True where it was derived, used where it was not

“A pendulum's period does not depend on how far it swings.”

The small-angle substitution plotted against what it replaces. sin θ and θ are drawn on one pair of axes with the gap between them marked, so the amplitude at which the approximation stops being free is read off rather than asserted.

Tested in The pendulum, and the small lie that makes it simple · the pendulum reading path

“Forty-five degrees is the angle that throws furthest.”

It is, in vacuum, and the drawn ranges follow sin 2θ exactly. Add one velocity-dependent force and the privilege evaporates: with quadratic drag there is no closed-form trajectory at all, and the optimum falls from forty-five toward the low thirties.

Tested in The angle that throws furthest, and why nobody notices · the projectile reading path

“Kinetic energy is conserved in a collision.”

Only when the restitution is exactly one. The opening figure is drawn at e = 0.6, the restitution of a tennis ball on a hard court, and prints the energy before and after. Momentum survives every collision; energy is the quantity deliberately given up, and giving it up is what makes the method work.

Tested in Collisions are easier than forces, and momentum is the reason · the momentum reading path

“Every force has a potential energy.”

The work done against friction depends on the distance travelled, so a long route costs more than a short one and a round trip does not return to zero. No function of position can describe that, and the landscape picture — which is the whole subject of the essay — simply does not exist for it.

Tested in The hill that gives it back, and the forces that do not · the energy reading path

“Light travels in straight lines.”

Through a narrow enough gap it does not, and no amount of ray tracing predicts the spreading. One ratio — the wavelength over the aperture — decides when the ray model may be used, and the essay draws the pattern at several values of it so the boundary is visible rather than stated.

Tested in Where rays stop being enough, and a shadow acquires a bright centre · the diffraction reading path

“Electric fields fall off as the inverse square of distance.”

A point charge does, a line gives an inverse, and a plane gives a constant — all three from the same law, drawn on one pair of axes. The exponent belongs to the arrangement of the charge and not to the force between charges.

Tested in The shape decides the falloff, and the force law never changes · the Gauss's law reading path

“The flux rule is Faraday's law.”

It agrees with it in every ordinary circuit and fails outright where the loop is ambiguous — a conducting disc rotating in a field with a sliding contact has no well-defined circuit to take the flux through. The law is about the field's rate of change; the flux rule is a consequence with narrower hypotheses.

Tested in The field that makes the other, and only while it is changing · the induction reading path

“Mass is converted into energy.”

The equation says something stricter: a mass *is* an energy, and the mass of a system already contains the energy of its parts. Burning coal converts about a ten-billionth of its mass — the same arithmetic, with no conversion event anywhere in it.

Tested in Mass is a form of energy, which is not the same as a source of it · the Mass-energy reading path

“Pressure is transmitted equally throughout a fluid.”

A CHANGE in pressure is. The pressure itself varies with depth by ρgh throughout, which the previous rung's figures draw explicitly — a press two metres tall carries a twenty-kilopascal difference between its ends that the principle says nothing about, and that is 6 per cent of a typical working pressure.

Tested in Force multiplied, and nothing gained · the hydrostatics reading path

“The righting arm is GM times the sine of the heel angle.”

That is the small-angle limit and it is conservative rather than merely approximate. The figure solves the waterline at each angle so that the displaced volume is conserved, measures the arm off the resulting geometry, and finds it larger than GM·sin φ from the first degree onward — because a wall-sided hull gains a second term growing as the square of the tangent.

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

“Flow rate is proportional to the pressure difference driving it.”

True while the flow stays orderly, and the proportionality fails once it does not — beyond which the flow rises roughly as the square ROOT of the pressure instead, so doubling the pressure buys about forty per cent more. Which regime applies is decided by a ratio this site does not own; what this page owns is the law that holds below it and the fact that it has a boundary.

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

“Every liquid has a viscosity.”

Only a Newtonian one does. The figure divides stress by shear rate — which is what an instrument reports — and plots it against the rate it was measured at: the Newtonian line is flat and the shear-thinning one falls by a factor of 7.1 across a single decade. Quoting one number for such a fluid is quoting the rate it happened to be stirred at.

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

“Waves on water travel at √(gh), set by the depth.”

That is the shallow-water limit, and it applies only when the wavelength is long compared with the depth. The figure draws the full relation at 4 m depth: a 2 m wave travels at 1.77 m/s, which is nothing like √(gh) = 6.26 m/s, and the two agree only once the wavelength is many times the depth.

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

“The mean free path is 1/(√2·nσ).”

The √2 belongs to the case where the TARGETS are moving too, and it comes from averaging relative speeds over the Maxwell distribution. For a fast particle crossing a field of effectively stationary scatterers the factor is absent — which is why the figure, which flies a probe through a fixed field of discs, is checked against 1/2nr and not against a formula with a √2 in it.

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

“The distance a diffusing particle covers is proportional to the time.”

It is proportional to the SQUARE ROOT of the time, so covering ten times the distance takes a hundred times as long. The figure fits a straight line through the origin to the measured mean square displacement of 700 walkers and returns a slope of 1 within a few per cent — a parabola is what a drifting particle would give, and that difference is how the two are told apart.

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

“Light bends because it falls, like anything else.”

Falling supplies half. The other half comes from the curvature of space itself, and it is invisible to anything moving slowly: the space term contributes in proportion to (v/c)², so it is negligible for a planet and equal to the falling term for a photon. The equivalence principle, which is a statement about a local frame, cannot see it at all.

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

“The Schwarzschild radius is where the escape velocity reaches the speed of light.”

The Newtonian calculation gives the right number and cannot be right: it uses a flat space, a Newtonian kinetic energy for a massless particle, and the idea of a body that rises, slows and falls back — which is not what light does. Two errors of the same size cancel. The correct statement is that √(1 − rs/r) reaches zero there, so no static observer exists at all.

Tested in The surface that only lets things in · the horizons reading path

“The pressure of light on a surface is the intensity divided by c.”

That is the value for a perfect absorber at normal incidence. A perfect mirror returns the momentum instead of taking it, so it receives twice as much; a real surface lies between, and a surface at an angle receives a component. The factor of two is the same one that separates a bouncing ball from a sticky one in an ordinary collision.

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

“Radiation depends on speed, so a fast charge radiates more.”

The Larmor formula contains the acceleration and not the velocity: a charge moving at any constant speed in a straight line radiates nothing whatever, in every frame. What makes a fast charge in a magnet radiate so much is that circular motion at speed v round radius R has acceleration v²/R, and the relativistic form adds powers of γ on top.

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

“The mean free path formula is a result about gases.”

It is a result about a straight line drawn through a random field of targets, and nothing in the derivation knows what the targets are. The same 1/nσ gives a molecule's path in air, a neutron's in lead, a photon's in a stellar interior and a neutrino's anywhere — over twenty-eight decades in the figure, on one straight line of slope minus one.

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

“Flux freezing explains the strong fields of collapsed objects.”

It supplies a floor and not a prediction. The estimate assumes perfect conductivity, no reconnection and no generation, and every one of those fails somewhere: the same argument applied to a laboratory plasma is wrong within milliseconds. What flux freezing gives is the field a collapse cannot avoid producing, which is a lower bound with a stated assumption attached.

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

“The potato radius is about 200 km, so anything larger is round.”

The crossing goes as the square root of the material's crushing strength, which is known to a factor of a few at best and depends on temperature, composition and how long the load has been applied. The figure computes 282 km at 200 MPa and 630 km at 1 GPa for the same density. It is an order of magnitude, and treating it as a boundary is precision the argument does not have.

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

“The Planck mass is tiny.”

It is 2.18 × 10⁻⁸ kg — about the mass of a flea's egg, and 10¹⁹ times a proton's. The figure finds it by bisection at 1.54 × 10⁻⁸ kg, which is the Planck mass divided by √2. It is the one Planck unit that lands in ordinary experience, and its enormity relative to every particle mass is the reason gravity between particles is negligible.

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

“The travelling twin ages less because of the acceleration at the turnaround.”

The acceleration marks the asymmetry, and does not cause the difference. Change the outbound distance while keeping the turnaround identical and the age gap changes in proportion to the distance — the difference accumulates during the coasting, not during the turn. A version with three inertial clocks and no acceleration anywhere gives the same answer.

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

“The transverse shift is always a redshift.”

It is a redshift when 'transverse' means the angle in the receiver's frame at the moment of reception, and a blueshift of γ when it means the angle in the source's frame at emission. The two definitions pick out different events, and quoting one without saying which is the standing ambiguity in the subject.

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

“Hawking radiation means a black hole slowly cools as it evaporates.”

It heats. The temperature goes as 1/M, so losing mass raises the temperature, which raises the loss rate — a negative heat capacity, and the opposite of every ordinary object. The lifetime goes as M³ and the process ends in a burst rather than fading out, which the figure's two straight lines of slope −1 and +3 show directly.

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

“The force of friction is μ times the normal force.”

True of a surface that is sliding and false of one that is not, which is most of them. The figure draws the response: while the block is stationary friction returns exactly the force applied to it, so the curve is the 45° line and μ appears nowhere. Only at the break — 30.0 N for a 50 N load at μs = 0.6 — does the coefficient enter, and after it the force is a different constant, μk·N = 22.5 N.

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

“Simple harmonic motion is what happens when the restoring force is proportional to the displacement.”

That is the definition, not the reason, and stated alone it makes the harmonic oscillator a special case rather than the generic one. No real system has a force exactly proportional to displacement. What every stable system has is a potential minimum, and the leading term of any function about a minimum is quadratic — so the proportionality is not a property of springs, it is the first term of an expansion, and the question worth asking is always how large the second one is.

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

“Fictitious forces are not real, so they can be ignored.”

They cannot be transformed away in any frame a person on a rotating planet can actually occupy, and the numbers are not small. The centrifugal term removes 0.0339 m/s² of apparent gravity at the equator, which is 0.35% of a weight and hundreds of times a laboratory balance's precision; it also tilts a plumb line by up to 0.099° from the direction of the Earth's centre. A survey that ignores either is wrong by amounts that matter.

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

“The cyclotron frequency does not depend on the particle's speed.”

True below about a per cent of the rest energy and false above it. The frequency is qB/2πγm, and γ is one plus the kinetic energy over the rest energy — so the classical statement fails at 5.2 keV for an electron and 9.5 MeV for a proton, a factor of 1836 between them. That single number is why cyclotrons accelerate protons and electrons are accelerated in machines that change their own frequency or run in a straight line.

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

“Ampère's law gives the magnetic field of a current.”

It gives one number about the field — the circulation round a chosen loop — and that is one equation for a function of three variables. It becomes a field only when a symmetry is supplied that makes the magnitude constant along the loop, and the symmetry is an assumption imported from outside the law. Four loops in the figure return the same μ₀I and none of them determines the field on its own.

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

“Maxwell derived the wave equation and predicted radio.”

He derived the wave equation and identified light, in 1865, on the strength of the speed coming out right. Radio was not predicted by anyone in the form it took; Hertz produced and detected the waves in 1887 to test the theory, described their practical use as nil, and died in 1894 before Marconi began. The prediction that mattered was that light is an electromagnetic wave, and its evidence was a number already measured for other reasons.

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

“Fermat's principle explains why light bends, because light 'wants' to spend less time in the slow medium.”

It is a rule for identifying the path, not a mechanism, and a ray has no way of surveying the alternatives before setting off. The mechanism is that light is a wave and the neighbouring paths interfere: near a stationary point the path lengths agree to first order, so the contributions add, and everywhere else they cancel. The principle is a summary of that cancellation, which is why it holds and why it fails in exactly the situations where the cancellation is incomplete.

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

“Two points are resolved when they are further apart than the Rayleigh criterion.”

The criterion is a convention about how much of a dip a detector can see, not a threshold in the physics. The figure measures the dip off the summed pattern: 26.5% at exactly the Rayleigh separation, 91.6% at 1.6 times it, and nothing at all at 0.7 times it. Where the transition is put depends on the noise, and a bright well-measured pair can be separated at a fraction of the criterion by fitting the known shape of the pattern.

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

“The construction shows that a wave's speed determines how much it bends.”

It shows that the *ratio* of the two speeds does. The angle in the figure follows from the wavelets in the second medium being smaller by a stated factor, and neither speed appears on its own — which is why refractive index is defined as a ratio, why it has no units, and why an interface between two media of equal speed and quite different composition bends nothing at all.

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

“A transmitted wave cannot be larger than the incident one.”

Its amplitude can and often is. At an impedance ratio of 0.25 the transmitted amplitude is 1.600 times the incident one, which the figure computes and which is not a violation of anything, because amplitude is not energy. The transmitted *power* is (Z₂/Z₁)t², which comes to 0.640 there, and the reflected 0.360, and the two add to exactly one at every ratio. Dropping the impedance factor gives 2.920 and makes an ordinary result look impossible.

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

“Waves obey an inverse-square law.”

Spherical ones do, and the exponent is a statement about the dimension of the spreading rather than about waves. The figure fits the slopes off the drawn curves: 0.000 along a line, −0.500 over a cylinder, −1.000 over a sphere, for the amplitude — so the intensity exponents are 0, −1 and −2. A ripple on a pond obeys an inverse-first-power law, a wave in a pipe obeys none at all, and neither is an exception to anything.

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

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

It is proportional to the average energy in each *quadratic term*, of which translation supplies three and everything else supplies more. A nitrogen molecule at 300 K has 3/2 kT of translation and kT of rotation; a solid has 3kT per atom, half of it potential. Identifying temperature with kinetic energy alone gives the right answer for a monatomic gas and the wrong heat capacity for everything else.

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

“A reaction speeds up when heated because the molecules move faster.”

They do, and the effect of that alone is a square root: the collision rate goes as the square root of the temperature, so a rise from 300 K to 310 K increases it by 1.7 per cent. The measured increase is often a factor of two. What produces it is the exponential — for a 0.35 eV barrier, the same 10 K raises the fraction of collisions with enough energy by a factor of 1.9, and the figure computes the rise needed for exactly a doubling: 16.2 K from 300 K.

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

“In relativity, everything is relative.”

