True as far as it goes, and silent on what decides

“T = 2π√(L/g) is the period of a pendulum.”

The exact period divided by the small-angle period, against amplitude. The textbook formula is the horizontal line at one; the truth is a curve that leaves it immediately and has no elementary form. The site's gate checks the plotted curve against Simpson's rule on the elliptic integral, by a route the generator does not take.

Tested in The period that depends on the swing, computed exactly · the pendulum reading path

“A rainbow is sunlight split by raindrops acting as prisms.”

The splitting is right and it predicts nothing about where the bow is. The angle comes from a minimum of the deviation function, traced ray by ray through the drop: nothing about water, light or weather picks forty-two degrees, and the same computation puts the secondary bow on the other side of its own minimum.

Tested in The angle the rainbow has to be, and why nobody chose it · the dispersion reading path

“A wave travels at its wave speed.”

Two speeds, and in deep water they differ by a factor of two: the crests run forward through the group, rise in the middle and vanish off the front. The figure draws the same packet in a medium with no dispersion as a control, where the two speeds coincide and the crests hold station.

Tested in The packet that moves at another speed than its own crests · the wave packets reading path

“The lines in an atom's spectrum are the atom's energy levels.”

They are the differences between them, which is why the lines come in families crowding onto a limit rather than spreading out. The figure draws the ladder and the transitions between its rungs, and the wavelengths fall out of the subtraction.

Tested in The spectrum is a subtraction, not a list of values · the atomic spectra reading path

“The interference pattern builds up as the particles interfere with one another.”

Only one is ever in the apparatus. The arrivals are drawn as a sample from a fixed distribution, and the check is statistical rather than pictorial: at a thousand arrivals the histogram correlates with the intensity above 0.9, and at twenty a χ² test against a flat screen fails to reject at 95 per cent.

Tested in One arrival at a time, and the pattern still appears · the matter waves reading path

“Pressure pushes down.”

The pressure at a point in a still fluid is one number with no direction attached, and the figure showing a submerged block draws it acting on every face at once — inward on the sides, down on the top, up on the bottom. What has a direction is the force on a surface, which is pressure times the area and points along that surface's own normal.

Tested in The pressure that only knows depth · the hydrostatics reading path

“Buoyancy is an upward force that fluids exert.”

There is no separate buoyant force in the physics. The figure draws the ordinary pressure on each face of a block and adds them: the sides cancel in pairs, and the top and bottom differ by exactly the weight of a column of fluid as tall as the block. Nothing was added to hydrostatics to obtain it.

Tested in The weight of the water that is not there · the buoyancy reading path

“A floating object is in equilibrium when the upthrust equals its weight.”

That fixes how deep it sits and says nothing about which way up. Equilibrium of forces is not equilibrium of moments, and the second is decided by where the two lines of action lie — which the previous rung's arithmetic cannot see, because it never asked where the buoyancy acted.

Tested in Why a ship comes back upright · the buoyancy reading path

“Small insects walk on water because they are light.”

Weight scales as the cube of size and the surface force as the first power, so the ratio goes as the square — but the comparison that decides is against BUOYANCY, not against nothing. The capillary length, about 2.7 mm in water, is where the two are equal, and it is a fixed property of the liquid rather than of the insect.

Tested in The skin that is not a skin · the surface tension reading path

“The pressure inside a bubble is 2γ/R.”

That is a droplet, which has one surface. A soap bubble is a film with two, so its excess is 4γ/R — and the figure draws both laws on one pair of axes, exactly a factor of two apart at every radius, so the distinction is visible rather than remembered.

Tested in The small bubble blows up the big one · the surface tension reading path

“A jet of water breaks into drops because of air resistance, or because the tap wobbles.”

It breaks in a vacuum, and it breaks when nothing shakes it. Surface tension alone does it: a cylinder has more surface than the row of drops holding the same volume, so the surface is not resisting the break-up but causing it. The figure draws the growth rate against wavelength, and it is positive without anything external in the calculation.

Tested in The thread that cannot stay a thread · the surface tension reading path

“The drops come off at whatever spacing the disturbance happens to have.”

Every wavelength longer than the circumference grows, so a random disturbance contains many growing modes at once. What decides is which grows FASTEST, and the growth rate has a clear maximum — found here by searching the drawn curve rather than quoted — at kR = 0.697, which is 4.51 diameters. Random input, non-random output.

Tested in The thread that cannot stay a thread · the surface tension reading path

“Liquids rise in narrow tubes.”

Some fall. The direction is set by the contact angle, which is fixed by three surface energies rather than by the liquid alone: mercury in glass has an angle near 140°, its cosine is negative, and the same formula returns a DEPRESSION. The figure takes the contact angle as an input for that reason.

Tested in How high water will climb · the capillarity reading path

“Fluid slides along a solid surface.”

It does not. The fluid in contact with a wall moves with the wall — the no-slip condition — which is why the figure draws zero velocity at the fixed plate and full plate speed at the moving one. It is an experimental fact rather than a consequence of anything above it, and everything about drag and pipe flow follows from it.

Tested in Momentum going sideways · the viscosity reading path

“Blood pressure rises when arteries narrow because there is less room.”

The dependence is far steeper than 'less room' suggests. Resistance goes as the inverse fourth power, so a vessel narrowed by 20 per cent has its resistance multiplied by 2.4 — and the body's control of blood distribution works by small changes in radius for exactly that reason. The exponent is the mechanism, not a detail of it.

Tested in The fourth power in a pipe · the viscosity reading path

“Stirring something harder makes it flow more easily.”

For most complex fluids, yes; for a dense cornflour suspension the opposite, and violently so — its apparent viscosity RISES with shear rate, which is why it can be run across and not stood on. The figure draws both curves on one pair of axes, and the sign of the slope is the entire difference.

Tested in The fluid that answers back · the rheology reading path

“A wave's speed is a property of the medium.”

For sound and for light in a transparent solid, near enough. For water waves it is a property of the medium AND the wavelength, which is why a single storm produces waves that arrive over several days sorted by period. The figure plots speed against wavelength and it is a curve, not a horizontal line.

Tested in The speed that depends on the length · the wave packets reading path

“A gas is a continuous fluid.”

It is not, and the reason it can be treated as one is a RATIO: air's molecules travel about 68 nm between collisions, against apparatus measured in millimetres, so a region small enough to be a point still holds a hundred million molecules. Below a few microns that ratio fails and the gas stops obeying the equations of fluid mechanics — the same gas, and a different description.

Tested in How far a molecule gets · the kinetic theory reading path

“The particle moves because more molecules hit one side than the other.”

True, and the size of the imbalance is the part that matters: a micron-sized grain is struck about 10^20 times a second, and the fractional imbalance is roughly one part in 10^10 — far too small to see directly. What makes it visible is that the imbalance never averages away completely, so displacements ACCUMULATE as the square root of time.

Tested in The jiggle that proved atoms · the diffusion reading path

“Superfluid helium is a Bose–Einstein condensate.”

The ideal Bose gas explains why a transition must exist and gets the shape of the curve wrong: its condensate fraction goes as 1 − (T/Tc)^3/2 and helium's superfluid fraction as roughly 1 − (T/Tλ)^5.6, which the figure draws together so the disagreement is visible. Helium's atoms interact strongly and the calculation's do not, and only about ten per cent of helium atoms are in the condensate even at absolute zero.

Tested in The liquid that will not slow down · the superfluidity reading path

“The equivalence principle says acceleration and gravity are the same thing.”

It says they are indistinguishable *locally*, and locally is a measurement rather than a word. Two balls released a metre apart in a box falling ten metres approach by 1.57 microns; make the box large enough, or the measurement good enough, and the difference is always there. What is genuinely equivalent is a uniform field, and no real field is uniform.

Tested in The floor that cannot be told from gravity · the equivalence principle reading path

“A satellite's clock runs fast because it is higher up.”

Higher makes it run fast and moving makes it run slow, and the two have opposite signs and comparable size. The total, (GM/c²)(1/R − 3/2r), is zero at r = 3R/2 — an altitude of 3,186 km, solved for in the figure rather than quoted — negative below and positive above. A low-orbit satellite's clock loses.

Tested in The clock that runs slow lower down · the gravitational redshift reading path

“A mass acts as a lens.”

It acts as a lens with no focal plane. A glass lens deflects in proportion to the distance from its axis and therefore has one focus; a mass deflects as 1/b, so rays at different distances cross the axis at different places — a focal line running outward for ever. The consequence is that the image of a point source is a ring or an arc rather than a point, which no ordinary optical instrument does.

Tested in The bend Newton got half right · the light deflection reading path

“The distant observer and the falling observer disagree, so one of them must be wrong.”

Neither is. They are computing different quantities: elapsed proper time along one worldline, and a coordinate label attached to events by a family of static observers. Asking which is correct is asking which of two rulers laid along different paths gives the true length, and the answer is both.

Tested in Two clocks that disagree about the fall · the horizons reading path

“Gravitational waves are emitted by any accelerating mass, as electromagnetic waves are by any accelerating charge.”

A single accelerating mass radiates nothing. The monopole term is forbidden by conservation of mass and the dipole term by conservation of momentum, so the first surviving term is the quadrupole — which requires the mass distribution to change shape, not merely to move. There is no gravitational analogue of a dipole antenna.

Tested in The wave that stretches one way and squeezes the other · the gravitational waves reading path

“Radiation pressure is negligible.”

It is negligible compared with gravity for large objects and larger than gravity for small ones, and the crossover is computed rather than asserted: for rock of density 2000 kg/m³ near a solar-luminosity source it is at a grain radius of 287 nm. Both forces fall as the inverse square of the distance, so the comparison does not depend on how far away anything is.

Tested in Light has a pressure · the radiation pressure reading path

“An accelerating charge radiates.”

True, and silent about the direction. The power goes as sin²θ about the acceleration, so nothing at all is emitted along the acceleration itself — and once the charge is moving relativistically the whole pattern sweeps forward into a cone of half-angle about 1/γ. The figure computes the peak at 13.4° for a charge at 0.9c, where the naive estimate gives 25°.

Tested in A charge that turns must glow · the radiating charge reading path

“Neutrinos pass through everything.”

They pass through almost everything almost always, which is a statement about a product. The figure computes the free path in solid lead at 1.55 light-years, so a slab a metre thick stops about one neutrino in 10¹⁶ — and a detector compensates with the other factor, running 10¹³ neutrinos a second through a target of many tonnes for months.

Tested in How far a neutrino gets · the kinetic theory reading path

“In a perfect conductor the magnetic field is constant.”

The field is not constant; the flux through a material loop is. A collapsing region has the same flux through a smaller area, so the field rises as 1/R² — computed here as a factor of 4.9 × 10⁹ for a collapse from 700,000 km to 10 km. Constant flux and constant field are the same statement only when nothing moves.

Tested in The field that cannot get out · the flux freezing reading path

“Large bodies are round because gravity pulls everything toward the centre.”

Gravity pulls on small bodies too, and they are not round. What decides is a comparison: the stress ρgh under a feature against the strength of the material it rests on. Since a uniform body's surface gravity is itself proportional to its radius, the tallest possible mountain falls as 1/R — so the argument is about a competition between two quantities, and the size at which one wins is computed rather than asserted.

Tested in The size at which a body becomes round · the Self-gravity reading path

“The Planck length is √(ħG/c³), so it is exactly 1.616 × 10⁻³⁵ m.”

The combination is fixed only up to dimensionless factors. Bisecting the Compton wavelength against the Schwarzschild radius gives 2.29 × 10⁻³⁵ m — larger by exactly √2, the factor of two in rs coming through a square root. Any argument of this kind fixes an order of magnitude, and quoting four figures suggests a precision the derivation does not contain.

Tested in Where every model runs out at once · the planck scale reading path

“Each twin sees the other's clock running slow.”

True while both coast, and silent about the turnaround, where the traveller's lines of simultaneity swing across a large stretch of the stay-at-home's worldline. Counting light pulses avoids the question entirely: the traveller receives them at √((1+β)/(1−β)) times the emitted rate for the inbound half and the reciprocal for the outbound, and the totals do not match.

Tested in The twin who comes back younger · the time dilation reading path

“A source moving across the line of sight is not Doppler shifted, because its distance is not changing.”

It is shifted by 1/γ — 0.866 at half the speed of light, computed in the figure. Nothing classical predicts it, because it is not about distance changing at all: it is the source's own clock running slow, read as a colour. The transverse shift is the term that makes the effect a test of relativity rather than of arithmetic.

Tested in The shift that survives at right angles · the doppler reading path

“Entropy is proportional to the amount of stuff, so a bigger system has proportionally more.”

For ordinary matter entropy is extensive — double the volume at fixed density and it doubles. A horizon's goes as the area, so doubling the mass quadruples it. That difference is not a technicality: it says the maximum entropy that fits in a region is set by the region's boundary rather than its volume, which is the holographic bound.

Tested in The entropy that lives on a surface · the entropy reading path

“Friction opposes motion.”

It opposes relative sliding at the contact, which is not the same thing. The friction on a driven wheel points forward and is the only thing accelerating the car; the friction on a rolling ball does no work at all, because the contact point is instantaneously at rest. Stated as opposing motion, the rule gets the sign wrong for every vehicle ever built.

Tested in The force that takes what it needs · the friction reading path

“Anharmonicity is a small correction.”

It is small in the period and not small in what it accounts for. Thermal expansion, thermal conductivity in an insulator, the overtone spacing of a molecular vibration and the fact that a solid can melt at all are consequences of the cubic and quartic terms and are exactly zero in the harmonic approximation. A correction that is numerically tiny and qualitatively load-bearing is worse than a large one, because it is easy to drop.

Tested in Every minimum is a parabola · the harmonic approximation reading path

“The Coriolis force explains which way water goes down a plughole.”

The term is real and its size is computable, and over a basin it loses. The figure gives the deflection over a stated flight: 2.3 m at the pole for something moving 20 m/s for 40 seconds, which is 0.29% of the range. Scaled to a sink — a tenth of a metre per second, a few seconds — the Coriolis deflection is microns, and it is swamped by the residual swirl left from filling. What the term does decide is systems large enough for the deflection to accumulate for hours: cyclones, ocean gyres, and long-range gunnery.

Tested in The forces that are not there · the circular motion reading path

“The radius of a track in a bubble chamber gives the particle's mass.”

It gives the momentum, and the mass only in combination with something else. The radius is mv/qB, so a light fast particle and a heavy slow one of the same momentum curve identically. A mass spectrometer gets round this by fixing the energy first — accelerating everything through the same voltage — after which the radius goes as the square root of the mass, and the figure computes 96.0 mm against 100.7 mm for neon-20 and neon-22.

Tested in The force that does no work · the magnetism reading path

“The field inside a solenoid is μ₀nI.”

For an infinite solenoid, exactly. For a real one the result is an approximation whose error is set by the aspect ratio: the field on the axis at the mouth of a long solenoid is half the central value, and the outside field is not zero but small. Quoting the formula without the length is how a coil design ends up a factor of two out at the ends, which is where the sample usually is.

Tested in The field that wraps a current · the ampere law reading path

“The electric and magnetic fields in a light wave are a quarter cycle out of step, each one feeding the other.”

They are exactly in step in a plane wave in vacuum — both peak together and both vanish together. The intuition behind the quarter-cycle claim comes from an LC circuit, where energy really does slosh between two stores. In a travelling wave the energy is not sloshing; it is moving, half in each field, and the fields are in phase. The out-of-phase case is real but belongs to a standing wave, where it is the same fact as a node in one field sitting at an antinode of the other.

Tested in The term that made light · the maxwell equations reading path

“Snell's law follows from Fermat's principle.”

It follows for a flat boundary between two uniform media with light treated as a ray. The figure locates the crossing point by golden-section search on the drawn optical path length, with no law of optics used anywhere, and the angles that come out — 55.80° and 33.46° — satisfy n₁sin θ₁ = n₂sin θ₂ to one part in 10⁸. What the derivation cannot supply is the reflected beam, its amplitude, its phase, or the polarisation dependence, all of which need the boundary conditions on a wave.

Tested in The path that does not change · the fermat reading path

“The resolution limit is a consequence of the wave nature of light.”

It is a consequence of a wave passing through a finite aperture, and light is not the only wave available. An electron accelerated through 100 V has a de Broglie wavelength of 123 pm, five thousand times shorter than green light, and an electron microscope's resolution follows the same formula with that wavelength in it. The limit is about apertures and wavelengths, and the choice of what is doing the illuminating is a free parameter.

Tested in How far apart two things have to be · the diffraction reading path

“Huygens' principle explains diffraction.”

It predicts that a wave spreads past an aperture, which is the qualitative fact, and in its 1690 form it gets the amplitude wrong and predicts a backward wave that does not exist. The repair — Fresnel's obliquity factor and a phase advance of a quarter cycle — was found by requiring agreement with experiment and only justified in 1882, when Kirchhoff derived the whole construction from the wave equation and the correction fell out.

Tested in Every front is a source · the huygens reading path

“A pulse inverts when it reflects off a fixed end.”

True, and the fixed end is the limiting case rather than a separate rule. A fixed end is a medium of infinite impedance, so the reflection coefficient (1−Z₂/Z₁)/(1+Z₂/Z₁) goes to −1 and the inversion is total. At a finite ratio of 3 it is −0.500, measured off the marched pulse in the figure; at a ratio below one it is positive and there is no inversion at all.

Tested in What happens where the medium changes · the impedance reading path

“Intensity falls as one over distance squared, so amplitude does too.”

Amplitude falls as one over distance, because intensity goes as the square of amplitude. Confusing the two is a factor-of-two error in every decibel calculation and a factor-of-four error in every energy one. The rule worth keeping is that the conserved quantity is power, so it is intensity that the geometry acts on directly and amplitude that follows.

Tested in How a wave thins out · the wave motion reading path

“The heat capacity of a diatomic gas is 5/2 R because it has five degrees of freedom.”

It has seven, and two of them are frozen. The vibration contributes two quadratic terms — kinetic and potential — and would take the total to 7/2 R at every temperature. The figure computes what actually happens: hydrogen sits at 3/2 below 60 K, 5/2 through room temperature, and reaches 3.38 by 5,000 K, because a degree of freedom whose first energy step is large compared with kT cannot be excited at all.

Tested in Half a kT for every way of moving · the equipartition reading path

“The Boltzmann factor gives the probability of a state of energy E.”

It gives the probability of one *state*, not of one energy, and the difference is a degeneracy. An energy level with many states at it is that many times more likely, which is why an atom's excited populations follow g·exp(−E/kT) and why the entropy of the system itself — as opposed to the reservoir's — appears in every free-energy argument. Forgetting the degeneracy is how a spectrum comes out with the wrong line ratios.

Tested in The exponential that decides everything · the entropy reading path

“The twin who travels comes back younger because of time dilation in the outbound and return legs.”

That gets the answer and hides the geometry. Proper time is the length of a worldline in this metric, and the figure sums √(dt² − dx²) leg by leg along each drawn path: 2.000 for the twin who stays and 1.600 for the one who leaves, a ratio of 0.8000 against √(1−β²) = 0.8000. The bent path is longer on the page and shorter in time, because the minus sign makes the straight path the longest — which is the whole answer and needs no acceleration in it.

Tested in The quantity nobody argues about · the spacetime diagram reading path

“The wire is neutral, so it has no electric field.”

It is neutral in the laboratory frame, and neutrality is not frame-independent. In the frame moving with the drifting electrons the lattice is contracted and the electrons are not, so the two densities no longer cancel and the wire carries a net charge of 1.113×10⁻²⁰ C/m — small, and exactly enough to produce the force that the laboratory frame calls magnetic.

Tested in Magnetism is electricity seen sideways · the field transformation reading path

“The kinetic energy of a rolling ball is ½mv².”

That is the translation only. The energy figure prints 71 per cent translation and 29 per cent rotation for a sphere and 67/33 for a solid cylinder, so after a drop of 1.00 m the sphere reaches 3.74 m/s rather than the 4.43 m/s that ½mv² alone predicts. The two shares add with no cross term between them, which is König's theorem and is why the split can be quoted as two numbers at all.

Tested in The point that keeps moving as if nothing had happened · the momentum reading path

“A chain of masses and springs is an approximation to the wave equation.”

The relation runs the other way, and the difference is measurable. A chain of N masses has frequencies 2√(k/m)·sin(nπ/2(N+1)), proportional to n only at the bottom — within a tenth of the straight line up to mode 12 of 24 — and stopping dead at 2√(k/m). The wave equation is the limit of the chain as the spacing goes to zero, and it has no ceiling, so it permits a copper crystal vibrations of any frequency whatever when the measured spectrum ends near 7 THz.

Tested in The two pendulums that will not stop swapping · the harmonic approximation reading path

“The Poynting vector gives the energy flow at every point of space.”

Only its flux through a closed surface is determined: adding the curl of any vector field to S leaves ∇·S untouched, so the theorem fixes the total and not the route. A capacitor gap at 84.7 kV/m sitting in a 1 T field carries S = EB/µ₀ = 67.4 GW/m², fifty million times the 1.36 kW/m² of sunlight, and delivers nothing anywhere, because the divergence is zero.

Tested in Where the energy of a field actually is · the field energy reading path

“Water has a large dielectric constant because its molecules turn to line up with the field.”

They barely turn. At 300 K in a field of 10⁶ V/m the alignment energy pE is 6.2 × 10⁻²⁴ J against a thermal 4.14 × 10⁻²¹ J, so the mean alignment ⟨cos θ⟩ is 5 × 10⁻⁴ — five parts in ten thousand. Counting only that gives a relative permittivity of 12.7 where the measurement is 80, and the missing factor of 6.3 comes from neighbouring molecules being correlated rather than independent.

Tested in The attraction that needs no charge · the the field concept reading path

“The sodium doublet is resolved once the grating has more than a thousand lines.”

Two things are missing: the order, and what counts as resolved. The requirement is R = mN ≥ λ/Δλ = 982, so 600 lines in second order do it while 1200 lines in first order do no better. And the criterion is a convention — the dip measured at exactly R = 982 is 18.9 per cent, a dip first appears at 0.83 of the Rayleigh separation, and the figures report 0 per cent at R = 300 and 53.4 per cent at R = 1200.

Tested in What a thousand slits buy that two cannot · the diffraction reading path

“Coherence means having a single wavelength.”

That is half of it. Temporal coherence follows from bandwidth, spatial coherence from the source's angular size, and the two are independent. Sunlight spans 300 nm of spectrum yet is spatially coherent across 1.22λ/θ = 73 µm, because the Sun subtends 0.53° — which is why a pinhole was needed before a wavelength filter was.

Tested in Why two lamps never interfere · the coherence reading path

“A wave on a real string obeys the wave equation.”

It obeys it to order slope-squared. Replacing sin θ by tan θ costs a factor of cos θ, which is 1.5 per cent at a 10° slope and 13 per cent at 30°, and a real string's bending stiffness adds a fourth-derivative term that has no place in the equation at all: with an inharmonicity coefficient of 2×10⁻⁴ the eighth harmonic of a piano string lands about 11 cents sharp of eight times the fundamental, which is why the top octave of every piano is deliberately stretched.

Tested in The equation that lets a shape travel · the wave motion reading path

“The set of frequencies a membrane produces is a fingerprint of its shape.”

The ratios are decided by the boundary alone, so the converse looks inevitable, and it is false. Kac asked in 1966 whether the shape could be recovered; in 1992 Gordon, Webb and Wolpert exhibited two eight-sided polygons with identical spectra — every one of infinitely many frequencies matching. The spectrum does fix the area and the perimeter, and it does not fix the shape.

Tested in The drum that has no harmonics · the standing waves reading path

“The energy available in a collision is the energy of the beam.”

What is available is the invariant mass of the whole system, √s. A 7 TeV proton striking a stationary proton gives √(2mc²E + 2m²c⁴) = 114.6 GeV; two 7 TeV protons meeting head-on give 14 TeV. The same accelerator, the same beam, a factor of 122 in what can be made — which is the entire reason high-energy machines collide.

Tested in The invariant that survives a boost · the Mass-energy reading path

“A rigid rod is a sound idealisation that relativity corrects only at high speed.”

