True where it was derived, used where it was not — part 3 of 5

“A more powerful car accelerates harder from a standstill.”

Below the speed P/μmg the force at the wheels is capped by the grip, not the engine. For a 1,500 kg car on tyres with μ = 0.9 that speed is 27 km/h at 100 kW, and the first 0.86 s of the run would be exactly the same with any bigger engine. At 368 kW the crossover reaches 100 km/h, and the time to 100 km/h stops falling: 3.2 s at 400 kW is the grip floor of 3.21 s.

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

“The Lorentz transformations have to be assumed linear, or derived from homogeneity, before anything can be said about them.”

With two or more space dimensions, requiring only that a one-to-one map of spacetime preserve every light cone forces it to be a Lorentz transformation followed by a shift and a stretch — Zeeman's theorem, which assumes neither linearity nor continuity. The figures bend the light-cone coordinates by amounts from 0.05 to 0.8 and every nonzero bend changes the order of some pairs, from 0.55% to 5.65%; a boost followed by a stretch changes none of 4,000.

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

“Infinity in spacetime is a single limit, reached by going far enough in any direction.”

It is at least five distinct places. Every observer slower than light ends at the point i⁺ whatever its speed, checked at 0, 0.5, 0.9 and 0.99 of light; every instant runs out to i⁰; light rays end not at a point but along the edge ℐ⁺, each at its own place; and the past versions i⁻ and ℐ⁻ are where the same things began.

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

“An engine's efficiency is limited by the Carnot factor between the hottest and coldest temperatures it reaches.”

That factor is a ceiling, and a real cycle sits well under it for a reason that can be computed. An ideal Otto cycle at compression ratio 9 reaches 58.5%, not the 88.8% Carnot allows between 300 K and its 2,672 K peak, because its heat arrives at an entropy-weighted mean of 1,491 K and leaves at a mean of 619 K — and 1 − 619/1,491 is 58.5% exactly.

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

“When two bodies come to a common temperature their energy has simply been shared out between them.”

Shared out by contact, the two kilograms of water meet at the arithmetic mean, 323.1 K. Drawn together by the best engine they meet at the geometric mean, 320.7 K. The 2.5 K difference, times the two bodies' heat capacity, is the 20.8 kJ of work — so a common temperature records how much work was taken out on the way, and contact takes none.

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

“Two lattices with the same energy bands have the same physics.”

The chains with v = 0.5, w = 1 and v = 1, w = 0.5 have bulk bands identical to rounding at every wavenumber. One holds edge states and the other does not, because the loop traced by (v + w cos k, w sin k) across the zone winds once round the origin in the first case and not at all in the second — a property of the bulk that the bands do not show.

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

“A mirror that reflects perfectly at normal incidence reflects well at every angle.”

At 0.764 of the design frequency the same stack is inside its gap at normal incidence and still reflects above 99.9% for one polarisation at every angle, but for light polarised in the plane of incidence the reflectance falls to zero by 78°, because the gap moves up in frequency and narrows as the angle grows and leaves that frequency behind.

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

“Boltzmann's and Gibbs's entropies are interchangeable definitions for any system large enough to have a thermodynamics.”

They are interchangeable below half filling, where their inverse temperatures differ by an amount falling as one over the size — fitted slopes of −0.98 for two-level units and −1.00 for oscillators. Above half filling of a bounded spectrum they are not: for 1,000 units three-quarters excited, Boltzmann's inverse temperature is −1.097 and Gibbs's is 3.0 × 10⁻⁵⁹, opposite in sign.

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

“Synchrotron radiation from a storage ring is drawn from the electrons' near field.”

On a circle the velocity and acceleration are perpendicular, their product is zero at every instant, and the Schott term vanishes identically. Every watt a storage-ring electron radiates is taken from its motion as it is radiated, and replaced by the accelerating cavities.

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

“Torsion balances test the equivalence principle for every kind of energy a body contains.”

They test it for the kinds of energy two laboratory masses differ in, which include nuclear and electromagnetic binding and exclude gravitational binding, because a 10 kg lead sphere is bound by its own gravity by 7.5 × 10⁻²⁶ of its mass. The best composition test in the world bounds a violation proportional to self-energy only below η of about 10¹².

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

“The self-bias of an etching tool is set by the power supply's settings.”

The supply sets the amplitude; the bias follows from how the plasma collects the two species, and in a chamber with two electrodes from their relative areas. A grounded wall five times the area of the powered electrode puts 5, 56 or 625 times as much voltage across the powered sheath, depending on how the sheath's thickness grows with voltage, and measured discharges sit between the first two.

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

“The uniqueness theorem is an existence result that has to be taken on trust.”

Read through the walkers it is a construction. The potential at a point is fixed by where walkers from that point stop, and where they stop depends only on the shape of the region, so two potentials that agree on the boundary give the same average everywhere inside. On a disc the walkers' exit angles match the Poisson kernel to a Kolmogorov distance of 0.005.

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

“A liquid's resistance to being stretched is its viscosity.”

For a Newtonian liquid stretched along one axis it is exactly three times the viscosity, and four and six times for a sheet and a blown film — Trouton's ratios, which follow from the stress tensor with free lateral surfaces and contain nothing about the liquid. For a polymer solution the ratio climbs from 3 at low stretch rates to several hundred above λε̇ = ½, while the shear viscosity has not changed at all.

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

“The osmotic pressure of a polyelectrolyte solution counts all of its counterions.”

It counts roughly the fraction that are not condensed. For DNA the computed osmotic coefficient falls from 0.154 to 0.122 as the solution is diluted, towards Manning's limiting 1/2ξ = 0.119, so a mole of DNA counterions exerts about an eighth of the pressure a mole of free ions would. A sodium polyacrylate gel in 1 mM salt exerts 430 kPa rather than the 1,235 its fixed charge would predict.

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

“The frequency at which a tide can shed its energy is a property of the tide.”

It is a property of the tide and of the latitude. The half-frequency waves' horizontal group speed falls as √(1 − f²/ω²) and reaches zero at the critical latitude: for a 500 m vertical wavelength they move 14 km a day at the equator, 10 at 20° and 3 at 28°, so near the critical latitude the energy handed to them stays where it was handed over.

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

“The waiting time between bursts is random in the way radioactive decay is.”

A decay's waiting times are exponential, with no longest wait. At 10⁻⁵ below the window, 19,620 calms pile up just below the channel's longest passage of 358 iterates — the tallest bin lies between 329 and 343 — and none lasts longer than 342. The waiting time has a ceiling fixed by geometry, and a record of calm lengths shows it.

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

“A cloud looks grey underneath because it absorbs the sunlight.”

A cloud of pure water droplets absorbs almost nothing in visible light. At an optical depth of 30 it sends 71 per cent of the light back up and 29 per cent down, and every per cent missing from its base is in its top: the base is grey because the top is white. Adding a tenth of a per cent of absorption per scattering does make a thick cloud absorb 18 per cent — because a photon that crosses it has scattered about a thousand times.

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

“The energy of light in a lossless medium is conserved.”

Only while the medium does not change in time. A lossless medium switched from index 1 to 2 keeps 0.625 of the field's energy, and one switched to 0.4 of its index multiplies it by 3.63; the difference is work done by the switch. What such a switch conserves is momentum, because the medium stays the same everywhere in space.

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

“Decay angles are where a polarisation shows most clearly.”

In the parent's rest frame, yes; in the laboratory, not once the parent outruns the product. Then the cone folds a forward and a backward direction onto each angle and their asymmetries cancel. At a Lorentz factor near a thousand, a lambda's proton angle keeps 5.7 per cent of the information its energy keeps, so a measurement from angles alone would need about eighteen times as many decays.

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

“A peak or a bump in a decay product's energy spectrum is evidence of a force between the products.”