Almost nothing is. The interval c²t² − x² is the same in every frame, and so is every quantity built from it: the proper time along a worldline, the rest mass of a particle, the phase of a wave, the number of ticks a given clock records between two given events. What is frame-dependent is the *split* of the interval into a time and a distance, and calling that 'everything' is what makes the theory sound like a licence rather than a constraint.

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

“A magnetic field is just an electric field seen from a moving frame.”

Only in situations where a frame exists in which the field is purely electric, and for most configurations none does. The two frame-independent quantities built from the fields are E²−c²B² and E·B; a field with the first negative is magnetic in every frame there is. The wire on this page happens to be a case where the trick works, which is why it is the one always shown — and generalising from it is the error this refutation exists for.

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

“A body's weight acts at its centre of mass.”

It acts at the centre of gravity, and the two coincide only in a field uniform across the body. For an elongated body of length L at distance r from the source the centre of gravity sits nearer the source by about L²/6r: 85 nanometres for a standing person on Earth, 2.4 micrometres for a ten-metre satellite in low orbit. That second offset is what gravity-gradient stabilisation lives on, and a uniform sphere is exactly exempt.

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

“Coupling two oscillators together shifts the frequency of both.”

It shifts one and leaves the other exactly alone. In the in-phase mode both masses move the same way by the same distance, the coupling spring never changes length, and its stiffness therefore cannot enter the restoring force. The eigensolver returns ω₁ = 1.0000√(k/m) at kc/k = 0.1 and again at 0.6, while ω₂ climbs from 1.0954 to 1.4832√(k/m). One of the two frequencies has no memory of the coupling at all.

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

“Energy travels to a resistor along the wire, carried there by the current.”

At the surface of 2 m of copper wire 1.6 mm across carrying 10 A, the electric field is axial at 83.6 mV/m and the magnetic field circumferential at 2.50 mT, so E×B/µ₀ points radially inward at 166 W/m² everywhere. Over the 101 cm² of lateral surface that is 1.67 W, which is exactly I²R. The flux is inward through the sides, not along the axis, and the same 1.67 W crosses a coaxial surface at three times the radius where the flux density has fallen to 55.4 W/m².

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

“A dipole placed in an electric field is pulled towards the source of the field.”

In a uniform field the two forces are equal and opposite and the net force is exactly zero; what is left is a torque, at most 6.2 × 10⁻²⁴ N·m for water in 10⁶ V/m. Force requires a gradient. In the parallel-plate figure the field is uniform through the middle and has fallen to 69 per cent of its central value at nine-tenths of the way to the edge, and that edge region is the only place in the picture where a dipole would feel a net pull at all.

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

“Ruling a grating twice as finely doubles its resolving power.”

Resolving power is R = mN, and substituting N = W/d and m = d sinθ/λ gives R = W sinθ/λ — the illuminated width and the diffraction angle, with the pitch cancelled out. In a mount that works at a fixed angle, halving d changes nothing at all, and the ceiling W/λ is fixed however the glass is ruled: 84,800 for a 50 mm beam at 589 nm. The figures show the same thing twice over — 1200 lines in first order and 600 in second both give R = 1200 and a dip of 53.4 per cent.

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

“A laser is coherent and a lamp is not.”

Coherence is a property of a comparison at a stated delay, not a label on a source. A tungsten lamp filtered to 10 nm gives near-unity visibility for path differences under 30 µm, which is why Newton's rings exist; a helium–neon laser with a 1 MHz linewidth gives none beyond 300 m. Each source is coherent below its own coherence length and incoherent above it.

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

“The solution of the wave equation is a sine wave.”

Sines are a basis, not the solution. The general solution is f(x−vt) + g(x+vt) for any twice-differentiable f and g, which is d'Alembert's 1747 result and is what makes a pulse possible: a triangular kink released from rest immediately becomes two copies of half the height running apart at 283 m/s on the string above, and no single sinusoid does that. Treating sines as the solution rather than as an expansion is exactly the step d'Alembert and Euler refused in 1753, and they were right to refuse it until Fourier justified it in 1822.

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

“Tightening a drumhead brings its overtones into tune with each other.”

The tension and the areal density enter every mode frequency only through the factor √(T/σ), which cancels out of every ratio. Raising the tension from 1,900 N/m to 2,100 N/m raises all six frequencies by 5.13% together and changes no ratio by anything at all. A tuning gauge moves the whole spectrum and never its shape.

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

“Mass increases with speed, so a fast particle is a heavier particle.”

The quantity that grows is γm, which is E/c² under another name and carries no information the energy does not. For an electron accelerated through 100 kV the total energy is 611 keV and the momentum is 335 keV/c, while 611² − 335² = 511², so the invariant is still the same 511 keV/c² it was at rest. Nothing was added to the electron except energy and momentum.

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

“If the doors are shut and kept shut, the pole is trapped, which shows that length contraction is a real compression.”

The trapping is real and the compression comes from the doors, not from the contraction. News that the near end has stopped cannot outrun the pole's own sound speed; across 20 m at even 0.5c that is 133 ns, during which an end still moving at 0.866c covers 34.6 m. The pole is compressed because it was struck, and the same conclusion follows in Newtonian mechanics with any finite sound speed.

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

“A system settles into its state of lowest energy.”

Ammonium nitrate dissolving in water absorbs 25.7 kJ/mol — the energy goes up — and it dissolves anyway, because the mixing term is worth more. Two ideal gases mixing change their energy by exactly nothing and mix regardless: R ln 2 = 5.76 J/K/mol of entropy, or 1.73 kJ/mol of free energy at 300 K. Energy alone gets the direction wrong in both cases.

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

“Ice is slippery because the pressure of a skate melts it.”

Clapeyron's relation gives water's melting line a slope of −13.5 MPa/K, so one atmosphere shifts the melting point by 0.0075 K, and the entire 70 MPa pressure axis of the first figure moves it by 5.2 K. Skating at −10 °C would need 135 MPa, and the melting line does not reach it: it ends at 209.9 MPa and 251.2 K, where ice III begins.

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

“A water-repellent surface is one with the largest possible contact angle.”

What decides whether a drop moves is the difference between the advancing and receding angles, not either angle. At 85° advancing and 55° receding the retention is γ(cos θr − cos θa) = 35.4 mN per metre of contact line, so a 10 µL drop 4 mm across is held by 142 µN against its own weight of 101 µN — it clings to a vertical window at an angle under 90°. A rough surface at 150° with 40° of hysteresis holds drops that a flat 110° surface with 5° sheds.

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

“More angular momentum always buys a wider barrier, so a body can be held out of any orbit it likes.”

The centrifugal barrier is L̃²/2r² and the term relativity adds is −GML̃²/c²r³. Both carry L̃², so multiplying the angular momentum scales them together and the 1/r³ term still wins at small r. That is why the stationary points vanish below L̃ = √12 GM/c rather than merely moving inward: the figure at L̃ = 3.20 GM/c has no stationary point outside the horizon at all.

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

“The quantum effect is that two distant outcomes are strongly correlated.”

The shared-list model drawn beside the cosine is perfectly anticorrelated at equal settings too, and its four-setting combination measures 2.0304 ± 0.0487 at 5,000 pairs — on the classical bound, not below it. The two models differ in shape rather than in strength, most widely by 0.2105 at 19.77°.

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

“The flip must be caused by something — a nudge at the moment of release, air, or the hand.”

The integration has no torque in it anywhere: the right-hand sides are the three products of angular velocity components and nothing else. The reversals repeat with a fixed period, and halving the initial departure adds a fixed interval to that period rather than removing the flips, which is the signature of an exponential and not of a disturbance. What the hand supplies is the initial departure, and no hand can supply zero.

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

“A heavy object is harder to tip over than a light one of the same shape.”

The overturning moment comes from the applied push and the restoring moment from the weight, so a heavier object does need a larger push. It needs a proportionally larger one, and it also needs a proportionally larger push to slide, so the ratio is unchanged and the failure mode is unchanged. What weight buys is a larger force before anything happens at all; what it does not buy is a different answer to which thing happens.

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

“Since the force is the Coulomb force between the charge and an equal opposite charge 2d away, the energy is their interaction energy too.”

The energy is exactly half of it, and the figure draws both curves. Integrating the attraction from infinity gives −q²/16πε₀d, while two real charges 2d apart would store −q²/8πε₀d. The image moves as the charge approaches, so half the work goes into rearranging the induced charge rather than into the interaction — which is the point at which the fiction stops being free.

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

“The force between two magnets falls off as the inverse square, like the force between two charges.”

It falls as the inverse fourth power. A dipole's field already falls as 1/r³ because two opposite poles nearly cancel; the force on a second dipole is a gradient of that field, which costs one more power. The figure differences the drawn dipole field numerically and measures the slope of the result on logarithmic axes as −4.00, against −3.00 for the field and −2.00 for a point charge — all three read off the same arithmetic.

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

“Polarising sunglasses work because reflected glare is polarised, so they cut out glare completely.”

The reflected beam is completely polarised at exactly one angle and nowhere else. The figure plots the degree of polarisation against angle: it is 1 only at 56.3° for water-like indices, falls to 0.62 at 30°, and is zero at normal incidence, where there is no plane of incidence to distinguish the two polarisations at all. A pair of glasses removes most of the glare from a road at a shallow angle and much less of it from a puddle underfoot.

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

“Refraction needs a surface, so an atmosphere with no layers in it does not bend light.”

The invariant along a ray in a stratified medium is n cos θ, with θ measured from the layers, and it is conserved whether the index steps or varies smoothly. Snell's law is what that invariant becomes when the variation happens over a distance short compared with a wavelength. The standard atmosphere has no surfaces at all and bends a horizontal ray at 2.62 × 10⁻⁸ per metre — seventeen per cent of the Earth's own curvature, which is why the optical horizon is further away than the geometric one.

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

“The coherence length is λ²/Δλ, so a source of a given linewidth has a definite one.”

λ²/Δλ is a convention that rounds three different behaviours to one number. Three line shapes with the same width at half height are drawn: the visibility of a flat band halves at 0.60 of that length and then *revives*, a Gaussian line halves at 0.44 of it and never revives, and a Lorentzian halves at 0.22 of it and keeps a long tail of visibility far beyond. The convention names a scale, not a distance at which anything happens.

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

“Any amount of pumping will eventually get a swing going, since energy is being added on every stroke.”

Below a threshold nothing happens at all, and the threshold is set by damping. At a damping ratio of 0.02 the first wedge opens only at a modulation depth of 8.0%; below that the losses take more per cycle than the modulation gives, and the amplitude decays exactly as if nothing were being done. A driven oscillator has no such threshold — any force however small produces a response — and the difference is the sharpest way to tell the two mechanisms apart.

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

“The factor √γ is an empirical correction to make the formula fit.”

γ read off each pair of speeds — the measured value squared, divided by Newton's squared — comes to 1.400 for air, 1.665 for helium, 1.666 for argon and 1.290 for carbon dioxide. Those are 1 + 2/f with f equal to 5.0, 3.0, 3.0 and 6.9: five ways of holding energy for a diatomic molecule, three for a monatomic one, and nearly seven for a triatomic whose bending modes are active. A stopwatch and a tube count the degrees of freedom of a molecule.

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

“A short pulse spreads slowly because it has less far to spread.”

It spreads fastest, and by a large factor. Being short means having a wide spectrum, and the spreading time τ = σ₀²/|β| goes as the square of the initial width — so halving the packet's length divides its spreading time by four. The three packets drawn differ only in bandwidth, and the shortest of them has a spreading time of 13 s against the longest one's 102.

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

“The forward cone has half-angle 1/γ.”

That is the small-angle limit of arccos β and is good only when the speed is genuinely relativistic. At 0.99c the exact angle is 8.11° and 1/γ gives 8.09°; at 0.9c they are 25.84° and 25.02°; at 0.5c they are 60.0° and 49.6°, which is 17% wrong. The figure draws both so the approximation can be seen arriving rather than assumed.

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

“The satellite is moving, so its clock runs slow.”

It runs fast, by six times the amount its motion slows it. The velocity term is −GM/2rc² and the potential term is +GM/c²(1/R⊕ − 1/r); at a 20,200 km altitude they are −7.2 and +45.7 µs a day. The sign of the total is set by the larger, and which is larger is a question about the orbit rather than about relativity.

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

“An electric heater is 100% efficient, so nothing can beat it.”

It is exactly 100% efficient and it is the worst curve on the figure. Efficiency measures how much of the energy consumed appears as heat; the coefficient of performance measures how much heat is delivered per unit of work, and those are different questions. The resistive heater sits at exactly one at every outside temperature; a real heat pump at a quarter of the ideal delivers 5.3 at 7 °C and 2.6 at −7 °C, and both are more than one.

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

“The shape depends on the liquid — a denser one is flung further out.”

No property of the liquid appears in z = ω²r²/2g. Density, viscosity and surface tension are absent, so mercury, water and oil in the same dish at the same rate settle into the same surface. Density does decide the *pressure* under that surface and the force on the dish; it decides nothing about the shape, because the condition is that the surface be an equipotential and a potential per unit mass carries no mass.

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

“Tidal forces are stronger near a more massive body, so falling into a large black hole is worse.”

The tidal parameter at a horizon is GM/r_s³ = c⁶/8G²M², which falls as the square of the mass. At the horizon of a ten-solar-mass hole it is 5.15 × 10⁷ s⁻² and at a billion-solar-mass one it is 5.15 × 10⁻⁹ s⁻² — smaller than the Earth's surface value of 1.54 × 10⁻⁶. The figure draws the curve for the giant hole above the Earth's, not below it: an experimenter could cross that horizon with a metre-sized laboratory and a very good interferometer and detect nothing at all.

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

“More damping always means a quicker return to equilibrium.”

The two exponents of the characteristic equation are real and distinct above critical damping, and the slower of them goes to zero as the damping grows: at ζ = 4 it is 0.127 ω₀ against ω₀ at critical, so the return takes about eight times as long. The figures show a heavily damped trace still visibly away from zero when the critical one has arrived and stopped.

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

“The braking is friction between the magnet and the tube wall.”

Friction depends on contact and barely on speed; this force never touches the wall and is strictly proportional to speed. The distinction is measurable rather than philosophical: friction would give a constant deceleration and a fall time proportional to the square root of the distance, while a linear drag gives a terminal speed and a fall time proportional to the distance. The integrated descent here reaches a constant speed in about a fiftieth of a second and holds it for the rest of the metre.