Rigidity needs an infinite sound speed, so it fails at every speed. Steel carries a push at 5,100 m/s, which is 196 µs per metre; a 1 ms hammer blow is only five crossing times of a one-metre bar, and a 3 km drill string does not feel a torque change for 0.59 s. Relativity's contribution is a ceiling 59,000 times above the speed steel actually manages.

Tested in The pole that fits and does not fit · the length contraction reading path

“A system with a lower free energy available to it will go there.”

The criterion is about equilibrium and supplies no rate. Graphite is lower than diamond by 2.90 kJ/mol at 298 K, and a diamond lasts indefinitely because the rearrangement has a barrier of several electronvolts — at 300 K a 0.35 eV barrier alone already costs a factor of 1.3 × 10⁻⁶, and several eV is a factor no timescale reaches.

Tested in What a system actually minimises · the entropy reading path

“One latent heat is enough to draw a substance's whole vapour-pressure curve.”

It is enough over a modest interval and visibly not enough beyond it. Water's curve integrated at the constant 43.32 kJ/mol that its triple point and boiling point imply reaches 37.4 MPa at the critical temperature against a measured 22.1 — 70 per cent high, because the latent heat falls from 45.05 kJ/mol at the triple point to zero at the critical point.

Tested in A boiling point is a pressure, not a temperature · the phase change reading path

“Young's equation predicts the contact angle from the three surface tensions.”

Two of the three are not independently measurable, because a solid cannot rearrange to relieve its own surface stress, so the equation is almost always run backwards from a measured angle. The tensions drawn in the figure come from a stated closure — the geometric-mean rule, which returns γsv = 68.48 mN/m for the 20° solid — and clean soda-lime glass is in the hundreds of mJ/m². Below about a micrometre the equation also needs a line-tension term: at τ ≈ 10⁻¹¹ N and a 1 µm base radius that shifts cos θ by 0.14, which is 9° at 60°.

Tested in The angle a liquid makes with what it sits on · the capillarity reading path

“The innermost stable circular orbit sits at three Schwarzschild radii.”

That is the non-rotating answer alone. Around a maximally rotating hole a co-rotating orbit's innermost radius falls to GM/c², one sixth of 6GM/c², and the efficiency rises from 5.719% to 42.3%; a counter-rotating one goes out to 9GM/c² and the efficiency falls to 3.77%. The quoted range of efficiencies exists because 3 rs is one point in it.

Tested in The orbit that cannot be made smaller · the orbit stability reading path

“Experiments have broken Bell's inequality.”

A run is a statistical statement about a bound, not a demolition of it. At 5,000 simulated coincidences the estimate is 2.8528 ± 0.0396, which is 22 standard errors above 2; at 100,000 it is 2.8202 ± 0.0090, or 91. Real apparatus at a visibility of 0.97 reaches 2.744 rather than 2.828, and the first loophole-free run reported 2.42 ± 0.20 from 245 trials.

Tested in The correlation no instructions can produce · the entanglement reading path

“A body spinning about its greatest moment of inertia is stable, and that is the end of the matter.”

It is stable against small departures, which is what the linear analysis asks. It is not stable against *dissipation*, which the analysis has no term for: an energy-losing body at fixed angular momentum slides to the state of lowest energy at that momentum, which is rotation about the greatest moment. That is why the intermediate-axis flip is a rigid-body phenomenon and why Explorer 1, which flexed, ended up tumbling about an axis nobody chose.

Tested in The axis that will not hold · the rotation reading path

“The normal force acts at the centre of the base.”

It acts there only when nothing else is applied. The reaction of the floor is a pressure distributed over the whole contact patch, and its resultant sits wherever moment balance requires — a distance Py/W from the centre, walking toward the leading edge in proportion to the push. Tipping is not a separate phenomenon that begins at some angle; it is that point arriving at the edge of the base and having nowhere further to go. Every figure here plots its position, and the figure for a floating hull plots the same quantity for a body with no edge to reach.

Tested in Slide or topple · the Free-body reading path

“The induced charge on the plane is −q, spread out somewhere near the foot of the perpendicular.”

It is −q, and where it sits is computable and wider than that suggests. The density is −qd/2π(r² + d²) to the three-halves power, so half the induced charge lies inside a radius of √3 d — 34.6 mm for a charge 20 mm up — and a tenth of it is still outside 3d. The running total in the figure is that density integrated over the surface and reaches 99.999% of −q only at the edge of the arithmetic.

Tested in The charge that has to be somewhere else · the conductors reading path

“A small current loop is a magnetic dipole.”

Only at a distance, and the distance is larger than 'small' suggests. On the axis the exact field is μ₀Ia²/2(a² + z²) to the three-halves power and the ideal dipole's is μ₀m/2πz³; their ratio is (1 + (a/z)²) to the minus three-halves, so the approach is algebraic rather than exponential. A loop is still 28% below the dipole value at twice its own radius, 5.7% below at five times, and reaches one per cent only at 12.2 radii.

Tested in The loop that behaves like a needle · the magnetism reading path

“A pile of glass plates polarises the transmitted beam.”

Slowly, and the number of plates is the price. Each surface removes about 15% of the s-polarised light at Brewster's angle and none of the p, so after m plates — two surfaces each — the transmitted degree of polarisation is (1−R)^(2m) differenced against 1. Eight plates reach 85.7%; one plate reaches 26%. Reflection polarises perfectly and throws away most of the light; transmission keeps the light and polarises it by attrition.

Tested in The angle at which reflection picks a side · the polarisation reading path

“The gradient above a hot road must be enormous, since the effect is so obvious.”

The index deficit at the surface is 3 × 10⁻⁵ — thirty parts in a million — over a layer five centimetres deep. What makes it visible is not its size but its steepness: 6 × 10⁻⁴ per metre, which is 3,800 times the Earth's curvature, so a ray bends back up over a horizontal distance of a couple of hundred metres. The critical angle for the whole effect is 0.444°, a fiftieth of the width of the Moon.

Tested in The ray that bends without a surface · the fermat reading path

“Fringes fade because the two beams get out of step.”

They stay exactly in step for a monochromatic source at any delay whatever. What fades is the *agreement between the components*: each wavelength in the source produces its own fringe pattern with its own spacing, and at a large path difference those patterns are displaced relative to each other by more than a fringe and add to a uniform glow. The sodium doublet makes this visible by having only two components — its visibility does not decay at all but oscillates, returning to one every 0.58 mm.

Tested in How far a wave can remember · the coherence reading path

“The instability sits at twice the natural frequency.”

It sits at 2ω₀/n for every integer n, and the wedges lean as they are climbed. The figure locates each one by measuring rather than by assuming: the second opens at a depth of 40.0% and the third at 62.5%, against 8.0% for the first, so the higher ones are unreachable by anything a rider can do. The third has also drifted from 0.667 to about 0.63 by the time the modulation depth reaches 0.9 — an assumption that it stays under its own small-amplitude apex found nothing at all when it was made.

Tested in The swing that is pumped, not pushed · the resonance reading path

“At high enough frequency the compressions would become isothermal and Newton's speed would be right.”

There is nowhere in a gas it is right. The ratio of heat's reach in a period to the wavelength goes as √(Df)/c, so it reaches one at f = c²/D — 5.8 GHz for air at atmospheric pressure. But the sound wavelength falls to the mean free path at 4.9 GHz, and past that there is no continuum left to carry a wave. The two frequencies land in the same decade at every pressure drawn, and they must: the diffusivity is about a third of the mean free path times the molecular speed and the sound speed is about the molecular speed, so both are the collision rate in disguise.

Tested in The correction that took a century · the wave motion reading path

“The spreading is σ₀√(1 + (t/τ)²).”

That expression keeps only the curvature of ω(k) and drops everything beyond, so it is exact for a narrow spectrum and progressively wrong for a wide one. Measured against envelopes traced from the full superposition, it agrees to 0.3% at 10% bandwidth, 3.3% at 18% and 8.5% at 28% — an ordering that is itself the size of the cubic term, and the reason the closed form is drawn as a dashed check rather than as the curve.

Tested in The packet that will not keep its shape · the wave packets reading path

“Beaming makes an approaching source look brighter.”

By the fourth power of the Doppler factor, which is a larger statement than 'brighter'. One power comes from each photon's energy, one from the rate they arrive at, and two from the solid angle they are squeezed into. Front against back, that is ((1+β)/(1−β))², which is 9 at 0.5c, 361 at 0.9c and 3.96 × 10⁴ at 0.99c — so a source pointed away is not dimmed but effectively deleted, and any catalogue of such sources is a catalogue of the ones pointed at the observer.

Tested in The sky that crowds into a cone · the doppler reading path

“A higher orbit means a bigger correction.”

It means a bigger *potential* term and a smaller velocity term, and the two vary differently with radius — the potential term saturates because there is only so much potential to climb out of, while the velocity term falls as 1/r. Setting the sum to zero gives r = 1.5 R⊕, an altitude of 3,186 km, and that radius contains neither G nor the Earth's mass nor the speed of light. Below it a satellite's clock loses time; above it, gains.

Tested in The clock that is wrong in two directions · the time dilation reading path

“The heating and cooling coefficients are two separate performance figures.”

They differ by exactly one, at every temperature, and the figure measures the gap off the drawn curves at 1.8 × 10⁻¹⁵ across the whole range. Whatever is taken from the cold side is delivered to the warm side *plus* the work that moved it, so COP_heat = COP_cool + 1 is energy conservation and not a coincidence. A single machine quoted as a refrigerator and as a heat pump differs in specification by one.

Tested in The engine that pays back more than it takes · the heat engines reading path

“A parabolic mirror focuses better than a spherical one.”

It focuses perfectly, for rays parallel to its axis, and the figures check that on the drawn curve rather than quoting it: a vertical ray reflected at a quarter, a half, three quarters and the whole of the dish radius crosses the axis at the same height to better than one part in ten thousand. What a parabola does not do is focus rays arriving off-axis, which come to a comet-shaped blur — so a liquid mirror has a perfect image on its axis and a rapidly worsening one away from it, and can only ever point straight up.

Tested in The surface a spin decides · the hydrostatics reading path

“The equivalence principle holds locally.”

'Locally' is not a qualifier but a number, and it has two arguments. The box that stays undetectably flat has a size 2δ/[(GM/r³)t²] fixed by the instrument's resolution δ and by the duration t — so the principle is local in time on the same footing as in space, and with a worse exponent. Patience is more expensive than room.

Tested in The term free fall cannot remove · the equivalence principle reading path

“The three regimes — under-damped, critically damped and over-damped — are three different behaviours.”

They are one solution. Every trace here comes from a single fourth-order integration of ẍ + 2ζω₀ẋ + ω₀²x = 0 with no branch in the code at all, and what changes across ζ = 1 is the discriminant of the characteristic equation rather than anything in the physics. Nothing in the motion is discontinuous there: the number of zero crossings in a finite time falls to zero continuously, and the trace at ζ = 0.999 is indistinguishable from the trace at ζ = 1.001.

Tested in The three ways of coming to rest · the pendulum reading path

“The energy goes into the magnetic field.”

The field is the same at the end as at the beginning, so it holds the same energy. Every joule of the released gravitational energy appears as resistive heating in the wall, and the accounting closes exactly: with the magnet at terminal speed the rate of work done by gravity equals the rate of dissipation in the tube, since the kinetic energy is not changing. For the drop drawn here that is 118 millijoules, deposited in a few centimetres of copper travelling with the magnet.

Tested in The magnet that falls slowly · the induction reading path

“The residual colour error of a doublet is a defect that better manufacturing would remove.”

It is a property of the available glasses rather than of the workshop. Cancelling the second derivative as well needs a third condition and therefore a third glass with an anomalous partial dispersion, which is what an apochromat is and why it costs what it does. In the meantime the residual can be made irrelevant without being reduced, by stopping the lens down against the floor diffraction puts under any aperture: the depth of focus grows as the square of the focal ratio while the residual grows only in proportion to focal length, and the two cross at an aperture of about 12 mm for the lens drawn here.

Tested in Two glasses that cancel a derivative · the dispersion reading path

“Light scattered at ninety degrees is completely polarised.”

True only in the Rayleigh limit, where a dipole cannot radiate along its own axis. The full computation shows the polarisation at ninety degrees falling to a half by a size parameter of about 1.5 and oscillating about zero thereafter, so a droplet's ninety-degree scattering is essentially unpolarised and can even be polarised the other way. A polarising filter turned against the blue sky darkens it markedly; turned against a cloud in the same part of the sky it does almost nothing, and that difference is a measurement of particle size — a direction of shaking read as a length.

Tested in When the particle is the size of the wave · the scattering reading path

“Adding more layers to a quarter-wave stack makes its reflection band wider.”

It makes it deeper and its edges sharper, not wider. The half-height width of a titanium-dioxide and magnesium-fluoride stack measured on its own computed reflectance runs 0.517, 0.475 and 0.394 of the design wavelength at two, four and eight pairs, narrowing toward the 0.327 the index contrast alone predicts for an infinite stack. What sets the width is the ratio of the two indices; what the number of pairs sets is how close to unity the peak gets.

Tested in The layer that makes a reflection vanish · the impedance reading path

“Anomalous dispersion is a rare curiosity of exotic media.”

It happens wherever a medium absorbs, which is everywhere. The index rises with frequency between absorption lines and falls across each one, and the falls are what Kramers and Kronig showed must accompany any absorption at all. Ordinary water is anomalously dispersive in the infrared, and glass is in the ultraviolet. What is rare is not the anomaly but a place where it can be exploited without the accompanying absorption destroying the pulse — which is why the experiment is done with gain lines rather than absorption lines.

Tested in The speed that carries no signal · the wave packets reading path

“Dividing the count of arrangements by N! is a mathematical fix for making the entropy extensive.”

It is a physical claim with a measurable consequence, and it was made forty years before there was any justification for it. Without the division, entropy per particle grows as ln N: two identical flasks joined would have more than twice the entropy of one, and opening a tap between them would produce entropy from nothing. With it, the entropy per particle is flat to three parts in a hundred over a factor of forty in sample size, the residual being the Stirling correction ln(2πN)/2N. The vapour pressure of a solid depends on the absolute entropy of its gas, and the measured values agree with the divided count and not the undivided one.

Tested in Mixing what is already mixed · the entropy reading path

“The equal-area rule is a graphical trick for locating the tie line.”

It is the statement that the two phases have the same Gibbs energy, written as an area. Along an isotherm dg = v dp, so the difference in Gibbs energy between the two ends of the tie line is the integral of v dp round the loop, and setting it to zero is exactly the equal-area condition. The construction is a thermodynamic equilibrium condition drawn geometrically, and the same argument would locate the tie line for any equation of state whatever.

Tested in The part of the curve no fluid follows · the phase change reading path

“The uncertainty relation gives a lower bound on the ground-state energy, so the true value must be somewhat higher.”

For the harmonic oscillator the bound is attained exactly. Minimising p²/2m + ½kx² subject to ΔxΔp ≥ ħ/2 gives ħω/2, which is the exact ground-state energy — because the true ground state is a Gaussian and a Gaussian is the one shape that saturates the uncertainty relation. For other wells the argument gives an estimate rather than the answer: the same reasoning applied to hydrogen gives −13.6 eV as well, and applied to a linear potential it is out by a few per cent.

Tested in The motion that cannot be stopped · the uncertainty reading path

“Iron is the most tightly bound nucleus, which is why stellar fusion stops there.”

Nickel-62 is the most tightly bound per nucleon at 8.795 MeV, marginally above iron-56 at 8.790, and the semi-empirical formula evaluated at its best proton number puts the peak at mass number 58. That stellar burning ends near iron rather than nickel is a statement about which reactions are available at the temperatures reached, not about the curve. The peak's position and the endpoint of stellar nucleosynthesis are two different facts that agree approximately and are often stated as one.

Tested in The mass that is missing · the Mass-energy reading path

“Nothing can travel faster than light.”

Plenty of things can, and none of them carries information. The intersection point of two closing scissor blades, the spot of a searchlight sweeping a distant wall, the phase velocity of light in a waveguide below cutoff and the peak of a pulse in an amplifying medium all exceed c and none is a signal. The correct statement restricts what can be *sent*, not what can *move*: no influence propagates outside the light cone, and everything else is a pattern whose parts were arranged in advance.

Tested in Which came first, and who decides · the simultaneity reading path

“Archimedes' principle gives the buoyant force, so the problem is solved once the displaced volume is known.”

The principle is exactly right and says nothing about stability, because it is a statement at one depth rather than a statement about a derivative. Everything in this essay follows from differentiating it: the force is ρgV(z) − W, the weight does not depend on z, and the whole question is the sign of dV/dz. A principle that gives a force correctly still has to be differentiated before it says whether an equilibrium holds.

Tested in The depth past which it must sink · the buoyancy reading path

“Marangoni flows are a laboratory curiosity.”

They set the shape of every drying drop, they are why a soap film drains in the pattern it does, and they are the reason welding pools change shape when the sulphur content of the steel changes by a few dozen parts per million — because sulphur reverses the sign of the temperature coefficient, turning an outward surface flow into an inward one and doubling the penetration of the weld. Every process with a free surface and a temperature or composition difference has one.

Tested in The surface that pulls toward the stronger side · the surface tension reading path

“A distant observer sees the falling body freeze at the horizon, so it never actually crosses.”

Both statements are true in their own coordinates and neither is the whole account. In the falling clock's own time the crossing happens at a definite finite moment and the centre is reached shortly after; in the distant observer's coordinate time it takes for ever, while the light received redshifts exponentially with an e-folding time of about 2 × 10⁻⁵ seconds for a ten-solar-mass hole — so the image fades to nothing in well under a millisecond rather than lingering. What 'never crosses' describes is a coordinate, and the last photon is emitted at a finite proper time.

Tested in The horizon that nothing marks · the horizons reading path

“An orbit closes when its apsidal angle is a rational fraction of a full turn.”

That closes one orbit, not all of them. The apsidal angle generally depends on the eccentricity as well as on the force law, so a rational value at one amplitude becomes irrational at another: measured on integrated orbits, the angle drifts by 1.7 to 3.7 degrees across the range of eccentricities drawn for the exponents that do not close. Bertrand's theorem is about the laws whose angle does not move, and only two do.

Tested in The orbit that does not come back to itself · the orbit stability reading path

“A skater spins faster with her arms in because angular momentum is conserved.”

The conservation law fixes the rate and says nothing about the energy, which rises by the same factor — 4.33 in the figures here, from 24 joules to 104. A statement that accounts for the rate and leaves a fourfold increase in energy unexplained has answered half the question. Integrating the force needed to hold each arm on its circle, over the distance it is pulled in, gives exactly the increase, to seven decimal places in the drawn figure.

Tested in The quantity that survives a change of shape · the rotation reading path

“The precession rate is Mgl divided by Iω.”

That is one root of a quadratic that has two. Setting the tilt's second derivative to zero gives I₁cos θ φ̇² − I₃ω₃φ̇ + Mgl = 0, whose roots are a slow precession — of which Mgl/I₃ω₃ is the leading term — and a fast one that rises in proportion to the spin. The figures draw both, and the discriminant gives a third statement the usual formula cannot: below 1245 rpm at 30° for the top drawn here, neither root is real and no steady precession exists at any rate.

Tested in The push that comes out sideways · the rotation reading path

“A released top precesses steadily.”

Released from rest it does not: it falls first, and the fall is what generates the sideways motion. The path traced by the axis comes to a cusp each time it returns to the starting tilt, because at that instant the precession rate is momentarily zero. Steady precession is the one initial condition in which the axis is already moving at exactly the right rate, and it has to be arranged by hand.

Tested in The push that comes out sideways · the rotation reading path

“Diffraction is what happens when light goes through a small aperture.”

The same aperture diffracts differently at different distances, and the parameter is the Fresnel number — the number of half-period zones the aperture holds. The figures draw one slit 0.7 mm wide at four screen distances: at N = 12 the pattern fills the geometrical opening and is covered in ripples, and at N = 0.25 it is a single broad lobe with the aperture no longer recognisable. Nothing about the aperture changed.

Tested in The spiral that says how much light arrives · the huygens reading path

“Adding an absorber removes the resonance.”

It replaces one resonance with two, at 0.854 and 1.171 times the original frequency for a ten per cent absorber, and in an undamped system both are infinite. The device converts a machine that is dangerous at one frequency into a machine that is dangerous at two and safe between them, which is an improvement only if the drive stays where it was.

Tested in The mass that makes another stand still · the resonance reading path

“The extraordinary index is a property of the crystal.”

It is a property of the crystal and the direction. n(θ) runs from the ordinary value along the optic axis — where the two waves are identical and the crystal behaves exactly like glass — to its extreme value at right angles, following an ellipse. Quoting one number for it is quoting the endpoint of a curve.

Tested in The crystal that answers twice · the polarisation reading path

“The evanescent field reaches about a wavelength past the boundary.”

It reaches a wavelength in the middle of the range and diverges at the critical angle, where the exponent √(n₁²sin²θ − n₂²) goes to zero. For 550 nm light from glass into air the 1/e depth is 1017 nm one fifth of a degree past critical, 248 nm at 45°, and 83 nm at 75°. There is no single answer to how far, and the range over which it changes by a factor of ten is under four degrees.

Tested in The reflection that happens where the glass is not · the total internal reflection reading path

“An emf is produced by a changing magnetic flux.”

Two different mechanisms produce it. Charges dragged sideways through a field feel qv × B, which acts only on charge that is moving; charges at rest in a changing field feel a genuine electric field, which acts on charge that is not moving. The figures compute the two contributions separately for a loop that is both moving and in a changing field, and neither of them is the emf — their sum is, and it matches a numerical derivative of the flux to a fraction of a nanovolt.

Tested in The rule that is two laws wearing one coat · the induction reading path

“Viscosity is friction between layers of fluid.”

In a gas there is no contact between layers to have friction at. What crosses the surface between two layers is molecules, and what they carry is momentum: a molecule from the fast layer arriving in the slow one speeds it up. The prediction that follows is the density independence, which a friction picture cannot produce and does not survive.

Tested in The viscosity that does not care how much gas there is · the kinetic theory reading path

“The electrons in a metal carry a heat capacity of 3/2 k each, by equipartition.”

They carry about a hundredth of it — 1.2 per cent for copper at room temperature — and the shortfall is not a small correction but a different functional form: the electronic heat capacity is proportional to temperature rather than constant. Equipartition assumes every degree of freedom can take up energy, and an electron deep in the sea cannot, because every state it could be promoted to is occupied. This discrepancy stood over the classical theory of metals from 1900 to 1927.

Tested in The pressure that is not a temperature · the exclusion reading path

“A sphere moving at close to light speed is flattened into an ellipsoid.”

Its outline photographs as a circle of its proper radius at every speed, because a rotated sphere is the same sphere. The figure traces the markings on it and shows them turned while the outline never moves. In the simultaneous-positions sense the sphere really is an oblate ellipsoid, and no camera at any exposure can record that shape.

Tested in The contraction no photograph shows · the length contraction reading path

“The spin–orbit splitting is the energy of the electron's magnetic moment in the magnetic field it sees in its own frame.”

That calculation gives twice the measured splitting. The frame it is done in is being carried round the nucleus, and a frame carried round a closed path in velocity space comes back rotated — by 2π(γ − 1) per orbit, which is 34.5 arcseconds for hydrogen's ground state. Subtracting that precession halves the answer and makes it right, and it is not a new interaction but a statement about how the calculation's own frame behaves.

Tested in The turn that two pushes leave behind · the velocity addition reading path

“The ring forms because the drop is thinnest at the edge and so dries there first.”

The thinness is not what decides it. The evaporation rate per unit area is set by how vapour diffuses away from a lens-shaped object, which is the same boundary-value problem as the electrostatic field around a charged lens, and it *diverges* at the contact line for any contact angle below ninety degrees. Above ninety it is smallest at the edge and the flow reverses, and a drop above ninety degrees is still thinnest at its rim.

Tested in The ring the drop leaves behind · the surface tension reading path

“The maximum height of a siphon is 10.3 metres.”