A decay with no forces at all gives a π⁰ energy spectrum from D⁰ → K⁻π⁺π⁰ that rises from nothing at 135.0 MeV, peaks near 665 MeV and falls to nothing at 829.8 MeV — purely the width of the kinematic boundary. Only in the plane of two invariant masses is the force-free density flat, and only there does a departure mean dynamics.

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

“A light sail is inefficient because light carries so little momentum per unit of energy.”

It is inefficient only while slow. The share of each arriving joule that becomes kinetic energy is 2β/(1 + β): 2 per cent at 0.01c, 33.3 per cent at 0.2c and 94.7 per cent at 0.9c, with the rest leaving as redshifted reflected light. Integrated from rest, a mirror needs 0.1124 of its rest energy to reach 0.2c, half what a photon rocket carrying the same energy aboard would.

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

“Whether a magnetic field is weak or strong is a matter of how many tesla it is.”

It is a comparison with each atom's own fine-structure splitting. The crossover field is 0.78 T for hydrogen's n = 2 level, 36.8 T for sodium's 3p and 1187 T for caesium's 6p, so the strongest steady laboratory field, about 45 T, is strong for hydrogen and lithium, near the crossover for sodium, and weak for potassium, rubidium and caesium.

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

“The energy–time uncertainty relation says energy can be borrowed for a short time ħ/ΔE.”

No derivation says that, and time has no operator to make a commutator relation of. What holds is a limit on change: a state with energy spread ΔE keeps a survival probability of at least cos²(ΔE t/ħ), so it cannot become orthogonal to itself before πħ/2ΔE. Four states with the same spread all stay above that floor, and two equally weighted levels run along it and reach orthogonality at exactly π/2.

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

“Diffusion up a concentration gradient would violate the second law.”

The second law constrains the total entropy production, a sum over species of each flux times its driving force. In the two-bulb run the total stays positive throughout, falling from 1.929 µW/K at half an hour, while nitrogen's own share is negative for 6.4 hours and reaches −0.0208 µW/K: hydrogen and carbon dioxide, running down their gradients, pay for it many times over.

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

“A resonator made from lossy parts cannot hold energy longer than its lossiest part allows.”

Energy put into the non-leaking combination stays. Exciting the first resonator alone leaves exactly γ₂/(γ₁ + γ₂) = 0.3333 of the energy trapped indefinitely at the bound state, and two identical resonators each leaking at 0.08 have an antisymmetric mode that does not leak at all, reaching quality factors above 250,000 even with the leaks unequal by one per cent.

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

“A matching network improves a load's match.”

It redistributes it. The area under ln(1/|Γ|) across all frequencies is 1.5708 for the bare load and 1.5708 for the optimised three-element network, exactly π/RC both times: the network moves match out of frequencies where it is not needed and into the band, and any match it adds inside the band is paid for outside it.

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

“The diffraction limit is a limitation of lenses.”

It is a property of the field a wavelength or more away from what emitted it. Components finer than a wavelength decay as e to the minus the distance times √(kₓ² − k²). An ideal instrument recording everything above a hundredth of its starting amplitude keeps detail up to 14.69 k from 0.05 wavelengths, 3.10 k from 0.25 and 1.07 k from two — and from two wavelengths it rebuilds emitters 0.35 wavelengths apart as a single peak, however perfect its optics.

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

“A hard magnetic material has a high coercivity because its walls are hard to move.”

Its walls are narrow, which is a different statement with a different consequence. Width and energy are tied by wδ = 4πA identically, so a material cannot have both a narrow wall and a cheap one, and the same anisotropy that narrows the wall raises the field at which a uniformly magnetised particle will rotate. In samarium cobalt the wall is 2.6 nm wide and a particle below 1,100 nm has no wall at all, so the coercivity of a fine powder of it owes nothing to wall motion.

Tested in The first length that belongs to the substance · the magnetisation reading path

“Coercivity is a property of a magnetic material.”

It is a property of a material and a measurement time together, and the two cannot be separated. Solving the switching condition across nineteen decades of time gives a particle with a ten-year barrier a coercivity of 0.41 of its anisotropy field in a second and 0.18 if the field is left on for ten years. Two laboratories sweeping at different rates measure different coercivities for the same powder, and neither has made an error.

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

“A long member develops a larger thermal stress than a short one, so length is what has to be controlled.”

The stress is EαΔT and has no length in it, which the first figure checks by computing it for a bar half a metre long and one thirty-seven metres long and getting the same answer to every figure carried. What length changes is the movement a joint has to accommodate, not the stress if it is prevented. Welding rails into a continuous line did not raise the stress in them at all; it removed the joints that had been taking the movement.

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

“Prestressing a concrete beam works by making it stronger.”

Nothing about the concrete changes. What moves is where the beam's stress state sits: under the load alone the bottom fibre is in tension at 11.1 MPa, which is four times what concrete carries, and with 1,500 kN of prestress 120 mm below the centroid the same section under the same load runs from 9.8 to 3.6 MPa of compression everywhere. The material is identical and it is now working in the half of its range where it is fourteen times stronger.

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

“Since the load sharing in a redundant structure cannot be computed, its strength cannot be either.”

The two are different quantities and only the first depends on the unknowns. Four tables whose leg tolerances differ enough to change the first-yield load by a factor of four collapse at exactly the same load, checked here to a part in a million, because at collapse every leg holds its own capacity and the total is the sum of them. No stiffness, no tolerance and no temperature appears in that sum.

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

“A fibre's V-number and an illumination designer's étendue are different quantities from different subjects.”

They are the same quantity in different units. Counting phase-space cells in a fibre's core and acceptance cone gives 2π²a²NA²/λ², which is V²/2 exactly — checked here at four radii on each of three fibres to twelve figures. The single-mode condition V < 2.405 is the statement that the core's étendue has fallen to about one square wavelength.

Tested in The invariant that is a count · the etendue reading path

“The ceiling on solar energy conversion is the Carnot efficiency between the Sun's surface and the surroundings, 95 per cent.”

That figure assumes the light arrives undiluted, which it does not. Sunlight at Earth carries 1,361 W/m², the flux of a 394 K body, while its spectrum belongs to a 5,762 K one — a hundred-thousandfold spread in solid angle whose entropy a converter has to carry. An absorber and a Carnot engine reach 5.4 per cent unconcentrated and 84.9 at the geometric concentration limit, both computed here by scanning for the optimum absorber temperature.

Tested in The work a diluted beam will not do · the etendue reading path

“A graded-index fibre works by accepting light over a wider range of angles.”

It accepts less, not more. The local numerical aperture falls with radius as √(1 − (r/a)^α), so integrating the acceptance over the core's area gives α/(α+2) of what a step-index core of the same peak index would take — exactly a half for a parabola, integrated here rather than substituted. What the grading changes is the arrival time, and the two quantities are almost independent.

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

“The mismatch is tiny, so it can be ignored.”

Its size relative to the coupling is the whole question, and the coupling is proportional to the amplitude. At small amplitude the mismatch wins and the exchange beats; at large amplitude the coupling wins and the exchange accumulates. That is the threshold measured empirically for the chain, restated as a comparison of two numbers rather than as an empirical knee.

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

“A resistor's noise depends on what it is made of — carbon is noisier than wire.”

The expression is √(4kTRΔf) and contains the resistance, the temperature and the bandwidth and nothing else. Nyquist's derivation gets it from the modes of a line joining two resistors in equilibrium, and the line's material, length and impedance all cancel: the mode count is proportional to the length and the delivery time is too. Real resistors do differ, and the difference is an excess noise that appears only when current flows through them.

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

“The virial theorem is a result of statistical mechanics and needs a system in thermal equilibrium.”

It needs neither a temperature nor thermal equilibrium. The relation 2⟨T⟩ = n⟨V⟩ is measured here off single eccentric orbits time-averaged over more than a hundred radial periods, for seven exponents, and it lands on the exponent to two parts in a thousand. One particle going round for long enough satisfies it; a crowd of particles averaged over members satisfies it for the same reason, which is an average over a phase space rather than over a heat bath.