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

“Achromatism needs one glass of high index and one of low.”

The condition contains only the Abbe numbers, not the indices. N-BK7 and F2 differ in index by 0.10 and in Abbe number by 27.8, and it is the second difference that does the work: the element powers go as V/(V₁ − V₂), so two glasses of identical index and different dispersion would achromatise perfectly while two of identical dispersion and wildly different index could not achromatise at all. The Abbe diagram here is drawn with index on one axis and dispersion on the other precisely because they are independent.

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

“Scattering goes as the inverse fourth power of the wavelength.”

That is the leading term of an expansion in the size parameter 2πa/λ and holds only while that parameter is small. Summing the full Mie series for a sphere of index 1.333 shows the ratio of blue to red scattering falling from 4.35 in the small limit to 1.09 by a radius of one micrometre — so a cloud droplet has no colour preference worth measuring. The law is not approximately right for a large particle; it is wrong by a factor of four.

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

“A quarter-wave layer of any intermediate index reduces the reflection to nothing.”

Two conditions must hold at once and the thickness supplies only one. The layer's impedance must be the geometric mean of the two sides, so that the two reflected amplitudes are equal; a quarter-wave layer of the wrong index leaves the difference. Magnesium fluoride at 1.38 on glass at 1.52 needs 1.2329 and leaves 1.26% at the design wavelength against 4.26% bare — a reduction of three and a bit times, not a removal.

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

“The group velocity is the speed a signal travels at.”

Only where the dispersion is weak and the medium does not absorb or amplify. Between two gain lines the group index computed here is −3.85, which makes the group velocity negative; a pulse synthesised from its own frequency components and propagated through that medium has its peak emerge 616 time units before the vacuum pulse arrives, and 0.21 of a pulse duration before the input peak entered. The group velocity is exactly what it is defined to be — the speed of the envelope's peak — and that peak is not a signal.

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

“Entropy is a measure of disorder, so mixing two samples of the same gas must increase it — the molecules are more mixed up than before.”

Remove the partition between two halves of a box of nitrogen and put it back, and the state is the one it started in: every measurable quantity is unchanged, no work can be extracted, and the process is reversible in the strict sense. If the entropy had risen it could be exploited — a cycle that gained k ln 2 per particle for nothing would be a perpetual motion machine of the second kind. The counting says the same: swapping two identical molecules does not produce a new arrangement.

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

“The whole loop describes states the substance cannot be in.”

Only the middle section does. Between the turning points the compressibility is negative and no state exists; between each turning point and the tie line the compressibility is positive and the state is metastable — superheated liquid on one side, supersaturated vapour on the other, both routinely produced. Clean water can be heated to about 280 °C at atmospheric pressure before it flashes, and cloud chambers work by holding vapour in the corresponding state on the other side.

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

“Helium stays liquid because its atoms attract each other so weakly.”

Weak attraction alone does not decide it. Hydrogen's well is three and a half times deeper than helium's and hydrogen freezes at 14 K; what matters is the ratio of the zero-point energy to the well depth, which is what the de Boer parameter measures. Computed from published Lennard-Jones parameters it runs 3.08 for helium-3, 2.68 for helium-4, 1.73 for hydrogen and 0.19 for argon — and the two heliums, alone among all substances, do not solidify at any temperature under their own vapour pressure.

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

“Heavy nuclei release energy when split because they are unstable.”

Stability is a separate question from where a nucleus sits on the binding curve. Uranium-238 has a half-life comparable to the age of the Earth and is thoroughly stable on any human scale, and splitting it still releases about 200 MeV because the products sit higher on the curve. What decides whether the release happens spontaneously is a barrier, not a slope: the fission barrier of uranium is about 6 MeV, and supplying it — with a neutron, in the case of uranium-235 — is what starts the process.

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

“Relativity says the order of events depends on the observer.”

Only for spacelike-separated pairs. The order reverses at a boost of Δt/Δx in units of c, which is a legal speed exactly when the separation is spacelike; for a timelike pair that expression exceeds one and no frame achieves it. So every pair that could be causally connected has an order all observers agree on, and every pair whose order is negotiable is a pair light could not have crossed between. The claim is true of a specific class and false as a general statement.

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

“A body of exactly the same average density as water will hang motionless wherever it is put.”

Only if it is as incompressible as the water. A body containing gas gets denser as it descends, because the gas obeys Boyle's law while the water barely compresses at all, so the net upward force falls with depth. Computing that force for a body trimmed neutral at ten metres gives a curve crossing zero from above, and a crossing from above is an unstable equilibrium — a nudge downward produces a downward force that grows.

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

“A liquid moves when there is a pressure difference across it.”

That is one way and not the only one. A gradient in surface tension is a shear stress applied at the surface, with no pressure difference anywhere: the balance is τ = μU/δ rather than a pressure gradient, and it drives a ten-micrometre film at 0.42 mm/s for a gradient of a twentieth of a millinewton per metre per millimetre. Soap dropped behind a paper boat propels it, and no pressure difference exists in the water at any point.

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

“The horizon is a surface, so something must be there to detect.”

Nothing local marks it. The metric is perfectly regular there in coordinates adapted to a falling observer, the curvature invariant is finite and small for a large hole, and every quantity a falling instrument can measure varies smoothly across it. The horizon is defined by which future paths lead out, which is a statement about the whole of the future rather than about anything present, and no measurement made in a small region and a short time can locate it.

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

“A bounded orbit under a central force is a closed curve.”

It is confined to an annulus and need not close. Integrating orbits under F ∝ rⁿ from the same start gives apsidal angles of 180.0°, 145.9°, 126.4° and 90.0° for n = −2, −1.5, −1 and 1; only the first and last divide a full turn evenly, and the other two produce rosettes that never repeat. A rosette that nearly closes is not nearly a closed orbit — after enough circuits it fills the annulus.

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

“Angular momentum is Iω, so a body's spin axis follows the torque applied to it.”

Iω is a scalar shorthand that holds only when the rotation is about a principal axis. In general the moment of inertia is a tensor, L and ω point in different directions, and a torque-free body's angular velocity moves while its angular momentum stands still — which is what the tumbling figure here shows and what makes an unstable axis unstable.

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

“A bigger absorber gives better protection.”

The depth of the protection does not depend on the absorber's mass at all — the response is exactly zero at the tuned frequency for any mass ratio whatever. What the mass buys is the *width* of the useful band, through the separation of the two new peaks, and that grows only as the square root of the mass ratio: halving the peak separation costs four times the mass.

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

“Tunnelling has no classical analogue.”

Frustrated total internal reflection is that analogue and it obeys the same formula: transmission [1 + ¼(γ + 1/γ)² sinh²κd]⁻¹, with κ the decay constant of a field inside a region it cannot propagate in. The figures compute the optical version and the quantum one from the same expression. Newton demonstrated the optical case with two prisms, more than two centuries before anyone needed the same mathematics to explain a nucleus.

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

“The flux rule is Faraday's law.”

Faraday's law is a statement about a field at a point — the curl of E equals minus the rate of change of B — and it is one of Maxwell's four. The flux rule is a theorem that follows from it together with the magnetic part of the force law, and only when the circuit is a well-defined curve of identifiable material points. The homopolar generator drawn here has no changing flux through any surface its circuit bounds and produces 0.785 volts, which the flux rule gives as zero.

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

“The elementary kinetic-theory result η = ⅓ρv̄λ is the viscosity of a gas.”

It is 32 per cent low, and the shortfall is not experimental. The elementary derivation takes every molecule to travel exactly one free path perpendicular to the shear and to carry the mean momentum of where it started, and none of those three is right. A proper transport calculation replaces the factor of a third by 5π/32 times it, and the corrected value for nitrogen is 17.9 μPa·s against a measured 17.9.

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

“Pressure comes from thermal motion, so a cold gas exerts less pressure.”

The degeneracy pressure of copper's electrons is 38 gigapascals — 380,000 atmospheres — at absolute zero, and cooling does not reduce it. It is the cost of confining particles that cannot share states, and it survives to zero temperature because it was never thermal. The figures compute it from the full degenerate integral and show where its exponent changes as the electrons become relativistic.

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

“Adding a surfactant makes a drop dry more evenly by lowering the surface tension.”

What a surfactant does is make the surface tension *vary with position*, because evaporation concentrates it unevenly, and a gradient of surface tension is a stress along the surface. That stress drives a recirculating flow which can be strong enough to reverse the outward transport entirely. The lowering of the tension is incidental; the gradient is the whole mechanism, and it is the same one that makes tears of wine.

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

“A siphon works because atmospheric pressure pushes the liquid up the short arm.”

A siphon runs in a vacuum. Degassed water in a sealed tube siphons over a crown with no atmosphere on either surface, held together by its own tensile strength, and the flow is driven by the weight of the unbalanced column in the long arm exactly as it is at atmospheric pressure. What the atmosphere sets is the *ceiling* — how high the crown may be before the liquid boils — and a siphon below that ceiling runs without needing the atmosphere for anything.

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

“Radiation pressure matters near a star and gravity takes over further out.”

Neither takes over. Both forces fall as the inverse square of the distance, so their ratio is exactly the same at every distance — the same grain is blown out from one astronomical unit and from a thousand, or from neither. What decides is the grain's size and density, and nothing about where it is.

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

“A squeal is cured by making the surfaces more slippery.”

Lowering the whole friction curve without changing its slope leaves the instability exactly where it was, because the criterion contains the difference between two coefficients in the rate law and not their size. What removes it is a stiffer holder, a shorter overhang or a contact that resists harder the faster it goes — and the last of these is why brake pads are formulated for the sign of a slope rather than for a value.

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

“Doubling a car's speed quadruples the energy needed, because kinetic energy goes as v².”

Going from 0 to 10 m/s and from 10 to 20 m/s both take the same time at the same acceleration and the same force, so the engine delivers three times as much work in the second case for one times as much impulse — and an observer travelling at 10 m/s sees the first case as the expensive one. The v² is not a statement about fuel; the fuel is settled by the force and the distance in the frame where the road is at rest, and that frame is picked out by the friction rather than by mechanics.

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

“The phase velocity in a waveguide exceeds c, so something travels faster than light.”

Nothing does. The phase velocity is the speed of the crossing point of two plane waves each travelling at c at an angle to the axis, and a crossing point carries nothing. The figures compute the group velocity on the same curve and find it below c by exactly the factor that makes the product c², so a pulse launched into the guide arrives late rather than early.

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

“A nucleus with more energy decays faster because it has more energy.”

The energy enters through an exponent, not as a rate. Doubling the alpha's energy from 4.08 to 8.95 MeV multiplies the decay rate by 10²⁴, which nothing linear in energy can produce; the figures compute the barrier integral for seven emitters and reproduce every half-life to within half a decade with no adjustable number in the model. What is doing the work is the width of the classically forbidden region, and that width falls with energy far faster than the energy rises.

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

“The hardest direction to magnetise a particle is across its easy axis, so that is where the largest field is needed.”

The largest field is needed *along* the easy axis and the smallest at forty-five degrees to it, where half the anisotropy field suffices. The figures measure the switching field at forty-five angles by finding where a swept loop jumps and recover the astroid curve to within the field step. At ninety degrees there is no switching field at all, because the magnetisation leans continuously and never has to choose.

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

“Thermal expansion is a materials constant.”

It is a function of temperature that happens to be flat over the range most tables are printed for. The same Boltzmann average that gives a straight line at low temperature curves away from it at high, because the cubic term stops being the only correction; and every expansion coefficient goes to zero as the temperature goes to zero, because the classical calculation itself fails there and the quantum one has no thermal population left to be asymmetric with.

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

“Van der Waals' equation describes the critical point.”

It gets the topology right and every exponent wrong. The figures measure its order-parameter exponent as 0.500 against a measured 0.326, its critical-isotherm exponent as 3.000 against 4.8, and its compressibility exponent as 1.000 against 1.24. Those are not small errors of a good model — they are the signature of a whole class of theories, every one of which replaces a molecule's neighbours by an average and thereby throws away the fluctuations that decide the answer.

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

“A body cannot reach the speed of light because its mass becomes infinite.”

Its momentum grows without bound and its mass does not change at all. The figures follow a body under a constant force: the momentum rises exactly linearly for ever, with no sign of anything becoming infinite, while the speed saturates because speed is a decreasing function of momentum near c rather than a proportional one. The word mass in this claim is doing the work that the relation between momentum and velocity is actually doing.

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

“A strong enough boost can turn a weak field into an arbitrarily strong one.”

It can raise both magnitudes without limit and cannot change their combination. The figures track a field along a boost and its point slides along a hyperbola: E and cB each grow like γ, their difference of squares stays put. So a field that is nearly a light wave stays nearly a light wave however hard it is boosted, and the ratio of its two parts approaches one rather than departing from it.

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

“Rubber and honey are both examples of viscous materials.”

They resist for different reasons and on different schedules. Honey's stress decays to zero under a fixed strain, which is what it means to be a liquid; rubber's decays to a floor and stays there, which is what it means to be a solid. The figures compute both from the same superposition and the difference is a single term. A material that reaches a non-zero plateau has a permanent structure to remember, and one that decays to nothing does not.

Tested in The liquid that remembers · the rheology reading path

“Gravitational radiation is too weak to matter.”

It is too weak to detect from anything ordinary and it is the fate of every close binary. The figures compute the time to coalescence for four pairs of masses on the same axes: the Earth-Sun system has 10²³ years, thirteen decades longer than the age of the universe, while a pair of thirty-solar-mass black holes a thousandth of an astronomical unit apart has minutes. The same expression gives both, and what separates them is the fourth power of a separation.

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

“Two bodies in thermal contact come to a common temperature.”

Not if one of them is self-gravitating. A body with a negative heat capacity in contact with an ordinary one runs away: give it heat and it expands and cools, so it takes more; take heat and it contracts and heats, so it gives more. There is no equilibrium to approach, and the ordinary argument for one assumes both bodies get hotter when given energy. That failure is the reason a self-gravitating system has no proper thermodynamic limit.

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

“Grains are a fluid, so hydrostatics applies with the bulk density in place of the liquid's.”