It is (p_ambient − p_vapour)/ρg, which is 10.11 m for water at 20 °C, 5.65 m at 80 °C, 12.32 m for ethanol and 0.76 m for mercury. Every one of those is a property of the liquid and the ambient pressure, and none is a property of the tube. Quoting a single number for it hides both dependences.

Tested in The height a siphon cannot pass · the hydrostatics reading path

“The Shapiro delay is a test of general relativity like the deflection is.”

It is a test of a different part of it. The deflection is set by the sum of the metric's time and space curvature terms; the delay, to leading order, is set by the time term alone. A theory could get the deflection right and the delay wrong. The parametrised comparison of theories was built for exactly this reason, and the delay is currently the sharpest constraint on it — measured by the Cassini spacecraft to about two parts in a hundred thousand.

Tested in The delay that is not a bend · the light deflection reading path

“Radiation pressure blows small grains away and gravity holds large ones.”

Only between two sizes. The ratio β = 3LQ/16πGMcρa rises as grains are made smaller only while the efficiency Q stays near one, and Q falls as the fourth power of the size below the wavelength of the light. So β turns over — for silicate grains in sunlight it peaks at 1.87 near 115 nm — and the blow-out band runs from 48 nm to 574 nm, with grains on both sides of it staying.

Tested in The size the light cannot blow away · the radiation pressure reading path

“A grain that is not blown away stays in its orbit.”

It spirals in. In the grain's own frame the starlight arrives from slightly ahead of the star — aberration by v/c — so it delivers momentum against the motion, and the grain re-emits that energy isotropically in its own frame and does not get the momentum back. The resulting drag takes a 10 μm grain from 1 AU into the Sun in 13,900 years, which is short compared with anything astronomical.

Tested in The size the light cannot blow away · the radiation pressure reading path

“Stick-slip happens because static friction is larger than kinetic friction.”

That gap is real and it is not the mechanism. A model with two coefficients and nothing else contains no length and no stiffness, so it predicts a sawtooth at every stiffness a spring can have — and the figures here show a real contact sliding silently the moment its holder passes a computed threshold, with every friction parameter unchanged. What is missing from the two-coefficient account is how far the surface has to move to lose its grip.

Tested in The chatter a stiffer holder removes · the friction reading path

“Energy is conserved, so the books balance.”

They balance in each frame separately, and the entries differ between frames. A rifle firing a bullet gives the bullet almost all the kinetic energy in the frame of the ground and gives the recoiling shooter most of it in a frame moving with the bullet. Both accounts are correct and neither is a correction of the other; what is common to them is the energy the propellant released, which is the only figure the chemist could have supplied.

Tested in The energy that depends on the observer · the energy reading path

“Below cutoff the wave is reflected.”

It is reflected in the sense that almost no energy gets through, and the mechanism is not reflection at a surface. The field enters the guide and decays exponentially along it, over a length that diverges as the cutoff is approached from below. That is the same solution — a real exponential where a travelling wave would have an imaginary one — that appears past a critical angle and inside a quantum barrier, and it is the reason a short enough length of undersized guide passes a signal it has no travelling mode for.

Tested in The pipe that will not carry a low note · the guided waves reading path

“Fermat's principle explains why light reflects at equal angles.”

It picks out the equal-angle point and says nothing about why light should care. The reason lives one level down, in the sum over paths: neighbouring routes that differ in length by much more than a wavelength arrive with scrambled phases and cancel, and only near a stationary point do a band of them agree well enough to survive. Which is why the criterion is stationarity — a first derivative vanishing is exactly the condition for a band of paths to share a phase, and the sign of the second derivative does not enter it.

Tested in The path that takes the longest time · the fermat reading path

“Quantum tunnelling lets particles pass through barriers.”

It lets a fraction pass, and the fraction is what the subject is about. For a 5 MeV alpha in a heavy nucleus the escape probability per attempt is about 10⁻³⁰, and the nucleus makes 10²¹ attempts a second — which gives a lifetime of years rather than the immediate escape the word suggests. The interesting content of tunnelling is always the exponent, and the exponent is an integral over the shape of the barrier rather than a property of its height.

Tested in A wall that a factor of two makes impassable · the tunnelling reading path

“Polarisation is the material's response to the field.”

It is the material's response to the field *inside*, which the polarisation itself has changed. That circularity is the whole of the subject: the bound charge appearing on the surface makes a field opposing the polarisation, which reduces the polarisation, which reduces the opposing field. Solving it is a single line for an ellipsoid and is not solvable in closed form for anything else, because only an ellipsoid polarises uniformly.

Tested in The field the matter takes away · the dielectrics reading path

“Hysteresis happens because the domains get stuck.”

Pinning is one mechanism among several and is not needed for the effect. The figures compute the loop of a single particle with no domain walls in it at all, purely from following a local minimum of its energy as the field is swept — and it is square, with a remanence and a coercivity. What produces hysteresis is a state that persists after it has stopped being the lowest one, and any system with more than one minimum has that available.

Tested in The magnet that has to fight its own field · the magnetisation reading path

“The harmonic approximation is good enough for a solid at ordinary temperatures.”

It is good enough for the frequencies and useless for everything else. It gives the vibrational spectrum, the heat capacity and the speed of sound to a few per cent, and it gives an expansion coefficient of exactly zero and a thermal conductivity of exactly infinity — both of them qualitatively wrong rather than numerically off. Any property that requires modes to be coupled requires the term the approximation drops.

Tested in Why heating a perfect spring changes nothing · the harmonic approximation reading path

“Critical opalescence happens because the fluid becomes cloudy.”

The fluid remains a single homogeneous substance with no droplets in it. What grows is the *correlation length* — the distance over which a density fluctuation extends — and when it reaches the wavelength of light divided by 2π the fluid scatters strongly at every angle. The figures put that crossing at twenty millikelvin from carbon dioxide's critical temperature, which is why the effect appears to switch on rather than to develop.

Tested in The point at which the two become one · the phase change reading path

“A steadily accelerating rocket eventually outruns any signal sent after it.”

It does, and calling that outrunning misdescribes it. A body under constant proper acceleration has a worldline whose asymptote is a light ray, so signals emitted from its starting point after a certain moment never catch it — not because the rocket is faster than they are, but because it started early enough. The figures draw that asymptote and the signals on either side of it, and the boundary is a horizon produced by acceleration alone, with no mass anywhere near it.

Tested in The push that does not point where the body goes · the relativistic dynamics reading path

“Electric and magnetic fields are two aspects of one field, so the distinction is arbitrary.”

The split is observer-dependent and the classification is not. Two numbers built from the fields — E² − c²B² and E·B — take the same value for every observer, and between them they decide which of three kinds a field is: one that can be made purely magnetic, one that can be made purely electric, and one that can be made neither. A light wave is permanently in the third class, and no boost in any direction moves it out.

Tested in The field nobody can transform away · the field transformation reading path

“Whether something floats is decided by comparing its density to the fluid's.”

That comparison decides whether a fully wetted body rises, and it does not decide whether an object floats. A steel needle denser than water rests on it, held by surface tension, and a paraffin-coated block on a wet floor stays down. Density settles the question only when the surface has no say — which is most engineering cases and almost none of the interesting ones.

Tested in The block the water does not lift · the buoyancy reading path

“Viscosity measures how runny a fluid is.”

For a Newtonian fluid it does. For every other kind the number an instrument reports depends on how fast it was stirred, and for a viscoelastic one it depends on how long the measurement lasted as well. The figures compute the apparent viscosity of four fluids across a decade of shear rate and it varies by more than a factor of ten for three of them — so a single quoted viscosity is a quoted measurement condition as much as a material property.

Tested in The liquid that remembers · the rheology reading path

“A gravitational-wave signal measures the masses of the two bodies.”

The inspiral measures one combination of them and not the two. The rate at which the frequency sweeps upward depends only on the chirp mass, a particular weighted product, so a pair of 10 and 40 solar masses and a pair of 20 and 20 produce nearly identical sweeps. Separating the two masses requires the later part of the signal, where post-Newtonian corrections enter and the degeneracy is broken — which is why published mass estimates have much larger uncertainties than the chirp mass does.

Tested in The orbit that has to shrink · the gravitational waves reading path

“Gravitational contraction could power the Sun.”

For nine million years, which is the number Kelvin computed and it is 480 times too small. The figures divide the energy given up in reaching a given radius by the rate of radiation and place the Sun's Kelvin–Helmholtz time at 9.4 million years against an Earth 4.5 billion years old. Nothing in that arithmetic was wrong; a term was missing, and no amount of care with the terms present could have found it.

Tested in The ball of gas that heats up as it cools · the Self-gravity reading path

“The load in a packing is distributed among the grains.”

It is distributed, and the distribution is the interesting part. A model in which every grain weighs the same and passes its load to two neighbours in a ratio drawn at random produces chains carrying several grains' worth past their neighbours and quiet regions between them, with the heaviest grain in the bottom row of a twelve-row packing at nearly twice the mean and the lightest at a seventh of it. Nothing in the rule is heterogeneous; the heterogeneity is what a random split does when it is iterated.

Tested in The silo that does not weigh what it holds · the granular matter reading path

“Reverse osmosis pushes water through a filter fine enough to stop salt.”

A filter has a pore size and a pressure drop that scales with flow; this has a threshold. Below 27 bar for seawater nothing crosses at all, in either direction — the flux is not small, it is absent — and the threshold is thermodynamic rather than geometric, being the same osmotic pressure the solution would generate on its own. The figures locate that intercept on the plotted flux line, and no improvement in the membrane will move it.

Tested in The pressure that comes from counting · the osmosis reading path

“A pendulum measures g because its period is 2π√(L/g).”

That formula is for a point mass on a weightless string, which no apparatus is. A real bar has its mass spread out, and its period is 2π√((k² + h²)/gh) with k the radius of gyration about the centre of mass and h the distance from that centre to the pivot. Both of those are properties of the mass distribution, both are hard to measure to five figures, and the figures show that the naive formula and the true one differ by tens of per cent for any real bar.

Tested in The length nobody has to measure · the pendulum reading path

“The gap comes from the difference between the two materials.”

The materials set the width and the periodicity sets the position, and only the second is structural. The figures fold a uniform medium's dispersion into an imagined unit cell and the branches cross at the zone centre and boundary; switching on a twelve per cent index contrast opens gaps at those crossings and moves nothing else perceptibly. A weak modulation does nothing whatever except where two travelling waves already shared a frequency.

Tested in The gap a repeat opens · the periodic media reading path

“A waveguide has a cutoff frequency below which nothing propagates.”

A hollow pipe does, because its walls impose a boundary condition that a long wavelength cannot satisfy. An index-guided channel does not: the transverse problem is a potential well, a symmetric one-dimensional well always has at least one bound state however shallow it is, and the fundamental mode of a fibre is guided at every frequency. The figures show the lowest mode's transverse profile spreading into the cladding as the contrast falls, and never unbinding.

Tested in The channel with no walls · the guided waves reading path

“Concentrating light means collecting a large area onto a small one.”

It means trading area against angle, and the product is what is conserved. The figures take a beam of half-angle 12° through an aperture three times smaller and the exit half-angle is 38.6°, not 12° — the area and the sine of the angle multiply to the same number at both ends. A concentrator that shrank the area without opening the angle would be increasing the radiance, which is the thing no passive system can do.

Tested in The brightness no lens can increase · the etendue reading path

“An isolator is a polariser that only lets light through one way.”

A polariser is perfectly symmetric — it passes the same component in both directions — and no combination of polarisers, waveplates, lenses and mirrors can pass light one way and stop it the other, because every one of them looks identical run backwards. The device needs a component that does not, and the only one available in ordinary optics is a magnetic field. The figures show the transmission forwards and backwards against the rotator's angle, and the backward leak is exactly zero at 45 degrees.

Tested in The rotation a return trip doubles · the polarisation reading path

“A 360° rotation returns a spin-½ to its original state, up to an unobservable sign.”

The sign is unobservable on the whole state and perfectly observable on half of one. Split a neutron beam, rotate the spin in one arm only, and recombine: the intensity is ½(1 + cos θ/2), at its minimum after one full turn and back to full brightness after two. The period is 720 degrees, the experiment was done, and the measured period is 720 degrees.

Tested in The angular momentum that is not a rotation · the spin reading path

“The interaction-free measurement works one time in four, and that is the limit.”

That is the limit of asking the whole question at once. Ask it in N gentle pieces, rotating the state by π/2N each time, and the chance of success is cos^2N(π/2N), which the figures plot: 53 per cent at four interrogations, 96 per cent at sixty-four, with the shortfall falling as π²/4N. There is no ceiling in the physics; the quarter was the price of impatience.

Tested in The measurement that never touched it · the measurement reading path

“An accelerating charge radiates because it is accelerating.”

True and silent on the mechanism. What radiates is the *join*: outside a sphere of radius ct the field still points at where the charge would have been had nothing changed, inside it points at where the charge is, and the two must be connected because a field line cannot stop in empty space. The connection is transverse and falls as 1/r rather than 1/r², and the figures measure its ratio to the radial field as a·r·sinθ/c² — from which Larmor's formula follows with no wave equation used anywhere.

Tested in The field that points where the charge is now · the retardation reading path

“The field is a convenient way of bookkeeping the forces between distant charges.”

A bookkeeping device does not have a conserved quantity of its own that can be handed to matter on demand. Here the field holds angular momentum in a completely static configuration and gives it up when the configuration changes, with the transfer traceable instant by instant. The same argument runs for energy and for linear momentum, and in each case the alternative — action at a distance with a delay — fails to conserve the quantity during the delay.

Tested in The angular momentum that is in nothing at all · the field energy reading path

“Entropy is a measure of disorder.”

It is a count of the microscopic arrangements consistent with what is known, which is why it depends on what is known. Szilard's engine is the sharpest demonstration: the gas is identical before and after the measurement, and its entropy falls by k·ln2 because an observer has learnt which half the molecule is in. Nothing about the molecule's arrangement changed. The figures compute the work that reduction makes available, and it is entirely real.

Tested in The bit that has to be paid for · the entropy reading path

“The second law limits how good an engine can be.”

It limits the work per unit of heat and says nothing whatever about the rate. Every quantity in Carnot's argument is a state function and no time appears in it, so the bound holds equally for an engine that completes a cycle in a millisecond and one that takes a century. What limits a real plant is a different optimisation — the most power rather than the most work — and it produces 1 − √(Tc/Th), a bound the second law does not mention.

Tested in The engine that has to finish · the heat engines reading path

“A rigid rod moved from rest to speed v ends up contracted by γ.”

It does, and only if it is moved in a particular way. Holding the proper length fixed — Born's definition of rigidity — requires the trailing end to accelerate harder than the leading end, by the ratio of their distances from the horizon the acceleration creates, and to go on doing so for ever. Give both ends the same acceleration instead and the proper length grows as γ. The two programmes differ by γ², and a rod made of matter follows the first only because its own elasticity supplies the difference.

Tested in The string that breaks between two rockets · the length contraction reading path

“Lensing magnification can be arbitrarily large.”

In the point-mass model it diverges at exact alignment, and the divergence is the model's rather than the world's. Every real source has a finite angular size, so the magnification is averaged over its face and comes out large and finite; a real lens has structure, which breaks the perfect symmetry the divergence needs. The figures show the total brightness running away as the alignment closes, and the divergence is the honest signal that the idealisation has been carried past its range.

Tested in The lens with no focal length · the light deflection reading path

“A measured Curie law confirms that the moments are independent, and the constant gives their size.”

It gives J(J+1) and no more. The figures separate the four Brillouin curves by their initial slope (J+1)/3J alone: a spin-half moment is three times as responsive per unit saturation as the classical dipole of the same size, and every curve saturates at one. Distinguishing them needs the bend, which needs a field over the temperature that a room-temperature magnet never reaches.

Tested in The magnetism classical physics forbids · the magnetisation reading path

“Absorption and dispersion are two properties of a material, related by the fact that both come from the same resonances.”

They are one property. The Kramers–Kronig reconstruction here computes ε′ at nine frequencies from the absorption alone — no resonant frequency, no damping, no oscillator strength — and lands on the directly computed curve to a part in ten million. Knowing where a substance absorbs fixes how it refracts everywhere, including where it is perfectly transparent, and the reason is causality rather than a shared origin.

Tested in The constant that depends on how fast it is asked · the dielectrics reading path

“The fine-structure constant measures the strength of the electromagnetic interaction.”

It does, and the statement leaves out what the number is doing in this calculation. Here α is the speed of the innermost electron in units of c, so (Zα)² is how relativistic the atom is — and the figure of (Zα)² against nuclear charge shows the same quantity passing a half for hydrogen-like uranium, where the splitting stops being a correction and the atom it corrects stops being a starting point.

Tested in The line that is really two · the atomic spectra reading path

“Equipartition fails at low temperature because quantum mechanics freezes modes out.”

That is one of its two failures and the better known one. The other has no quantum mechanics in it at all: equipartition also fails whenever a term in the energy is not quadratic, which is a purely classical fact and is why the figure of kT/n against n is drawn with no ħ anywhere on it. A classical gas of ultrarelativistic particles obeys equipartition exactly and gets 3kT per particle.

Tested in The share that is not half a kT · the equipartition reading path

“Kirchhoff's law says a good absorber is a good emitter.”

It says a body's emissivity equals its absorptivity at each wavelength, each direction and each polarisation separately, and the totals only follow when the spectra overlap. A selective solar absorber has absorptance 0.95 at half a micrometre and emittance 0.05 at ten, breaks no law, and reaches 761 K in sunlight where a grey surface reaches 364 K. The stronger, spectral, form is the one that does the work.

Tested in The glow that says nothing about the surface · the blackbody reading path

“The quantum of circulation is h over the mass of a helium atom, so it is a property of helium.”

It is a property of whatever carries the phase. In helium-4 that is a single atom and the quantum is h/m; in helium-3 and in every superconductor it is a pair, and the quantum is h/2m. The factor of two is a measurement of how many particles are bound into the object whose phase is winding, and it is how pairing was confirmed in both cases.

Tested in The whirlpool that comes in one size · the superfluidity reading path

“General relativity is needed to predict the gravitational redshift.”

The redshift follows from the equivalence principle and energy conservation alone, and this collection derives it that way at the rung below. What needs more is the converse: the argument here runs the implication backwards, from a measured redshift to the impossibility of a flat spacetime, and it establishes that some curved geometry is required without saying which one. Choosing among the curved geometries is what a field equation is for.

Tested in The parallelogram that will not close · the gravitational redshift reading path

“Magnetic reconnection converts magnetic energy into heat and motion.”

It does, and the sentence omits the only difficult part. The conversion needs the frozen-in condition to fail, and it fails only in a layer of thickness L over the square root of the Lundquist number — three metres inside a flare ten thousand kilometres across. What decides the rate is not how much energy is available but how fast plasma can be pushed through a sheet of negligible volume, and the answer computed here is five orders of magnitude too slow.

Tested in The knot the field cannot untie · the flux freezing reading path

“The self-image happens because the light from adjacent slits arrives in step at that distance.”

That is the two-slit account and it gives the wrong distance. The revival is a property of the whole spectrum at once: propagating a distance z multiplies the m-th order by exp(−iπλzm²/d²), and what makes the pattern reassemble is that every one of those phases is a multiple of 2π together, for every m, at z = 2d²/λ. Two orders would revive at four times as often as they do.

Tested in The grating that photographs itself · the diffraction reading path

“Staging helps because a smaller rocket is easier to build.”

It helps because a spent stage is dead mass carried by everything that follows, and dropping it changes the mass ratio of what is left — which is the only quantity the logarithm contains. The figure computes payload fraction against stage count for a fixed velocity budget and finds one stage impossible at any size whatever, two giving 2.13 per cent and three giving 2.76.

Tested in The push that needs nothing to push against · the momentum reading path

“The defect state sits inside the gap because the defect layer has a different resonant frequency.”

It sits inside the gap because the surrounding stack is the only thing preventing it from radiating away, and its own frequency has to be one the stack forbids. The figures show the field decaying away from the defect at 0.693 nepers per cell, which is the attenuation the perfect crystal's own band structure gives at that frequency — the same evanescent wave, measured two ways, and the reason a crystal with no gap can trap nothing.

Tested in The mode that lives in the mistake · the periodic media reading path

“A steady electric field accelerates a band electron, which is why a metal conducts.”

A steady field makes the electron *oscillate*. The integrated trajectory at 10 MV/m turns round after 0.69 ps and returns to its starting point 1.379 ps after release, over an amplitude of 180 nm, because the force acts on k and the band is periodic in k. Conduction exists because that orbit is interrupted: a collision every 10 femtoseconds arrives 138 times before one Bloch period is complete, and a metal has a resistance instead of an oscillation.

Tested in The mass a curve decides · the bands reading path

“Radioactive decay is exponential, which is what it means for a nucleus to have no memory.”

Exponential decay requires an energy distribution running over the whole real line, and no bound system has one. Transforming a resonance's own spectrum gives a survival probability that is flat below Γτ_z² — 1.57 × 10⁻² lifetimes for the case drawn — and an inverse square beyond 23 lifetimes, with the exponential in between good to a part in ten thousand.

Tested in The exponential that is only true in the middle · the decay reading path

“A solenoid confines its field, which is why it is used where a magnet must not disturb its surroundings.”

A solenoid with no external field would have no magnetic moment, and a coil with no moment cannot be a magnet. Beyond forty solenoid lengths the summed field agrees with a point dipole of moment NIπR² to 0.08 per cent and falls as the inverse cube with a fitted exponent of −3.001. The confinement is a matter of degree, and the degree is what the aspect ratio buys.

Tested in The field outside the solenoid, which is not zero · the ampere law reading path

“Air is blue because each molecule scatters short wavelengths more strongly, and there are a lot of molecules.”

N molecules with random phases scatter N times one molecule's worth; N molecules in phase scatter N². Both happen at once, in the same gas, to the same light: the sideways sum is incoherent and gives Rayleigh's answer, and the forward sum is coherent and gives the refractive index. Air's n − 1 = 2.78 × 10⁻⁴ fixes a polarisability that returns a 550 nm optical depth of 0.0877 against a published 0.0973, with no scattering measurement used anywhere.

Tested in Why a litre of water is not blue for the reason the sky is · the scattering reading path

“Adiabatic demagnetisation gets colder every time it is repeated, so enough repetitions get arbitrarily close.”

In the model, yes: at a field ratio of four, five stages take 1 K to 0.98 mK. In a real salt the spins make a field of a few millitesla for each other, and below μb/k they are ordered by it whatever the coil is doing — 3.4 mK for 5 mT — so the entropy is already spent and there is nothing left to pump. The refrigerant runs out before the arithmetic does, which is why a multi-stage machine changes salt rather than repeating with one.

Tested in The staircase that never reaches the floor · the third law reading path

“An accelerating charge radiates energy at the Larmor rate, and that is the whole of the classical theory of radiation.”

Larmor's formula is a statement about the field far away and contains no force on the charge at all. Requiring the charge to pay for what leaves gives F = mτȧ with τ = μ₀q²/6πmc = 6.27 × 10⁻²⁴ s, and that equation has solutions Larmor's does not: with no applied force at all, an acceleration growing by thirteen decades in 2 × 10⁻²² seconds.

Tested in The force a charge exerts on itself · the radiating charge reading path

“Reflection happens at the surface.”

It happens over a thickness c/ωp, which is 12.5 nm for aluminium and 5.3 m for the F2 layer, and diverges as the cutoff is approached — 17 m at 0.95 of the plasma frequency for the ionosphere. The field enters as an evanescent tail, stores energy in that layer and returns every joule of it, and the layer is a property of the medium rather than of the wavelength.

Tested in The frequency below which nothing gets in · the plasma oscillation reading path

“A distant sound is a quiet sound — the difference is the level.”

Spreading takes every frequency alike and absorption does not. Over a kilometre of air at 20 °C the loss to absorption is 3.7 dB at 1 kHz and 117 dB at 8 kHz; over ten kilometres, 37 dB and 1,170. A flat broadband source ten kilometres away has nothing left above about 500 Hz, so what arrives is not the same sound made smaller.

Tested in The distance that takes the treble out · the attenuation reading path

“The cone angle depends on the shape of the aircraft and the strength of the shock.”