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

“The vacuum energy problem is a disagreement between speculative quantum gravity and observation, so it may go away with a better theory of gravity.”

The largest contributions are not speculative. The quark condensate of the strong interaction is required by physics that is tested, its size is fixed by measured hadron masses, and it alone exceeds the measured density by 10⁴⁵. Cutting the zero-point sum off at one electronvolt — below which nothing is in doubt — still overshoots by 10⁸. The problem exists entirely inside physics nobody disputes.

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

“The Planck energy is where gravity becomes strong, so nothing about quantum gravity can be tested.”

It is where gravity becomes strong if the gravitational field lives in three spatial dimensions at every scale. With n compact extra dimensions the four-dimensional Planck mass is not fundamental — it is M_*^(2+n)Rⁿ — and the true scale can be at a TeV with R of order a millimetre for two dimensions. That version is testable, has been tested, and is excluded; three or more dimensions require R below a nanometre and are untouched.

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

“Since swaps can be chained, a quantum network can be extended indefinitely.”

The pairs' qualities multiply rather than add. Two Werner states of parameter p swap to one of parameter p², so a chain of n swaps gives p^(n+1) and the threshold for any Bell violation rises from 0.7071 for a direct pair to 0.9457 for three swaps — located here by bisecting the drawn curves. A chain long enough to be useful needs either pairs better than anyone can make or a way of improving them in the middle.

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

“A state of N particles takes 2^N numbers to write down, so nothing with many particles can be computed.”

That is true of a general state and false of every state anything prepares. A gapped ground state's entanglement across any cut is bounded, so it can be written as a matrix product with a fixed bond dimension needing about 2Nχ² numbers — linear in N. The exact description exhausts a large machine at about forty spins and the matrix product does not run out at any length worth drawing.

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

“The effect is unobservably small, so it is a formal curiosity.”

It has been observed twice, in the only two places the fields are not human. The ATLAS and CMS experiments measured light scattering light in the near-misses of lead nuclei, whose own fields supply photons of tens of gigaelectronvolts. And a magnetar's field of 10¹¹ tesla is twenty-three times the critical field, so the vacuum around one is strongly birefringent and the polarisation of its X-rays carries the signature.

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

“The coupling strength is set by how strongly the guides are coupled, which is a material property.”

It is set by the overlap of two evanescent tails, so it falls as the exponential of the gap divided by the tail's own decay length. The slope measured off the computed curve is 11.93 per micrometre, which is the mode's decay rate to every figure carried. Between a hundred-nanometre gap and a seven-hundred-nanometre gap the coupling length runs from 6.3 micrometres to 8 millimetres.

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

“A denser fluid holds a floating body more firmly.”

What holds it is the density difference across the surface it is sitting in, not either density. A ten-centimetre cube at an air–water boundary is held at 98 newtons per metre and the same cube at an oil–water boundary at 14.5 — both read off the computed force curve. Mercury under water holds at 1,229, and the reason is the contrast rather than the mercury.

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

“The relativity of simultaneity is too small to matter outside particle physics.”

Carrying a clock synchronisation once around the Earth's equator fails to close by 207 nanoseconds, which is sixty metres of light travel, and every satellite-navigation fix corrects for it. What changed between 1905 and now is not the size of the effect but the resolution of clocks: the patch shrinks as the square root of the precision demanded, so a millionfold better clock shrinks it by a thousand.

Tested in How big now is · the simultaneity reading path

“A peak in the transform sits at the distance to the neighbour that produced it.”

Every peak sits short, here by about half an ångström, and the offset is not an experimental error. The electron's phase is shifted by the potential of the atom it left and again by the atom it bounced off, both shifts are close to linear in the wavenumber, and a term linear in k displaces a Fourier peak. Reading the peak position as a distance underestimates every bond in the sample by a fifth.

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

“The exponential tail is absorption by defect states inside the gap.”

Defect absorption exists and is a separate, usually flatter, feature further down. The exponential tail is produced by a model with a clean gap and no states in it: the gap is fluctuating, a photon slightly below its mean value is absorbed wherever the gap is momentarily narrower, and the exponential comes from the Boltzmann-like rarity of a large fluctuation. The slope tracks temperature, which a fixed population of defects would not.

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

“Streaming is a viscous effect and vanishes as the fluid is made less viscous.”

Rayleigh's expression for the slip velocity is minus three quarters of U dU/dx over the angular frequency, and there is no viscosity in it. Halving the viscosity halves the thickness of the layer that generates the streaming and leaves the streaming it generates unchanged. The limit of small viscosity is not the same thing as no viscosity.

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

“Barus's exponential gives the viscosity inside the contact.”

It gives the pressure at which the liquid stops being one and nothing after that. Extrapolated to the centre of the contact it returns 6 × 10¹⁶ pascal-seconds for a traction fluid, which describes no material: the liquid passes its own glass transition at about a gigapascal and thereafter shears at a limiting stress like a soft solid, which is a different law with a different variable in it.

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

“The quark–gluon plasma is the least viscous fluid known.”

Its viscosity is about ten to the eleven pascal-seconds, which is a hundred million times that of pitch. What is small is its viscosity divided by its entropy density, which is the quantity a bound can be written for — and by that measure it is roughly one and a half times the floor against water's four hundred. A statement about viscosity alone would be the opposite of the truth.

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

“A ringing black hole is like a struck bell.”

A bell sustains for thousands of cycles and a black hole's fundamental mode gives one. At 62 solar masses and 0.68 of maximum spin it rings at 274 hertz and has decayed by a factor of e in 3.7 milliseconds, which is a quality factor of three. Nothing about the sound of a bell — its long decay, its sharp pitch, its sensitivity to where it was struck — carries over.

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

“The memory is undetected because it is too small.”

It is a fixed fraction of the peak strain — about a fifth at the best orientation — for every equal-mass merger, because both quantities scale as the total mass over the distance and the ratio contains neither. It is undetected because it is a step rather than an oscillation, so its power sits at frequencies below where a suspended mirror can be held still, and not because of its size.

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

“An iron core attracts the magnetic field into itself.”

Nothing attracts anything. Two boundary conditions — the normal part of B continuous, the tangential part of H continuous — give the tangent of the angle to the normal on one side as the tangent on the other divided by the relative permeability, so a line at 89 degrees to the normal inside a core leaves the surface within a degree of straight out. The shape of the pole face decides where the field goes, and the iron only makes the line forget the angle it had.

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

“The scalar potential is invalid because it is multivalued, so the vector potential must be used instead.”

The multivaluedness is the physics, not a defect. Going once round a wire carrying ten amps lowers the potential by ten amps exactly, which is Ampère's law. The scalar potential is the standard tool in magnetostatic field solvers for regions with no current, because it carries one unknown per point where a vector potential carries three.

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

“The work-energy theorem says the change in kinetic energy equals the net work done.”

It says that for a particle, and a particle is what it says it for. For a body that changes shape, integrating the net external force over the centre of mass gives the change in the centre of mass's kinetic energy, and that integral is not a work because each force is dotted with the wrong displacement. The two agree for a rigid body sliding and disagree for a jumper, a car, and every inelastic collision.

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

“Energy is always conserved.”

It is conserved for a system whose rules do not depend on time, which is Noether's statement and is exact. A ball in a box with a moving wall has rules that do depend on time, and its energy rises without limit: with the wall at a fiftieth of the ball's speed, two thousand head-on collisions multiply it by six and a half thousand. What is conserved is the energy of the ball and whatever is moving the wall, which is a different system.

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

“A larger telescope has fainter diffraction rings.”

Its rings are closer to the star and are exactly as bright relative to it. The pattern is the transform of the pupil, so scaling the pupil scales the pattern and leaves every ratio alone: the first ring sits 13.3 decibels below the peak for a hard-edged aperture of any size whatever. What changes with size is where the rings are, not how bright they are.