Hydrostatics follows from a fluid having no shear strength, and grains have one: friction between them and against the wall. That single difference introduces a length, R/2μK, which hydrostatics does not contain and cannot contain, and past a few of those lengths the stress stops growing with depth entirely. The two descriptions agree only for a fill shallower than a silo's own radius.

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

“Osmosis is what lifts sap to the top of a tall tree.”

The arithmetic refuses it. Supporting a hundred and fifteen metres of water needs 1.13 megapascals, which is 457 moles of dissolved particles per cubic metre — a strong brine, and roughly twenty times what xylem sap actually carries. The figures solve for the concentration each height requires and mark the measured sap value, which reaches five metres. Root pressure is real and it is a shrub's mechanism, not a redwood's; what lifts sap in a tall tree is tension applied from the leaves.

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

“A precision instrument should be made as symmetric as possible.”

Here the opposite is true, and quantitatively so. Bessel's expression splits g into a term divided by the sum of the two knife-edge distances and a term divided by their difference, and only the second carries the mismatch between the two periods — the one adjustment a workshop cannot make exactly. The figures show a nearly symmetric pendulum turning a one-part-per-million period mismatch into nine parts per million in g, and a deliberately lopsided one turning it into 1.5.

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

“The light inside the stop band is absorbed by the layers.”

Every layer in the calculation is lossless — the indices are real numbers and no absorption appears anywhere. The transmitted intensity falls as e^(−2Nκ) in the number of cells, with κ = 0.693 nepers per cell at mid-gap for a two-to-one index ratio, and the missing light is reflected rather than dissipated. A perfect stop band reflects a hundred per cent and heats nothing.

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

“A concentrator falls short of the theoretical limit because of aberrations and manufacturing tolerance.”

A perfect paraboloid, figured exactly, falls short by a factor of four. The figures compute its concentration against rim angle from the geometry of the Sun's image and it peaks at 11,434 against a ceiling of 46,165 — not because the mirror is imperfect but because forming an *image* wastes étendue. A shape that gives up imaging reaches the ceiling exactly, and the figure that draws one measures its aperture ratio against 1/sin θ to a part in a million.

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

“Reflecting a beam back through a rotator will undo the rotation.”

It undoes it for a naturally active medium and doubles it for a Faraday rotator, and the figures draw both. A handed crystal is handed with respect to the beam, so reversing the beam reverses the sense; a magnetic field defines a direction in the laboratory and does not care which way the light is going, so the second pass adds. The round trip is 0 in one case and 2 in the other, and there is nothing in between.

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

“Any moving charge radiates, so an electron in orbit must spiral in.”

A charge in *uniform* motion radiates nothing, and its field falls as 1/r² everywhere, carrying no energy away. The figures compute the ratio of the radiation field to the velocity field and it is a·r/c², so the two are equal only at a distance of c²/a — 9.2 × 10¹⁵ metres, about a light-year, for an object accelerating at one gravity. That is why nothing in ordinary experience visibly radiates, and why an antenna is a charge whose motion changes fast rather than one that moves far.

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

“Information is physical, so computation has a fundamental energy cost.”

Only irreversible computation does. A logic gate that discards a bit must dissipate at least kT·ln2; one that does not — and any computation can in principle be arranged not to — has no lower bound at all. And the figures put the bound in context: it is 2.87 zeptojoules at room temperature, against about 10⁻¹⁷ joules for a transistor switching in a modern processor, a factor of three thousand. Nothing anybody has built is limited by it.

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

“A power station is inefficient because it falls short of the Carnot limit.”

It falls short of a limit that corresponds to producing no electricity at all. The figures compute the power of an engine with finite thermal contact against its efficiency: the curve is zero at zero efficiency for the obvious reason and zero again at Carnot's value, because reaching the ceiling requires the working fluid to sit at the reservoir temperatures and then no heat crosses the contacts. Three measured plants lie within a few points of the maximum-power efficiency and 16 to 28 points below the ceiling.

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

“The string does not break, because in the rockets' frame everything including the string is contracted equally.”

There is no single frame in which both rockets are at rest for more than an instant, so the phrase 'the rockets' frame' names nothing. Taking the momentarily comoving frame at each instant and measuring the gap in it gives a growing number: at 0.9c the drawn worldlines are 42 per cent further apart on their own slice than they were at the start, while the laboratory sees no change whatever.

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

“Langevin's derivation of the Curie law is classical statistical mechanics, and its agreement with measurement was a success for the classical theory.”

Langevin's derivation assumes each molecule carries a magnetic moment of fixed magnitude free to point anywhere. Classical mechanics does not permit such an object — the moment of a classical charge distribution changes when the field does, and it is that change which cancels the answer. The Langevin curve drawn here is the J → ∞ member of the Brillouin family, and it is a quantum result with the discreteness taken to zero, not a classical one.

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

“Bell's theorem needs statistics, so any single experimental run is consistent with pre-agreed answers, and the conclusion is only ever probabilistic.”

It is true of two particles and false of three. The sixty-four lists of pre-agreed answers for a three-particle state are enumerated here in full: thirty-two reproduce three of the four certain predictions and thirty-two reproduce one, and not one reproduces all four. So measuring three of the rows determines the fourth by pure logic, and the fourth comes out the other way. There is no inequality and no counting.

Tested in The disagreement that one run settles · the entanglement reading path

“Fine structure is the spin–orbit interaction.”

Spin–orbit is one of three terms of the same size, and taking it alone gives the wrong answer. The relativistic correction to the kinetic energy and the Darwin term are both of order α² times the level, and only their sum reproduces the measured splitting and the degeneracy of levels with the same j. The figure computes the Dirac result, which contains all three, and checks it against Ry·α²/16.

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

“Every degree of freedom carries half a kT.”

Only quadratic ones do. The mean energy in a coordinate entering the energy as |x|ⁿ is kT/n, integrated numerically here and agreeing with the closed form to a part in ten million: a whole kT for a linear term, half for a quadratic one, a quarter for a quartic. Counting coordinates rather than quadratic terms gets an ultrarelativistic gas wrong by a factor of two.

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

“A black body is an idealisation nothing achieves.”

A small hole in a closed box achieves it to any accuracy wanted, and the figure of attenuation through repeated interactions is why: light that enters has to be reflected many times before it can find its way out, and each reflection removes the same fraction. A hole of one per cent of a cavity's area with walls of reflectivity 0.5 has an effective emissivity above 0.999, whatever the walls are made of.

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

“Energy is conserved, so all of the beam's energy is available to make new particles.”

It is conserved and most of it is not available. Momentum is conserved too, and the products of a fixed-target collision must between them carry the beam's whole momentum, so they must between them carry the kinetic energy that goes with it. What is left over is the invariant mass of the initial pair, and that is the quantity the threshold arithmetic uses — 5.63 GeV of kinetic energy to make an antiproton out of a beam and a stationary target, against the 1.88 GeV the pair itself weighs.

Tested in The collision that wastes most of the energy · the relativistic dynamics reading path

“E = mc² says that mass can be converted into energy, as it is in a nuclear reactor.”

Nothing is converted. The invariant mass of an isolated system never changes at all, whatever happens inside it: a sealed reactor weighs exactly what it weighed before, because the energy released is still inside it. What falls is the mass of the *products* once the heat has left — the reaction moves energy out of the system, and the mass follows the energy rather than turning into it.

Tested in The box of light that weighs something · the Mass-energy reading path

“Gravitational time dilation is an effect of acceleration, so it is really special relativity in a lift.”

A uniformly accelerating frame in flat spacetime does show a shift between its floor and its ceiling, and that is why the effect can be derived from the equivalence principle. What it cannot do is hold for two clocks *at rest* in a static field over a region large enough to compare — the argument here uses only that the field does not change with time, and it concludes that the spacetime containing those clocks is not flat, which no acceleration in flat spacetime can arrange.

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

“Resolution is limited by the quality of the optics, so a better-made lens resolves more.”

A perfect lens of a given aperture resolves exactly λ/2NA and no better, and the figures show why: the object's first diffracted order has to fit inside the aperture alongside the zeroth, and an order that misses the lens never reaches the image at all. Manufacturing quality decides how close a lens gets to that bound and cannot move it.

Tested in The image that is a diffraction pattern twice · the imaging reading path

“Near the object the pattern is a shadow, and far away it is a diffraction pattern; the interesting physics is far away.”

The Talbot planes are all in the near field, at distances where the Fraunhofer treatment does not apply at all, and there is nothing shadow-like about them: the plane at half the Talbot distance is a perfect image of the grating displaced sideways by half a period, and the plane at a quarter is either uniform or at twice the object's frequency depending on the duty cycle. The near field has as much structure as the far field and it is structure of a different kind.

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

“The rocket equation is Newton's second law with a variable mass, F = d(mv)/dt.”

Written that way it is wrong and gives the wrong answer, because a system whose mass changes is not a system. The correct treatment keeps the vehicle and its exhaust together as one system of fixed total mass and applies conservation of momentum to it: the figure integrates both halves through a burn and requires their sum to stay at zero, which it does to a part in ten million.

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

“A band gap is useful because it reflects — a photonic crystal is a very good mirror.”

It is a very good mirror, and every application that matters uses the *state a defect is allowed to hold inside it*. The figure computes a stack with one wrong layer: at the middle of the forbidden band the transmittance is exactly one, and the linewidth falls by a factor of sixteen for every two extra pairs of layers. A better mirror does not exclude the state more thoroughly; it traps it more tightly.

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

“An effective mass is a convenient fudge factor that absorbs the lattice's complications.”

It is a second derivative and it is measured as one. The tight-binding chain drawn here has ħ²/(d²E/dk²) equal to 0.470 m_e at the bottom of a 3.60 eV band, computed by central difference on the drawn curve and by the closed form, agreeing to a part in 10⁶ — and the same expression returns −0.470 m_e at the zone boundary and diverges at ka = π/2. A fudge factor does not have a pole in it.

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

“A watched pot never boils is a joke, but the quantum Zeno effect really does freeze a decay by measuring it often enough.”

It does, and only because the head of the curve is quadratic. Under a strictly exponential law the survival after a fixed interval is e^(−ΓT) whatever the number of measurements, which the figure checks to a part in 10¹⁵ — repeated measurement of a memoryless process buys precisely nothing. The freezing is a measurement of the departure from exponential, and the interval it must beat is Γτ_z², not τ_z.

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

“Ampère's law plus symmetry proves the field outside a solenoid is zero.”

It proves the field outside an *infinite* solenoid is zero, and the symmetry step is where the infinity enters — a finite winding has ends, so no translation along it is a symmetry and no rectangular loop can be argued away. Summed turn by turn, a 24:1 winding has a mid-plane field two radii out of 8.6 × 10⁻⁴ of its centre value, not zero, and a 1:1 winding has 7.3 × 10⁻².

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

“A liquid scatters far more than a gas because the molecules are ten thousand times closer together.”

It scatters far less per molecule, because a liquid is held together and it is the fluctuations that scatter, not the molecules. The factor is ρkTκ_T, which is exactly one for an ideal gas and 0.062 for water at 20 °C — computed from water's own compressibility. Mercury, sixty times stiffer again, gives 0.0066.

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

“Absolute zero is unattainable because there is always some heat leaking in.”

A leak is an engineering problem and this is not one. With a perfectly insulated, perfectly reversible cycle, each stage of magnetise-then-demagnetise multiplies the temperature by B_low/B_high — a ratio read back off the drawn treads and equal to the field ratio to a part in 10¹² — so the sequence is geometric and no finite number of stages reaches zero. Removing every leak changes nothing about that.

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

“Radiation reaction acts like a drag, opposing the motion.”

It is proportional to the rate of change of acceleration, not to the velocity or the acceleration. In steady circular motion ȧ points opposite to a and the reaction does oppose the motion; in a uniformly accelerated charge ȧ is zero and the reaction vanishes while the radiation does not, which is the case the drag picture gets exactly backwards.

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

“Radio waves bounce off the ionosphere, so the ionosphere is a reflecting layer.”

It is a layer with a cutoff, and which waves bounce depends on the frequency. The F2 layer at 10¹² electrons per cubic metre cuts off at 8.98 MHz, so a 1 MHz broadcast is turned back and a 100 MHz one goes straight out — which is why one is heard across an ocean at night and the other stops at the horizon. The same layer is a mirror and a window at the same instant.

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

“Since absorption goes as f², halving the frequency quarters the loss, so the exponent fixes everything.”

The exponent of the measured curve is 1.42 across the audible range, not 2, because a relaxation process gives f² only below its own relaxation frequency and a constant above it — and air has two such processes at frequencies that depend on the humidity. The f² law is the shape the classical calculation predicts and the shape real air does not have.

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

“Faster means the boom arrives sooner.”

It arrives later. The delay along the track is h√(M²−1)/(Mc), which rises with Mach number and approaches h/c: 22 seconds at Mach 1.2, 32 at Mach 1.6 and 38 at Mach 3 for an aircraft at twelve kilometres, against a limit of 41 seconds, which is the plain travel time of sound straight down.

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

“Friction is independent of load because the asperities flow plastically, so the real contact area is the load divided by the hardness.”

That gives the law, and so does an elastic surface that never yields. Integrating Greenwood and Williamson's model — many asperities with an exponential spread of heights, each obeying Hertz — gives a real area growing as W^1.0000, because the area and the load carry the same exponential factor. Plasticity is sufficient and not necessary; roughness is what is doing the work.

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

“Chaos means a system is unpredictable, so there is no point computing it.”

It is predictable for a definite length of time, and the length is computable. With an exponent of 2.048 per second the separation doubles every 0.34 s, so knowing the release to a thousandth of a radian buys 3.0 seconds and knowing it to a part in 10¹⁵ buys 16.5. The forecast is short and it is not zero, and every extra decimal place is worth the same 1.12 seconds.

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

“The water is sucked up, so the limit is the 10.3 m a perfect vacuum can lift.”

That limit applies to a pump at the top of a column open to the atmosphere at the base, where the driving pressure difference cannot exceed one atmosphere. A continuous liquid column under tension is not being pushed by anything: the hydrostatic line crosses zero at 10.3 m and simply continues, reaching −0.88 MPa at a hundred metres, which is nine atmospheres of tension and not a pressure a pump could produce.

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

“A stable layer returns a displaced parcel to where it belongs.”