The envelope of the wavefronts has sin θ = 1/M and nothing else in it — no pressure, no density, no size, no shape. Measuring the tangent from the apex to each of six constructed wavefronts at one speed gives the same angle to a part in 10¹², because the ratio of a circle's radius to its distance from the apex is c/v for every one of them. A finite-strength shock is slightly steeper than the Mach angle, and it tends to it as the disturbance weakens.

Tested in The cone the source leaves behind · the doppler reading path

“A coefficient of friction cannot exceed one, because that would mean friction stronger than the force pressing the surfaces together.”

Nothing forbids it. A clean metal pair in vacuum seizes, with an effective coefficient of several; a rubber tyre on dry asphalt runs at 1.0 to 1.5; a gecko's foot holds against gravity with no normal load at all. Each of those is a case where the friction is not shear of load-bearing junctions, so the load has stopped being the quantity it is divided by.

Tested in The grip that is not a coefficient · the friction reading path

“Any system with a nonlinearity in it will be chaotic.”

The same code, the same integrator and the same nudge at 3° instead of 120° return an exponent of 2.1 × 10⁻⁴ per second — four orders of magnitude smaller, and consistent with zero. A double pendulum near its rest position is two coupled normal modes and is as predictable as anything in mechanics. Nonlinearity is necessary and it is a long way from sufficient.

Tested in The error that doubles on a schedule · the chaos reading path

“Liquid water cannot be at a negative pressure, because it would boil.”

It would, if a bubble could form, and forming one costs energy: a nucleus of radius r has a surface energy going as r² and a volume term going as r³, so there is a barrier, and below a critical radius any bubble collapses. At −1 MPa the critical radius is 146 nm and the barrier is enormous; homogeneous nucleation needs around −140 MPa. What sets the real limit is a pre-existing gas pocket, which is why the pores of the conduit walls are the relevant length.

Tested in The column that is pulled, not pushed · the capillarity reading path

“Buoyancy is decided by density, so a parcel denser than its surroundings sinks.”

It is decided by the density a parcel would have if it were moved, which is not the density it has where it is. The whole classification is a comparison of two slopes with no density in it anywhere — the environment's lapse rate against the adiabatic one — and the same cold, dense air is stable under one profile and unstable under another.

Tested in The layer a parcel cannot leave · the stratification reading path

“A fibre gyroscope works because light takes longer to go round the coil the way it is turning.”

It does, and the useful part is what the expression does not contain. The refractive index of the fibre cancels out, so a glass gyroscope and a vacuum one of the same enclosed area read the same; the shape does not appear, so a coil can be wound to any outline; and the axis need not be inside the loop's centre or even inside the loop's plane. What is measured is the enclosed area times the rotation rate, and nothing else about the instrument.

Tested in The ring where the two beams disagree · the simultaneity reading path

“Uniform acceleration means the speed increases at a constant rate.”

It means the acceleration measured on board is constant, which is what an accelerometer reads and what a passenger feels as weight. The speed measured in the launch frame does not rise linearly — it approaches c — while the quantity that does rise linearly with proper time is the rapidity, whose hyperbolic sine and cosine are the two coordinates of the worldline.

Tested in The wall of silence behind a rocket that never stops · the accelerated frames reading path

“Chirikov's overlap criterion gives the point at which a system becomes globally chaotic.”

It gives π²/4 = 2.467 for the standard map and the measured value is 0.971635, so it is too large by a factor of 2.54. The scaling it predicts is right — resonance widths really do grow as the square root of the coupling — and the coefficient is wrong because the criterion counts the two largest resonances and ignores the infinitely many smaller ones that have already eaten most of the gap.

Tested in The last curve to go · the chaos reading path

“The energy released by the falling chain ends up as kinetic energy of the pile.”

None of it does. The chain arrives at rest, so the pile has no kinetic energy at all, and the ledger shows the released potential energy and the dissipated energy agreeing to four decimal places at 0.5 λgL². Every joule is destroyed at the impacts. A chain lowered gently releases the same energy and destroys none of it.

Tested in The pile that lands heavier than it weighs · the momentum reading path

“A resonance curve's width tells you the damping.”

Only if the width is homogeneous. A collection of resonators with slightly different frequencies has a broad response and a long ringdown at the same time, so the two measurements disagree — and the disagreement is the measurement, because its size is what says how much of the width is spread rather than loss.

Tested in The width that is a lifetime · the resonance reading path

“Reflections happen at surfaces.”

Reflections happen where the impedance changes, and how much reflects depends on how fast it changes compared with a wavelength. A boundary a hundred wavelengths thick reflects nothing at all and is still a boundary; a boundary a thousandth of a wavelength thick reflects the full Fresnel amount. What a surface supplies is abruptness, and abruptness is relative to the wave.

Tested in The taper that matches every note · the impedance reading path

“Diffusion is the same process in any number of dimensions, with a different constant.”

The mean square displacement grows as t in every dimension, so on that measure they are identical. The recurrence property is not a matter of a constant at all: it changes from probability one to probability 0.3405 between two dimensions and three, and every rate that depends on two diffusing objects meeting inherits the discontinuity.

Tested in The walk that comes home · the diffusion reading path

“The third law is a statement about entropy.”

Its working content is a statement about heat capacity. Entropy is the integral of C/T from absolute zero upward, and that integral converges only because C goes to zero — for a solid as T³, which the Debye model computes and which is what makes an absolute entropy definable at all. A heat capacity that stayed at 3R would leave every absolute entropy infinite.

Tested in The entropy that is still there at zero · the third law reading path

“The plug in the middle of the pipe shrinks away as the driving is increased.”

It shrinks and never disappears. The stress in a pipe is proportional to the distance from the axis and is therefore exactly zero on the axis, so there is always a core below the yield stress. The computed plug is 67 per cent of the diameter at 1.5 times the threshold and 20 per cent at five times, and the fraction is exactly the reciprocal of the overdrive.

Tested in The paste that holds up its own hill · the rheology reading path

“Driving a fluid harder makes a bigger wave.”

Driving it faster than its own buoyancy frequency makes no wave whatever. There is no angle whose cosine exceeds one, so above N the response is trapped at the source and decays away from it rather than radiating. The generator refuses such a request rather than drawing a beam that does not exist.

Tested in The wave that picks an angle · the stratification reading path

“Aberrations come from the surface not having the right shape.”

The paraboloid has exactly the right shape for the axis and the wrong one for everything else, and no single surface has the right shape for both. Abbe's condition is a constraint on the *mapping* between entrance height and exit angle, which one surface has too few degrees of freedom to satisfy alongside stigmatism — which is why an aplanatic system has two.

Tested in The condition a lens must meet · the imaging reading path

“An interferometer measures the diameter of a star.”

It measures one Fourier component of the brightness distribution per baseline. Three sources drawn here — a uniform disc, an equal double star and a Gaussian — lose their fringes at the same separation and are completely different objects; an observer who measured only the first null would report the same size for all three and be wrong about two of them.

Tested in The fringe that measures a star · the coherence reading path

“Optical fibre is used at 1,550 nanometres because that is where it works best.”

That is where the loss is least, at about 0.2 decibels a kilometre. It is not where the dispersion is least: silica's material dispersion crosses zero at 1,273 nm and a standard fibre's total at 1,310, so the operating wavelength has the lowest loss and 18 ps per nanometre per kilometre of dispersion. The two optima are at different wavelengths and every system design is a negotiation between them.

Tested in The wavelength a fibre does not smear · the dispersion reading path

“Gauge freedom is a mathematical convenience with no physical content.”

It has content wherever a gauge-dependent quantity appears in a physical argument. The canonical momentum mv + qA is what Hamiltonian mechanics conserves and what quantum mechanics turns into a derivative, and it is gauge dependent — the two curves drawn here for one trajectory differ completely. What is gauge invariant is its integral round a closed loop, which is the enclosed flux, and that is why the Aharonov–Bohm phase is observable.

Tested in The potentials that are not unique · the maxwell equations reading path

“A thicker conductor carries alternating current better.”

Past about a skin depth it does not. The exact slab calculation gives the resistance ratio as the half-thickness divided by the skin depth once the slab is thick, so the alternating resistance climbs as the square root of the frequency for ever regardless of how much metal is added. That is why heavy radio-frequency conductors are hollow and why litz wire exists.

Tested in How far a field gets into metal · the conductors reading path

“The infinite-wire formula is a good approximation for laboratory wires.”

It is, and the figure says how good: a wire ten times as long as the distance to the field point gives 98 per cent of the infinite answer, one as long as the distance gives 45, and one half as long gives 24. The approximation is excellent where it is used and collapses quickly, and the crossover is at a length ratio of about five rather than at a length.

Tested in The law that is always true and rarely useful · the ampere law reading path

“Interstellar travel is an engineering problem.”

The mass ratio of a photon rocket — the best any rocket can be — is e raised to the total rapidity, which is 38 times the payload for a round trip to Proxima and 10^18 for one to the galactic centre. No improvement in exhaust velocity is possible past a photon rocket and no improvement in anything else changes an exponent, so the difficulty is arithmetic rather than technological.

Tested in The ship that never arrives at c · the accelerated frames reading path

“The experiment confirmed time dilation.”

It confirmed the sum of two effects with opposite signs, which is a sharper test than either alone: a theory getting one term right and the other wrong fails on the eastward flight even while passing on the westward one. What it did not do is reach high precision — the quoted uncertainties are twenty to thirty per cent — and the modern confirmations come from satellite navigation, where the same sum is corrected for continuously.

Tested in The two clocks that flew in opposite directions · the time dilation reading path

“The Chandrasekhar limit is where the star's gravity overcomes the electron pressure.”

There is no such crossing at a finite radius. The limit is where the equation of state's exponent reaches four-thirds, at which the mass of a self-gravitating polytrope is independent of its radius — so the mass is fixed and the radius drops out. Squeezing a star at that exponent buys no additional support at any size, which is a different statement from a force balance failing.

Tested in The mass no cold matter can hold up · the Self-gravity reading path

“A whistler is a descending tone, so low frequencies travel slowest.”

The group velocity is zero at zero frequency and zero again at the gyrofrequency, with a maximum at a quarter of the gyrofrequency — which is found by searching the drawn function here rather than quoted. So the arrival curve has a nose: everything either side of that peak arrives later. The descending tone is the branch below the nose, and the rising branch above it is observed too.

Tested in The whistle that arrives sorted · the plasma oscillation reading path

“4s fills before 3d because 4s is lower in energy.”

It is lower for the neutral atom at the start of the fourth row and not thereafter. The crossing depends on the nuclear charge and on how many electrons are already present, so it moves as the row is filled — which is why the transition metals ionise from 4s rather than 3d, and why a simple ordering rule gets the filling order right and the ionisation order wrong.

Tested in The order the shells fill · the atomic structure reading path

“Nuclear spin is a small correction, since nuclear magnetic moments are a thousandth of electronic ones.”

The magnetic effect is small and the statistical effect is not. Hydrogen's rotational lines alternate three to one in strength, exactly, because the two protons are spin-halves and the molecule's state must change sign when they are swapped — and its low-temperature heat capacity is that of a frozen 3:1 mixture rather than of an equilibrium one, a discrepancy that stood unexplained for fifteen years.

Tested in Four states, and one of them is odd · the spin reading path

“A refractive index below one would let light travel faster than c, so it cannot happen.”

It happens for every material at high enough frequency, and X-ray mirrors are built on it. What travels at the phase speed is a crest, which carries no news; the sum rule fixes the total absorption a medium can have and forces the index to approach one from below above every resonance, and nothing in that arrangement moves a signal faster than light.

Tested in The answer that cannot come first · the attenuation reading path

“Huygens' principle says every point on a wavefront is a source of wavelets, and the next wavefront is their envelope.”

That construction gives the right answer in three dimensions and the wrong one in two, where the disturbance behind the front is not zero and cannot be an envelope of anything. The principle in its strict form — that a disturbance is confined to the front — holds only in odd dimensions above one, and Huygens' own argument had no way to see that.

Tested in The arrival that keeps arriving · the wave motion reading path

“A spectrum determines the shape that produced it.”

It determines the area and the perimeter, and in general not the shape. Pairs of differently shaped drums with identical spectra were constructed in 1992, settling a question open since 1966. What the leading terms of the count give is the area, from how fast the count grows, and the perimeter, from how far it lags.

Tested in How many ways there are to vibrate · the standing waves reading path

“The best angle for a driven golf ball is about thirty degrees, because that is what the drag calculation gives.”

Drag alone gives 38° for a golf ball's ballistic parameter, and the real answer is between ten and fifteen. The difference is lift: a backspinning ball generates an upward force that a drag-only model has no term for, so the calculation here is the right answer to a question that is not quite the one a golfer is asking.

Tested in The angle that drag moves · the projectile reading path

“The pendulum stands up whenever the shaking is fast enough.”

Fast enough and not too violent. The integration finds a band: below aΩ = √(2gL) the inverted position is not an equilibrium at all, and above an upper edge measured here at a fifth to a third of the pendulum's length the fast motion stops being small and the pendulum is thrown out. Stability is a band, not a threshold.

Tested in Held up by a force that averages to nothing · the pendulum reading path

“The principle is an elegant restatement, but the equation of motion is what does the work.”

They are equivalent for a point particle and not interchangeable in practice. The action is a single scalar, so it makes coordinate changes trivial, extends to fields and to relativity unchanged, and connects symmetries to conservation laws directly. And it is the classical limit of the sum over paths, which the equation of motion has no route to.

Tested in Least action, except that it is not least · the least action reading path

“The angle of repose is a property of the material.”

It is a range, not a number. A slope steeper than the angle at which flow stops will keep flowing once started, and one shallower than the angle at which flow starts will not begin — so the same sand has stable slopes anywhere between about 31° and 35°, and where a particular heap sits in that band depends on how it was built.

Tested in The angle that does not know the size of the heap · the granular matter reading path

“The energy needed is the osmotic pressure times the volume produced.”

That is the answer for the first drop only. Taking a fraction of the feed leaves the rest saltier, so the last drop is pushed against a higher pressure than the first: at half recovery the reversible cost is 1.10 kWh/m³ against 0.79, and a single stage held at the final pressure throughout costs 1.58.

Tested in What it costs to take the salt out · the osmosis reading path

“The fountain effect is a curiosity of a laboratory liquid.”

The same relation ΔP = ρSΔT is used as an instrument. Fountain pumps with no moving parts circulate helium in space cryostats, and thermomechanical pressure is how a superfluid's entropy is measured — the effect is the standard way of getting at S for a liquid that no calorimeter can be inserted into without heating it.

Tested in The fountain a lamp can drive · the superfluidity reading path

“The Jeans analysis derives the instability of a uniform self-gravitating gas.”

It perturbs a background that cannot exist. A uniform self-gravitating medium is not static — it is already collapsing — so the unperturbed state does not solve its own equations, which Jeans knew and which is called the Jeans swindle. Redoing it in an expanding background gives the same criterion with a growth that is a power of time rather than an exponential.

Tested in The disturbance that grows instead of travelling · the Self-gravity reading path

“Debye screening means the electric field inside a plasma is zero.”

It means a static excess charge is screened over a length λ_D, which is not the same as no fields. Fields on scales larger than λ_D exist and dominate the dynamics — that is what plasma waves, sheaths and instabilities are made of — and the screening length is precisely the scale below which charge separation is expensive and above which the plasma is neutral.

Tested in The long-range force that does not reach · the plasma oscillation reading path

“The limit explains why there is a maximum stellar mass.”

It sets the scale and does not do the work alone. A 100-solar-mass star sits at 0.83 of its limit, so radiation pressure dominates its structure, drives a strong wind and makes it unstable — but the observed upper limit near a few hundred solar masses also involves how such stars form and how they shed mass, and none of that is in the ratio.

Tested in The brightness a mass cannot exceed · the radiation pressure reading path

“Supernumeraries are rare because they need unusual conditions.”

They need small drops and nothing else. Their angular spacing goes as the two-thirds power of wavelength over drop radius, so drops larger than about 0.65 mm crowd them closer together than the sun's own half-degree width and the contrast washes out. The physics is always there; what varies is whether a source half a degree wide can show it.

Tested in The fringes below the rainbow · the diffraction reading path

“The phase depends on the sequence of optical elements used to produce it.”

It depends only on the circuit those elements trace on the sphere of states. Six circuits of quite different shapes are computed here and every one gives minus half its own enclosed solid angle, to 10⁻¹⁶ radians — so two entirely different sets of components tracing the same circuit give identical phases.

Tested in The phase that is only a shape · the polarisation reading path

“Isotopes can be separated by effusion because lighter molecules escape faster.”

They do, and the factor is the square root of a mass ratio, which for uranium hexafluoride is 1.00429. Reaching 90 per cent enrichment from natural uranium therefore needs about 1,665 ideal stages — which is why gaseous diffusion plants covered square kilometres, and why centrifuges, separating by mass rather than by its square root, replaced them.

Tested in The gas that leaves is not the gas inside · the kinetic theory reading path

“Classical nucleation theory gives the rate of condensation.”

It gives the exponent well and the prefactor badly, and measured rates depart from it by many orders of magnitude for some substances. The theory survives because the exponent dominates: an error of 10⁵ in the prefactor moves the predicted onset supersaturation by under one per cent, which is smaller than the experimental scatter.

Tested in The barrier a new phase has to climb · the phase change reading path

“The effect has never been observed, so it is speculative.”

It has not been observed directly, and it is a prediction of ordinary quantum field theory in flat spacetime with no new physics in it — the same framework that gives the Lamb shift. What is speculative is not the effect but the interpretations attached to it; the calculation is as standard as any in the subject.

Tested in The temperature of an acceleration · the accelerated frames reading path

“This means relativity is only a convention.”

The conventional part is the synchronisation and nothing else. The invariance of the interval, the round-trip constancy, time dilation between clocks brought back together and every measured prediction are unaffected by the choice — which is why the theory can be written in an anisotropic convention and still make identical predictions, only with uglier equations.

Tested in The speed that cannot be measured one way · the simultaneity reading path

“The theorem is about electrostatics.”

It is about any potential obeying Laplace's equation, which includes the magnetic scalar potential in a current-free region and the gravitational potential in empty space. That is why there is no stable point in the Earth–Moon system's gravity alone except where the rotation supplies something Laplace's equation does not.

Tested in Nothing can be held still by a static field · the potential reading path

“Decoherence needs a measuring apparatus, or at least a macroscopic environment.”

It needs one scattered photon. A single air molecule bouncing off a dust grain carries away enough information to distinguish two positions a micrometre apart, and the coherence is gone. In the emptiest achievable vacuum the microwave background left over from the early universe still does it in about a hundred seconds.

Tested in Where the interference goes · the measurement reading path

“The theorem is a tidy restatement of things already known from the force law.”

It is the only route to most of them. There is no way to read a conserved quantity off a field theory's equations by inspection, and the conserved currents of the Standard Model were all obtained this way; the theorem also delivers quantities nobody would have guessed, including the eccentricity vector of an inverse-square orbit, whose symmetry is a rotation in four dimensions and is invisible in three.

Tested in The conservation law a symmetry hands over · the least action reading path

“Treating the legs as springs fixes the problem.”

It closes the arithmetic and moves the difficulty. The answer then depends on stiffnesses that are rarely known to better than a factor of two, on the floor, which is not four independent springs, and on the manufacturing tolerance of the legs — which the figures here show dominating the load sharing at a shortfall of a hundredth of the leg's own compression.

Tested in The table statics cannot settle · the Free-body reading path

“The equation breaks down at the shock, so the physics stops there.”

The physics is unchanged; the approximation stops. A real front steepens until the gradient is large enough for viscosity and heat conduction to matter, which happens at a thickness of a few mean free paths, and there it settles into a travelling structure of finite width. What is lost is not the physics but the assumption that the medium is a continuum with no dissipation, and the shock thickness is precisely where that assumption is repaid.

Tested in The front that steepens until it cannot · the wave motion reading path

“Russell saw a solitary wave on a canal, so the phenomenon is a curiosity of shallow water.”

The same balance appears wherever a nonlinearity and a dispersion of the right signs meet: in optical fibres, where it carries data at rates that a spreading pulse cannot; in Bose–Einstein condensates; in Josephson junctions; and in the atmosphere as the Morning Glory cloud. The equation is the same equation, and the sech-squared profile with amplitude fixing width is the same profile.

Tested in The pulse two failures keep alive · the wave packets reading path

“A liquid's viscosity follows an Arrhenius law with a single activation energy.”

It follows one over a modest range and nowhere else. Water is measurably non-Arrhenius over a wide interval, and a glass-forming liquid approaching its transition curves away from any straight line so steeply that no single activation energy describes it — the viscosity rises through thirteen decades over a temperature interval in which nothing structural is happening that a diffraction pattern can see.

Tested in The thickness that goes both ways · the viscosity reading path

“Maxwell's counting rule settles when a structure is rigid.”

It gives a necessary condition and not a sufficient one. Counting says a pack is rigid above four contacts per grain in the plane; taking the rank of the actual rigidity matrix shows local regions that are over-braced sitting beside regions that are still floppy, so a pack can satisfy the average and have a mechanism. The rank is what decides, and the count is what predicts the rank when nothing is special about the arrangement.

Tested in The heap that becomes a solid · the granular matter reading path

“The Roche limit is 2.44 planetary radii.”

That is the fluid case with the two densities equal, and both conditions matter. A rigid body of the same density comes apart at 1.26 radii; a body denser than the primary survives closer in; and the drawn curve varies as the cube root of the density ratio, so an iron satellite of a gas giant has a limit less than half an icy one's. Quoting the number without the ratio is quoting a coefficient as if it were a law.

Tested in The distance that forgets the moon · the Self-gravity reading path

“The escape time follows from the mean density and the opacity.”

That gives about two thousand years and the accepted figure is nearer 170,000. The gap is the density profile: the Sun is a hundred times denser at its centre than on average, the free path there is correspondingly shorter, and most of the journey time is spent in the innermost fraction of the radius. The uniform estimate gets the exponent and the mechanism right and the number wrong by two orders, and says which.

Tested in The light that takes a hundred thousand years to leave · the kinetic theory reading path

“The 46° halo is the same effect at the other prism angle, so it should be equally common.”

It is the same effect and it is rare, for a reason the same calculation gives: the 90° prism transmits over a window of incidence only 32° wide against the 60° prism's 76°, so far fewer orientations contribute. Its minimum deviation is also more sensitive to the crystal's orientation out of the plane, which smears it further, and it needs cleaner crystals to be seen at all.

Tested in The ring at twenty-two degrees · the dispersion reading path

“The intensity in a speckle pattern fluctuates about its mean.”

It does not fluctuate about anything. The distribution is a decaying exponential with its maximum at zero, which the histogram here matches to eleven per cent in every well-populated bin with no fitted parameter. There is no typical brightness and no peak near the mean; the most probable intensity at any point is nothing at all, which is why the pattern looks like grain rather than like noise on a grey.

Tested in The grain that is in the light · the coherence reading path

“Grangier's experiment showed g² of zero, so a single emitter is all that is required.”

The observed dip was to about 0.18, not to zero, and the shortfall is mostly the number of atoms in the observation region at once. N independent emitters give exactly 1 − 1/N, so two atoms give 0.5 and ten give 0.9; the experiment needed a beam thin enough that two atoms were almost never present together, and the residual is a measurement of how well that was achieved.

Tested in The experiment a wave cannot pass · the photon reading path

“A system swept slowly through an avoided crossing follows the lower branch.”

It does if the sweep is slow compared with the gap squared over ħ times the rate, and the crossover is sharp in a way worth quantifying. The two-state equation integrated here gives a probability of jumping that matches Landau and Zener's exponential to 0.012 across a hundred-fold range of sweep rates — so doubling the gap makes the system four times harder to shake off the adiabatic path, and that quadratic is why a small gap is far more permeable than it looks.

Tested in The crossing that never happens · the bands reading path

“Exchange effects are small corrections to the Coulomb energy.”

The exchange splitting in helium's lowest excited configuration is 0.8 electronvolts, which is 9,000 kelvin, and the exchange coupling that makes iron ferromagnetic corresponds to 1,000 kelvin — against a magnetic dipole interaction between neighbouring atoms of about 0.1 kelvin. The whole of magnetism above liquid-helium temperature is exchange, and the actual magnetic interaction is four orders of magnitude too weak to matter.