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

“An optic is diffraction-limited if its surface is accurate to a quarter of a wavelength.”

A quarter wave peak-to-valley of defocus is a fourteenth of a wave root-mean-square, and it is the root-mean-square that decides the peak: the Strehl comes out at 0.80. A quarter wave peak-to-valley of some other error shape has a different root-mean-square and a different Strehl, so the rule is a convention attached to one aberration and quoted for all of them.

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

“Massless particles are the simple case and massive ones are a complication.”

The counting runs the other way. A massive particle has a rest frame in which its spin can point anywhere, which is where 2j+1 comes from; a massless one has no rest frame, so only the two projections along the motion survive. It is the massive case that has more states, and the massless one is a degenerate limit of it rather than a starting point.

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

“The obstruction is the uncertainty principle, so it is a statement about what can be known.”

The uncertainty principle enters only to say how narrow a beam can be before it diffracts. What does the blocking is classical: a magnetic field whose z-component varies with z must, by the divergence-free condition, have a transverse component varying across the beam, and that exerts a Lorentz force on a charge. The argument would work with no quantum mechanics in it except the size of the magnetic moment.

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

“An absorption refrigerator is less efficient than a compression one, so it is a worse machine.”

They are not comparable as quoted. A compression fridge's coefficient counts work in and an absorption fridge's counts heat in, and the two differ by the efficiency of whatever made the work. Multiply a compression fridge's coefficient by the efficiency of the power station feeding it and the product is close to the absorption figure, because the absorption machine is a compression machine with the power station built in.

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

“A thermoelectric generator is inefficient because it has no cycle.”

It has the same Carnot ceiling as anything else and reaches a fraction of it set by one dimensionless group. With that group equal to one, a couple between 600 and 300 kelvin converts 10.8 per cent against a Carnot ceiling of 50; with the group equal to four it converts 19.1. Nothing about the absence of a cycle appears in the expression.

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

“The problem is that temperature is hard to define for a moving body.”

It is not defined, which is different. A temperature is a property of a system in equilibrium, and radiation that is hotter ahead than behind is not in equilibrium in the frame it is being asked about. The question has no answer for the same reason 'what is the temperature of a room with a fire at one end' has none.

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

“Entropy is invariant because it is a thermodynamic potential, and those transform simply.”

The other thermodynamic potentials do not transform simply at all — the free energies contain a temperature, which has no transformation law. Entropy is invariant for a reason that has nothing to do with thermodynamics: it is the logarithm of how many arrangements there are, and different observers counting the same arrangements get the same number.

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

“Radiation drag is negligible, so it is of no consequence.”

It is negligible for slow, small things and decisive for fast ones, because the rate rises as the square of the Lorentz factor while the energy rises as its first power. An electron at a Lorentz factor of ten loses its energy to the microwave background in 200 billion years and one at 100,000 does so in 23 million, which is the dominant loss for the fast electrons in radio galaxies.

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

“A freely falling clock always accumulates more time between two events than any other clock present at both.”

An orbiting clock two Earth radii out loses 7.47 µs per revolution against a distant clock, a clock held at that radius by a rocket loses 4.98 µs, and a clock thrown straight up to return in the same time loses 3.37 µs. The orbiting and thrown clocks both fall freely; the held clock does not, and it still beats the orbit. The statement is true only for free paths with no focus between their ends.

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

“An isothermal star is at the same temperature throughout.”

An isothermal star in relativity has a uniform redshifted temperature, so a local thermometer reads more towards the centre. For a uniform star of compactness 0.34, roughly a neutron star, the centre reads 1.13 times the surface; at compactness 0.6 it reads 1.41 times.

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

“The logarithm in the chemical potential is a property of ideal gases.”

It is a property of counting. Choosing which of N molecules have reacted, and weighting each choice by a Boltzmann factor, gives a distribution centred exactly on the Gibbs minimum with a width of √(x(1−x)/N) — 9.4 per cent for 20 molecules, 3.0 per cent for 200 — and the logarithm of the number of choices per molecule is the mixing term.

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

“A thermal camera measures the temperature of what it looks at.”

It measures a brightness, and a reverse voltage lowers the brightness below the thermal value without changing the temperature. A 0.25 eV semiconductor held at 300 K reads 220 K at −0.1 V and 144 K at −0.3 V, because its light's chemical potential is negative.

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

“Shot noise is the fundamental limit on any measurement made with light.”

It is the limit for light whose noise is the same in every quadrature. Squeezed light keeps the product of its two quadrature noises at the vacuum's value and moves the noise out of one of them: light squeezed by 12 dB and measured along its quiet quadrature is 12 dB below shot noise, and the same light measured 14.1° away is back at shot noise.

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

“Since every body falls the same way, the mass cannot appear in any effect of uniform gravity.”

It does not appear in the trajectory and it does appear in the phase. For the same interferometer held vertical, the gravitational phase is 57.4 rad for a neutron, 904 rad for a helium atom at the same wavelength and 4.3 × 10⁵ rad for a rubidium atom, while every one of them falls the same 0.67 nm over a 3.2 cm leg. The phase is the action divided by Planck's constant, and the action carries the mass.

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

“The Earth's dense core was inferred from its gravity.”

Gravity supplies J₂ = 1.08263 × 10⁻³, which fixes the difference of the moments of inertia. The precession of the spin axis supplies their ratio, 3.27379 × 10⁻³. Only the two together give C/MR² = 0.3307, well below the 0.4 of a uniform ball — and a uniform Earth with the same gravity would precess once every 31,173 years rather than 25,772.

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

“The soliton heights from an arbitrary starting shape can only be found by solving the evolution.”

They are set by the levels of a linear problem solved once at the start. A Gaussian hump of height 8 has levels at κ = 2.356 and 1.252, predicting solitons of 11.10 and 3.14; integrating the nonlinear equation to t = 0.6 gives 11.10 and 3.14.

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

“Integrated optical couplers need the guides to be made exactly as designed.”

A cut coupler does: a 10 nm width mismatch between its guides caps its transfer at 67 per cent, and 20 nm at 15 per cent. A tapered coupler 800 µm long transfers 99.7 and 99.8 per cent at the same errors, because a mismatch only moves the point along the taper where the guides' constants cross.

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

“A medium whose refractive index oscillates in time behaves like a Bragg mirror and reflects some band of light.”

A spatial repeat gaps frequencies and a temporal repeat gaps wavenumbers. With the permittivity modulated as 1 + 0.2 cos Ωt, wavenumbers between 0.475 and 0.521 of Ω/c have no real frequency at all, and a wave there grows by up to 0.159 in the logarithm of its amplitude each modulation period rather than being turned back.

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

“The average transmission of a disordered sample is what a typical sample transmits.”

For 160 random stacks of 120 layers the logarithms of transmission spread 1.5 powers of ten either side of a typical value of 6.8 × 10⁻⁴, while the plain average is 0.037, 1.7 powers of ten higher, dominated by the rare stacks that hold a resonance at that wavelength.

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

“Faster than the waves it makes, a ship feels more wave drag.”

For waves on a hidden interface the lee wave exists only below the fastest interfacial wave speed. Above it the drag falls to a small remnant — the steady train needs a wave that can keep pace with the hull, and none can — and a hull at 1.2 times the wave speed only raises a local bulge on the interface.

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

“A reversed toroidal field at the edge of a pinch must be imposed by coils outside it.”

It is the relaxed state's own shape. The minimum-energy field at fixed helicity in a cylinder has an axial component J₀(λr), which passes through zero at r = 2.405/λ; at λa = 3 the axial field reverses at 0.80 of the wall radius. A uniform-current pinch started at a pinch parameter above 2.405 relaxes to such a state with nothing done at the edge.

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

“Multivalent ions attract two like-charged surfaces by bridging them, sitting between the two and pulling on both.”