It returns it and does not stop it. Nothing damps the motion, so a parcel released 300 m above its level passes through that level and overshoots by 300 m below it, every time — the integrated trajectory has a period of 9.95 minutes against 9.95 from 2π/N. What a stable layer does is oscillate, and the oscillation is what an internal wave is made of.

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

“The Sagnac effect shows that the speed of light depends on the motion of the apparatus, so it contradicts relativity.”

It is derived here entirely in the inertial frame, where light goes at c both ways and the *starting point* moves. One beam chases a receiver running away and the other meets one coming toward it, so the transit times are 2πR/(c − RΩ) and 2πR/(c + RΩ) and their difference is 4AΩ/c² to first order. Nothing in that argument uses a variable light speed and the result is first order in Ω, which is why it was measurable in 1913.

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

“The rocket outruns the light, which is why the signal never arrives.”

It never reaches the speed of light and never gains on the ray. The gap between the hyperbola and its asymptote shrinks without limit — the signal closes on the rocket for ever and arrives at no finite time. Nothing outruns anything; what happens is that an infinite amount of the rocket's coordinate time fits into a finite amount of its own.

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

“A positive Lyapunov exponent means the system behaves like a random walk.”

It does not. The measured spreading rate divided by the estimate that assumes uncorrelated kicks runs from 0.07 to more than two hundred across the range drawn, and near a coupling of 2π there exist orbits that accelerate steadily rather than diffusing at all. Exponential separation of neighbours and statistical independence are different properties and only the first is what the exponent measures.

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

“F = ma applies to the moving part of a chain, with m the mass currently moving.”

It does not, and using it gives the wrong answer by a factor of order one. Newton's law is a statement about a fixed set of matter, and the moving part of a falling chain is not one — it is losing members. Writing F = d(mv)/dt with a changing m is a different equation from F = ma and the difference is the whole subject.

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

“A sharper resonance means a system that responds faster.”

It means the opposite. The response of a resonator is slow in exactly the proportion that it is sharp — the number of cycles it takes to reach its steady amplitude is the same Q that sets the narrowness — which is why a high-Q filter rings, why a sharply tuned circuit cannot pass a fast pulse, and why selectivity and speed are traded against each other rather than obtained together.

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

“A longer taper is always better.”

It lowers the cutoff and buys nothing above it. The computed cutoffs here scale in proportion to the length — 398 nm for a 150 nm taper and 1062 nm for a 400 nm one — so the only question a design has to answer is what the longest wavelength of interest is. Past that, extra length is material spent on nothing.

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

“A random walk's typical distance is what says how much ground it has covered.”

It says how far it got, not how much it saw. In two dimensions a walk of N steps reaches about √N and visits about N/ln N distinct sites, so it covers nearly all of a region it can reach; in three it reaches √N and visits about 0.66N distinct sites out of a volume of order N^1.5, so almost all of the region is untouched.

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

“Residual entropy means the substance has not reached equilibrium, so thermodynamics does not apply.”

Thermodynamics applies and gives the right answer. The residual entropy is measured by integrating a heat capacity and is predicted by counting arrangements, and the two agree to a per cent for ice. What has failed is ergodicity, not thermodynamics: the system explores its frozen-in states on no accessible timescale, so those states are a permanent multiplicity rather than a temporary one.

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

“A material either has a yield stress or it does not.”

Measured yield stresses depend on how long the material has been left alone and on how the measurement is made, sometimes by a factor of several. Many pastes build structure at rest and lose it under shear, so the stress needed to start them exceeds the stress needed to keep them going — and a single number is a summary of a rate-dependent process rather than a material constant.

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

“The crests of a wave move in the direction the wave is going.”

In a stratified fluid they move across it, exactly at right angles. Because the frequency depends only on the direction of the wavevector, the group velocity has no component along it, so energy travels perpendicular to the phase — the drawn dot product is zero to twelve decimal places — and a photograph of the crests shows them sliding sideways through a beam that is going somewhere else.

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

“The sine condition is a small-angle approximation to something exact.”

It is the opposite: the paraxial relation h = f tan u is the small-angle statement and h = f sin u is the condition that has to hold at all angles for the image of a nearby off-axis point to be sharp. A system built to the tangent condition is the one that has been approximated, and it is comatic by exactly the difference between the two functions.

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

“Resolving a star requires a telescope large enough to form its image.”

Michelson and Pease measured Betelgeuse in 1920 with two small mirrors on a beam, and no telescope resolved that star for another seventy years. The measurement is of a contrast, not a picture: what it needs is a baseline long enough for the contrast to fall, and the aperture that would be needed to form the image never has to exist.

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

“A fibre's dispersion is a property of the glass.”

Half of it is. The other half comes from the waveguide — a mode confined by a core spreads into the cladding by a wavelength-dependent amount — and that half is set by the core's diameter and index step rather than by the material. It is negative, it can be engineered, and shifting the zero from 1,273 to 1,550 nm by changing the core profile is a standard product.

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

“In the Coulomb gauge the potential propagates instantaneously, which contradicts relativity.”

It propagates instantaneously and contradicts nothing, because the potential is not measurable. The figure shows the Coulomb-gauge scalar potential at full value beyond the light cone while the Lorenz-gauge one is zero there; the fields are zero outside the cone in both, because the Coulomb gauge's vector potential carries a term that cancels the instantaneous part of −∇φ exactly.

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

“A metal enclosure shields against magnetic fields.”

One millimetre of copper gives 0.9 decibels of absorption at mains frequency, which is a factor of 1.1. At a megahertz the same sheet gives 133 dB. Magnetic shielding at low frequency is done with permeability — mu-metal offers the flux a path it prefers — rather than with conductivity, and no thickness of copper anybody would build fixes the fifty-hertz case.

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

“Ampère's law gives the field of a current.”

It gives the circulation of the field round a loop, which is one number, and the field only where symmetry already supplies the direction and constancy along that loop. For a finite wire it supplies neither: the field is not constant on a circle in any useful sense and the enclosed current is ambiguous. The law is still exactly true and computes nothing.

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

“The traveller sees the journey take less time because the clock has slowed.”

The traveller's clock is the only clock they have and it does not appear to them to run slow. What is short is the *distance*: in the ship's frame the galaxy is length-contracted by γ, and at turnover γ is in the tens of thousands. Describing it as a slowed clock imports the home frame's description into a frame where it does not belong.

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

“The eastbound and westbound aircraft are symmetric, so the effect should be the same size either way.”

They are not symmetric, because the ground is not at rest. In the non-rotating frame a clock at 30° latitude is already travelling east at 402 metres a second, so an aircraft flying east adds to that and one flying west subtracts from it — and the kinematic term is quadratic, so adding and subtracting the same amount from a large number changes it by different amounts.

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

“A star above the limit collapses to a black hole.”

It collapses, and what stops it is a separate question. Neutron degeneracy takes over at nuclear density and holds up stars to about two solar masses; above that nothing known does. And the collapse begins slightly below the limit, because at those densities electrons are captured onto nuclei, which removes the very pressure holding the star up.

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

“Dispersion in a plasma is a property of the plasma.”

The whistler dispersion is a property of the whole path — the integral of the electron density along a field line, weighted by the local gyrofrequency. A single measured delay curve gives one number for tens of thousands of kilometres of magnetosphere, and separating what happens where requires many paths or an independent constraint.

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

“A d orbital is further out than an s orbital of the same shell, so it is less bound.”

It is nearer in, not further out — at fixed n the mean radius falls as the angular momentum rises. What decides the energy is not the bulk of the orbital but the small amount of probability inside the innermost shell, where the nuclear charge is unscreened: 0.88 per cent for 3s, 0.11 for 3p and 0.00 for 3d in the solved orbitals here.

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

“Electrons of parallel spin repel each other more strongly.”

They do not repel more strongly; they are further apart before any repulsion is considered. The computed pair correlation for the antisymmetric spatial state is exactly zero at zero separation with no interaction in the calculation at all, and the root-mean-square separation differs from the symmetric case by ten per cent. The energy difference appears only when a repulsion is switched on, and it is a consequence of the separation rather than a force.

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

“Anomalous dispersion is an anomaly — a peculiarity of some materials near their absorption lines.”

It is neither rare nor optional. The refractive index falls with frequency across a band exactly as wide as the absorption line, in every medium that has one, and the integral that connects the two makes that region compulsory. What is unusual is not the falling index but being able to see it, because the same band is where the material is opaque.

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

“The reverberation after a sound is the room — echoes from the walls.”

A room supplies most of it and is not required for any of it. The tail computed here is in an unbounded, uniform, lossless medium with no boundary anywhere, and it is a property of two dimensions rather than of anything the wave met. A struck plate rings on for the same reason before its edges are considered.

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

“The modes of a region depend on its shape, so counting them requires solving for that shape.”

Counting them does not. Three regions of exactly the same area — a square, a right triangle and an oblong four times as long as it is wide — have completely different spectra and mode counts that agree to leading order and differ only through their perimeters, with measured slopes matching −L/4π. The individual frequencies need the shape; the count needs an area and a length.

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

“Air resistance reduces the range, so the best angle is still 45° and everything just falls short.”

Loss and asymmetry are separate effects and only the second moves the optimum. Integrating the trajectory at a drag force equal to 2.4 weights at launch puts the best angle at 38.1°, not 45° — and the best of five angles drawn at that drag is 32° — and the range curve leans: ten degrees under the optimum costs little and ten degrees over it costs a great deal.

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

“This is parametric resonance, which is how a child pumps a swing.”

It is the same equation in a different corner of its chart. Parametric resonance drives a mode unstable by modulating at about twice its frequency; this drives a fixed point stable by modulating at many times any natural frequency. One takes energy in at resonance, the other never resonates with anything, and the boundary between them is on the same stability diagram.

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

“The principle says nature is economical: it minimises effort, or time, or energy.”

The action is not effort, time or energy, and it is not minimised in general. It is the time integral of kinetic minus potential energy, a quantity with no independent meaning, and the content of the principle is that its first variation vanishes. Nothing in it is thrifty; the same formalism written with a plus sign describes nothing.

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

“Damp sand is stronger than dry, so a large enough damp pile can stand at any angle.”

Cohesion supplies a stress, not an angle, and the weight it must hold grows with the pile. The ratio c/ρg is a height: a sand with 200 Pa of cohesion stands vertically at a centimetre and is back within a degree of the dry angle by seventy centimetres. That is why a sandcastle works and a sand cliff does not.

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

“Desalination is expensive because membranes are imperfect; a better membrane would make it cheap.”

A perfect membrane changes nothing about the floor. The minimum work is the free energy of mixing read backwards — 0.79 kWh per cubic metre for the first drop from seawater and 1.10 at half recovery — and a good plant already uses about three, so the whole of the remaining margin is a factor of three rather than the factor of a thousand a technological framing implies.

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

“Helium II is a mixture of two substances, one superfluid and one normal, in some proportion.”

There are no two kinds of atom and nothing that could be separated. The two-fluid description is a decomposition of one liquid's motion into two velocity fields, and its test is that quantities computed from it — a fountain height of 19 mm per millikelvin at 1.8 K, a counterflow speed of 0.2 m/s per watt per square centimetre — come out right, not that anybody can find the two components.

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

“A gas cloud collapses if its gravity is stronger than its pressure.”

Neither is a force until a size is chosen, and the comparison is between two times rather than two forces. Gravity collapses a region in a free-fall time that contains no radius at all; pressure can only answer on a sound-crossing time that grows with the region. So every gas is stable at small scales and unstable at large ones, and the criterion is a wavelength.

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

“Screening is the plasma's charges physically covering the test charge, like a coat.”

Nothing is stuck to anything. The electrons are moving at thermal speed and any individual one is near the charge for a moment; what is steady is the average excess density, which the Boltzmann factor gives as a small fractional enhancement over a region of size λ_D containing a hundred thousand particles. It is a statistical rearrangement, not a shell.

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

“Whether radiation can blow a star apart depends on how close to it you are.”

It does not depend on distance at all. Radiation pressure goes as L/4πr²c and gravity as GM/r², so the ratio has no r in it — computed here at radii spanning four decades the ratio is identical to a part in 10¹⁵. What the comparison yields is a luminosity, and a body either is above it everywhere or below it everywhere.

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

“The supernumerary arcs are interference between light from different drops.”

Every drop produces them independently. They are the interference of the two rays that a fold caustic joins — the pair that emerge from the same drop at the same angle with different impact parameters — and the calculation for a single drop reproduces their positions from that pair alone.

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

“A phase shift in optics is an optical path length — a thickness times a refractive index.”

Not all of it. Turning a half-wave plate changes the phase of transmitted circular light by exactly twice the angle turned, with the plate's thickness, material and the wavelength all unchanged and none of them in the answer. The measured slope is 2.000 radians per radian, and it is a property of the geometry of the path through the space of states.

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

“Two vessels connected by a hole come to the same pressure.”

Only if the hole is larger than the mean free path. If it is smaller, what equalises is the flux each way rather than the pressure, and equilibrium sits at P₁/√T₁ = P₂/√T₂ — so a vessel four times hotter holds twice the pressure indefinitely. Every low-pressure gauge at a different temperature from its vessel reads wrong by that factor.

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

“A supersaturated vapour condenses because condensing lowers its free energy.”

It lowers the free energy of the finished droplet and raises it for every droplet on the way there. At a supersaturation of 3 the barrier is 73 kT and the critical droplet is 0.98 nm across; below that radius a droplet is more expensive than the vapour it came from and evaporates again. Thermodynamics says where the system is going and nothing about whether it can start.

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

“An accelerating detector heats up because it is being shaken.”

Nothing is shaking it and no energy is being supplied by the acceleration itself. The detector's excitation is a property of how the field's ground state decomposes into modes for an accelerated observer: the inertial vacuum contains no quanta of the inertial modes and a thermal population of the accelerated ones, and the two descriptions are of the same state.

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

“The speed of light has been measured to nine decimal places, so of course it is the same in both directions.”

Every one of those measurements is a round trip — an interferometer, a time-of-flight to a mirror, a resonant cavity. The round-trip speed is c under every synchronisation convention: computed here across five of them the round-trip times agree exactly. What varies between conventions is the split of that round trip into two halves, and no measurement reads the split.