Tested in The force with no force in it · the exclusion reading path

“The bending is caused by the polarisation charge that appears on the interface.”

That is a correct account and not an alternative one. The surface bound charge is what the discontinuity in the normal E *is*, and computing the field from it reproduces the same answer; the boundary-condition route reaches it without solving for the charge. Both are needed in practice — the conditions give the field and the bound charge says where the force on the dielectric acts.

Tested in A refraction with no wave in it · the dielectrics reading path

“Mutual inductance is symmetric, so a transformer works the same in both directions.”

The mutual inductance is symmetric and a transformer is not. The voltage ratio depends on the two self-inductances as well, so a step-up transformer run backwards steps down; what is symmetric is the coupling coefficient and the power transfer, not the voltage. Confusing the two is the commonest way the theorem is misused.

Tested in The coupling that is the same both ways · the induction reading path

“A superconductor expels magnetic flux from its interior.”

It expels flux from a region a penetration depth thick, which is the whole of a bulk sample and almost none of a thin film. The profiles here give 95 per cent expulsion for a slab forty depths thick and 7.6 per cent for one a single depth thick — same material, same field, same constant. That is why films stay superconducting in fields that destroy the bulk, and why the critical field of a film rises as its thickness falls.

Tested in The field that is pushed out · the superconductivity reading path

“A laser medium is at a negative temperature, which is a metaphor for population inversion.”

It is the definition applied. Putting the measured populations into the Boltzmann ratio returns a negative T, and the same number comes out of the slope of the entropy computed independently — the two agreeing here to a part in a million. What makes it a limited statement is that only the inverted levels are at that temperature; the lattice they sit in is not, which is why the state relaxes and why it can be prepared at all.

Tested in Hotter than any temperature there is · the third law reading path

“Van der Waals's equation predicts the inversion temperature at 6.75 times the critical temperature.”

It does, and real gases give between 4.9 and 8.7. The model gets the mechanism exactly right — the competition between an attraction that costs energy to separate and a volume that costs work to compress — and the coefficient to within a factor well under two. Quoting 6.75 as a prediction rather than as a model's answer is the usual way van der Waals is over-read.

Tested in The gas that cools by being let go · the phase change reading path

“The causal limit on the sound speed is a curiosity with no observable consequences.”

It caps the maximum mass of a neutron star at a little under three solar masses, whatever nuclear physics turns out to be, because a star is held up by stiffness and the stiffness has a ceiling. That bound is what makes a compact object above it a black hole by elimination, and it is how the 2.6-solar-mass companion in GW190814 was argued about before anything else was known.

Tested in Nothing is allowed to be rigid · the relativistic dynamics reading path

“The mass ratios here settle whether interstellar travel is possible.”

They settle the propellant, which is one of four difficulties and not the worst. Stopping at the far end squares the ratio and returning squares it again; the power required to reach a tenth of light speed in a human lifetime is thousands of times humanity's output; and a gram of dust at 0.1c arrives with the energy of a hundred kilograms of explosive. The chart is a lower bound on the difficulty.

Tested in The fuel a starship needs · the Mass-energy reading path

“Chaos is sensitive dependence on initial conditions.”

Sensitive dependence on its own is an instability, and an unstable system runs away. The figures here measure both halves at once on the same orbit: the separation of two trajectories grows at a rate of 0.4188 per step, and every one of the twenty-four thousand points lies inside a box 2.56 by 0.77. A system that does the first without the second flies apart and has no attractor; chaos is the first *subject to* the second, and the fold is what the constraint forces.

Tested in The fold that has to be there · the chaos reading path

“Huygens' cycloidal cheeks solved the pendulum clock's amplitude error.”

They solved it exactly, in theory, and the clocks kept worse time. The construction requires the string to wrap against a shaped surface, which adds friction where a plain suspension has almost none, and a real string has stiffness so it does not lie on the cheek as an ideal one would. The error removed was second order in the amplitude and the errors introduced were first order in the contact, so the trade was a bad one — which is a statement about the mechanism rather than about the mathematics, and the mathematics is exact.

Tested in The curve that does not ask where it started · the pendulum reading path

“On the edge of the geometrical shadow the light is half the unobstructed value, because half the wave has been blocked.”

Half the *amplitude* gets through, and intensity is amplitude squared, so the value on the boundary is a quarter: 0.250000 in these units, exactly, and it follows from the split rather than being measured. The half-and-half intuition is right about what is blocked and wrong about what is being counted, and the same slip turns a straight edge's 25 per cent into 50 wherever it is made.

Tested in The wave that comes from the rim · the huygens reading path

“The surface wave is confined to the surface.”

It is confined to a skin whose thickness is a wavelength, and the wavelength is the wave's own rather than the material's, so a long-period wave reaches deep and a short-period one does not. The vertical displacement at one wavelength down is 19.3 per cent of its value at the surface, and the horizontal displacement changes sign at 0.193 wavelengths — a reversal rather than a node. Because different periods sample different depths, a surface wave is dispersive in a layered Earth even though it is not dispersive in a uniform one, and that dispersion is how the crust is measured.

Tested in The wave a surface is enough to hold · the guided waves reading path

“A minimal surface is one that minimises area subject to its boundary.”

It is one whose area is stationary, which admits saddle points as well as minima. The two rings have two catenoid solutions at every separation below the critical one — one stable and one not — and they are both minimal surfaces in the technical sense. They merge and annihilate at 0.6627 ring radii, which is exactly the shape a fold in a family of solutions has, and which is why the film's disappearance is sudden rather than gradual.

Tested in The angles a film has no choice about · the surface tension reading path

“Polymer solutions climb rotating rods.”

The same solution at the same speed climbs a thin rod and is thrown off a thick one, and the changeover is computable: the hoop tension falls as the fourth power of the distance from the axis and the centrifugal term as the second, so the first wins only inside a radius the fluid's own constants fix. For the solution drawn here that radius is 34.6 millimetres — 10 and 25 millimetre rods climb by 2.21 and 1.16 millimetres, and a 60 millimetre one dips by 4.83. A demonstration that works with a glass stirring rod fails with a broom handle.

Tested in The liquid that climbs the rod · the rheology reading path

“The size of a black hole is its event horizon, at two masses.”

That is the radius nothing returns from, and it is not the radius of anything anybody can see. Light can orbit at three masses, on an unstable circle; and the smallest impact parameter a ray can have and still escape is root twenty-seven, 5.196 masses, which is what fixes the angular size of the dark patch a distant observer sees. The shadow is therefore 2.6 times the horizon radius across, and quoting the horizon as the observed size understates it by that factor.

Tested in The circle light cannot leave · the horizons reading path

“The damping rate is Landau's formula, root pi over eight times the exponential.”

That is the asymptotic form and its accuracy varies enormously across the range it is quoted for. Integrated exactly, the rate at a wavenumber of half a reciprocal Debye length is 0.1534 and the formula gives 0.1514 — within one and a half per cent — while at 0.3 the exact answer is 0.0126 and the formula gives 0.0203, sixty per cent high. The error is invisible unless the exact problem is solved beside it, which is why the figure here draws both.

Tested in The wave that dies with nothing to rub against · the plasma oscillation reading path

“A zone plate has a focal length of r₁²/λ.”

It has that one and an infinite series of others, at every odd fraction of it. At a third of the design distance each open ring spans three half-period zones instead of one, two of the three cancel, and a focus remains; at a half it spans two, the cancellation is complete, and there is nothing. The odd-order peaks reach 100, 99.9, 96.3 and 93.8 per cent of the main peak's on-axis intensity — not a ninth and a twenty-fifth, because only a ninth of the light goes into the third order and its focus is three times tighter, and the two effects cancel.

Tested in The lens that is a set of rings · the diffraction reading path

“Total internal reflection sends all the light back, so nothing happens to it.”

Nothing happens to its amplitude — the coefficients have modulus exactly one past the critical angle — and something happens to its phase, differently for each polarisation, which is the only thing left that can distinguish one angle from another. For a glass-to-air boundary the difference between the two phases rises from zero at the critical angle to 46.53 degrees at 51 degrees of incidence and falls to zero again at grazing. That difference is a retardation, and it is what a device can be built out of.

Tested in The retarder with no crystal in it · the polarisation reading path

“The instrument measures current, so it measures height.”

It cannot, because the current is far too steep a function of height to be read as one: ten picometres — a tenth of an atomic radius — changes it by 24 per cent, and a factor of two in current is nine picometres. The measurement is made the other way round, by a feedback loop that holds the current fixed and records the voltage the piezoelectric scanner needed. The recorded quantity is the correction, and the instrument's precision is the precision of a feedback null rather than of a current reading.

Tested in The last atom does all the seeing · the tunnelling reading path

“Entanglement cannot signal because relativity forbids it.”

It cannot signal because of an arithmetical fact about the state, which the figure computes: summing over the far measurement's outcomes leaves the near half's density matrix exactly unchanged — Bloch length 0.0e+0 and trace one to a part in 10¹⁴ — whatever axis the far analyser is turned to, while the correlation swings by two. Relativity is not used anywhere in that computation. What is worth noticing is the converse: a working copier would break it, since two copies measured along two axes would reveal the far setting.

Tested in The state that cannot be copied · the measurement reading path

“Flux through a superconducting ring is conserved.”

It is conserved and it is also quantised, and the second is the stronger statement. A ring cannot hold an arbitrary flux at all: it holds a whole number of quanta of 2.0678 × 10⁻¹⁵ webers and drives a current to make up whatever the applied field left over. Conservation follows from zero resistance and is a classical statement; quantisation follows from the condensate's wavefunction having to come back to itself round the ring, and no classical argument supplies it.

Tested in The two in the flux quantum · the superconductivity reading path

“The constraints hold automatically.”

One of them does, unconditionally: nothing can change the divergence of B, whatever the fields do. The other holds if and only if charge is conserved, because the rate of change of the divergence of D is minus the divergence of the current. The same integration run twice, with a current whose charge is accounted for and one whose is not, gives residuals of 1.1 × 10⁻¹⁴ and 7.7 — from the same equations, the same grid and the same current. Conservation is a requirement of the theory rather than an extra observation about the world.

Tested in The two equations that are not laws of motion · the maxwell equations reading path

“The number of photons in a cavity depends on how much light was put in.”

In equilibrium it does not depend on it at all. The walls emit and absorb until the free energy is least, and the minimum fixes the number: 16πζ(3)(kT/hc)³ per unit volume, which at the microwave background's 2.7255 kelvin is 4.11 × 10⁸ per cubic metre. Photon number is not conserved, so it is not a variable of state — which is the same statement as the chemical potential being zero, and is where every difference from an ordinary gas comes from.

Tested in The gas that nobody counted · the blackbody reading path

“Onsager's relations say the matrix of transport coefficients is symmetric.”

They say it is symmetric provided nothing breaks the time-reversal symmetry of the underlying motion. A magnetic field does break it, and then the relation becomes L(B) = Lᵀ(−B) — the coefficient measured in one field equals its transpose measured in the reversed field, and the symmetric statement fails. That is not a caveat: the Hall effect and the whole family of thermomagnetic cross-effects live in exactly the part the symmetric version excludes.

Tested in The second experiment that cannot disagree · the diffusion reading path

“The superfluid density falls continuously to zero at the transition, as an order parameter does.”

It falls discontinuously, from a value that is not adjustable. Integrating the renormalisation-group flow gives a stiffness that drops to zero from 0.643 in units where the universal prediction is 2/π = 0.6366, and that number depends on nothing: not on the material, not on the vortex core energy, not on the bare stiffness. Measurements on helium films spanning a wide range of thicknesses all lose their superfluidity on that one line, which is why the prediction is called a universal jump rather than a critical exponent.

Tested in The transition with nothing to order · the phase change reading path

“The effect is a projection, so it tells you nothing about the source.”

It tells you two things at once, and they are the reason it is used. Seeing an apparent speed of β_app requires a true speed of at least β_app/√(1 + β_app²), so an observed 7.02 puts a floor of 0.990 under the true speed and a ceiling of about 16 degrees on the angle to the line of sight. Nothing else about a distant jet gives both a lower bound on its speed and an upper bound on its orientation from one measurement.

Tested in The motion that measures faster than light · the velocity addition reading path

“A moving source is Doppler shifted, so its light changes colour.”

It changes colour and it also changes brightness, by a much larger factor, and the two are not separate effects. The one quantity all observers agree about is the specific intensity divided by the cube of the frequency, so a Doppler factor of δ multiplies the intensity by δ³ per unit frequency and δ⁴ in total. For a source at β = 0.99 that is 1.6 × 10⁹ between the approaching and receding directions, which no measurement of a colour would suggest.

Tested in The brightness that is not the same for everyone · the doppler reading path

“The rotating frame's geometry is non-Euclidean, so the disc is curved.”

The disc is not curved and neither is the spacetime it sits in, which is exactly flat. What is non-Euclidean is the three-dimensional geometry a rotating observer builds out of local measurements, and it is non-Euclidean because there is no consistent way to slice a rotating frame into simultaneous surfaces — go once round the rim synchronising clocks and the last one disagrees with the first by an amount proportional to the enclosed area, which is what a Sagnac interferometer measures.

Tested in The disc that cannot be spun · the length contraction reading path

“Lagrangian mechanics is better than Newtonian mechanics because constraint forces drop out.”

They do drop out, and that is a loss as well as a saving. A rope has a breaking strain and a track has a limit on what it can push; neither quantity appears anywhere in the equation of motion written in the angle, so a solution can be perfectly correct and say nothing about whether the apparatus survives. The multiplier drawn here restores the number — 3.000 weights at the bottom of a quarter-circle swing — and it has to be asked for.

Tested in The force a coordinate cannot see · the least action reading path

“A target within range can be hit, so knowing the range tells an artillerist what they need.”

It tells them half. Every point strictly inside the boundary is reached by two launches, not one: the range equation is a quadratic in the tangent of the launch angle, and its discriminant is positive there. The two roots are the flat trajectory and the lobbed one, with different flight times, different impact angles and different clearances — and choosing between them is most of the craft.

Tested in Everywhere a throw can reach · the projectile reading path

“Two modes of a system are orthogonal, so multiplying them together and integrating gives zero.”

Only against the right weight. The modes of the lumpy string here are not orthogonal in the plain sense; multiply two of them and integrate and the answer is not zero. They become orthogonal when each element of the string is counted in proportion to the mass it carries, and the largest off-diagonal overlap against that weight is at the level of the integration's own error. A uniform string hides the distinction, because there the weight is a constant.

Tested in The count that cannot be cheated · the standing waves reading path

“A Doppler radar measures the speed of the target.”

It measures the component along the beam and has no access to the rest. A target crossing at 45° reads 71 per cent of its speed, at 60° it reads half, and at 90° it reads nothing at all — and nothing in a single beam's return says which case it is. Every such instrument therefore reads low, never high, by a factor that cannot be recovered from the measurement it made.

Tested in The shift a mirror gives twice · the doppler reading path

“Reciprocity means a system looks the same from both sides.”

It means the transmission does. The reflections do not: the same stack returns the same fraction from either side — only because nothing here absorbs — and the reflected *phases* differ by 1.6 radians, because a wave meets a different first surface from each side. The theorem constrains the pair of ends and says nothing about what happens between them.

Tested in Swap the ends and nothing changes · the wave motion reading path

“Light refracts because it travels more slowly in the denser medium, so it bends towards the normal.”

That is true and it does not say which angle. The condition is that the component of the wavevector along the surface is unchanged, because the two sides must agree on the phase at every point of the boundary; the magnitude of the wavevector is fixed by the medium at n times the vacuum value, so the direction is whatever makes those two things compatible. Snell's ratio of sines is the projection of two circles of radii n₁ and n₂ onto one line.

Tested in The law that only asks about one component · the refraction reading path

“Intensity interferometry is better than amplitude interferometry because it is immune to the atmosphere.”

It is immune and it is much less sensitive. The correlated signal is smaller than the shot noise by the degeneracy parameter — about 10⁻³ photons per mode for a bright star through a narrow filter — so the measurement is recovered only by averaging, and reaches a few hundred standard deviations after hours where a fringe measurement would take seconds. That is why it works on the brightest few dozen stars and on nothing fainter.

Tested in The correlation that survives what the phase does not · the coherence reading path

“The energy method gives the force, so the mechanism does not matter.”

The energy method gives the total force and says nothing about where it is applied, which is what a designer needs. The stress is concentrated at the slab's leading edge, over a region of order the plate separation, and that is also where the field is largest and where breakdown starts. A part designed on the total force and silent about its distribution fails at the edge.

Tested in The force that lives where the model is not · the dielectrics reading path

“A drop can hold as much charge as is put on it; eventually it will just repel harder.”

It has a limit and the limit is sharp. Electrostatic pressure grows as the square of the charge over the fourth power of the radius while the Laplace pressure holding the drop together goes as one over the radius, so the two are equal along Q² = 64π²ε₀γR³ — 0.635 nC for a one-millimetre water drop. Past that the drop does not merely deform: it draws itself into a cone and throws off a jet of droplets each below its own limit.

Tested in The pressure a charge puts on its own metal · the conductors reading path

“Deep ground is at the annual mean temperature because heat takes a long time to get there.”

It is at the mean because the oscillation has been attenuated, and the attenuation and the delay are the same length. At πδ — seven metres in damp soil — the swing is down to 4.3 per cent of the surface's and the phase is inverted, so that ground is coldest in August. Below about four δ there is no measurable annual signal at all, and the temperature there is set by the geothermal gradient rather than by the sky.

Tested in The summer that reaches the cellar in December · the diffusion reading path

“The triple point is where the three coexistence curves happen to meet.”

It is where they are *forced* to meet and where they can only meet at a point. Counting gives F = C − P + 2: one phase leaves two numbers free and fills an area, two leave one and make a curve, three leave none. Nothing about water enters the count, and the same arithmetic says four phases of one substance would need minus one degree of freedom, which is not a small number but an impossible one.

Tested in Why the triple point is a point · the phase change reading path

“Ehrenfest's theorem shows that quantum averages obey Newton's laws, so classical mechanics is recovered for large objects.”

The theorem gives d⟨p⟩/dt = ⟨−V′(x)⟩, which is the *average of the force over the state*. Newton's law needs −V′(⟨x⟩), the force at the average position. The two agree only when V′ is linear in x, and for a quartic well the computed centroid departs from the classical orbit by more than the starting displacement within a couple of periods — with the theorem holding exactly throughout.

Tested in The average that obeys Newton · the correspondence reading path

“The resonances sit at the energy levels of the well between the barriers.”

They sit near them and not at them. The infinite-well levels of the gap here are 2.467 and 9.870; the resonances are at 6.438 and 13.915. A well with leaky walls has states that are not bound, whose wavefunctions extend into the barriers, and whose energies are correspondingly shifted — and the shift is of the same order as the amount by which the walls are penetrable.

Tested in Two walls that let more through than one · the tunnelling reading path

“A diode conducts one way because current can flow from p to n and not the other way.”

Current flows both ways and the two are wildly unequal. Carriers going uphill over the junction barrier are suppressed by the Boltzmann factor of that barrier, which the applied voltage changes; carriers rolling downhill are not held back by the barrier at all and their supply is set by thermal generation. The ratio at ±0.5 V is 2.5 × 10⁸, and it comes from one exponential and one constant rather than from a passage that opens and closes.

Tested in One level, and the field that bends the bands · the bands reading path

“Special relativity follows from the Lorentz transformation, so understanding it means understanding that transformation.”

The whole theory can be built from one measured ratio and two multiplications. Flashes sent every second by one observer arrive every k seconds at another; the Lorentz factor is (k + 1/k)/2, the speed is (k² − 1)/(k² + 1), and velocities compose because k factors multiply. No coordinates are assigned to any event in the derivation, and the transformation appears at the end as a summary rather than at the start as a postulate.

Tested in Everything from an exchange of pulses · the doppler reading path

“Proper time along a worldline is the integral of dt/γ, which follows from special relativity.”

It follows from special relativity *plus* the clock hypothesis, which is a separate assumption: that an ideal clock's rate depends on its instantaneous speed and on nothing else. Nothing in the two postulates says what an accelerating clock does, because the postulates are about inertial frames. The hypothesis is eminently testable and has been tested to 10¹⁸ gravities, and it remains an input.

Tested in The clock that does not feel the turn · the accelerated frames reading path

“Second sound is a temperature wave, so it is heat behaving like sound.”

It is one of the two wave modes of a two-component fluid, and calling it a temperature wave describes what it carries rather than what it is. In the first mode the two components move together and the total density oscillates; in the second they move oppositely with the density constant, so the *ratio* of the components oscillates — and since that ratio is set by temperature, the wave carries temperature. Both modes come out of the same pair of equations.

Tested in The heat that arrives as a wave · the superfluidity reading path

“The Brazil nut effect is understood: small grains fall through the gaps.”

That is one mechanism of at least three, and it is not always the dominant one. Convection of the whole bed lifts everything and returns only what fits in the narrow downward stream at the walls; interstitial air changes the answer enough that the same experiment in a vacuum can give the opposite result; and at some density ratios a large intruder sinks instead. There is no single criterion that predicts which way a given mixture will go.

Tested in The big one comes to the top · the granular matter reading path

“Ions are pulled into the sheath by its field, so their entry speed is whatever the field gives them.”

They have to arrive at the sheath edge already moving at the ion sound speed or no steady sheath exists at all. Integrating the sheath equation from a sub-sonic entry gives an oscillatory solution that turns back through zero rather than a monotone potential; the threshold is exactly Mach one. So the plasma must accelerate its own ions in a much wider, much gentler presheath before the sheath begins, at a cost of half an electron temperature.

Tested in The wall a plasma builds against itself · the plasma oscillation reading path

“An optical trap holds a particle at the brightest point of the beam.”

It holds it a little past the brightest point. The gradient force vanishes exactly at the waist, where the intensity is stationary, and the scattering force does not — so the equilibrium sits downstream, where the restoring pull has grown to match the push. For the beam computed here that is 0.066 micrometres past the focus, and every optical trap is offset this way.

Tested in The light that pulls rather than pushes · the radiation pressure reading path

“A spinning top precesses at Mgl divided by its spin angular momentum.”

Setting the tilt's second derivative to zero gives a quadratic in the precession rate rather than a value, so there are two steady rates at every spin — a slow one, whose leading term is Mgl/I₃ω₃, and a fast one that rises in proportion to the spin. The quadratic has real roots only above a critical spin, and a top released from rest is at neither of them, because its initial precession rate is zero and zero is not a solution.

Tested in The top that nods before it settles · the rotation reading path

“A Newton's cradle transfers the striking speed to the far ball.”

Integrating the equations of motion for five touching balls with the contact force two steel spheres actually exert gives 0.989 of the striking speed to the far ball, with the remainder distributed among the others. The transfer is exact only when the balls are separated so that each contact is a two-body collision; with them touching, several contacts overlap in time and no two-body argument applies.

Tested in Five balls, and the law that does not choose · the momentum reading path

“Adding turns adds holding power.”

Adding turns multiplies it, and the difference is the whole practical content. Each turn at a coefficient of 0.25 multiplies by 4.8, so the first turn adds 3.8 times the hand's pull and the third adds 87 times it. A rule expressed as an addition would predict that ten turns are twice as good as five; they are twenty-three times as good.

Tested in The part of the wrap that is actually gripping · the friction reading path

“The principle of least action says the true path minimises the action.”

There are two actions and two comparison classes. Hamilton's action compares paths of the same duration and is the time integral of kinetic minus potential energy; Maupertuis' abbreviated action compares paths of the same energy and is the integral of momentum along the path. Both are stationary at the true trajectory and the figures show them rising at different rates from it. A statement naming neither has named neither.

Tested in The principle that fixes the energy instead of the clock · the least action reading path

“A physical system period-doubles because it resembles the logistic map.”

It period-doubles because sampling it once per drive cycle makes it a map, and that requires the dissipation to contract phase-space volume onto an attractor thin enough that one coordinate along it suffices. Where the contraction is weak the sampled dynamics is genuinely two-dimensional, the cascade is modified and the constants shift, so the resemblance is a consequence of the damping rather than an analogy.