No bridging is needed. Between two equal planes a single counterion feels their two fields cancel and is equally likely anywhere in the gap. The attraction appears because the counterions are too few at the walls: the pressure is the contact density minus σ²/2ε, and with strongly correlated ions the contact density is that of a uniform layer, 2/d in natural units, which falls below one once the gap exceeds 2μ.

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

“Where the magnetic field is not uniform, the guiding-centre drifts carry the plasma's pressure-balancing current.”

They carry a fraction −μ₀p/B² of it, which is negative. In a slab at a vacuum-field β of 0.8 the drift down the gradient of the field's strength runs backwards at 47.1 per cent of the total where the gradient is steepest, and the curl of the magnetisation carries 147.1 per cent. The sum is exactly b × ∇p / B at every point.

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

“A circuit that is nearly a harmonic oscillator is the ideal quantum bit, since it is well understood and easy to drive.”

A harmonic oscillator has every transition at the same frequency, so a drive tuned to the lowest one drives them all. The transmon's use depends on its departure from harmonic: at EJ/EC = 50 the second transition is 6.1 per cent below the first, 303 MHz for a 5 GHz qubit, and that difference is what a control pulse must resolve.

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

“The extra bending from the curvature of space is a small relativistic correction that only matters for fast particles.”

It is the same angle at every speed — 2GM/bc², 0.876 arcseconds at the Sun's limb — for a comet, a proton and a photon alike. What depends on speed is the Newtonian part, 2GM/bv², which grows as the body slows until the space part is a part in a hundred million of the whole at the Earth's orbital speed. The space term is not small; it is outvoted.

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

“The precession of an orbit depends only on the inverse-square part of the force, since the rest is small.”

The precession is itself a second-order effect, so it depends on second-order terms in the field as much as first-order ones. Writing a scalar theory's field as 1 − GM/rc² or as e^(−GM/rc²) changes nothing a first-order experiment can see and doubles the backward precession, from −1 to −2 units; a theory is not fixed by its Newtonian limit.

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

“A particle attracted to strong field can be trapped at a field maximum in the space between electrodes, like a charge in a potential well.”

The square of a static field in empty space has no local maximum anywhere — each component is harmonic, so |E|² is subharmonic — which is the same fact that stops a static field holding a charge still. A particle pulled up the gradient therefore ends on an electrode surface; only a particle pushed down it can be held in free space, at a zero of the field.

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

“Clumping in a gas needs an attraction — gravity, cohesion, or a sticky surface.”

The discs here attract nothing; they only lose a fraction of their relative speed on each collision. A patch 20 per cent denser than its surroundings collides more often and cools faster, and after 1.83 collision times its pressure has fallen below its neighbour's, so grains are pushed into it. The instability is a runaway of dissipation, not of attraction.

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

“A damped oscillator and a bouncing ball both lose a fixed fraction of their energy each cycle, so both take for ever to stop.”

The oscillator's period does not depend on its amplitude, so a geometric decay of amplitude takes an infinite number of equal periods. The bouncing ball's period is proportional to its speed, so the cycles shorten as they shrink and their total is finite. The difference is isochronism, not the loss.

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

“The turning of a Foucault pendulum's plane is a clean measurement of the Earth's rotation at any length.”

It is clean only if the pendulum's own turning is small beside the Earth's, and that depends steeply on length. At Paris the Earth turns the plane 11.3° an hour; for Airy's turning to stay under a tenth of that, a 67 m pendulum swinging 3 m may have an ellipse up to 57 mm wide, a 1 m pendulum swinging 10 cm only 0.05 mm.

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

“Perfect imaging means a faithful, undistorted image.”

The fish-eye's images are points, so the image of every point is sharp, but each point lands at its inversion through the centre, R²/r on the opposite side. A source near the centre images far out and a straight row of sources images onto an arc. Sharpness and fidelity are separate properties, and this instrument has the first everywhere and the second nowhere.

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

“A focus on the surface of the lens is useless, since there is nowhere to put a detector.”

It is the reason the lens is used. A feed antenna or detector placed on the sphere's surface receives a perfectly focused beam from the direction diametrically opposite, and moving it round the surface steers the beam with no loss of focus; a mirror on the back half turns the sphere into a reflector that returns every beam along its own path.

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

“An oxygen carrier with a higher affinity for oxygen is a better carrier.”

Myoglobin binds oxygen half at 2.8 mmHg and so is nearly full both in the lungs and in tissue; between 100 and 40 mmHg it gives up 4 per cent of what it holds. Haemoglobin, half full at 26 mmHg with its four sites helping one another, gives up 21 per cent over the same range. A carrier has to fill in one place and empty in another, which a single independent site does only over a hundredfold change of pressure.

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

“The band gap sets the number of carriers through a Boltzmann factor e^(−Eg/kT).”

The gap is the energy of making a pair, so it fixes the product of two concentrations, and each concentration goes as e^(−Eg/2kT) — half the gap. The same is true of water, whose ions each rise as a Boltzmann factor in half the 0.58 eV that ionising one molecule costs. The factor of two in the exponent is the arithmetic of a square root, not a detail.

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

“Gold is yellow because it reflects yellow light, as a yellow dye does.”

It reflects nearly all light of long enough wavelength and absorbs strongly above a photon energy of about 2.4 eV, 517 nm. Everything redder than that edge is reflected well and everything bluer less well, so the reflected mixture is weighted towards yellow and red. Silver's edge is at 318 nm, in the ultraviolet, so it reflects the whole visible range alike.

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

“Superfluid helium flows without friction up to the speed of sound, since phonons are its softest excitations.”

Phonons are the softest in energy, not in energy per momentum. The ratio ε/p falls from 238 m/s at the phonon end to its minimum just past the roton, and the tangent from the origin to the measured spectrum touches at 1.98 Å⁻¹ with a slope of 57 m/s — about a quarter of the sound speed.

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

“The narrowing is a relativistic correction to momentum conservation.”

Momentum is conserved exactly in both descriptions. What changes is energy: kinetic energy is (γ − 1)mc² rather than p²/2m, and that one change turns the Thales circle of possible outgoing momenta into an ellipse squashed along the beam, every chord of which subtends an acute angle. The product tan θ₁ tan θ₂ comes out 2/(γ + 1) to a part in 10⁹ from the boosted kinematics.

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

“A stronger spring or a larger swing always makes the mass move faster.”

It makes the mass carry more momentum and energy; the speed saturates. At an amplitude of 3 c/ω₀ the velocity flattens at 0.983 c for most of each swing while the Lorentz factor keeps climbing to 5.5 through the middle — the spring's work goes into inertia, and the phase portrait becomes a lens rather than an ellipse because momentum, unlike velocity, has no ceiling.

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

“Any change to the relative phases of the modes changes the shape of the pulse.”

A phase that rises in equal steps from mode to mode leaves the pulse exactly as it was and moves it later in time; for a train of modes a phase proportional to frequency is a delay and nothing else. Only a phase that curves with frequency, such as one rising as the square of the mode number, reshapes it: a quadratic phase of 0.025 turns per mode squared drops ten modes' peak from 100 to 29 and smears it across half the round trip.

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

“The rule is about how many beams are used, so a cubic lattice from six beams must need its phases locked.”

Only beams that interfere count. Three retro-reflected pairs whose frequencies differ from axis to axis by tens of megahertz give three standing waves whose cross terms average to zero, so the pattern is three independent one-dimensional lattices, each of which a phase can only slide.

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

“Once the right mixture of waves is known, a superoscillating spot is as robust as an ordinary one.”

Its components are large and cancel almost completely at the spot. For zeros every 0.15 wavelength the largest component is 127 times the field at the centre, so errors of 0.1 per cent in the mixture already distort the spot, 1 per cent — ordinary for a fabricated mask — swamps the field around it, and 5 per cent leaves nothing of the design.

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

“Protons in a collider radiate like electrons, only less, because they are heavier.”