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

“Earnshaw's theorem forbids stable levitation.”

It forbids it for a fixed charge in a static electric field in a charge-free region, which is four conditions. Diamagnetic levitation breaks the sign of the response, a Paul trap breaks the staticness, optical tweezers break it with dissipation and induced dipoles, and a maglev train breaks it with feedback. All four are real and none contradicts the theorem.

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

“Interference is destroyed because measuring the particle disturbs it.”

Disturbance is not the mechanism and is not needed. What suppresses the interference is the existence of a correlation between the particle and something else, whether or not anything is read out: the visibility equals the overlap of the environment's two states, and V² + D² = 1 exactly. An interaction that leaves no distinguishing record leaves the fringes untouched however violent it is.

Tested in Where the interference goes · the measurement reading path

“Noether's theorem says every symmetry gives a conservation law.”

Every *continuous* symmetry of the *action* does. A discrete symmetry gives none: reflection is a perfect symmetry of electromagnetism and there is no conserved quantity attached to it, only a selection rule. And a symmetry of the equations of motion that is not a symmetry of the action gives none either — the scaling that leaves Kepler's equation invariant changes the action by a factor and supplies nothing.

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

“Adding supports makes a structure safer, because the load is shared more ways.”

It makes the load-sharing undetermined, which is a different thing. A four-legged table on an uneven floor puts more than half the load on two legs and can lift a third off entirely; the redistribution from a leg one part in a thousand short is computed here and does not depend on the load at all. Redundant structures are safer against a member failing and less predictable in what each member carries.

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

“Waves add, so two sounds together are the sum of the two apart.”

Superposition is a property of linear equations, not of waves. In a nonlinear medium two tones produce their sum and difference frequencies and every combination of them, which is exactly how an acoustic parametric array makes a narrow beam of audible sound out of two ultrasonic ones. The figure of harmonic amplitudes here shows a single input tone growing five harmonics that were never present.

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

“Solitons pass through each other because the equation is linear enough for superposition.”

During the overlap the sum of the two is emphatically not what is drawn — the merged profile is lower than the taller of the two constituents and is not their sum anywhere. What emerges is unchanged, which is a much stranger statement than superposition and is true of almost no nonlinear equation. The price is a phase shift, measured here at +0.55 for the fast one and −1.10 for the slow one.

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

“The kinetic theory predicts viscosity proportional to the square root of the temperature.”

It predicts that for hard spheres, and real gases follow a steeper law — 0.77 for air at 300 K rather than 0.5. The excess is the attraction between molecules, which deflects a slow pair more than a fast one and so shrinks the effective cross-section as the gas warms. The exponent falls back towards a half by four thousand kelvin, which is the hard sphere being recovered when the kinetic energy is far above the well depth.

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

“The forces in a pack of grains can be found from the loads by statics, given enough care.”

Above the threshold the contacts are redundant, and the count of redundant contacts is measured here rather than argued: a fully packed lattice patch of 37 grains carries 19 contacts whose forces no balance of forces or moments determines. Two silos filled identically carry different force networks, and which one a given pack has depends on how it was built.

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

“Anything passing inside the Roche limit is torn apart.”

Only a body held together by its own gravity is. A body small enough for its material strength to beat its self-gravity ignores the limit entirely: Phobos orbits well inside Mars's and is intact, and a spacecraft or a person would be unaffected. The limit is a statement about self-gravitating bodies, and the crossover size below which strength wins is set by the material rather than by the orbit.

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

“A random walk of N steps of length λ covers a distance Nλ.”

It covers √N λ, and the difference is the whole subject. Measured over 240 independent walks here the exponent is 0.503 against an exact half, so a walk of 900 steps ends about 30 lengths from where it began rather than 900. Escaping a body therefore costs (R/λ)² steps rather than R/λ, and shrinking the free path by a thousand lengthens the journey by a million.

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

“The ring appears at 22° because that is the deviation an ice prism produces.”

An ice prism produces every deviation from 21.8° upwards, over a window of incidence 76° wide. What puts a ring at 21.8° is that the deviation is *stationary* there, so a wide band of orientations delivers light within a fraction of a degree of one angle while the rest is spread over forty. The histogram drawn here has a hard cliff at the minimum and a long tail beyond it, and the cliff is the ring.

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

“Speckle is noise and can be averaged away.”

It averages as the square root of the number of *independent* patterns, and independence has to be bought. The measurement here gives a slope of −0.509 against an exact −½: halving the contrast costs four patterns and dividing it by ten costs a hundred. Each independent pattern has to come from a separate polarisation, a separate wavelength or a separate moment, and every one of those was wanted for something else.

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

“A single photon splits at a beam splitter and half of it goes each way.”

The simulated run here gives a coincidence rate 0.014 of the accidental one, seventy standard deviations below the classical floor of one. A field that divided would trigger both detectors at the accidental rate at best and more often at worst; a wave of any intensity distribution whatever is bounded below by one, which is proved here by trying two hundred of them.

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

“Levels of the same symmetry cannot cross, so a crossing in a spectrum means the states have different symmetry.”

That is right for a one-parameter family and wrong in general. With two parameters to vary, two levels of the same symmetry can be made degenerate at isolated points — conical intersections — and those points are where molecular photochemistry happens. The non-crossing rule is a statement about how many knobs are being turned, not about symmetry alone.

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

“Two electrons in a helium atom repel, so the triplet state is lower because the electrons are further apart.”

The direction is right and the reasoning is inverted. Antisymmetry of the *spatial* part is what keeps the electrons apart, and that spatial antisymmetry requires the *spin* part to be symmetric — a triplet. The spins are not doing anything; they are a label that records which spatial symmetry is available, which is why an effect with an energy of electronvolts is written as if it were magnetic and is not.

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

“The field is discontinuous at the boundary because the permittivity is.”

Neither field is discontinuous in the way that suggests. The component of E along the surface is exactly continuous, because a line integral round a vanishing loop must vanish; the component of D across it is exactly continuous, because the flux out of a vanishing pillbox must equal the free charge inside, which is none. What jumps is the normal component of E, by the permittivity ratio, and the tangential component of D by the same factor.

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

“Reciprocity is a consequence of the loops being identical or symmetrically placed.”

The loops here are a circle and a tilted, offset rectangle — as asymmetric an arrangement as fits on a page — and the agreement is unaffected. What the theorem needs is that the medium be linear and reciprocal, not that the geometry be symmetric; it fails in a magnetised ferrite, where the permeability tensor is not symmetric, and that failure is exactly what a circulator is built to exploit.

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

“A superconductor screens magnetic fields better than any normal metal.”

Above about nine terahertz an ordinary copper sheet has the shorter screening length, computed here from the two expressions with no fitting. The superconductor's advantage is not that its screening length is small but that it contains no frequency at all: copper's diverges as the frequency falls and is infinite for a steady field, while the London depth is the same twenty-two nanometres at every frequency including zero.

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

“Any system can in principle be brought to a negative temperature by adding enough energy.”

Only one with an upper bound on its energy can. A gas has no ceiling on its kinetic energy, so its entropy rises without limit and its temperature never turns over; nuclear spins in a lattice do have one, which is why the effect was first demonstrated on them in 1951 and why every subsequent realisation is a spin system, a bounded optical lattice, or a set of internal levels.

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

“The Joule–Thomson effect is a cooling.”

It is a temperature change whose sign is set by where the gas starts. The inversion curve computed here divides the plane, and outside it every gas warms. Nitrogen's inversion temperature is 621 kelvin so air cools at room temperature; hydrogen's is 202 and helium's 43, so both warm — and a Linde machine fed room-temperature hydrogen heats it up.

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

“Since nothing is rigid, rigid-body mechanics is wrong.”

Rigid-body mechanics is excellent wherever the transit time is short compared with the timescale of the motion, which covers essentially all of engineering. What it cannot be used for is an argument about simultaneity — the case it is invariably reached for. Bell's spaceship paradox, the pole and barn, and every 'push the rod and signal instantly' proposal are the same misuse.

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

“A photon rocket is the ultimate drive because light carries the most momentum per unit energy.”

It carries the least. A photon carries momentum E/c, and a massive exhaust carries γmv, which for the same energy is larger at every speed below c — so a photon rocket has the highest exhaust speed and the worst momentum per joule. What makes it the limiting case is the exhaust speed, which decides the logarithm's coefficient; what makes it impractical is that converting mass to a collimated beam at any useful power is beyond anything known.

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

“Dissipation destroys structure, so a damped system ends up somewhere simple.”

This map contracts area by exactly 0.3 at every point, without exception, so a region of any size shrinks to nothing under iteration — and what it shrinks onto is more intricate than what it started as, not less. The contraction is what makes the layers thin enough to be distinguished at every magnification; a volume-preserving system has no attractor to be fractal. Dissipation is a necessary ingredient of a strange attractor rather than an obstacle to one.

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

“The cycloid is the isochronous curve because of a special property of the cycloid.”

It is the isochronous curve because it is the curve whose height above the lowest point is exactly proportional to the square of the arc length from it: y = s²/8a, with no approximation. That condition is a differential equation, its solution is a cycloid, and the physics never mentions rolling wheels at all. The same condition read as a statement about restoring force says the force along the curve is proportional to the displacement along it, which is the definition of simple harmonic motion.

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

“The edge wave is a convenient fiction — nothing is actually radiating from the rim.”

It is a decomposition rather than a claim about atoms, and it makes quantitative predictions no fiction could. Its amplitude falls as the distance to the power −1.0000, measured on the computed field over a decade, which is what a line source and nothing else does; its magnitude is the same at equal distances into the light and into the shadow; and its strength on the boundary is exactly half the incident amplitude, fixed by nothing but the requirement that the total field be continuous where the geometrical one is not.

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

“Surface waves do most of the damage in an earthquake because they carry most of the energy.”

They do not necessarily carry most of it at the source. They arrive largest because they spread over a circle rather than a sphere, so their amplitude falls as the inverse square root of distance where a body wave's falls as the inverse first power — a ratio that grows as the square root of the distance and reaches 31.6 at a thousand kilometres. The dominance is geometry acquired on the way, not a property of the source, and it is why a distant earthquake is felt as a long roll and a nearby one as a jolt.

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

“Foam geometry is complicated, so its angles vary with the liquid and the bubble size.”

Two angles are available and no others, and neither contains anything about the liquid. Three films meeting along a line make 120.00 degrees with one another because three equal coplanar vectors sum to zero only at that angle; four such lines meeting at a point make 109.47 degrees because four equal vectors in space sum to zero only at the tetrahedral angle. Both are solved here from the balance rather than quoted, and a junction with any other number of films rearranges within milliseconds because the tensions there cannot balance at all.

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

“The Weissenberg number is a Reynolds number for elastic fluids.”

A Reynolds number compares inertia with viscosity and is a ratio of two terms in the same equation. The Weissenberg number compares two stresses the fluid itself produces — the elastic against the viscous — and is a shear rate multiplied by the fluid's own relaxation time, so it can be large in a flow whose Reynolds number is a millionth. Rod climbing happens in creeping flow, where inertia is negligible everywhere and the only reason anything moves outward at all is the small centrifugal term the elasticity has to beat.

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

“Strong-field light bending is a different theory from the 1.75-arcsecond deflection measured at the Sun.”

It is the same equation integrated further in. The null geodesic solved here — d²u/dφ² + u = 3Mu² — gives capture below 5.196 masses and, run at an impact parameter of sixty masses, returns a deflection of 0.0666 radians against 4M/b = 0.0667. One integration covers both, and the weak-field formula is what its first term gives; there is no boundary between the regimes, only a term that stops being small.

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

“Landau damping produces entropy, since a wave has become particle motion.”

The Vlasov equation is exactly reversible and conserves every functional of the distribution, entropy included. What the damping does is wind the perturbation into finer and finer oscillations in velocity — ten sign changes at ten plasma periods, twenty-four at twenty-four, forty at forty — and the field, being an integral over velocity, stops being able to see it. Entropy appears only when a collision rate, however small, smooths that structure away, and until then the loss is recoverable: reversing the velocities brings the wave back.

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

“A zone plate is a poor lens because of its chromatic aberration.”

Its focal length varies by 62 per cent across the visible where a fused-silica singlet's varies by 2.7 — a hundredfold worse, and of the opposite sign, since the plate's focus moves *in* as the wavelength grows. That is disqualifying for white light and irrelevant for the application it is actually used in: at X-ray wavelengths there is no glass to make a lens out of, the source is monochromatic, and a set of rings that absorbs is the only focusing element available.

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

“A rhomb is achromatic because glass has low dispersion.”

It is achromatic because its retardation depends on the index rather than on a thickness measured in wavelengths, which is a difference of kind and not of degree. Across the visible the rhomb's retardation moves by 4.05 degrees and a quartz quarter-wave plate's by 62 — a factor of fifteen — and the reason is that the plate's retardation is proportional to one over the wavelength, which changes by a factor of two across the band, while the rhomb's depends on an index that changes by one per cent.

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

“A scanning tunnelling microscope resolves single atoms because its tip is atomically sharp.”

The tip is not made sharp and cannot be. What supplies the resolution is that the current falls by 0.94 decades for every angstrom, so an atom of the tip standing 2.5 angstroms further back than the apex carries 1/230 of the apex's current: for the three tip shapes computed here the apex carries 99.52, 96.29 and 81.92 per cent of the total, the last of them being a tip whose neighbours are only 0.9 angstroms behind. A tip is prepared by crashing it into the surface until some protrusion sticks out furthest, and the exponential does the rest.

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

“No-cloning means an unknown quantum state carries no usable information.”

It carries information that can be extracted once, and the best universal copier reaches a fidelity of five sixths on every state. The best universal copier reaches a fidelity of five sixths on every state — better than the two thirds a measure-and-prepare machine averages, and short of one — so an approximate copy is available and is quantitatively limited. Quantum key distribution rests on precisely that number: an eavesdropper who copies gets 0.8333 rather than 1, and the shortfall shows up as errors in the legitimate channel at a rate that can be measured.

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

“Flux quantisation is a small quantum correction to a classical effect.”

It is a quantum condition holding over centimetres, and its consequences are macroscopic and periodic. The transition temperature of a thin-walled cylinder oscillates with the flux threading it, with a period of exactly one quantum and a cusp at every half — a shape that no smooth field effect produces, and a periodicity in a steady applied field that nothing classical does at all. The size of the effect is a few millikelvin and the period is what carries the physics.