Tested in The map a dripping tap turns out to be · the chaos reading path

“A nonlinear oscillator is a linear one with a correction to its amplitude.”

The correction is not to the amplitude but to the set of frequencies present. Driven at one frequency, the oscillator here returns a component at three times it whose size grows as the cube of the drive — a component the linear equation cannot produce at any amplitude. The figures project that component out of the integrated motion rather than assuming it, and its growth exponent comes out at 3.00 against 1.00 for the fundamental.

Tested in The oscillator that answers at three times the question · the harmonic approximation reading path

“The energy stored in an inductor is ½LI².”

It is, and the statement leaves out where it comes from and how it is delivered. Integrating the power that flows against the coil's own back emf during the rise gives ½LI² to four parts in ten thousand, and over the same interval the resistor dissipates the same amount again. Half the energy the supply delivers while the current is arriving never reaches the field.

Tested in The circuit that fights its own change · the induction reading path

“Contact resistance can be reduced by using thicker conductors.”

Ninety-four per cent of a constriction's resistance lies within ten spot radii of the contact, and more than ninety-nine per cent within a hundred. The current has forgotten where it came from long before it arrives, so the size and shape of the bodies either side do not enter the answer as long as they are large. What reduces it is a larger contact spot, or more of them.

Tested in The resistance that is a length · the conductors reading path

“The size of the jumps is set by how disordered the material is.”

The disorder sets the small end. The large end is set by the restoring term — the field a displaced wall sets up against itself — and the figures show the cutoff moving when only that term is changed. That is why the same material in a different shape crackles differently, and why the distribution below the cutoff is the same power law regardless.

Tested in The curve that is really a staircase · the magnetisation reading path

“Maxwell's equations are asymmetric between electricity and magnetism.”

Only the source terms are. With the charges and currents removed, the four equations are unchanged by a rotation taking E into cB and cB into minus E, through any angle. The figure computes the two Lorentz invariants at each angle from the rotated fields and finds them moving on a circle: the pair is conserved and neither member is. The asymmetry is entirely the absence of one kind of source.

Tested in The symmetry one missing charge would complete · the maxwell equations reading path

“A superconductor expels magnetic field.”

Only one kind does. Above a ratio of penetration depth to coherence length of one over root two the energy of a boundary between normal and superconducting material is negative, so the material makes as much boundary as it can: flux enters as a lattice of tubes, each carrying one quantum, with a normal core at the centre of each. The figures compute that surface energy by relaxing the Ginzburg–Landau equations and find it changing sign at 0.72.

Tested in Two lengths, and which one is longer · the superconductivity reading path

“Momentum is conserved because the field carries some.”

For this arrangement the field's momentum alone gives the wrong answer: the system is static and its total momentum must be exactly zero, and the field's is not. The books balance only when the mechanical momentum hidden in the current is included. Conservation is a statement about the total, and leaving out a term that nothing is visibly doing makes it fail.

Tested in The momentum of something that is not moving · the field energy reading path

“Numerical aperture measures how much light a fibre gathers.”

It measures the cone only. The gathering is the etendue — the core's area times π times the square of the aperture — and the two can be traded: a large-core fibre with a small aperture and a small-core fibre with a large one can gather the same light and behave completely differently, in bandwidth, in bending loss and in how many modes they carry.

Tested in The cone a fibre will accept · the etendue reading path

“Stopping down a lens increases depth of field.”

It increases the geometric depth in proportion to the f-number and shrinks the aperture, so the diffraction blur grows in proportion too. Past the aperture where the two are equal, stopping down makes everything worse at once. The wave-optical depth goes as one over the square of the numerical aperture, so the exchange rate between sharpness and depth is fixed and cannot be improved by any design.

Tested in The focus that is a slab, not a plane · the imaging reading path

“A rotating polariser measures the direction of polarisation.”

It measures the depth of a modulation as well, and that depth is the degree of polarisation. Every beam gives the same average — a polariser passes half of anything over a whole turn — and the four beams in the figure differ only in how deeply the transmitted power swings. A beam with no preferred direction gives a flat line and no angle at all.

Tested in The light with no direction of shaking · the polarisation reading path

“The refractive index is a property of a medium and scattering is a property of particles in it.”

Both are the same amplitude. The transmitted wave is the incident wave plus everything scattered forward; the imaginary part of that scattered amplitude removes energy and gives the extinction, and the real part retards the sum and gives the index. Computing the index of air from a molecule's forward scattering gives the measured value to a per cent.

Tested in Everything a scatterer removes, from one direction · the scattering reading path

“A sphere is a poor imaging surface.”

It is a perfect one for exactly two points, at R/n and nR from its centre. Rays from the first extrapolate back through the second with no spread the arithmetic can find, at any angle. Those aplanatic points are what the front element of an oil-immersion objective is used at, and they are the reason such a lens can collect a numerical aperture above one without introducing the aberration a strongly curved surface normally brings.

Tested in The surface that images one point exactly · the fermat reading path

“A spectrometer separates wavelengths.”

A dispersive one does. An interferometer measures the same spectrum with nothing that separates anything: it records the total power against path difference, which is the cosine transform of the spectrum, and the wavelengths are separated arithmetically afterwards. The two panels of the first figure carry identical information and neither is more fundamental.

Tested in The fringe and the spectrum are one measurement · the coherence reading path

“Interference always redistributes energy rather than removing it.”

That holds when the sources cannot feel each other. Two sources closer together than half a wavelength and driven in opposition radiate essentially nothing, and the missing power is not redistributed anywhere: it is never emitted. Each source finds itself working against a load the other has changed, and the effect is measurable at the driver rather than in the field.

Tested in What adding does to the energy · the superposition reading path

“A resonance is a peak.”

It is a peak when the resonant path is the only one. Where a smooth path exists alongside it the two amplitudes add before squaring, and the response is Fano's profile: asymmetric, with a maximum on one side and an exact zero on the other, at positions whose product is minus one. A symmetric peak is the special case in which the resonant path overwhelms the other.

Tested in The resonance with a zero in it · the resonance reading path

“Huygens' construction explains reflection, refraction and diffraction.”

It gets all three right and predicts a fourth thing that does not happen. A spherical secondary wavelet radiates backwards as strongly as forwards, so summing the wavelets over a plane gives a wave going back the way the original came. The figure does that sum and finds it as large as the forward one for an isotropic wavelet, and essentially nothing for Kirchhoff's.

Tested in The backward wave Huygens had to remove · the huygens reading path

“A band gap is opened by the strength of the periodic modulation.”

It is opened by the repeat. A chain of alternating masses has a gap whose edges are the frequencies of each mass oscillating alone against fixed neighbours, and the gap closes as the two masses become equal — at which point the repeating unit is really half as long and the second branch is the first folded back. The springs are unchanged throughout.

Tested in The frequency a lattice cannot carry · the periodic media reading path

“A pattern with deep minima means a bad match.”

It means a large reflection, which is a bad match only if power transfer is the object. A deliberately large reflection is how a resonant cavity, an antenna trap and a quarter-wave transformer all work, and in each the depth of the pattern is a design target rather than a fault. What a deep pattern always means is that the two waves are of comparable size.

Tested in The node that is not standing still · the standing waves reading path

“A bend loses a fixed fraction of the light.”

It loses a fraction that depends strongly on which mode is carrying it. A higher-order mode is held less tightly, its tail reaches further, and its caustic matters at a much larger radius — so a bend removes the higher modes first and leaves the fundamental. A coil of a few turns is therefore a mode filter, and is used as one.

Tested in The mode that will not turn a corner · the guided waves reading path

“Temperature is a measure of the average kinetic energy of the particles.”

It fixes the average and also fixes how much the total wanders about it, by kT² times the heat capacity — a relation between a fluctuation and a response that is not contained in the statement about averages. For a mole the wandering is a part in 10¹² and can be ignored; for a nanoparticle it is per cent and is measured; for a molecule it is of order the energy itself.

Tested in The temperature a molecule does not have · the equipartition reading path

“Pressure melting is irrelevant.”

It is irrelevant to a skate and decisive under a glacier. Three kilometres of ice produces about 270 atmospheres and depresses the melting point by two kelvin, which is why the base of an ice sheet is at its pressure melting point and wet, and why a glacier slides on water it made itself.

Tested in The melting curve that leans the wrong way · the phase change reading path

“The Soret effect is too small to be useful.”

A single stage separates by a few per cent, and stages compound. A thermogravitational column stacks the effect using convection, so a fraction of a per cent per gap becomes a complete separation over a few metres — which is how Clusius and Dickel separated chlorine's isotopes in 1938 with a glass tube and a hot wire.

Tested in The gradient that drives the other thing · the diffusion reading path

“The entropy of an isolated system never decreases.”

It never decreases on average, and for a small system it decreases on individual runs with a probability that is computable. Crooks' theorem gives the ratio of the probability of a trajectory producing entropy σ to that of its reverse producing −σ as e raised to σ over k, so an entropy decrease of ten k has a probability of about one in twenty thousand — small, non-zero, and observed.

Tested in The second law, with a probability attached · the entropy reading path

“Van der Waals' equation explains the departures from ideality.”

It reproduces the first correction with two fitted constants. The second virial coefficient is computable from the pair potential alone by one integral, with no fitting, and its temperature dependence is a curve rather than the straight line van der Waals' constants imply. Measuring the coefficient over a range of temperature is therefore a measurement of the intermolecular potential, which no two-parameter equation can be.

Tested in The first correction to the gas law · the kinetic theory reading path

“Two observables fail to commute because measuring one disturbs the other.”

Disturbance is a consequence and not the reason. The commutator is a property of the two operators computed with no apparatus anywhere in the calculation, and it is what decides whether a state with definite values for both exists at all. A relation about which states exist cannot be explained by what an instrument does to one.

Tested in The questions that can be asked together · the uncertainty reading path

“The shift is caused by a residual field leaking out of the solenoid.”

A leak would displace the fringes by an amount depending on how far out the paths run, and it would not be periodic. The measured displacement is periodic in the enclosed flux with period h/e exactly, and the definitive experiments used a closed toroidal magnet with no ends for the flux to leave by, and a superconducting shield thicker than the penetration depth.

Tested in The phase a magnet leaves on a path it never touched · the matter waves reading path

“A classical oscillation has a definite amplitude, so it has a definite energy.”

A coherent state of mean quantum number n has a spread of √n in that number, which is a definite classical amplitude and an indefinite quantum energy at the same time. The relative spread falls as 1/√n, so the two descriptions agree in the limit without ever agreeing exactly.

Tested in The state that swings like a pendulum · the correspondence reading path

“Attosecond ionisation experiments have measured the tunnelling time.”

They measure a delay between an ionising field and an electron's appearance, which requires a model to convert into a time spent under a barrier — and the answer depends on which of the definitions above the model implements. Different groups analysing similar data have reported both a substantial delay and a delay consistent with zero.

Tested in How long the crossing takes · the tunnelling reading path

“Sharing an entangled state among many parties gives each pair a share of it.”

The pairwise concurrence of an N-party W state is 2/N, computed here through Wootters' formula and checked against the closed form. Past about six parties no pair can violate a Bell inequality at all, although the state as a whole remains fully entangled.

Tested in What two have they cannot give a third · the entanglement reading path

“The size of an atom is a property of the electron.”

It is a property of the reduced mass of the pair. Positronium is twice the size of hydrogen and muonic hydrogen is a hundred and eighty-six times smaller, both computed here from the same expression with only the masses changed, and the muonic value matches the 285 femtometres the proton-radius experiments are built around.

Tested in Why an atom is the size it is · the atomic structure reading path

“A spacetime diagram shows what is happening.”

It shows one frame's coordinates on a Euclidean sheet, and that choice makes one worldline vertical and the others tilted. The symmetric drawing puts two observers at equal and opposite tilt with identical page units — the same physics, and none of the impression that one of them is at rest.

Tested in The diagram a ruler cannot read · the spacetime diagram reading path

“A spinning body has a centre, even if finding it needs care.”

Its centre of energy is displaced by S/Mc times the frame's speed, perpendicular to both the spin and the motion, so sweeping over all frames sweeps the centre over a disc. For an electron that disc is half a reduced Compton wavelength across, which is larger than any structure the electron is known to have.

Tested in The centre that is not a place · the relativistic dynamics reading path

“The neutrality of a current-carrying wire is the evidence for charge invariance.”

It is evidence, and it is weak. Conduction electrons drift at a fraction of a millimetre a second, the effect goes as the square of that, and the predicted field beside a 40 A wire is a hundredth of a microvolt per metre. An atom's inner electron moves ten million times faster, which makes an atom some 10¹⁴ times the instrument a wire is.

Tested in The one quantity a boost leaves alone · the field transformation reading path

“Rapidity is a convenient substitution that makes boosts add.”

It is the arc length of a genuine geometry. The circumference of a circle of rapidity radius r is 2π sinh r, and reading the curvature off the small-radius behaviour gives exactly −1 — which is why the rotation left behind by two boosts equals the triangle's area rather than some multiple of it.

Tested in The space that speeds live in · the velocity addition reading path

“A thrown ball follows a parabola because gravity pulls it down.”

Trial paths that join the same two events are compared by the proper time along them, and the maximum is at the real trajectory to better than a part in five hundred — found by searching, not assumed. Climbing higher runs the clock faster and moving faster runs it slower, and the balance point is Newton's parabola.

Tested in The longest way round is the shortest clock · the time dilation reading path

“Flow in a pipe has a parabolic profile.”

Only when viscosity reaches the axis within a cycle. Above a Womersley number of about five the core moves as a plug and the fastest fluid is not on the axis but near the wall — at 0.85 of the radius at a Womersley number of 12, computed from Bessel functions of complex argument summed from their series.

Tested in The shear that only reaches so far · the viscosity reading path

“Treating pressure as uniform around a small body is an approximation that introduces error.”

It introduces an error of about ρgh over the body's own height, which is a part in ten million for a cell at three metres and a part in a few thousand for a fish. The second is not an error to be minimised — it is the buoyancy, and the whole reason the fish floats.

Tested in The push that has no direction · the hydrostatics reading path

“Beverloo's law is a power law with exponent five halves.”

The local slope is measured on the drawn curve and reaches five halves only for wide openings. Near the jamming end the subtracted annulus makes the apparent exponent four or five, which is what a measurement over a narrow range returns — and the historical disagreements about the exponent are disagreements about which part of this curve was measured.

Tested in The hourglass that keeps time · the granular matter reading path

“The osmotic pressure of a solution is given by van 't Hoff's law.”

That is the pressure a perfectly selective membrane would develop. A real one develops σ times it, with σ between zero and one and dependent on the solute as much as on the membrane. The figures compute the pressure that stops the flow, which is σ times the ideal value, and it is how σ is measured.

Tested in The membrane that almost holds · the osmosis reading path

“A square tube and a round tube of the same size behave alike.”

Only above a contact angle of forty-five degrees. Below it the square tube's corners wick without limit while the round tube stops at Jurin's height, and the filament reaches a metre before it has narrowed to a micrometre where the round column has risen twenty millimetres.

Tested in The corner a liquid never stops climbing · the surface tension reading path

“An orbit is determined by its energy and its angular momentum.”

Those two fix the semi-major axis and the eccentricity and leave the orientation free — every rotation of an orbit about the centre has the same energy and the same angular momentum. What fixes the orientation is a third conserved quantity, and it exists only for an exactly inverse-square force.

Tested in The arrow that says which way the orbit points · the orbit stability reading path

“Two detectors confirm a detection, and a third improves it.”

Two give a ring on the sky, not a point: an arrival-time difference fixes only the angle to the baseline, and the figure computes the ring for a given delay. Three narrow it to two patches and a fourth chooses between them, so a network is what makes a detection pointable rather than merely credible.

Tested in What the instrument actually hears · the gravitational waves reading path

“A freely falling charge does not radiate, because it is locally inertial.”

Local inertiality settles what a co-falling detector finds and nothing else. A detector held stationary nearby is accelerating relative to the charge and does register radiation, and the two statements are consistent because they are statements about different detectors — the same structure as the Unruh effect, where one observer finds a thermal bath and another finds a vacuum.

Tested in Whether a charge on a table glows · the radiating charge reading path

“The Planck length is where quantum gravity becomes important.”

It is also where the standard procedure for measuring a length stops returning an answer. Adding a gravitational term to the uncertainty relation gives a sum with a minimum, and scanning the drawn curve over four hundred thousand momenta finds it at ℓ_P√2 — a floor rather than a regime boundary.

Tested in The length no experiment can resolve · the planck scale reading path

“The 4π period was measured, so a rotation of a spin has been shown to require 720 degrees.”

What was measured is the period of an interference pattern against a magnetic field. Reading that as a rotation angle requires the identification of Larmor precession with the action of the rotation group on the state, which follows from the Hamiltonian being minus the magnetic moment dotted into the field. That identification is well tested elsewhere and is an input here rather than an output.

Tested in The turn that has to be made twice · the spin reading path

“A bead-pull measures the field in a cavity.”

It measures a frequency shift proportional to the square of the field at the bead, so it recovers the magnitude and not the sign. It also depends on the bead: a dielectric bead reports the electric term alone, while a metal bead reports the electric term minus half the magnetic one and changes sign along the resonator. Which quantity has been measured is a property of the bead, not of the cavity.

Tested in The dent that raises the note · the standing waves reading path

“The force on the contents of a surface is the integral of the stress over it.”

That holds for static fields. In general the integral equals the force plus the rate of change of the electromagnetic momentum stored inside, and the second term is not always small: it is what makes a static-looking system carry momentum, and it is why an antenna recoils. The static statement is a special case in which one term happens to vanish.

Tested in The force read off a surface that touches nothing · the field energy reading path

“A guiding centre is where the particle really is.”

It is the leading term of an expansion in the Larmor radius over the length the field changes across. The figures here work at ratios of a few parts in ten thousand to a few per cent, and the last one measures the ratio explicitly. Near a magnetic null the ratio is unbounded and the guiding centre is not defined at all, which is why reconnection layers and shock fronts need the full orbit.

Tested in The drift that does not care what the charge is · the magnetism reading path

“Anti-lock braking works by keeping the wheel rolling.”

It works by holding the wheel near the maximum of the force-slip curve, which is at ten per cent slip or so rather than at zero. A wheel held at zero slip delivers no force at all. The controller is a maximum-seeking device operating on a curve whose right-hand branch is unstable, and the instability rather than the loss of force is what makes the problem hard.

Tested in The grip that needs a little slipping · the friction reading path

“A small leak makes a spin about the long axis impossible.”

It makes it temporary. The time scales as the reciprocal of the leak, which the figures verify across a decade, so a smaller leak buys proportionally more time and never buys immunity. That is the reason the failure mode is dangerous: an object can be spun on the ground, tested, launched, and only then flip.

Tested in The axis a leak of energy chooses · the rotation reading path

“There is a curved surface of best focus.”

There are three. Astigmatism splits the off-axis focus into two line images at different distances, lying on a sagittal and a tangential surface, with the Petzval surface between them at a fixed ratio: the tangential departure is exactly three times the sagittal. A lens can be made to place either one on a flat sensor and never both.

Tested in The flat scene that comes back curved · the imaging reading path

“The sky is unpolarised only towards the sun and away from it.”

Those are the only zeros the single-scattering model has, and three more were found by measurement: Arago above the antisolar point in 1809, Babinet above the sun and Brewster below it. They sit fifteen to twenty degrees away and move with the sun's height and the dust in the air, which is what makes them a measurement rather than a curiosity.

Tested in The pattern the sky is written in · the polarisation reading path

“The ratchet reaches Carnot efficiency at stall, so it is a reversible engine.”

The rate balance does give exactly 1 − T₂/T₁ at the stall load, which is computed rather than quoted here. But the pawl and the paddle are joined by a shaft, so the two baths are in permanent thermal contact, and that leak runs whether or not the wheel turns. With any leak at all the efficiency peaks below Carnot at a finite load and falls to zero at stall.

Tested in The engine a fluctuation cannot run · the heat engines reading path

“The recurrence is explained by the chain being close to an integrable system.”

That is one of two explanations and they describe different regimes. Near-integrability accounts for the trajectories staying on tori at low energy density; the solitary-wave picture accounts for the recurrence itself, since pulses that pass through one another unchanged bring the initial shape back. Both are right, and neither predicts the threshold's position without a calculation.

Tested in The energy that refuses to be shared · the equipartition reading path

“One opaque layer explains the thirty-three kelvin.”

It gives forty-eight, which overshoots. The atmosphere is not opaque across the whole infrared: about a third of the surface's radiation falls in a band where it leaves more or less directly, computed here from the Planck curve. A model with a fractional number of layers reproduces the number by being tuned to it, which is a different thing from explaining it.

Tested in The height a planet is seen from · the blackbody reading path

“The area increase in GW150914 confirms the theorem.”

It is a test rather than a confirmation, and its strength is a number. The areas before and after are measured from different parts of one signal — the inspiral fixes the masses, the ringdown fixes the remnant — and the increase is a few times the uncertainty on the difference. That is enough to have failed and not enough to be called a precision measurement.

Tested in The area that is not allowed to shrink · the horizons reading path

“The shift is a test of general relativity.”

It is a test of the equivalence principle and of little else. The rate difference follows from a photon losing energy climbing, which requires only that gravity acts on energy; every metric theory of gravity that respects the equivalence principle predicts the same first-order result. Distinguishing general relativity from its rivals needs the deflection of light or the perihelion advance, not this.

Tested in The clock that measures a height · the gravitational redshift reading path

“A null result means nothing was learned.”

Each limit excludes a class of theories that would have produced a departure at that level — a long-range force coupling to baryon number, a scalar partner to the graviton, a violation arising at a particular energy scale. The measurement is worth repeating because it has always come out the same way, which is exactly what would make any departure decisive.

Tested in The fall that does not depend on what is falling · the equivalence principle reading path

“The drag coefficient is 1 − 1/n².”

There is a further term, of a few per cent, because the medium sees the light Doppler-shifted and therefore responds with a slightly different index. Lorentz predicted it and Zeeman measured it. It is the part of the result that cannot be obtained from any argument about a dragged medium, because it is a statement about how two frames are related.

Tested in The drag that was only an addition · the velocity addition reading path

“The invariant mass identifies the parent.”

It identifies the four-momentum of whatever was combined. Combining two products that came from different decays gives a value too, distributed smoothly — the combinatorial background every such measurement fights. A peak is evidence of a parent; a single event's invariant mass is evidence of nothing.

Tested in The cone a decay cannot leave · the relativistic dynamics reading path

“The negative case is mathematically impossible.”

It is perfectly consistent: the transformations form a group, the relativity principle holds, and the geometry is Euclidean. What it does is let a chain of ordinary boosts reverse the time order of any pair of events, which the figure computes explicitly. It is excluded by an argument about causality, and that argument is an additional physical assumption rather than a consequence of the four.

Tested in The transformation that never mentions light · the spacetime diagram reading path

“A cell reaches osmotic balance by letting water in until the concentrations match.”

It cannot. The mobile-ion excess is positive for every positive fixed charge, so diluting the charge by swelling reduces the excess without reaching zero, and the cell swells until it bursts. Balance requires a second impermeant species outside, which is what a sodium pump manufactures by throwing sodium out as fast as it leaks in.

Tested in The swelling a membrane cannot stop · the osmosis reading path

“The coloured bands of a draining film are its equilibrium thickness profile.”

The equilibrium profile computed here spans seventy-five to fifty-eight nanometres over three centimetres, which is black throughout. The colours belong to a film thicker than equilibrium and still draining, and describing them needs the flow rather than the balance. The two are different calculations of different things.

Tested in The film that goes black before it bursts · the surface tension reading path

“The packing a column settles at is a function of how hard it is tapped.”

It is a function of how hard it is tapped and of what the column has been through. A freshly poured column follows one curve; once shaken hard it moves onto another, which it then retraces in either direction. The two differ by several per cent in packing fraction at the same tap intensity.

Tested in The pile that is never finished settling · the granular matter reading path

“A magnetic field exerts a pressure B²/2μ₀ across its lines and a tension B²/μ₀ along them.”