Mass enters as the cube of the Lorentz factor. In the same tunnel at nearly the same radius, LEP's 104.5 GeV electrons had a critical energy of 818 keV and the LHC's 7 TeV protons 43.8 eV — 18,652 times lower although the protons carry 67 times the energy.

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

“The size of a resonant scatterer is set by the size of the particle doing the scattering.”

At resonance, with only radiative damping, the cross-section is 3λ²/2π whatever the charge and mass: 1.66 × 10⁻¹³ m² at 589 nm for an electron, a proton and a caesium ion alike, 2.5 × 10¹⁵ times the Thomson cross-section. The charge and mass set only the width, which runs from 2.0 × 10⁻⁸ of the frequency for the electron to 8.3 × 10⁻¹⁴ for the ion.

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

“Gravitational waves have no dipole term because there is no negative mass.”

Electromagnetism has negative charge and still radiates no dipole from two charges with the same charge-to-mass ratio: their dipole moment is that ratio times the centre of mass, which a closed system cannot accelerate. Gravity's dipole vanishes for the same reason — every body's gravitational charge is its inertial mass — and would return, at the orbital frequency, if the ratio differed between two bodies.

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

“Minkowski's momentum nE/c is the light's share of the total momentum inside a medium.”

The total momentum of pulse and slab together is E/c throughout, to rounding error, while Minkowski's momentum rises to 1.50 E/c inside glass of index 1.5 — more than there is. It is a different quantity, which in the same accounting comes with a backward 0.50 E/c assigned to the slab.

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

“The electron's mass is electromagnetic energy, which is why its size can be read off as the classical electron radius.”

A shell's field energy equals mc² only at 1.41 fm; the electron shows no structure down to 10⁻¹⁸ m, where the classical field energy would be 1,409 times its mass. In quantum electrodynamics the self-energy grows only logarithmically — 4.5 per cent of the mass at 10⁻¹⁸ m and 18.0 per cent at the Planck length — and the observed mass is an input, not an output.

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

“A charge distribution's quadrupole moment is a fixed property of it, like its total charge.”

Only when the total charge and the dipole moment both vanish. About an origin moved a distance s along the axis, the zz component of a single charge's quadrupole grows as 2qs² and a dipole's changes with slope −4, while the linear quadrupole's stays at 4.00 qd² wherever the origin is put.

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

“Water is incompressible, so the pressure at depth is exactly ρgh.”

Integrated down a column of 0 °C seawater from its equation of state, the density rises 1.67 per cent by the ocean's mean depth and 4.71 per cent by 10.99 km, where the pressure is 113.5 MPa against 110.8 for an incompressible column — the weight of about 266 m of extra water packed into the same depth.

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

“A slow enough capillary rise converts the surface energy it releases into the potential energy of the column without loss.”

Half is always lost. At equilibrium the column's potential energy is exactly half the surface energy released — 0.499 and 0.501 in a 1 mm tube that rings, 0.500 and 0.500 in a 0.2 mm tube that creeps — because Jurin's height is where the weight equals the pull and a column's potential energy is half its weight times its height.

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

“The half in the harmonic oscillator's (n + ½)ħω is a special feature of quadratic potentials.”

It is two quarters, one for each smooth turning point. Integrated from the forbidden side, the exact wave at a smooth turning point leaves with a phase of 0.7881 rad, against π/4 = 0.7854; a hard wall costs half a cycle; and the rule (n + μ/4)h with μ counting those gives n + 1 for a box, n + ¾ for a half well and a bouncer, and n + ½ for any smooth single well.

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

“The centre of percussion and the centre of oscillation are two different points that happen to be near each other.”

They are the same point. For a rod pivoted anywhere from 0.02L to 0.44L from its end, the centre of percussion found from the impulse balance and the end of the equal-period simple pendulum found by integrating the swing agree to 0.6 parts in ten thousand of the length — the lengthening of the period by a one-degree swing.

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

“Gold is used for plasmon sensors because it gives the sharpest resonance.”

Silver gives a far sharper one: at 633 nm, each at its best thickness, silver's dip is 0.79° wide at half depth, gold's 5.54° and aluminium's 18.38°, in the order of each metal's losses. Gold is used because it does not tarnish.

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

“A mirror sends light back the way it came.”

Only when it is square to the beam. A flat mirror returns light 2θ away from its arrival direction, 20° for a ray 10° off its normal; three mutually perpendicular faces return every traced ray, up to 45° off the axis and in every azimuth, to within rounding error of exact reversal.

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

“The brightness of an emission line measures the temperature of the gas emitting it.”

Not for the 21 cm line while it is thin. Its levels sit at their statistical weights, three to one, above a kelvin — 0.74987 of the atoms are in the upper level at 100 K — so the emission counts atoms: at 10²⁰ per square centimetre the brightness is 5.20, 5.34 and 5.47 K for spin temperatures of 50, 100 and 1000 K.

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

“Excited atoms decay in nanoseconds, so a highly excited atom is too short-lived to use.”

The circular state's lifetime rises as n⁵: from the Larmor loss on its orbit, 32.2 ms at n = 51, where about 30 ms is measured, and 0.9 s at n = 100. What shortens it in practice is thermal radiation, which at 300 K leaves only 0.13 ms at n = 51, and 9.6 ms at 4 K.

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

“At absolute zero the atoms of a solid stop moving, so their clocks stop running slow.”

Their zero-point motion remains. For iron-57 nuclei in a crystal with a Debye temperature of 470 K, half the vibrational energy per atom over Mc² is 4.30 × 10⁻¹³ at absolute zero; heating adds 2.23 × 10⁻¹⁵ per kelvin near room temperature.

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

“A moving charge's field is a Coulomb field carried along, so its effect on a passing atom is weaker at higher speed because the encounter is shorter.”

The encounter is shorter by γ and the field stronger by γ. At a distance b the transverse field peaks at γ times the Coulomb value and lasts about 1.53/γ times b/c — 0.0511 b/c at γ = 30 — so the sideways impulse, 2q²/4πε₀bv, is the same at every speed; what changes is that the same impulse now reaches far higher frequencies.

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

“A battery's voltage is set by how much charge it holds.”

It is set by the free energy of its reaction per charge moved. A zinc–copper cell's 1.102 V is −213 kJ per mole divided by two moles of electrons' charge, and a hundred-million-fold change in the concentrations moves it by only 0.237 V, because two electrons move per ion and the slope is 29.58 mV per decade.

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

“The heat a molecular machine dissipates can only be found by measuring the heat.”

A lower bound on it can be read from the fluctuations of its motion alone: the entropy it produces while making a count with mean N and variance σ² is at least 2k·N²/σ². None of 1,500 three-state machines with random rates over four orders of magnitude violates the bound.

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

“A resistive sheet absorbs best when it is laid directly on the metal it is meant to hide.”

On the metal it sits at a node of the standing wave, where there is no electric field to drive a current in it, and absorbs nothing. A sheet of 376.7 Ω per square absorbs everything at a quarter-wavelength, 7.5 mm at 10 GHz, and nothing again at half a wavelength.

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

“Driving the wall harder is the only way to pump more energy in.”

Frequency matters as much as amplitude, and only at whole multiples of the fundamental. After thirty round trips at 1% amplitude the energy is 3.37, 10.9, 145 and 977 times its start at 1, 2, 3 and 4 times the fundamental, but never more than 1.27 times anywhere more than a tenth of the fundamental away from a whole multiple.

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

“A parcel displaced downward in a stable column must come back, because it is lighter than its new surroundings.”

A wide parcel does come back: 2 m across, it oscillates with the 23.6-minute buoyancy period. A parcel 6 cm across loses its excess heat in 11 minutes while keeping its salt for 18 hours, becomes heavier than the water around it, and after an hour is five times further down than it was pushed.

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

“Slow viscous flow ignores inertia, so inertia plays no part in a slow body's drag.”