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

“Maxwell's equations are four laws that all have to be solved.”

Two of them contain no time derivative, so there is nothing in them to solve forward. Integrating only the two that do — Faraday's and Ampère's — for a hundred and ten steps on a two-dimensional grid leaves the other two satisfied to 2.4 × 10⁻¹⁶ of the field's own size, while the field itself changes by a factor of 8.6. Neither divergence is computed during the run, neither is imposed, and neither drifts, because taking the divergence of a curl gives zero identically.

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

“Radiation pressure is negligible, so the adiabatic index of radiation is a curiosity.”

It is negligible in a room and it is not in a star. The index computed here by compressing a box at constant entropy is 1.333333, which is exactly the value at which a self-gravitating sphere stops being stable against collapse — so a star whose support comes mostly from radiation rather than from gas is marginally stable, and how close to four thirds its effective index sits decides whether it can exist at all. The index is not a curiosity; it is the reason there is an upper limit on stellar mass.

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

“Kelvin proved the relation between them in 1854.”

He derived it and did not believe he had proved it, and he was right to be uneasy. The argument treats the thermoelectric parts of the process as reversible while the ordinary heat conduction going on alongside them is not, and there is no justification within thermodynamics for separating the two. The relation was verified experimentally for seventy-seven years before Onsager supplied a reason for it, and the reason turned out to come from the microscopic equations rather than from thermodynamics at all.

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

“Below the transition the system is ordered and above it disordered, as in any other transition.”

Below the transition the correlations decay as a power law — slowly, but to zero — so there is no long-range order there either. What the low-temperature phase has is *quasi*-long-range order, with an exponent that varies continuously with temperature from zero up to a quarter at the transition. The renormalisation flow shows why: the whole low-temperature phase is a line of fixed points rather than a single one, so every temperature below the transition is critical and none of them is ordered.

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

“The illusion can be made arbitrarily large by choosing the geometry.”

Only by making the source fast. The maximum over all angles is γβ, which is a function of the speed alone, so an apparent speed of ten requires γ of at least ten and a true speed within half a per cent of light's. And below β = 1/√2 there is no angle at all that produces an apparent speed above one, so the effect has a threshold rather than being available to any source seen from the right direction.

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

“Beaming is an astrophysical effect about jets.”

It is a kinematic fact about any moving source, and its most precisely measured instance involves no jet at all. The microwave background's dipole is the same Doppler factor applied to a thermal spectrum: a speed of 369 kilometres a second gives a sky hotter ahead and colder behind by 3.35 millikelvin, against a measured 3.362. That is a hundred times larger than every other feature of the microwave sky, and it is the only part of it that is about the observer rather than about the universe.

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

“The rotating disc paradox is about whether the circumference or the radius contracts.”

Both answers are known and neither is in doubt: the circumference contracts because it lies along the motion and the radius does not because it lies across it. The difficulty was never which contracts but what to make of the result — a ratio of circumference to radius that exceeds two pi, in a frame where nothing has changed but the state of rotation. That is a statement about the geometry the rotating observer measures, and it cannot be removed by any choice of coordinates.

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

“Higher modes have more nodes because the wavelength is shorter, so the nodes are evenly spaced.”

The count is right and the spacing is not. On a string whose density rises fivefold over part of its length the nodes bunch where the medium is heavy and spread where it is light, because the local wavelength follows the local speed — and the fourth mode still has exactly three of them. The count survives an arbitrary medium; the spacing does not survive any departure from uniformity at all.

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

“Make the pulse shorter to measure range better; the velocity measurement is unaffected.”

A pulse of duration T has a bandwidth of about 1/T, and a frequency shift smaller than that bandwidth cannot be resolved. Halving the pulse halves the range uncertainty and doubles the smallest measurable speed, exactly. It is the same trade as any other between a duration and a spectral width, and no waveform escapes it — the sophisticated ones spread the product over a longer transmission rather than beating it.

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

“An optical isolator proves that light can be made to travel one way, so reciprocity is only a rule of thumb.”

An isolator contains a magnet, and the magnetic field is what breaks the time-reversal symmetry the theorem rests on. Without it the same device is exactly reciprocal. The other physical exception is a moving medium, which is why an ultrasonic flowmeter can read a difference of 91 nanoseconds between an upstream and a downstream pulse over ten centimetres of water — and why neither exception can be built out of shape alone.

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

“Photons arrive independently, so the arrival times at two detectors cannot be correlated.”

They are correlated, and the excess is exactly |γ|². For thermal light the correlation is a statement about a classical fluctuating wave — the beat between the many independent emitters producing intensity that wanders — and the photons are simply reporting where the wave was large. The independence that fails is the independence of the *classical intensities*, not any interaction between photons.

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

“Negative charge is attracted to a positive conductor, so a negatively charged surface is pulled inward.”

The pull is outward whatever the sign, because the pressure goes as the square of the field. What is happening is like charges repelling within the layer, and two negative charges repel exactly as two positive ones do. A charged soap bubble expands; it does not matter which way it was charged, and reversing the charging makes no measurable difference to the radius.

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

“The temperature wave in the ground travels downward at a speed, so the seasons arrive at depth after a delay set by that speed.”

There is no speed. A diffusive disturbance is nonzero everywhere the instant it starts, so nothing arrives at any time and there is no front to time. What the solution has is a phase lag proportional to depth — which looks like a speed and is not one, because it is proportional to the square root of the period. Halving the period does not halve the delay at a given depth; it multiplies it by the square root of two.

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

“Water's melting curve leans backwards, which is unusual, so the phase rule must work differently for it.”

The rule counts and does not care. The backward lean is Clapeyron's relation with a negative volume change, because ice is less dense than water, and it gives −138 bar per kelvin — a slope so steep that it is drawn vertical on any diagram wide enough to show the other two curves. The counting that makes the triple point a point is untouched: it never asked which way anything leaned.

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

“The harmonic oscillator shows that quantum and classical predictions coincide, which is why the classical limit works.”

The harmonic oscillator is the one potential in which the average force and the force at the average cannot differ, because the force is linear. It is therefore the worst possible example from which to conclude anything about the classical limit: the agreement there is exact for every state, however wide, however far from classical — including states with a single quantum in them.

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

“Making the barriers thicker makes the device work less well.”

It makes the peak sharper without lowering it. Every resonance reaches exactly one however opaque the barriers are, provided they are identical; what changes is the width, which falls from 5.2 × 10⁻¹ to 3.7 × 10⁻² as the barriers go from 0.6 to 1.2 wide. A better barrier buys selectivity and pays in time, because the linewidth is the escape rate of the state trapped between them.

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

“The built-in potential can be measured by putting a voltmeter across an unbiased diode.”

It reads zero, and must. The junction's built-in potential is exactly cancelled by the contact potentials at the two metal–semiconductor contacts, because the whole loop is at one Fermi level and a voltmeter measures a difference in Fermi level. Getting energy out of a junction requires something that shifts the Fermi levels apart — light, or a temperature difference — and the impossibility of doing it with a loop of wire is the second law wearing a circuit diagram.

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

“The twin paradox needs general relativity, or at least the details of the acceleration.”

It needs neither. Counting signals settles it: over a round trip each twin receives the same number the other sent, and the asymmetry is that the traveller switches from receiving at 1/k to receiving at k halfway through the journey while the stay-at-home does so much later. The arithmetic gives the ratio of elapsed times as (k + 1/k)/2 with nothing about the turnaround in it beyond the fact that it happened.

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

“Acceleration has no effect on a clock, so an accelerating observer sees the same physics as an inertial one.”

The clock hypothesis is about the rate of an ideal clock, not about everything an accelerating observer encounters. An accelerating observer has a horizon behind them, cannot receive signals from beyond it, and finds the vacuum to be a thermal bath at a temperature proportional to the acceleration. All of that is acceleration-dependent physics; none of it is the clock running at a different rate.

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

“The two-fluid model says helium II is a mixture of two kinds of atom, one superfluid and one normal.”

Every atom is identical and none of them is either kind. The two densities are a bookkeeping device for a single quantum fluid whose excitations carry the entropy and the viscosity while the ground state carries neither. The test that the model is more than bookkeeping is that its one parameter, the ratio of the two densities, can be measured by a torsion pendulum and then predicts the fountain pressure, the counterflow velocity and this wave's speed with nothing further adjusted.

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

“Shaking a mixture must mix it, because mixing increases entropy.”

Entropy arguments require a temperature, and a granular material does not have a useful one. Lifting a one-millimetre sand grain by its own diameter costs about 10¹² times kT at room temperature, so thermal energy cannot rearrange anything and the equilibrium the second law describes is never explored. What decides the arrangement is the driving and the geometry, and both of those have a direction in them.

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

“A plasma is quasineutral, so the electron and ion densities are equal everywhere in it.”

Everywhere except where it meets something, which is the only place a plasma is ever used. Inside the sheath the ion density exceeds the electron density by tens of per cent of itself, and it has to: the layer's whole job is to hold a potential of several electron temperatures, which requires an uncompensated space charge. Quasineutrality is a statement about scales larger than the Debye length, and the sheath is a few of them thick.

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

“A stronger trap needs a more powerful laser.”

It needs a tighter focus at least as much. The gradient force depends on the *gradient* of the intensity, which for a Gaussian beam scales as the power over the cube of the waist, while the scattering force scales as the power over the square. Halving the waist quadruples the stiffness and doubles the ratio of gradient to scattering force; doubling the power doubles both and improves the ratio not at all — which is why optical tweezers waited for high-aperture objectives rather than for bigger lasers.

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

“The contact between the balls is a spring, so the details do not matter.”

The same five balls with a linear spring between them send 0.942 of the speed to the end rather than 0.989, and the residual left behind is nearly five times larger. The figures sweep the exponent of the contact law from one to 2.4 and the far ball's share rises throughout. The details are the answer: everything else in the problem is identical between those runs.

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

“The capstan equation gives the tension in a wrapped rope.”

It gives the largest ratio the wrap can hold, which is a limit rather than a state. A belt transmitting a quarter of its capacity is not slipping over three quarters of its wrap and creeping over the rest at a quarter of the exponential rate; it has an idle arc where the tension is constant and an active arc where it climbs at the full rate. The figure computes how the wrap divides, and the active arc reaches the whole wrap only at capacity.

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

“Newton's corpuscular theory of light was refuted by the discovery of interference.”

Interference made a wave theory necessary; it did not by itself decide the corpuscular prediction about speed. Written as Jacobi did, mechanics is optics with an index proportional to the momentum, so a corpuscle entering glass must speed up while a wave entering it must slow down. Foucault measured the speed of light in water in 1850 and found it slower, which is the measurement that settled the sign of the exponent.

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

“Chaos in a mechanical system arrives by period doubling.”

That is the route for dissipative systems, which have attractors for a cycle to inherit stability from. A frictionless mechanical system has none: its route is the destruction of invariant curves as a coupling grows, which produces no cascade of doublings and no Feigenbaum constant. The two routes are drawn in different essays of this collection because they are different phenomena.

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

“Even harmonics come from a stronger nonlinearity than odd ones.”

They come from a different symmetry, not a different strength. A potential that is even in the displacement can only produce odd harmonics, at any amplitude, because the equation is unchanged under reversing the sign of both the displacement and the drive. The figures show a symmetric well producing even components below one part in ten thousand of the fundamental, and a well with a quadratic term producing a large second harmonic at the same drive.

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

“A solenoid's inductance is μ₀N²A/ℓ.”

That is the limit for a coil much longer than it is wide, and it is not a formula for a coil. Computed as a sum of Maxwell's mutual inductances between every pair of turns, a coil as long as its diameter has 68% of it and one a quarter as long has 36%. The ratio is Nagaoka's coefficient, tabulated since 1909, and the computation here agrees with the table at three aspect ratios.

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

“The resistance of a joint is the resistivity times its length over its area.”

A constriction has neither. The current arrives from everywhere and leaves in every direction, so there is no path length to divide by and no cross-section that is being crossed. Solved exactly, the resistance of a circular spot of radius a between two half-spaces is ρ over 2a — a resistivity divided by a length. Doubling the size of the contact halves it; making the conductors ten times thicker does nothing at all.

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

“Charge quantisation is explained by the standard model.”

It is an input to it rather than an output. Dirac's argument remains the only known reason electric charge must come in multiples: the string attached to a monopole is undetectable exactly when a charge carried round it gains a whole number of turns of phase, which requires the product of the two charges to be a multiple of Planck's constant. One monopole anywhere quantises every charge everywhere.

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

“The critical field of a superconductor is where superconductivity stops.”

For a type-II material there are three fields. Flux begins to enter at the lower critical field, which falls as the ratio of the two lengths rises; the thermodynamic field is where the condensation energy equals the expulsion energy and marks nothing observable in the magnetisation; and superconductivity ends at the upper critical field, which for niobium–titanium is about eighty-five times the thermodynamic one.

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

“The hidden momentum depends on the shape of the circuit.”

The figures compute it by summing the electric potential round each loop's own perimeter for four different shapes scaled to the same enclosed area — a square, a tall rectangle, a circle and an L — and the four agree with each other and with the closed form to the accuracy of the polygon. Only the area enters, because only the area enters the dipole moment.

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

“A lens in front of a fibre increases the light coupled into it.”

A lens conserves the product of area and solid angle. Demagnifying a source to fit the core opens its cone by the same factor, so what the core gains in area it loses at the wall, and the coupling is unchanged for any source already larger than the core. The figure computes the fraction coupled against source size and it is a ceiling, not a design target.

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

“A microscope's depth of focus can be computed from the blur circle.”

The ray formula predicts a depth that goes to zero linearly with the aperture, and the real one goes as the inverse square, which at NA 1.4 is smaller by a factor of several. The correct criterion is a quarter of a wavelength of path error across the pupil, which the figure shows leaving eighty-one per cent of the peak intensity — a wave statement with no ray counterpart.

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

“A polarisation state is a two-component complex vector.”

That describes a fully polarised beam only. A partially polarised one cannot be written as any such vector, because the four measurable quantities it has do not satisfy the relation a pure state's do — its Poincaré vector is shorter than its intensity. The set of beams is a solid ball rather than its surface, and the interior is exactly what a two-component vector cannot reach.

