurement of light bending by a planet would measure the speed of gravity.”

Precision is not the obstacle; order is. At the 2002 pass the bend was 1,140 µas and the velocity term 46 µas, both above the 10 µas the experiment reached, while a term of order (v/c)² — the first place a speed of gravity different from light's would enter — is seven orders of magnitude below the velocity term.

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

“The charge density on a conductor is proportional to the curvature of its surface.”

On a conducting spheroid the density follows the fourth root of the Gaussian curvature exactly, so a tip 25 times more curved than the waist carries 5 times the charge per unit area; the mean curvature rises 9.6 times there. On two spheres joined by a wire it follows the square root. No single power of any curvature fits both.

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

“Magnetostriction is a property of iron, and its size is roughly the same in every magnetic material.”

It spans more than three decades and both signs: grain-oriented transformer steel under one part per million, polycrystalline iron −7, nickel −33, cobalt −62, Galfenol about +250 and Terfenol-D about +1600. The largest values come from rare-earth atoms whose non-spherical 4f electron clouds turn with the spin.

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

“Steel's resistance to the coil only increases as it gets hotter, because its resistivity rises.”

It rises as the square root of the resistivity, from 0.99 mΩ at room temperature to 3.2 mΩ near 700 °C, and then collapses to 0.37 mΩ above the 770 °C Curie point, where the permeability falls to one and the current spreads ten times deeper.

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

“The size of a trapped cloud is its thermometer: measure the size and the temperature follows.”

For a classical cloud the mean square size is proportional to the temperature. For a degenerate Fermi cloud the size stops changing below about a quarter of the Fermi temperature, and the temperature has to be read from the shape — the flat top and steep shoulders of the profile — rather than the width.

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

“Bloch oscillations are never seen in metals because they do not exist in real materials.”

They exist wherever an electron stays coherent for a Bloch period. In a metal with a 0.3 nm lattice at 10⁵ V/m the period is 138 ps against a scattering time of about 10 fs, so a swing is never a thousandth complete. A 10 nm superlattice at 10⁶ V/m has a period of 0.41 ps, shorter than its scattering time, and the oscillation was observed in 1992.

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

“Every proton synchrotron must cross transition on its way to high energy.”

A ring's transition is set by its focusing. The CERN PS Booster (γ 1.17 to 3.13, transition about 4.1) stays below its own; the SPS (27.7 to 480, transition about 18) and the LHC (480 to 7460, transition 55.7) are injected above theirs; only the PS and RHIC, among these, must cross.

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

“Every nuclear reaction confirms E = mc², so no special experiment is needed to test it.”

A reaction's energy balance is usually computed from masses, which makes the agreement a definition. The 2005 comparison measured both sides independently — the mass loss of ²⁸Si capturing a neutron, 9.097 thousandths of a mass unit, as a ratio of cyclotron frequencies, and the 8.47 MeV of gamma rays as diffraction angles from a silicon crystal — and found 1 − Δmc²/E = (−1.4 ± 4.4) × 10⁻⁷.

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

“A drop on a shaken bath bounces in time with the shaking, once a cycle.”

Only in a window of the shaking's strength. In the model, the drop rests on the film below a peak acceleration of about 0.7 g, lands once a cycle at a fixed phase up to about 2 g, alternates high and low bounces near 2 g, lands once every two cycles near 3 g, and lands irregularly beyond — a period-doubling cascade.

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

“A plant wilts when it has lost most of its water.”

With a wall of volumetric stiffness 10 MPa, a cell at full turgor in pure water holds 0.73 MPa; losing only 6.8% of its water takes the turgor to zero, and the tissue goes limp long before it is dry. Below that point the cell can lose more water only by concentrating its sap.

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

“A fair coin, fairly tossed, lands on either face with probability exactly one half.”

Only if it spins exactly about a diameter. A real toss tilts the angular momentum from the coin's plane, and the normal then precesses about it, keeping the starting face up for more than half the flight: ½ + arcsin(cot²ψ)/π, which is 50.8% at a tilt of 81°. Tossers in 2023 found 50.8% over 350,757 flips.

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

“Making a station larger makes it Earth-like in proportion to its size.”

The miss of a ball tossed up a metre falls only as the inverse square root of the radius: 96 cm at 10 m, 36.8 cm at 100 m, 11.9 cm at 1 km and 3.8 cm at 10 km. Each tenfold enlargement buys only about a threefold improvement.