For a cylinder it sets the drag. With no wall, the only length available to cut off the cylinder's logarithmically growing disturbance is the distance ν/U at which inertia becomes comparable with viscosity, and unbounded fluid drags exactly like a wall at 10.07 ν/U.

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

“Sorting fast particles into one box and slow ones into another requires a Maxwell's demon, which the second law forbids.”

It is forbidden for elastic particles in a closed system. Shaken grains are neither — energy is pumped in by the floor and lost in every collision — and at B = 6 the sparse compartment's grains are 50 times hotter than when the grains share equally and the crowded compartment's are 0.29 times as hot.

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

“The Gulf Stream is on the western side of the Atlantic because the prevailing winds push water there.”

The wind pattern is roughly symmetric across the basin. In Stommel's gyre, the same wind and friction give a symmetric circulation when the Coriolis parameter does not change with latitude, and a return flow crowded into the western 8 per cent of the basin, three times faster, when it does.

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

“An exact computation of a chaotic map must give a typical orbit.”

The doubling map computed in floating point is exact — doubling a binary number and dropping its integer part introduces no rounding — and every orbit reaches exactly zero within 53 steps. Every number a computer stores is a fraction with a power of two below it, and those are precisely the rare starting points whose orbits end at zero.

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

“Isotopes of a molecule have the same bond energy, because the bond is made by the same electrons.”

They have the same well depth. The energy to break the bond from the lowest level differs, because the zero-point energy falls as the square root of the mass: in one Lennard-Jones well, reduced masses in the ratio 1 : 1.33 : 2 sit 0.9474, 0.9544 and 0.9627 of the depth below dissociation and hold 27, 31 and 38 levels.

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

“Optical interferometers should record each telescope's light and combine it later, as radio interferometers do.”

Recording the phase requires a heterodyne receiver, which adds at least hν/k of noise: 0.068 K at the 21 cm line and 16.6 K at 345 GHz, but 1,358 K at 10.6 µm and 26,156 K for green light. Below about 6 THz the floor is under room temperature; at optical frequencies it swamps most stars, and the light has to be combined directly.

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

“The coloured fringes in the corners of a photograph are a blur and cannot be undone after the picture is taken.”

Lateral colour is a shift, not a blur: the red, green and blue images of an edge 15 mm off centre are each sharp and are centred 25.8 µm apart for a singlet with its stop 5 mm in front. Rescaling each colour channel by its own magnification removes it, which is what camera software does.

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

“Fire — the flashes of colour — measures how pure or valuable a stone is.”

It measures dispersion, amplified by light leaving a face near the critical angle. A degree short of the critical angle, red and violet leave a diamond face 3.2° apart and a cubic zirconia face 5.0° apart, because zirconia disperses more; at 10° from the normal the spreads are 0.48° and 0.65°.

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

“The universe's neutrinos should be as hot as its light, since both were in equilibrium in the early universe.”

They were, until the neutrinos stopped interacting. Afterwards the electron–positron pairs annihilated and handed their entropy to the light alone, raising the entropy of light from 2 units to 11/2's worth, so light ended hotter by (11/4)^⅓: a 2.7255 K photon background implies a 1.945 K neutrino background.

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

“Relativity forbids any experiment from detecting absolute motion.”

It forbids any local experiment from doing so. In a closed universe a twin who sends light both ways round space finds the two flashes return 2γvL apart by the twin's own clock — 15 years for 0.6c in a ten-light-year universe — and zero only for one state of motion.

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

“The stars with the strongest hydrogen lines must contain the most hydrogen.”

The strength of the Balmer lines tracks the fraction of hydrogen that is neutral and in its second level, which at an electron pressure of 20 Pa peaks at 9,900 K and falls on both sides — too few atoms excited below, too few left neutral above. Nearly all stars are mostly hydrogen; the ones with the strongest lines are merely at the right temperature.

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

“Temperatures below a kelvin are measured with a thermometer calibrated against some other thermometer.”

Between 0.9 mK and 1 K the international scale, PLTS-2000, is defined by the melting pressure of helium-3 itself: a polynomial in T whose fixed points include the minimum at 315.24 mK and 2.93113 MPa, the superfluid A transition at 2.444 mK and the solid's magnetic ordering at 0.902 mK.

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

“A 'forbidden' transition is one that cannot happen.”

It happens slowly. Hydrogen's 21 cm line, a magnetic-dipole transition, has a lifetime of 11 million years and is the brightest radio line in the Galaxy; the auroral green line of oxygen lasts 0.8 s and the nebular [O III] line about 40 s. They are forbidden only where collisions end the state first.

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

“Anything with a quantum coordinate will tunnel if it is made cold enough.”

Dissipation suppresses it. Caldeira and Leggett showed that a coupling to the environment multiplies the tunnelling exponent by 1 + 0.87/Q, so at 0.99 of the critical current the rate falls to 88 per cent of its undamped value for Q = 100, 53 per cent for Q = 20 and 8 per cent for Q = 5.

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

“The Trojan asteroids sit fixed at the Lagrange points.”

They swing round them. The slow libration round Jupiter's L4 takes 12.4 of Jupiter's orbits, 147 years; round the Sun–Earth L4 it would take 222 years, and the gentler the hill, the more easily a passing perturbation carries a body off.

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

“Tides would long ago have circularised Io's orbit and ended its heating.”

They would, in about 102 thousand years, if Io were alone. The 1 : 2 : 4 resonance with Europa and Ganymede forces an eccentricity of 0.0041 that tides cannot remove, and the energy dissipated is replaced from Jupiter's rotation through the tides Io raises on the planet.

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

“An atom's size is set by the strength of the electric attraction, so a stronger attraction always makes a smaller, more tightly bound atom.”

Only in three dimensions, where the kinetic cost of confinement, 1/a², and the attraction, −1/a, have different powers and balance at a finite size. In four dimensions both go as 1/a² and a weak attraction binds nothing while a strong one binds infinitely; in five the attraction wins at small sizes whatever its strength.

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

“The ionosphere is a vast reservoir of charge, so the fair-weather current could run on for a long time without the thunderstorms.”

The circuit's discharge time is its resistance times its capacitance, which for a layer whose field sits where its conductivity is lowest comes to ε₀/σ at the ground: about seven minutes in the model. The storms have to be running all the time, and it is that short memory that lets the field track them hour by hour.

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

“A slightly unequal supply makes the rotating field slightly weaker and a motor slightly slower, in proportion to the imbalance.”

The imbalance adds a field turning backwards. With the third coil's current a fraction k of the others the forward part is (2 + k)/2 and the backward part |k − 1|/2 — 11 per cent of the forward part at k = 0.7 — and a rotor running forwards sees the backward part sweep past at nearly twice the supply frequency, inducing currents out of proportion to its size.

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

“A source drifting behind a lens gains or loses its images one at a time.”

Traced across the caustics of the drawn lens, the count runs 1, 3, 5, 3, 1. Every change is by two, at a caustic, where a pair of opposite parity is created or destroyed with formally infinite magnification.

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

“Two waveforms with the same harmonic amplitudes are the same drive as far as the plasma is concerned.”

They deliver the same power spectrum, but the sheaths respond to the waveform's extremes, which depend on the relative phase. At 0° the driven electrode's ions arrive with about 80 eV and the grounded one's with 36 eV; at 90° those numbers swap, with every amplitude unchanged.

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

“Bigger particles make a stronger electrorheological fluid.”

In the point-dipole picture the yield stress, (9/8)φε₀ε_fβ²E² times a number fixed by the chains' geometry, contains no particle size, because a larger sphere's bigger dipole is exactly offset by there being fewer chains per unit area. Size matters at the two ends instead: below about 0.1 µm thermal motion breaks the chains at any usable field, and above about 10 µm the particles settle out.

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

“A magnet can only be levitated over a conductor if the conductor is a superconductor.”