The two numbers are right and the sentence hides the structure. The traction on a surface is T·n, and computing it for a normal at any angle to the field gives a vector of magnitude B²/2μ₀ — the same at 0°, at 37°, at 90°, checked to machine precision at seven angles. What changes is the direction, and it changes by reflection in the field rather than by rotation with the surface. The tension quoted as B²/μ₀ is the difference between pulling out at one face and pushing in at the opposite one, not the traction on either.

Tested in The same force whichever way the surface faces · the flux freezing reading path

“Alfvén waves heat the corona.”

The flux is there and the dissipation is the open question. Measured transverse amplitudes of twenty to thirty kilometres a second, at coronal densities and a ten-gauss field, give 460 to 1,040 watts per square metre — enough for the quiet Sun's 300 and a coronal hole's 800, short of an active region's 10,000 by a factor of ten. And a wave that is not damped deposits nothing: the Alfvén wave's own dissipation length at coronal resistivity is longer than the solar system, so every candidate mechanism is a way of getting energy out of it that is not resistivity.

Tested in The wave that does not know what the gas is made of · the flux freezing reading path

“A short antenna is inefficient because it radiates poorly.”

It radiates poorly and that is not usually the binding constraint. At a hundredth of a wavelength the radiation resistance is 0.02 Ω and the capacitive reactance is tens of thousands of ohms, so almost all of the applied current goes into charging and discharging the near field rather than into the radiation resistance at all. The problem is a match between a resistor and a capacitor differing by six orders of magnitude, and it is narrowband for the same reason it is difficult.

Tested in The distance where a field changes its mind · the retardation reading path

“A phased array steers its beam by changing the phase of each element.”

True and it omits the failure. The steering law sin θ₀ = −α/kd puts the beam where the phase says, and the same law puts extra beams wherever the phase advances by another whole turn. For elements more than half a wavelength apart those extra beams land in real angles: the figure here counts three for a spacing of 1.5 wavelengths, which is 2⌊d/λ⌋+1, and they are why array elements are spaced at half a wavelength however inconvenient that is.

Tested in When the source is not heard all at once · the retardation reading path

“An excited atom decays at a rate fixed by the atom.”

The rate is a property of the atom and of what its field can couple into. In a resonator of quality factor Q and mode volume V the rate is multiplied by 3Q/4π²V in cubic wavelengths — ten thousandfold in a photonic-crystal defect — and between two mirrors closer than half a wavelength it falls to nothing, because there is no mode to emit into. Hulet, Hilfer and Kleppner measured that suppression in 1985. Whether a source radiates is not decided by the source.

Tested in The solution that is thrown away · the retardation reading path

“A mountain slows the wind that passes over it.”

It slows the atmosphere, mostly somewhere else. The wave carries a momentum flux of (π/4)ρNUh² upward without depositing any of it until it breaks, which for the largest ranges is in the stratosphere tens of kilometres up. Models that left the term out had stratospheric westerlies too strong by tens of metres per second, and the force is exerted on air that never came near the mountain.

Tested in The wave that is required to stand still · the stratification reading path

“Internal waves in a closed basin form standing modes, like sound in a room.”

For a rectangular basin they do. Tilt one wall and they do not: the reflection law contracts the ray on each circuit, so a trajectory traced through 260 reflections collapses from a spread of a third of the basin to one part in a million, measured off the traced path. The energy piles onto a limit cycle rather than filling a mode. The prediction is from 1995 and it was photographed in a tank two years later.

Tested in The reflection that changes the wavelength · the stratification reading path

“The Coriolis force deflects a moving body to the right in the northern hemisphere.”

It deflects it to the right continuously, which means the path is a closed circle rather than a bent line. Integrating the two-line system with no other force gives a trajectory that returns to its starting point after one inertial period, checked here to a millionth of its own radius. A description that stops at 'deflects' leaves out that the deflection has a period and that the net displacement over that period is zero.

Tested in The deflection that closes on itself · the circular motion reading path

“A pressure gradient pushes air from high pressure to low.”

It does, and in a rotating frame at small Rossby number the air ends up moving at right angles to the push. The trajectory here is integrated from rest under a steady eastward gradient: the parcel oscillates about a mean velocity that is northward, and the mean over a whole number of inertial periods matches the geostrophic value a/f to two per cent, measured off the path. Isobars on a weather map are streamlines, not directions of travel.

Tested in The ratio that decides whether the planet is turning · the circular motion reading path

“The refracted wave in a metal is absorbed after travelling a short distance.”

It is absorbed before travelling any distance worth the name. The ratio of the decay depth to the wave's own wavelength inside the material is 0.03 at most across the metals drawn, so the amplitude has fallen by a factor of e before a single oscillation is complete. Calling the result a wave is a courtesy, and the reflection is decided by a layer a few tens of nanometres thick.

Tested in The angle that is two angles · the refraction reading path

“Snell's law says the refracted ray is on the opposite side of the normal from the incident one.”

Snell's law says sin θ₂ = n₁ sin θ₁ / n₂, and the sign of n₂ decides which side. The construction has two intersections and the physical one is the one whose energy flows away from the interface, which for a negative index is the lower — so the refracted ray is on the same side, at an angle the signed law predicts and the geometry reproduces to nine decimal places.

Tested in The ray on the wrong side of the normal · the refraction reading path

“A neutrino mass would shift the endpoint of the beta spectrum down.”

It ends the spectrum early by exactly the mass and also bends the approach to it, and the bend is what is actually measured — an endpoint's absolute position is not known to a fraction of an electronvolt from any other source. The Kurie plot is exactly straight for a massless neutrino, verified here as a residual against a fitted line at a part in 10⁹, and a mass makes it curve down over the last few electronvolts.

Tested in The energy that did not all arrive · the decay reading path

“A daughter builds up until it reaches its parent's activity.”

Only if it is the shorter-lived. Three regimes follow from one ratio: a much shorter-lived daughter reaches exactly the parent's activity and stays there; a comparably short-lived one settles at λ₂/(λ₂−λ₁) times it, which is measurably above one; and a longer-lived daughter never reaches equilibrium at all, accumulating while the parent lasts and then decaying on its own schedule.

Tested in The chain that runs at its slowest member's rate · the decay reading path

“Stripping the electrons from an atom slows electron capture, because there is nothing left to capture.”

It does, and the more startling effect goes the other way. A beta decay whose available energy is too small to put an electron into the continuum can put one into an empty bound orbital instead — a channel that exists only when the orbital is empty. Rhenium-187 goes from 41.6 billion years as an atom to 32.9 as a bare nucleus, and dysprosium-163 is stable as an atom and decays in 47 days when stripped.

Tested in The half-life that chemistry can change · the decay reading path

“A current-carrying wire is neutral in the laboratory and charged in the frame of a moving charge, because the two lattices contract differently.”

That account is right and it is a story about a particular arrangement. The general statement is that (cρ, J) transforms exactly as (ct, x) does, so a wire with ρ = 0 and J ≠ 0 in one frame has ρ ≠ 0 in every other — at exactly one rapidity out of all of them, which the figure counts. No lattice appears in the derivation and the answer is the same.

Tested in Charge and current are one thing · the field transformation reading path

“E·B and E² − c²B² are invariant under a Lorentz transformation.”

They are, and the stronger statement is that they are the only two. An antisymmetric rank-two tensor in four dimensions has exactly two independent scalars — its contraction with itself and its contraction with its dual — so any other invariant anybody constructs is a function of those. That is why the three-way classification of fields is a classification and not a first attempt at one.

Tested in Six numbers, one object · the field transformation reading path

“Residual entropy is the entropy of a degenerate ground state.”

That is the ice case, and it is a count that nobody can change. A glass's residual entropy depends on the cooling rate — 15.6, 13.0 and 11.0 joules per kelvin per mole for the three rates computed here, for the same substance — because what is frozen in is whatever the liquid had when it stopped being able to relax. One number is a property of a spectrum and the other is a property of an experiment.

Tested in The entropy that depends on how fast it was cooled · the third law reading path

“The third law says the entropy of any substance approaches zero as the temperature approaches zero.”

It approaches k ln g, with g the degeneracy of the ground state, and that is zero only when the ground state is unique. Ice's 3.41, carbon monoxide's 4.6 and nitrous oxide's 5.8 joules per kelvin per mole are the values of R ln g for their respective counts, and the figures here compute each from the count and set it beside the calorimetry.

Tested in A law about spectra, not about heat · the third law reading path

“An absorption coefficient says how much energy a wave loses.”

It says how much momentum it loses as well, and it is the same coefficient. Absorbing a fraction of the momentum flux per unit length gives a body force 2αI/c in the medium, so the force and the heating rate are one measurement read twice — their ratio is ρc_p/c, which contains no intensity and is checked here at two intensities four decades apart.

Tested in Where the loudness goes · the attenuation reading path

“Iodine is used as a contrast agent because it is dense and has a high atomic number.”

Those help and they are not the reason. What matters is where its K edge is: two per cent by mass of iodine adds 47 per cent to water's attenuation just below 33.17 keV and 213 per cent just above it, from a change of a third of a kiloelectronvolt in the photon energy. An element of the same density whose edge sat at 5 keV or at 300 would be useless, because a diagnostic tube produces neither.

Tested in The steps in an absorption curve · the attenuation reading path

“Optical molasses traps the atoms it cools.”

The force depends on velocity and not on position, so nothing pulls a displaced atom back. A cooled atom random-walks through the light and eventually leaves it; it lingers only because its steps are short. Holding atoms in place needs a force that depends on where they are, which is what adding a magnetic field gradient to the same beams supplies.

Tested in The friction made of light · the radiation pressure reading path

“Turning the light down keeps lowering the temperature.”

Only while the light-shift wells can hold the atoms being cooled. Below about a hundred recoil energies of depth the straight line fails and the temperature climbs — 8.25 µK at 50 and 16.8 µK at 25 in the runs here — and underneath everything sits the recoil of one photon, ħ²k²/mkB, which is 198 nK for caesium and is not a property of any mechanism that absorbs and re-emits light.

Tested in The limit that belonged to a simpler atom · the radiation pressure reading path

“The best shot at a target is the one that reaches it with least effort.”

The least-speed shot is also the one whose landing point does not move to first order when the launch angle is wrong, because at that speed the two launches that reach the target have merged into one. A free throw at its least speed, 7.17 m/s, arrives 1.5 cm low for a 2° aiming error; at 8 per cent more speed the flatter of its two solutions misses by 11 cm for the same error. What it does not forgive is an error in speed.

Tested in One curve answers every slope · the projectile reading path

“Drag lowers the best launch angle because it steals range.”

Every angle loses range to drag, so the loss by itself moves nothing. What moves the answer is that drag bends the curves of equal range on the velocity map, outward most for steep launches, so the unchanged circle of throws touches them lower down. With drag at 2.4 times the weight the tangency found by integrating the curves is at 38.1°, the angle a direct search of the range also gives.

Tested in The best throw is a tangency · the projectile reading path

“The water condensed in a pore is ordinary liquid water at atmospheric pressure.”

It is under tension, at a pressure below the air's by (RT/Vₘ) ln RH, which is exactly the Laplace pressure of the meniscus holding it: −1.4 MPa at 99 per cent humidity, −14.5 MPa at 90, −95.1 MPa at 50. The homogeneous cavitation limit of about −140 MPa is reached at 36 per cent — and at that humidity the pores concerned are about a nanometre across, a few molecules of water.

Tested in The pore that fills from dry air · the capillarity reading path

“Raise the voltage far enough and any surface wets completely.”

The law says so, at 110 V for a 1 µm coating and 191 V for a 3 µm one. Real angles stop falling well before that, typically somewhere between 60° and 80°, and stay put however high the voltage goes. Charge trapped in the insulator, ionisation at the drop's sharp edge and the stability of the contact line have each been proposed as the cause, and none accounts for every experiment.

Tested in The angle a voltage can set · the capillarity reading path

“Texturing the surface of a solar cell helps by reflecting less of the sunlight.”

It also randomises the direction of the light inside, and light in a randomised slab can leave only through an escape cone holding 1/n² of its directions. For weak absorption the path is lengthened 4n² times — 51 for silicon — so a 100 µm silicon slab with a mirror behind it absorbs 65.7 per cent of 1100 nm light instead of 6.76 per cent.

Tested in The cone light has to find to get out · the etendue reading path

“A link designed to tolerate its fibre's mean polarisation delay will work.”

It will work most of the time. The Maxwell distribution has a tail: the delay exceeds twice its mean with probability 1.7 per cent and three times its mean with probability 4.2 × 10⁻⁵, about 22 minutes a year on a link in continuous use. Systems are designed to an agreed outage probability rather than to the mean, and three times the mean is the usual margin.

Tested in The delay that is a random variable · the dispersion reading path

“Falling through a hole in the Earth takes forty-two minutes, and the centre is at about 170 GPa.”

Those are the uniform-Earth figures: 42.2 minutes and 173 GPa. With the real density a body falls through a straight shaft in 38.2 minutes, passes the centre at 9.92 km/s rather than 7.91, and the pressure there is 364 GPa — because the dense core raises the gravity above it and is itself heavy.

Tested in The pull that grows on the way down · the Gauss's law reading path

“The Josephson effect is a delicate curiosity of superconducting physics.”

It is how the volt is realised. At 70 GHz one junction's first step is 144.7 µV, and 69,085 junction-steps in series give 10 V with an uncertainty set by the microwave frequency. Since 2019 the constant 2e/h has been exact by definition, and a Josephson voltage together with a quantum Hall resistance is how a Kibble balance weighs a kilogram against Planck's constant.

Tested in The voltage that is a frequency · the superconductivity reading path

“The larger the quantum numbers, the more classical the motion.”

At fixed anharmonicity a larger packet loses its classical look sooner — its collapse time falls as one over the square root of the number of quanta, from 30 periods at one quantum to 2.7 at a hundred — and the revival time does not change at all. What grows with the quantum number in a box or an atom is the number of orbits before the revival, 2n and 2n/3.

Tested in The return a classical cloud never makes · the correspondence reading path

“A superposition leaking energy into its surroundings loses its quantum character in proportion to the energy lost.”

The coherence between two packets decays at 2α² times the energy decay rate, so a superposition of amplitude 3 loses its negative regions eighteen times faster than it loses energy at first. And for every amplitude the last negative value disappears at exactly γt = ln 2, when half the energy is gone, whatever remains of the stripes.

Tested in The probability that goes below zero · the correspondence reading path

“Doubling a vehicle's power roughly doubles how fast it can go.”

At the top of the range the power needed grows almost as the cube of the speed, so doubling it raises the top speed by about the cube root of two. The car in the figures reaches 171, 220, 281 and 358 km/h at 50, 100, 200 and 400 kW, and a rider going from 250 to 400 W, 60% more, goes from 37.5 to 44.4 km/h.

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

“Knowing which events can influence which is enough to fix the geometry of spacetime.”

With one space dimension it fixes almost nothing. A warp of the two light-cone coordinates by different increasing functions preserves the causal relation of all 4,000 sampled pairs of events, yet bends straight worldlines and changes the proper time between timelike pairs by factors from 1.00 to 1.87. The claim becomes true, up to a uniform stretch, only with two or more space dimensions.

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

“The far future of flat spacetime is one moment that everything reaches.”

For anything slower than light it is one point, i⁺, and because the past of that point is the whole diagram, an inertial observer who waits long enough can receive a signal from every event. For light it is a whole edge: rays that set out from x = −2, −1, 1 and 2 end at four different points along ℐ⁺, which is why radiation is defined there rather than at i⁺.

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

“A regenerator improves a gas turbine by recovering heat that would otherwise be wasted.”

It does, and the size of the improvement follows from where it moves the heat. In the ideal cycle at pressure ratio 12 and 1,500 K, the regenerator means heat is bought only above 737 K, at a mean of 1,074 K rather than 989 K, and thrown away only below 610 K, at a mean of 437 K rather than 486 K. The loop is unchanged and the efficiency rises from 50.8% to 59.3%.

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

“Hot water stores useful energy in proportion to the heat it holds.”

It stores heat in proportion to its temperature above the surroundings, and work in a much smaller and growing share. With the surroundings at 20 °C, a kilogram at 40 °C holds 84 kJ of heat and 2.7 kJ of work, 3.3%; at 60 °C, 6.3%; at 80 °C, 251 kJ of heat and 22.7 kJ of work, 9.0%.

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

“Disorder destroys the states that live at the edge of a lattice.”

It depends on which symmetry the disorder keeps. Scattering every coupling by up to 30 per cent leaves the edge level at zero in 160 random chains, with a median below a hundred-thousandth of the coupling. Adding random site energies of up to 0.2 of the coupling, which couples each sublattice to itself, moves it off zero to a median of 0.041.

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

“An omnidirectional mirror reflects every angle.”

It reflects every angle from a particular medium. The same stack's window is 43.6% wide from air, 15.0% from an index of 1.34 like water, and gone from an index of 1.44 upwards, because a denser medium lets light bring a larger component of momentum along the layers and reach the places in the band diagram where the gaps move apart.

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

“Whether temperatures can be negative is a settled question.”

The count of states decides which way energy flows between large systems, and there it sides with Boltzmann. But Gibbs's entropy satisfies some exact relations that Boltzmann's satisfies only approximately — equipartition, and the invariance of the entropy under slow changes — and for small systems, or for defining the efficiency of an engine with one reservoir above infinite temperature, which definition to use is still argued.

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

“Radiation reaction is fully described by requiring the reaction force's work to equal the energy radiated.”

That requirement holds only between moments at which the product of acceleration and velocity takes the same value — over a whole cycle, or between two states of no acceleration. In between, the difference is the Schott term mτ a·v, which for an oscillating charge swings by the whole mean radiated power every quarter cycle.

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

“Lunar ranging's bound is set by how precisely the distance to the Moon can be measured.”

With phases spread evenly, six thousand normal points at 2 mm each would bound η near 6 × 10⁻⁶ by statistics alone. Published bounds sit near 10⁻⁴ because the fit has to separate the signal from reflector heating, atmospheric delay, the lunar interior and a monthly term in the Sun's own perturbation of the orbit — and because new and full moon, where the signal peaks, are the hardest phases to range at.

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

“The energy of the ions striking the wafer is the self-bias voltage.”

It is the sheath voltage averaged over each ion's transit, and the sheath voltage swings through a full cycle. Argon crossing a 1 mm sheath takes about 120 ns: at 1 MHz that is a tenth of a cycle and the ions land at two energies eighty electron temperatures apart; at 60 MHz it is seven cycles and they land within a few.

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

“Finding the potential at one point inside a region requires solving Laplace's equation everywhere in it.”

The potential at a point is the average boundary potential found by random walkers started there, so it can be estimated at that point alone. Four thousand lattice walkers from one point in a square give 0.405 ± 0.008 against 0.408 from relaxing the whole grid, and the error falls as one over the square root of the number of walkers whether the region is a square or a cube.

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

“The conformation of a chain in a flow is fixed by the flow rate.”

When a chain's drag grows as it unfolds, a coiled and a stretched chain are both stable over a window of rates — between 0.44 and 1 times the unfolding rate for a drag that grows fourfold. Raised slowly, the chain stays coiled up to 1 and jumps to 0.88 of its length; lowered, it stays stretched down to 0.44. Single DNA molecules showed the loop in 2003.

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

“Counterion condensation is a chemical binding of ions to the polymer.”

It is an electrostatic threshold with no chemistry in it. A single counterion beside a line charge has a Boltzmann weight that goes as distance to the power −2ξ, and whether the integral of that weight converges at the chain decides whether the counterion wanders off or is captured: at ξ = 0.9 it is within r₀ of a vanishingly thin rod with probability 0.063, at ξ = 1.1 with probability 0.996.

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

“An internal wave loses energy to smaller waves at whatever rate its amplitude sets, wherever it is.”

The fastest route, parametric decay into waves at half its frequency, needs those waves to exist, and an internal wave cannot oscillate below the local inertial frequency. For the semidiurnal tide the half-frequency waves are forbidden poleward of 28.8°. A pump at strength 0.2 grows a subharmonic by 0.157 e-folds per tidal period at 20° and leaves it bounded at 33°.

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

“Just before a stable cycle appears, a system is nearly periodic with small deviations.”

The deviations are not small; the bursts cover every value the map can take. What is nearly periodic is the calms between them, and their length grows without limit as the cycle's birth approaches: 245 iterates at a distance of 10⁻⁵, 804 at 10⁻⁶ and 8,125 at 10⁻⁸, a fitted slope of −0.506 against the −½ a parabolic channel requires.

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

“Strongly forward-scattering particles, like cloud droplets, let much more light through than particles that scatter evenly.”

Only when the slab is thin. Measured in transport mean free paths — the scattering length divided by 1 − g — slabs with mean scattering cosines of 0, 0.5, 0.85 and 0.95 transmit within 9.2 per cent of one another from four transport lengths onward, and on the same diffusion curve. A forward scatterer is an even scatterer with a longer step.

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

“Reflection happens at a surface.”

Reflection happens wherever the medium changes, and the change can be in time. Switching the index of a whole ring of grid cells from 1 to 2 at one instant, with no surface anywhere on it, sends back a pulse of amplitude 0.125 — the value continuity of D and B requires, to three decimal places.

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

“The laboratory energy spectrum of a two-body decay is flat.”

Only when the parent is unpolarised. Pions from taus moving at 0.8c fill the same interval, 313 to 2667 MeV, whatever the tau's spin, but a fully polarised sample is tilted: the slope read back from 60,000 decays is +0.993 for spin along the flight and −1.006 against it, where an unpolarised sample gives −0.010.

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

“The rate of a decay through several intermediate states is the sum of the rates through each.”

Only where their bands do not overlap. Three resonances whose rates on their own are 67.0, 25.5 and 33.2 per cent of the total add to 126 per cent, and where two bands cross, the share of decays landing there moves from 4.98 to 13.10 per cent as the relative phase of the two amplitudes changes, with every mass, width and strength held fixed.

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

“With enough laser power a sail can be driven as fast as desired.”

Power sets the push; diffraction sets how long it lasts. A beam focused from an aperture D spreads to the sail's radius at a distance πDa/2λ, 3 million km for a 1 km aperture and a 4 m sail, and beyond it the light caught falls as the square of the distance. The 1 g sail on 100 GW coasts at 0.267c, and quadrupling the aperture from 1 to 4 km raises that only to 0.441c.

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

“A magnetic field splits a spectral line into three: one unshifted and two shifted by the same amount either side.”

That is the pattern only when the field is strong compared with the atom's own fine-structure field. In 1 T, sodium's D1 line becomes four components at ±2/3 and ±4/3 of μB·B and D2 becomes six at ±1/3, ±1 and ±5/3. Only far above sodium's crossover of 36.8 T does the strength gather into three bands at −1, 0 and +1 times μB·B.

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

“A state with a large energy spread changes quickly.”

It may change quickly; the spread is permission, not obligation. Two levels driven off resonance by as much as the drive have the same spread as two driven on resonance, and their survival never falls below 0.500, while three equally spaced levels reach orthogonality only at 1.0887 times the limit and a coherent state never does, bottoming out at 0.0183.

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

“A substance always diffuses from higher to lower concentration.”

Only in a mixture of two. In Duncan and Toor's two bulbs, computed from the Maxwell–Stefan equations, nitrogen flows from the bulb with less nitrogen into the bulb with more from 0.1 to 6.5 hours, opening a difference of 0.1468, because carbon dioxide leaving one bulb drags it along. In a welded steel, carbon crosses from the side with 0.478 per cent into the side with 0.441, leaving 0.336 and 0.562 per cent either side of the weld after 13 days.

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

“A mode at a frequency where waves can propagate away must leak and have a finite width.”

Only if nothing cancels its leak. Two resonances leaking into one channel at rates 0.1 and 0.05, coupled with strength 0.2, have a combined mode whose decay rate is zero to 10⁻¹² at a detuning of 0.1414, while the other mode takes the whole 0.15. The mode sits in the continuum, and its two routes out cancel.

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

“Doubling the bandwidth doubles the reflection that has to be accepted.”