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

“A sensor tuned to water can tell rain from anything else on the screen.”

Anything touching the glass with an index above 1.07 lets the beam out at 45°: ice (critical angle 59.5°), water (61.3°), washer fluid (63.5°) and an oil smear (75.3°) all read as wet. The sensor detects optical contact, not water.

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

“A diving mask simply keeps water out of the eyes.”

It restores air in front of the cornea and so the cornea's power, but its flat window refracts every ray: near the centre angles are multiplied by about 4/3, so objects look a third larger and nearer, and the whole air-side hemisphere of view is filled by a cone of water only 97° across.

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

“Interdiffusion of two metals is described completely by a single diffusion coefficient for the mixture.”

A single coefficient describes how the composition profile spreads, but not how the lattice itself moves. Darken's analysis needs two intrinsic diffusivities: their weighted sum sets the spreading, and their difference sets the drift of the lattice planes, which moves the markers as √t at a rate proportional to the mismatch.

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

“Watt's separate condenser saved the heat that went into the cylinder walls.”

It saved more than heat: it removed the entropy made when steam at 100 °C condenses on a wall at a lower temperature. Condensing a kilogram on a wall at 45 °C throws away 307 kJ of work against surroundings at 20 °C, most of the 484 kJ a kilogram could yield between 100 and 20 °C — the quantity the second law counts, a century before it was named.

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

“Water is a good heat-pipe fluid at any temperature where it is liquid.”

Its merit ρσL/μ rises from 1.8 × 10¹¹ W/m² at 20 °C to a maximum of 5.1 near 148 °C and falls towards the critical point at 374 °C, where surface tension and latent heat vanish; and at 5 °C each pascal of vapour-pressure drop costs 17 mK against 1.6 mK at 50 °C, so a water pipe is sluggish when cold.

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

“Correlating two noise records gives the travel time between them at once.”

A single window of simulated noise gives a ragged correlation in which the pulse is lost; the background falls only as the square root of the number of windows averaged, and the pulse-to-background ratio grows from 2.6 after one window to 13.1 after a hundred. Seismologists stack days to months of records.

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

“A watch resonator is shaped like a fork only for convenience of mounting.”

The fork's two tines swing in opposition and leave the base nearly still, so little energy leaks into the mount. The quality factor rises from about 2,000 for a single clamped tine to 10,000 for a fork in air and 80,000 for one sealed in vacuum, and the ringdown time Q/πf from 19 ms to 0.78 s.

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

“Once a plasma ignites it burns steadily, like a flame.”

Integrating the balance at fixed confinement time gives two fates from one plasma: at 5 s it climbs without stopping from 14 keV and goes out from 6 keV. Near 10 keV the reaction rate rises as the square of the temperature, so a burn tends to run away, and a real machine needs active control.

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

“Twist in a tokamak stops the drift.”

The drift continues at the same speed. Twist carries each particle round the bore so that it drifts outward on one half of its circuit and inward on the other. The integrated path closes into a circle offset by the drift speed over the rate of circulation, about 1.3 cm for a 10 keV ion at a safety factor of 3.

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

“The 2022 shot produced more energy than it consumed.”

It released 3.15 MJ for 2.05 MJ of laser light, a gain of 1.5. The lasers drew about 300 MJ from the grid to deliver that light, so measured at the wall it returned about 1 per cent, and the figure for the fuel's kinetic energy, about 15 kJ, shows where most of the energy never reached.

Tested in The fuel held together only by its inertia · the radiation pressure reading path

“Hydrogen's 21-centimetre line comes from the magnetic interaction of two dipoles at the distance of the Bohr radius.”

The ordinary dipole interaction averages to zero over the spherical 1s cloud. The whole splitting comes from the part of the electron's cloud on top of the proton, where the interior term acts, which is why 2s is exactly an eighth of 1s, with 177.6 MHz measured, and why a 2p electron, with none at the centre, has no such term.

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

“An ion engine's thrust comes from the exhaust, so the force on the spacecraft is applied somewhere vague and the momentum balance is all that can be said.”

The force is on the far grid and it can be computed there. The Maxwell stress ½ε₀E² on that grid, with E four-thirds of V/d, equals the exhaust's momentum flux at every voltage, to the limit of the arithmetic.

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

“Decay inhibited between two mirrors means the atom holds its energy for ever.”