Any conductor repels a magnet moving fast enough over it. Lift over drag is exactly v/w for a thin sheet: 31 at 500 km/h over 10 mm of aluminium, 5 over 1 mm of copper. The price is a drag that levels off at a fixed power, F₀w, rather than vanishing.

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

“A shear-thickening fluid's stress rises steadily with the rate at which it is sheared.”

Only below a volume fraction near 0.569 in the model drawn. Above it the flow curve bends back: at φ = 0.574, a rheometer raising the shear rate sees the stress jump from 1.15 to 24.4 times the onset stress at a single rate, and on the way down it drops at a rate 1.3 times lower.

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

“Compressing a fluid always warms it.”

It warms it by αT/ρcₚ per unit of pressure, and the sign is the sign of the thermal expansion coefficient α. Fresh water is densest near 3.98 °C at the surface, so below that temperature α is negative and compression cools it — by 31 mK per kilometre of descent at 1 °C.

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

“The coefficient of restitution is a property of the two materials in contact.”

The same steel ball at the same speed gives 0.98 on a 40 mm steel plate, 0.77 on 10 mm and 0.19 on 4 mm. Zener's parameter depends on the square of the ball's radius over the plate's thickness, so geometry sets the number as much as the materials do.

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

“A softer planet should wobble faster, as a looser spring would let it swing more freely.”

The restoring torque comes from the part of the equatorial bulge that does not follow the displaced spin axis. A yielding body re-forms more of its bulge round the new axis, so the torque is weaker and the wobble slower: T = T(Euler)/(1 − k/kₛ), infinite for a fluid.

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

“A pinhole cannot concentrate light, since it has no lens.”

The best pinhole's image of a distant point is about twice as bright at its centre as the light arriving at the hole, and a hole that passes exactly one Fresnel zone makes it four times as bright: every part of that zone's light reaches the centre within half a wave of every other part, and adds.

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

“A surface that suddenly brightens when viewed from the light's direction must be made of reflective material.”

In a random layer of dull grains, 70 of 70 points seen from the light's own direction are lit, and 22 of 70 are in shadow when the observer moves 25° away. Nothing about the grains changed; the shadows came into view.

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

“A liquid cooled towards absolute zero becomes less viscous, as the gas it resembles does.”

A gas's viscosity falls as √T on cooling. A degenerate Fermi liquid's rises as 1/T², because the quasiparticles that carry momentum collide at a rate proportional to T²; liquid helium-3 thickens a hundredfold for each tenfold cooling below about a tenth of a kelvin.

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

“With enough channels, a sounder can measure the temperature at any height as precisely as a thermometer on a balloon.”

Each channel's weighting function is about 17 km wide at half maximum when the absorber falls with a 7 km scale height, whatever its absorption. An inversion 8 K strong and 2 km thick near 5 km changes the channel readings by at most 0.62 K, and its shape and height cannot be recovered.

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

“A force that is magnetic in one frame and absent in another cannot be real.”

In the beam's own frame there is no magnetic force and the two charges repel by the full Coulomb force; the laboratory sees the same separation grow γ² times more slowly because the beam's clocks run slow and its transverse inertia is γm. Computed both ways, the sideways accelerations agree exactly at every speed.

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

“A denser plasma, having a stronger field, gives a higher-energy beam.”

The field grows as the square root of the density, but the dephasing length falls as its −3/2 power, so the energy per stage, 4γₚ²ε mc², falls as one over the density: about 107 MeV in 1.8 mm at 10¹⁹ cm⁻³, 1.07 GeV in 5.8 cm at 10¹⁸, 10.7 GeV in 1.8 m at 10¹⁷.

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

“A cell must cover much of its surface with receptors to catch the molecules that reach it at the diffusion-limited rate.”

N absorbing patches of radius s on a sphere of radius R catch a share Ns/(Ns + πR) of what the whole sphere would. For a 5 µm cell and 1 nm receptors, half the maximum needs about 16,000 receptors covering 0.016 per cent of the surface; 157,000, covering 0.16 per cent, catch 91 per cent.

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

“Since thermal noise is set by temperature, only cooling can reduce it in a measurement.”

In a band from 10 to 100 times an oscillator's resonance, where gravitational-wave detectors observe above their pendulum suspensions, the rms thermal motion falls as 1/√Q: from 1.5 × 10⁻² √(kT/k) at Q = 1 to 1.5 × 10⁻⁴ at Q = 10⁴. Raising Q moves the noise into the resonance peak, outside the band.

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

“Hydrogen can only be ionised by heating it to a temperature comparable with its binding energy.”

Both of the ways it is ionised avoid that. A thin gas ionises far below the binding temperature because a free electron gains so much room; a dense one ionises at any temperature because the bound state no longer exists. The crowding line on the density–temperature map sits near 6 × 10²⁸ m⁻³ whatever the temperature.

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

“The thinner a layer, the weaker its echo.”

Only below the tuning thickness. From a full wavelength down to about a fifth of one, the two faces' echoes overlap more and more constructively and the combined echo grows, to 1.45 times one face's at 20 m for a 30 Hz pulse in 3000 m/s rock; below that it falls in proportion to the thickness.

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

“A wave several times higher than the waves around it must have been built from extra energy arriving from somewhere else.”

In the drawn evolution the whole train conserves its norm to a part in 10⁸ while one crest per modulation period rises to 2.41 times the background, the troughs beside it falling to 0.07: the crest's energy is drawn in from its neighbours. The Peregrine breather does the same, reaching exactly three times the background from an even train.

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

“An oscillator just outside the locking range beats against the drive at the difference between their frequencies.”

It beats more slowly, at √(Δω² − K²): 0.66 K for a detuning of 1.2 K. And the beat is not a smooth tone: at 1.05 K the phase lingers near a quarter turn and slips a whole turn in a burst every 19.6 time units.

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

“The most tightly bound nuclei are the hardest to convert into neutron-rich ones.”

Capture depends on the mass difference with the neighbouring nucleus, not on the binding per nucleon. Iron-56 needs a 3.7 MeV electron and captures at 1.1 × 10⁹ g/cm³. Helium-4 needs 20.6 MeV and holds out to 1.4 × 10¹¹ g/cm³.

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

“Thermal expansion at low temperature is a small anharmonic correction with no connection to the third law.”

Grüneisen's relation makes the expansion proportional to the heat capacity, so it vanishes the same way. Copper's coefficient falls from 1.61 × 10⁻⁵ per kelvin at 300 K to 5.1 × 10⁻¹⁰ at 1 K, as T³ from the lattice and then as T from the electrons below 3.8 K.

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

“Osmotic pressure always counts the dissolved molecules, so it can always be used to weigh them.”

Only while the coils are apart. Past the overlap concentration the pressure is one kT per correlation blob, and the chain length drops out. An osmometer reading 10⁵ and 10⁶ g/mol chains in a good solvent at 200 g/L would report about 6,000 and 6,300 g/mol, nearly the same number for chains ten times different.

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

“Using a rinse with half the surface tension doubles the height of pattern that survives drying.”

The critical height rises only as the inverse fourth root of the surface tension. Halving the tension gains 19 per cent. 30 nm lines survive to 92 nm from water, 124 nm from isopropanol and 143 nm from a fluorinated rinse of 12 mN/m. Doubling the height would need a sixteenth of water's tension, which is why production lines dry past the critical point instead.

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

“The damping in a bolted or riveted joint can be modelled as a linear viscous damper with a fixed damping ratio.”

The frictional loss per cycle grows as the cube of the amplitude (computed slope 3.00), against a viscous damper's square. So the fraction of energy lost per cycle is proportional to the amplitude: a joint is nearly lossless for small vibrations and strongly damping for large ones.

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

“A scattering length is a property of collisions and says nothing about bound states.”

Near threshold it is the bound state's size seen from outside. As a square well deepens past the depth that first binds, the scattering length jumps from minus to plus infinity. Just past it, at depth parameter 1.6, it is 22.4 R against a bound-state size 1/κ of 21.9 R.

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