It halves the logarithm of the match. The floor is exp(−π/RCω), so at RCω = 0.5 a perfect network could reach 0.0019 and three elements reach 0.0146, while at RCω = 8 the floor is 0.675 and three elements manage only 0.748. Broadband matching of a strongly reactive load becomes exponentially harder, and the difficulty is in the load.

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

“Near-field and far-field diffraction are two different theories, Fresnel's and Fraunhofer's.”

They are one multiplication at different distances. The same plane-wave spectrum of a five-wavelength slit, each component multiplied by the phase it gains, gives the slit's own shape at 0.5 wavelengths, a bright centre at 25 and the diffraction pattern at 100 — conserving the travelling power to 10⁻¹⁰ and agreeing with a direct Fresnel integral to 2.75 per cent where Fresnel's approximation holds.

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

“The mean field gets the Curie temperature, so it is the right description with the numbers slightly off.”

It is wrong at the other end of the temperature range by more decades than any fit could hide. Mean field permits one excitation — a whole spin turned over against its neighbours — so it predicts a magnetisation deficit that is exponentially small at low temperature. The real excitations are long spin waves with no energy gap at all, and at 20 K the two accounts differ by 31 decades, computed here over a Brillouin zone.

Tested in What holds a magnet together is not magnetism · the magnetisation reading path

“A particle below the single-domain diameter stays in one domain, and the diameter follows from comparing the wall's cost with the saving from dividing.”

That comparison is what the last figure computes and it answers itself for the soft materials. Permalloy's crossing comes out at 4.0 nanometres against an exchange length of 5.7, so the wall the comparison priced could not have existed at that size. The estimate is valid where the wall is much narrower than the particle, which is where the anisotropy is high, and it says nothing about soft particles — whose single-domain size is set by the exchange length instead and is tens of nanometres.

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

“The magnetic viscosity of a sample is a relaxation with a characteristic time.”

It has no characteristic time, which is the content of the observation rather than a gap in it. A single barrier gives an exponential with a definite time in it; a spread of barriers gives a sum whose logarithm is straight, matched here to 0.0125 of the magnetisation by a least-squares line across eight decades, against 0.3332 for a narrow spread. The straight line is a signature that no time exists, so a sample that has been watched for a year will drift as much again in the next ten.

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

“Thermal stresses are relieved by allowing a little flexibility at the supports.”

A support ten times stiffer than the member — which is a loose connection by structural standards — still passes ninety per cent of the stress, and halving it needs a support as flexible as the whole member. The relief curve is drawn across eight decades of stiffness ratio and is flat exactly where a designer would want it steep. Either the connection is a mechanism that lets the member move freely, or the member takes essentially all of EαΔT.

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

“Tightening a bolt harder always makes a joint safer.”

It reduces the fluctuation the bolt sees — from 20 kN to 4 kN for the joint computed here — and buys nothing beyond the separation load, which for a 30 kN preload and a load factor of 0.20 is 37.5 kN. Past that the clamping is used up, the bolt carries the whole external load, and the curves meet. A preload also consumes part of the bolt's own strength, so the useful range is bounded at both ends and is specified rather than left to whoever holds the spanner.

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

“The collapse load is found by assuming a mechanism and equating work.”

That gives an upper bound, and a wrong mechanism gives an answer that is too high and therefore unsafe. Scanning the hinge position for a uniformly loaded propped cantilever shows the work balance returning anything from 11.657 to over 20 times Mₚ/L² depending on where the hinge is assumed; the true value is the least. Putting the hinge at midspan, which is the natural guess, overestimates by 2.9 per cent.

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

“Étendue is conserved because rays cannot be squeezed together — it is a fact about geometry.”

Geometry gives no smallest value, because nothing in a ray has a size, so the geometrical statement permits an étendue of zero and a beam concentrated without limit. The floor comes from the wavelength: a patch of phase space smaller than λ in one dimension is forbidden by the same Fourier relation that gives a slit its diffraction pattern. Étendue divided by λ² is a count, and a count stops at one.

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

“A solar cell's 33 per cent limit is a thermodynamic one.”

It is a consequence of having a single threshold. Detailed balance on a 5,762 K spectrum diluted to the solar constant gives a peak of 30.5 per cent at a gap of 1.25 eV, against a thermodynamic ceiling of 5 per cent for an unconcentrated heat engine and 85 for a concentrated one. The cell beats the engine and falls far short of thermodynamics, and the whole of the shortfall is that one gap cannot suit photons of every energy at once.

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

“The best profile is a parabola.”

The best exponent measured by tracing six rays through each of seventy profiles is 1.96, not 2, because the cancellation is exact only to first order in the index contrast. The minimum is sharp: an exponent wrong by a tenth costs a factor of ten in delay spread, which is why the profile of a graded fibre is measured on every preform. In a real fibre the material's own dispersion moves the optimum again, by a wavelength-dependent amount.

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

“The chain's thermalisation rate is a power of the nonlinearity, and the power can be measured.”

A power can be measured over the range a computation reaches, and it is 1.90 here — found by scanning for the exponent that collapses four runs onto one curve, which reduces their spread fourfold. Those runs are all above the knee. The weakly nonlinear regime the 1955 calculation sat in is decades of amplitude below them, its predicted exponent is far higher, and no run of a length anybody will wait for reaches it.

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

“Averaging for longer always improves a measurement.”

It improves it as the square root of the time, so a hundredfold gain costs ten thousand times as long, and it only helps against noise that is white. Against a baseline that itself wanders, the total has a least value and averaging past it makes the measurement worse — computed here at 6.6 nanovolts after 16 seconds for a ten-kilohm source with a drift of three nanovolts per root second.

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

“The same method gives the mass of any gravitationally bound structure.”

It gives the mass of a structure that has had time to become virialised, which means several crossing times. A globular cluster has had fourteen thousand and Coma nine; a supercluster has had 0.14, so its parts have not completed one traverse, are still separating with the expansion, and their velocity spread is not a virial dispersion. The condition needs only a size and a speed and should be checked before the mass is quoted.

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

“The problem is that the vacuum energy is too small.”

There are two problems and only the first is about size. The second is that a density which does not dilute is comparable, now, with one that falls as the cube of the expansion: the crossing is located here by bisection at a redshift of 0.31, a few billion years ago out of fourteen. A hundredfold change in the constant would put that crossing far in the past or the future, and nothing about the estimate explains why it is here.

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

“Large extra dimensions solve the hierarchy problem.”

They remove the tuning and move the question. A Planck cutoff needs the Higgs mass's bare value and its top-quark correction to cancel to one part in 3 × 10³²; a cutoff at a TeV needs no cancellation at all. But the size of the extra dimensions then needs its own explanation — the two-dimension case requires a compactification radius thirty orders of magnitude above the fundamental length — and none has been given.

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

“Losses in a fibre can be handled the way they are for classical signals, by amplifying.”

A quantum state cannot be copied, so there is no amplifier. A photon survives 22 km of fibre with probability 1/e, so a direct link over a thousand kilometres delivers 1.8 × 10⁻¹⁰ pairs a second from a source firing ten thousand million times a second — one every hundred and seventy years. Eight segments joined by swaps divide the exponent rather than the rate and deliver four million a second, a gain of 2 × 10¹⁶.

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

“The area law is a statement about entanglement.”

It is also a statement about what is computable, about which correlations can exist, and about how sharply a phase transition announces itself. The block entropy of a gapped chain is a constant and of a critical one is a logarithm whose coefficient is the central charge — so measuring an entanglement identifies which universality class the transition belongs to, from a quantity that is not a correlation function of anything.

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

“Light waves in vacuum add exactly, because there is no medium to respond nonlinearly.”

Maxwell's equations are linear and quantum electrodynamics is not. A photon can make a virtual pair and the pair can absorb a second photon, so two beams scatter — with a cross-section of 10⁻⁶⁸ square metres at optical frequencies, which is the smallest in physics and is not zero. A magnetic field polarises the pairs and the vacuum acquires two refractive indices differing by 2 × 10⁻²¹ at sixteen tesla.

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

“Stratification makes a submerged body stable at its neutral depth.”

Only if the fluid's density rises with depth faster than the body's own does. For a body as compressible as water the critical gradient is 4.62 parts per million per metre, computed here, and that is close to real ocean stratification — a strong thermocline exceeds it and the deep ocean does not. Both cases are drawn and their slopes have opposite signs.

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

“A rotating observer has no notion of simultaneity.”

No global one. The failure of a synchronisation to close around a region is 2ΩL²/c², so a region is consistent to whatever precision is demanded provided it is small enough — 785 kilometres on the rotating Earth at the nanosecond level, 207 metres inside a laboratory centrifuge. The useful question is not whether such an observer has a now but over what distance, and the distance is computable.

Tested in How big now is · the simultaneity reading path

“The technique measures the structure of the material.”

It measures the first two or three shells around one chosen element and nothing else. The photoelectron travels about four and a half ångströms between inelastic collisions at the worst point of its range, so a neighbour six ångströms away contributes seven per cent of what one at two ångströms does. Beyond that the sample could be anything.

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

“A band gap is a number a material has.”

Four ordinary recipes applied to one spectrum of gallium arsenide return values spanning several tens of millielectronvolts — more than eight times its own Urbach energy — because the straight portion that is extrapolated is straight only well above the tail, and how far down the fit reaches decides the answer. The recipe is part of the number.

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

“Acoustic mixing works because the sound agitates the fluid.”

What mixes is the steady circulation, not the oscillation. A fluid parcel oscillating with amplitude U/ω travels about a hundred and sixty micrometres at a kilohertz and returns; the drift carries it away and does not bring it back. In water the drift beats diffusion for channels wider than sixteen micrometres at a metre-per-second amplitude, and that crossing is what the technique is bought for.

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

“Viscosity is a function of temperature.”

It is a function of temperature and of pressure, and over the range a machine actually reaches the pressure term is much the larger. Heating a mineral oil from twenty degrees to a hundred divides its viscosity by about thirty; squeezing the same oil to the 1.32 gigapascals computed here for a loaded line contact multiplies it by more than ten to the eleven.

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

“The bound is a theorem.”

It is a conjecture. It is proved within a family of strongly coupled theories that have a gravity dual, it is known to be violated in theories with enough species of field, and it has no proof for ordinary matter. What is not in doubt is the measurement: nothing anybody has measured lies below it, and the two systems that come closest reach it from far apart.

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

“Measuring the ringdown frequency measures the remnant's mass.”

It measures one combination of mass and spin, and the two cannot be separated by a pitch alone: a hole at 90 per cent of maximum spin rings 1.4 times higher than a non-spinning one of the same mass, so any pitch is consistent with a curve of masses. What breaks the degeneracy is a second number — how fast the mode dies, or the frequency of a second mode.

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

“The memory is the trace left by the masses that flew apart.”

That is the linear part and it is the smaller one for a binary that merges. The larger contribution is sourced by the gravitational waves themselves: the radiation carries energy away, energy gravitates, and an unbound flux of energy leaves the same kind of permanent imprint an unbound mass would. The wave is the source of its own memory, which is why the effect is nonlinear in a theory whose waves are usually treated as linear.

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

“A magnetic circuit is closed, so a small air gap makes little difference.”

A one-millimetre gap in a three-hundred-millimetre core of relative permeability five thousand leaves under six per cent of the field the closed core had. The circulation of H is fixed by the winding; H in the gap exceeds H in the core by the relative permeability; so a gap one part in three hundred of the path takes nearly all of the available circulation.

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

“The circulation of H depends on the path taken.”

It depends on one whole number: how many times the path winds round each current, with a sign. A path that goes round twice counts twice, one that goes round and returns counts nothing, and one enclosing two opposed currents counts zero however it is shaped. Neither the path's length nor its distance from the wire appears anywhere in the answer.

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

“Static friction accelerates a car, so it supplies the energy.”

It supplies the external force and none of the energy. The contact patch of a rolling tyre is instantaneously at rest, so the friction force acts through no displacement and its work is zero. The energy comes from the fuel. Both requirements are real and separate: an external force is needed to change the momentum, and an internal energy source is needed to change the kinetic energy.

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

“A slow compression and a fast one differ only in how long they take.”

They differ in the answer. Compressed to a quarter of its length, the ball drawn here ends at 15.7 times its energy when the wall creeps and 41.0 times when the wall moves at nine tenths of the ball's speed — the fast piston does extra work, which is why a fast compression is irreversible and why it and a slow expansion are not a cycle.

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

“Apodisation improves an image.”

It trades. Every decibel of sidelobe suppression is bought with a wider core: a Hann taper is 18 decibels quieter than a hard edge and its core is 1.63 times wider, so two equally bright stars that were just resolved are no longer. Which is wanted depends on whether the second object is comparably bright or ten million times fainter, and the two cases want opposite ends of the curve.

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

“A mirror polished to twenty-five nanometres is a good mirror.”

It depends entirely on the wavelength it is used at. The same surface gives a Strehl of 0.32 at 400 nanometres and 0.998 at five micrometres, because a surface error costs twice itself in wavefront and the cost in waves is that divided by the wavelength. A surface specification quoted without a wavelength says nothing at all.

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

“Whether a particle has two states or three is decided by whether its mass is exactly zero.”

Nothing observable turns on an exact zero. The third state's effect on any process scales with the rest energy over the total energy, which for a 91-GeV particle at 5 TeV is 0.018. So a very light massive photon would be indistinguishable from a massless one in any experiment above its mass, and every bound on the photon's mass is quoted with the length scale over which it was tested.

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

“An electron's magnetic moment therefore cannot be measured.”

It is measured to twelve significant figures, by a completely different arrangement: a single electron in a trap, where the spin's precession is compared with the cyclotron motion of the same electron in the same field. What is impossible is separating a free electron's spin states by deflection, which is a statement about one method.

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

“Raising the driving temperature raises the performance without limit.”

It approaches a ceiling. The engine factor is one minus ambient over the driving temperature, which tends to one, so the whole expression tends to the fridge coefficient alone — 10.1 for the case drawn. Doubling a 450-kelvin driving temperature buys 40 per cent, and there is nothing above the fridge coefficient at any temperature whatever.

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

“The figure of merit can be raised without limit by improving the material.”

The electronic thermal conductivity is tied to the electrical one by a ratio nearly the same for every conductor, so with the lattice contribution removed entirely the group can be no larger than the Seebeck coefficient squared divided by that ratio. Only the lattice part is genuinely free, which is why the last forty years of the subject have been about scattering phonons rather than about electrons.

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

“A moving body's temperature is lower by the Lorentz factor, as Planck and Einstein showed in 1907.”

Their derivation is internally consistent and so are the two later ones that give the opposite and no change at all. What the three differ over is what heat means under a boost, and since the first law ties heat to temperature, a choice about one is a choice about the other. All three agree at rest and none is refutable, because no thermometer reads any of them.

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

“The second law is frame-independent because entropy is.”

Entropy being invariant makes the statement well posed and does not prove it. What makes the second law frame-independent is that entropy is invariant AND that the events it compares are timelike-separated, so every observer agrees which came first. Two spacelike-separated entropy increases have no agreed order, and the law says nothing about their sequence — which is fine, because it makes no claim about it.

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

“The drag is radiation pressure.”

It is the anisotropy of radiation pressure created by the motion itself. A body at rest in isotropic radiation feels pressure from every side and no net force. The force appears only because moving makes the radiation anisotropic in the body's own frame, so it is proportional to the speed rather than to the intensity alone, and it vanishes at rest however bright the bath.

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

“The action is a number that belongs to a path.”

It is also a function of the endpoint. Fixing where the motion starts and recording the action of the true path to every place and time gives one function whose gradient is the arriving momentum and whose time derivative is minus the arriving energy. The figures check both against trajectories integrated separately, to a part in ten thousand.

Tested in The action that knows where every path ends · the least action reading path

“Solving the Hamilton–Jacobi equation gives the motion everywhere.”

A smooth solution exists only until two trajectories of the family reach the same point. Past the envelope of a fan of throws the function has two branches, and at the focus of an oscillator it has no value at all; the figures draw both, computed from the trajectories rather than from the equation.

Tested in The action that knows where every path ends · the least action reading path

“The best free throw is the one that needs the least speed, because there an error in angle costs nothing.”

At the least-speed launch, 51.4° for a release 0.95 m below and 4.19 m short of the hoop, an error of angle does cost almost nothing — and the ball comes down at 38.5°, leaving its centre ±3.7 cm of room. For a thrower scattering ±0.05 m/s and ±1° the chance of a clean pass is 37 per cent aimed there and 58 per cent aimed at 58.5°, computed both from the exact window of speeds at every error of angle and by counting over a grid of both scatters.

Tested in The throw most likely to go in · the projectile reading path

“Aiming well is a matter of hitting the middle.”

Every aim in the figures is at the middle. What changes with the aim is the shape of the region of launches that succeed and how it lies against the thrower's scatter, and the best aim is where the most of the scatter falls inside — which depends on whether that thrower errs more in speed or in angle.

Tested in The throw most likely to go in · the projectile reading path

“The difference between an orbiting and a thrown clock is a weak-field curiosity.”

The exact Schwarzschild calculation keeps the same order down to 6.5 GM/c², where over one period the orbiting clock keeps 73.4 per cent of a distant clock's time, a held clock 83.2 per cent and a thrown one 88.4 per cent. Far out the three losses times the radius settle to 3/2, 1 and 0.68, matching the weak-field figures.

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

“Light that climbs out of a gravitational field loses energy, so thermal light arrives with the right colours but dimmer.”

It arrives as a Planck spectrum at a lower temperature, which is a different shape from a dimmed one. Carrying a 3000 K spectrum up a field that lowers every frequency by 0.8 gives a spectrum that matches Planck's law at 2400 K to one part in 10¹⁴, and the bottom spectrum scaled to the same peak height visibly does not.

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

“The equilibrium constant is a property of the molecules alone.”

K is, but how far a reaction goes is not. Dinitrogen tetroxide with ΔG° = +4.77 kJ/mol at 298 K is 88.6 per cent dissociated at 0.01 bar, 18.8 per cent at 1 bar and 6.0 per cent at 10 bar, because each gas's chemical potential rises with the logarithm of its own partial pressure and the dissociation makes more molecules.

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

“Light from a light-emitting diode is not thermal radiation.”

Above its band gap an ideal diode at temperature T with a voltage V across it emits with exactly Planck's occupation, 1/(exp((ε − qV)/kT) − 1). The voltage multiplies every point of the thermal spectrum by exp(qV/kT) — 21.8 powers of ten at 1.3 V across a 1.42 eV gap at 300 K — and leaves its slope set by the lattice's kT.

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

“The coincidence dip at a beam splitter shows that light is made of photons.”

Only a dip deeper than one half does. Two independent laser beams with a random relative phase, sampled over 40,000 trials, empty the dip to 0.499 of its far value, and two thermal sources to 0.669; no classical field can go below one half. Single identical photons reach zero.

Tested in The outcomes identical photons refuse · the photon reading path

“Identical photons in an optical network simply bunch together.”

In a symmetric three-way splitter with one photon in each input, identical photons never produce six of the ten possible outcomes — those whose output labels do not sum to a multiple of three — and put a third of their probability into one photon per output. Distinguishable photons produce all ten.

Tested in The outcomes identical photons refuse · the photon reading path

“Injecting squeezed light into a gravitational-wave detector lowers its noise.”

Squeezing in phase lowers the photon-counting noise at high frequency and raises the radiation-pressure noise at low frequency by the same factor. In a model interferometer whose two noises cross at 70 Hz, 6 dB of phase squeezing is worse than none below 99 Hz; only squeezing that rotates with frequency lowers the noise everywhere.

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

“The neutron interferometer measures the neutron's weight.”

It measures a phase that is 2πm²gλA sin α / h², which contains the mass twice, Planck's constant twice and the wavelength once. With the wavelength measured separately the experiment tests the combination m²g/h², so it is a test of quantum mechanics in a gravitational field as much as of gravity.

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

“The flattening of a planet's gravity field tells how flattened the planet is.”

It fixes J₂, and very different bodies share a J₂. A uniform body flattened by 0.34 per cent and a perfect sphere carrying 0.268 per cent of its mass as an equatorial ring both have J₂ = 0.001339; they differ at the next order, where J₄ is −3.85 × 10⁻⁶ for one and −0.001 for the other.

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

“A disturbance that is not exactly a soliton spreads out and disappears.”

A hump of height 6 shaped as sech²x, integrated in the Korteweg–de Vries equation, leaves two crests of heights 8.00 and 2.00 running apart at speeds 16 and 4, with only a small ripple spreading behind. Read as a well, the same hump holds levels at κ = 2 and 1, and each level becomes a soliton of height 2κ².

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

“Light moves from one waveguide to its neighbour by leaking across the gap, and a longer coupler transfers more.”

Between identical guides the light beats: a coupler of two 0.5 µm silicon guides 250 nm apart transfers everything at 37.6 µm and gives it all back by 75 µm, oscillating with length for ever. Only between guides tapered so that their propagation constants cross does the transfer grow steadily with length — above 95 per cent from 195 µm.

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

“A photonic time crystal amplifies light without any source of energy.”

The energy comes from whatever modulates the index. A wave at the centre of the gap grows by a factor of 73 over thirty periods of a 0.2-deep modulation, and the equation contains nothing that limits the growth; every joule is work done by the drive changing the permittivity against the field already present.

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

“A thick disordered medium transmits in proportion to one over its thickness, whatever the scattering.”

Only when the waves' phases are thrown away. The same random stacks of layers with every surface's reflection added as intensities transmit 1.0 per cent at 400 layers, falling as one over the thickness; computed with interference kept, by multiplying transfer matrices, the typical transmission falls by a factor of e every 17 layers and is below 10⁻¹⁰ at 400.

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

“More engine power always makes a ship go faster.”

Where the resistance has a hump, a steady thrust meets it more than once. A thrust that crosses the resistance curve at 0.94, 1.00 and 1.82 knots leaves a ship accelerating from rest at the first crossing, below the hump, although a steady state at 1.82 knots exists; only a thrust above the top of the hump, or a push past it, reaches the fast branch.

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

“A perfectly conducting plasma conserves its field line topology, so it cannot relax at all.”

Perfectly, it cannot. A small resistivity lets field lines reconnect in thin sheets, which destroys the linkage of every individual flux tube while barely touching the total helicity: the helicity lost per unit time is bounded by the square root of the resistivity times the energy lost, so as the resistivity falls the energy can go while the helicity stays.

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

“Confirming the size of the gravitational redshift confirms that it is a property of time.”

It confirms that one kind of clock shifts by the predicted amount. Universality is a separate claim: that every clock shifts by the same amount. The Sun's potential at the Earth swings by 3.3 × 10⁻¹⁰ each year, so two unlike clocks side by side would acquire an annual ratio term of Δβ × 1.65 × 10⁻¹⁰ if they responded differently — a quantity no measurement of a single clock's shift can see.

Tested in The clocks that must all slow together · the gravitational redshift reading path

“Doubling the charge of the counterions doubles how strongly they correlate.”

The coupling parameter Ξ = 2πq³ℓ_B²σ goes as the cube of the valence. At 0.5 elementary charges per square nanometre it is 1.6 for monovalent ions, 12.8 for divalent, 43.2 for trivalent and 102 for tetravalent — which is why sodium never produces the attraction and calcium, spermidine and spermine do.

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

“A plasma in a magnetic field leaves the field as it found it.”

It weakens it. The current its pressure gradient drives is diamagnetic, and with pressure balance B²/2μ₀ + p constant, a peak pressure of 0.6 of the outside field's magnetic pressure lowers the field at the centre to 0.63 of its outside value. The flux change through a loop round the plasma is −μ₀W⊥/B₀ at low pressure: 33 milliwebers for a column at 100 kPa in 3 T, which is how the stored energy of a fusion plasma is weighed.

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

“Shielding a qubit from charge noise requires removing the stray charges.”

It can be done by making the circuit indifferent to them. The dependence of the levels on offset charge falls as exp(−√(8EJ/EC)), with EJ the Josephson energy and EC the charging energy: the ground level moves by 0.263 of a transition as the charge sweeps at EJ/EC = 1 and by 3.0 × 10⁻⁸ at 50, and the first excited level becomes 10⁵ times less sensitive.

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