Only if the mirrors are perfect and the dipole lies along them. A dipole across a pair of parallel mirrors has a mode at any spacing, and real mirrors leak. The Hulet experiment needed its atoms to be aligned by a field for precisely this reason, and a real cavity's lifetime is limited by what it lets through.

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

“X-ray lasers are hard to build only because there are no X-ray mirrors.”

Mirrors are a real difficulty, but the larger barrier is the ratio A/B, which goes as the cube of the frequency. With Doppler width and photon energy, the pump power per unit volume for a given gain rises about as ν⁴, some sixteen orders of magnitude from a carbon dioxide laser to a wavelength of 1 nm.

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

“A calorimeter that sees a single energetic gamma ray has seen a single photon.”

A pion at 50 GeV sends its two photons out only 0.31° apart at the smallest. In cells a few centimetres across at about a metre, those arrive as one shower, which is why two-photon searches had to build finely segmented calorimeters.

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

“A heavy nucleus takes no part in pair production; it only has to be nearby.”

It takes essentially all of the photon's momentum, for almost no energy: at threshold a lead nucleus moves off with only 2.7 eV. Kinetic energy goes as the square of speed and momentum as speed itself, so a heavy partner absorbs momentum free of charge, and the process is impossible without it.

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

“Water sticks sand together as glue does, by adhering to it.”

A bridge pulls in two ways: surface tension around its neck, and low pressure inside the curved water acting across the neck's area. As a bridge shrinks, the neck narrows and the suction rises, and the two changes cancel, leaving about 2πRγ cos θ. Nothing in that is adhesive strength.

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

“Degassing a lake means pumping the gas out, against the pressure of the water.”

A pump is needed only to start the flow. After that, bubbles forming in a vertical pipe from 200 m make the column lighter than the lake outside by the equivalent of tens of metres of water, and the flow drives itself. It stops of its own accord at about 0.036 mol/L, far below any danger.

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

“The tension in the cord slows the stage until the cord runs out, so the spin falls steadily.”

The cord unwinds at a constant rate, one radius per radian of the starting spin. The spin itself follows (c − φ²)/(c + φ²): gently at first and fastest near the end. Past the ideal length it reverses, so a cord 10 per cent long leaves about 10 per cent of the spin going the wrong way.

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

“A chain falling from a jar goes as fast as anything dropped from the same height.”

It runs at √(gh₂/(1 − α)), about 4.7 m/s for a 2 m drop with α = 0.12, against 6.3 m/s for a falling stone. The chain's falling leg has to keep accelerating new links from rest, and each link arriving at the floor loses its energy there.

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

“The rings are diffraction from the drops, like the corona around the Moon.”

A corona comes from light passing round a drop, and is in the forward direction. A glory's rings surround the backward direction, in a region where the surface wave and axial rays interfere. Their spacing, about 1.5° for 10 µm drops, varies inversely with drop size as a diffraction pattern's would, but what produces the source is a surface wave around the sphere.

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

“Facet size is set entirely by diffraction.”

It is set by a balance, D/R against λ/D, in which both terms matter equally at the optimum. Beyond that, nocturnal insects use wider facets than this rule for more light, and many eyes have a local acute zone with larger facets looking forward, so the rule describes the sharpest eye that a given head can hold.

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

“The ring-down time gives an absolute absorption directly, with no calibration.”

It gives the sum of mirror loss and absorption. The absorption is the difference from the empty-cavity loss, and mirrors drift with wavelength and with what settles on them. An absolute value needs that reference, and a line's depth is measured far more reliably as a difference against its neighbouring baseline.

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

“Radioactivity showed Kelvin was wrong, since it supplies the Earth's heat.”

Radioactivity supplies about half. Antineutrino measurements put uranium and thorium near 20 TW of the 47 TW leaving the surface, and the rest is still secular cooling. Perry's 1895 objection needed no radioactivity: with an interior a hundred times as conductive as its lid, integration gives about 3 thousand million years.

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

“A turntable evens out the heating, so food is cooked uniformly.”

It averages along circles and cannot average along a radius. A given radius receives one mean, different from other radii, and the plate's centre turns about itself and receives no averaging at all. This is why food is often coldest in the middle and why the packet says to let it stand.

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

“A headland takes more waves because it sticks out into their path.”

It takes more than its width's share, because refraction bends rays towards the shoal off the point from both sides. In this calculation, 21 rays evenly spaced offshore arrive with far more within 200 m of the point than in the middle of the bay. The bay's shelter is due to bending and not only to position.

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