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

“To find how much light a particle removes, add up what it scatters in every direction.”

That gives the scattering cross-section and misses absorption entirely. The optical theorem gives the sum of both from the forward amplitude alone, and the two routes disagree for any absorbing particle by exactly the absorbed part. The forward direction is the one place the scattered and incident waves are inseparable, which is why it is the only direction that knows about everything.

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

“An exact surface would remove aberration from an image.”

It removes it from one object point. The figure traces the same exact surface with the object moved sideways and the blur grows from nothing, close to the first power of the displacement — which is coma. A single surface has one shape, imaging one pair of points perfectly consumes it, and the rest of the field is left with whatever the shape happens to do.

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

“Apodising an interferogram improves the spectrum.”

It removes the side lobes a sharp cut-off produces and widens every line by about half. Neither result is more correct than the other: both are choices about what to do with information the scan does not contain, and the choice depends on whether a weak line beside a strong one matters more than the last of the resolution.

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

“Two equal waves in antiphase cancel, so their combined intensity is zero.”

Their combined intensity is zero at that point. The intensity is a function of position and the statement is about one place; the same pair is delivering four times one wave's intensity half a fringe away. An energy argument has to be made over a region, and over any whole number of fringes the answer is the sum of the two intensities exactly.

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

“At the transmission zero the system is absorbing everything.”

Nothing is absorbing. The figures decompose the amplitude into a constant background and a resonant part whose phase sweeps through half a turn, and at one detuning the two are equal and opposite. The energy is reflected or returned, not taken up, and a lossless system has a Fano zero in exactly the same place as a lossy one.

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

“The backward wave is removed by taking the envelope on the forward side only.”

That is an instruction to ignore the answer rather than a reason it does not occur. The removal that works is a factor, one plus the cosine of the angle over two, which falls to exactly nothing at a hundred and eighty degrees — and it is not invented for the purpose. It comes out of solving the wave equation with Green's theorem, along with a quarter-period phase advance and a factor of one over the wavelength that the geometric construction contains no trace of.

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

“A chain of masses and springs is a discrete model of a string.”

It is a different medium, and the difference appears at short wavelength. A string's frequency rises without limit as the wavelength shortens; the chain's stops at twice the frequency of one mass held by two springs, reached when neighbours move in exact opposition. The two agree to second order in the wavenumber and nowhere else.

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

“The pattern's depth measures how much of the wave is being absorbed.”

It measures the magnitude of the reflection, and the reflection tells nothing on its own about whether the missing power was absorbed or transmitted. The ratio of the extremes inverts exactly to the reflection magnitude, and the reflected power is its square; what happened to the rest is a separate question the pattern cannot answer.

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

“A bend loses light because rays hit the wall at a shallower angle.”

The ray account gives a threshold radius and no gradual behaviour, and it says nothing about why a fibre carrying a mode far from cutoff can be bent tightly with no measurable loss at all. The correct account is about the mode's evanescent tail: at a radius set by the bend, the field would have to travel faster than the cladding permits, and what leaks is the amount of field out there — which is exponentially small and exponentially sensitive.

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

“A molecule at 300 kelvin has an energy of about kT.”

It has that energy on average over time, and at any instant it has whatever it has. The distribution of a six-degree-of-freedom system's energy is wide — the standard deviation is more than half the mean — so a statement about one molecule's energy at one moment is a statement about a draw from a broad distribution rather than about a property the molecule possesses.

Tested in The temperature a molecule does not have · the equipartition reading path

“Water is anomalous because its melting curve slopes backwards.”

The thermodynamics is not anomalous at all: the slope is the latent heat over the temperature and the volume change, exactly as for everything else, and the latent heat is positive as it always is. What is unusual is a structural fact — ice is less dense than water — and the backwards slope is that fact read through an entirely ordinary relation.

Tested in The melting curve that leans the wrong way · the phase change reading path

“The cross coefficients between different transport processes are independent quantities to be measured separately.”

They are equal. Onsager's relations, which follow from the microscopic equations running the same forwards and backwards in time, fix the coefficient coupling heat flow to a concentration gradient equal to the one coupling matter flow to a temperature gradient. Measuring either gives the other, and the equality has been checked in a dozen systems.

Tested in The gradient that drives the other thing · the diffusion reading path

“Jarzynski's equality makes free energies easy to measure from fast processes.”

It makes them possible and not easy. The exponential average is dominated by rare trajectories in the low-work tail, so the number of runs needed grows exponentially with the dissipation — the figure shows an estimate still a kilotesla-worth of kT high after ten thousand runs at three kT of dissipation. The equality is exact and the estimator is poor.

Tested in The second law, with a probability attached · the entropy reading path

“A gas obeying the ideal gas law is behaving ideally.”

At the Boyle temperature the first correction vanishes identically, so a real gas with strong attractions and a finite molecular size follows pV = nRT to first order in its density while doing nothing ideal at all. The second correction does not vanish there, so the agreement is a statement about one term rather than about the gas.

Tested in The first correction to the gas law · the kinetic theory reading path

“The uncertainty principle says that two quantities cannot be measured at once.”

Most pairs of quantities can. The relation has the commutator on its right-hand side, and for a pair that commutes the right-hand side is zero and the relation permits both spreads to vanish together. The grid of commutators computed here has more zero cells than non-zero ones, and each zero is a pair a single measurement can pin down exactly.

Tested in The questions that can be asked together · the uncertainty reading path

“The effect proves the vector potential is physically real in the way a field is.”

The potential is not unique — a gauge change alters it everywhere and shifts nothing measurable. What is measurable is its line integral round a closed contour, which is gauge-invariant and equals the enclosed flux. The figures compute that integral round four different contours and get the same number, which is the invariant statement the effect actually makes.

Tested in The phase a magnet leaves on a path it never touched · the matter waves reading path

“The classical limit is the limit of large quantum numbers.”

A stationary state of large quantum number does not swing at all: its probability density is time-independent and spread across the whole classical range, which is nothing like a pendulum. What swings is a superposition of many such states, and the classical limit is a statement about which superposition rather than about which energy.

Tested in The state that swings like a pendulum · the correspondence reading path

“The transmitted packet's peak arriving early means something crossed faster than light.”

The peak that emerges is not the peak that went in. A barrier attenuates by a factor that grows through the packet, so the late half is suppressed far more than the early half, and what comes out is a reshaped remnant of the leading edge. No point on the transmitted wave lies outside the light cone of the incident wave's own front, and the transmission at the thicknesses where the effect is dramatic is smaller than a part in a million.

Tested in How long the crossing takes · the tunnelling reading path

“Entanglement is a quantity a state has more or less of.”

Two three-qubit states with the same total have it in different places. GHZ has no pairwise entanglement whatever and all of it three-way; W has all of it pairwise and none three-way. The figures compute both splits from reduced density matrices, and no local operation converts one state into the other.

Tested in What two have they cannot give a third · the entanglement reading path

“An atom's size is set by the orbit its electron happens to be in.”

The size is the radius that minimises the sum of a confinement energy rising as one over the square of the radius and an attraction falling as its first power. The minimum is located here by searching the drawn curve and agrees with the Bohr radius to a part in a million, with no orbit, no quantisation condition, and no angular momentum anywhere in the argument.

Tested in Why an atom is the size it is · the atomic structure reading path

“If each observer measures the other's metre stick as shorter, one of them must be wrong.”

Both measurements come out at exactly 1/γ, to twelve decimals, and they are made across different sets of events. Each observer measures the two ends of the other's stick at what they call one time, and the two notions of one time pick out different pairs of events on the diagram.

Tested in The diagram a ruler cannot read · the spacetime diagram reading path

“Every observer agrees on where a system's centre is, and disagrees only about its coordinates.”

The weights in the average are energies, and energies are frame-dependent. Boosting a two-body system moves the share of the total carried by each body, so different observers put the centre at genuinely different events — not at different labels for one event.

Tested in The centre that is not a place · the relativistic dynamics reading path

“Charge density is the same in every frame, since charge is.”

Charge density rises by the Lorentz factor, because the same charges occupy a contracted length. The invariant is the product of the density and the volume, and the figures verify the cancellation across the whole speed range. Confusing the two is what produces the paradox that a neutral wire cannot become charged.

Tested in The one quantity a boost leaves alone · the field transformation reading path

“The rotation two boosts leave behind is a separate effect that has to be derived from the transformation.”

It is the angle by which the boost triangle's interior angles fall short of two right angles. The figures compute both — one from hyperbolic trigonometry, one from composing two Lorentz boosts — and they agree to better than a nanodegree at every angle between the boosts.

Tested in The space that speeds live in · the velocity addition reading path

“A straight line is the shortest path, so an inertial worldline minimises something.”

It maximises. Proper time is computed along a family of routes between the same two events and is largest for the straight one, checked by scanning the drawn curve; the ordinary length of the same drawn curves is smallest there. Both extremes sit at the same route and point in opposite directions.

Tested in The longest way round is the shortest clock · the time dilation reading path

“Viscosity communicates a wall's motion to the whole fluid.”

Only for steady motion. With the wall oscillating, the amplitude falls as exp(−y/δ) with δ = √(2ν/ω), and the envelope is checked against the drawn profiles to a part in a hundred million. Four skin depths out there is essentially nothing, however long one waits.

Tested in The shear that only reaches so far · the viscosity reading path

“Pressure is a scalar because it is defined as force divided by area.”

That definition gives a different number for every orientation of the area until something makes them equal. What makes them equal is a force balance on a shrinking wedge, in which the surface forces go as the square of the size and the weight as the cube, so the weight drops out in the limit. The wedge's excess is computed here and falls in exact proportion to its size.

Tested in The push that has no direction · the hydrostatics reading path

“The discharge goes as the area of the opening, so as the square of its diameter.”

It goes as the five-halves power. The extra half comes from the exit speed, which is set by the size of the opening rather than by the head above it — grains fall freely through a region of about the orifice size, so they leave at about √(gD), and the rate is that speed times the area.

Tested in The hourglass that keeps time · the granular matter reading path

“A cell placed in a solution shrinks or swells to a new steady volume.”

Only if the membrane holds the solute. When the solute leaks in, the cell shrinks and then returns, because the gradient it was responding to is abolished by the solute that caused it. The integration here shows the dip and the recovery, and a measurement made in the first moments cannot tell a leaky membrane from a tight one.

Tested in The membrane that almost holds · the osmosis reading path

“Capillary rise is higher in narrower channels, so the narrowest part of a container fills first.”

A corner has no narrowest part, so there is no height to compute. What decides whether the liquid enters it at all is the sum of the contact angle and the corner's half-angle: below ninety degrees the liquid runs in without limit, above it the liquid stays out however sharp the corner is made.

Tested in The corner a liquid never stops climbing · the surface tension reading path

“The Laplace–Runge–Lenz vector is a formal device with no observable content.”

Its direction is the perihelion's direction and its length is the eccentricity, both verified here against the orbit itself. When an extra inverse-cube term is added it turns, at a rate that matches the perihelion's own motion measured independently from the integrated path to better than two per cent.

Tested in The arrow that says which way the orbit points · the orbit stability reading path

“Black holes are slowly evaporating.”

Every black hole anybody has observed is colder than the microwave background it sits in, so it absorbs more than it emits and is growing. The two temperatures cross at about 4.5 × 10²² kg — a hundredth of the Moon — and a stellar-mass hole will not begin to lose mass until the background has cooled below sixty nanokelvin, in something like 10¹² years.

Tested in The hole that outlives everything and then does not · the horizons reading path

“A charge radiates if and only if it is accelerating.”

Whether radiation is present is not a locally defined question. A detector riding with a uniformly accelerated charge finds none, because the radiation goes into the region beyond its own horizon — nearly a light-year behind it at one gravity. A detector at rest far away finds the full Larmor power. Both are right about what they measure.

Tested in Whether a charge on a table glows · the radiating charge reading path

“A higher-energy probe always resolves a smaller distance.”

Only until the probe's own energy makes a horizon wider than its wavelength. The two lengths cross at about 10⁻³⁴ metres, located here by bisecting the drawn curves, and beyond it more energy hides more. The achievable resolution is the larger of the two, and that has a minimum.

Tested in The length no experiment can resolve · the planck scale reading path

“The belt trick shows why the sign is there.”

The belt demonstrates that a 2π rotation of an object tied to its surroundings cannot be undone by deformation while a 4π rotation can — a fact about the topology of the rotation group, which is the same group whether the thing being rotated is a spin, a belt or a coffee cup. It makes the double cover memorable and predicts nothing: it does not say which systems pick up the sign, and integer-spin systems, obeying the same topology, do not.

Tested in The turn that has to be made twice · the spin reading path

“A perturbation formula can be used for any load by making the load small enough in the answer.”

The exact frequency of a loaded string is root-found here against the first-order prediction as the load grows. The linear estimate runs away without bound while the true frequency flattens onto the mode of the clamped pieces, one rung lower than the unloaded mode. The two agree to a per cent for a load a few per cent of the string's mass and disagree by everything at ten times that.

Tested in The dent that raises the note · the standing waves reading path

“Field lines pull like elastic bands, so a picture of the lines shows the force.”

The tension along a line is only half of the tensor; the other half is a pressure of the same size across it. Two like charges have no line running between them at all and repel, and the figure computes that repulsion as the pressure their sideways-lying lines exert on the plane between them. Reading only the tension gets the sign wrong for every repulsive configuration.

Tested in The force read off a surface that touches nothing · the field energy reading path

“The magnetic moment is conserved, so a trapped particle stays trapped.”

It is an adiabatic invariant, not a conserved quantity: its change is smaller than any power of the slowness, which is not zero. Anything that varies on the timescale of a gyration breaks it — a collision, a wave at the cyclotron frequency, a field null. Mirror machines leak for exactly this reason as well as through the loss cone, and the loss cone itself narrows only as the inverse square root of the mirror ratio.

Tested in The drift that does not care what the charge is · the magnetism reading path

“Grip is set by the coefficient of friction.”

The slope of the force-slip curve at the origin is computed here from the tread stiffness alone and contains no coefficient at all, because at vanishing slip nothing is sliding. The coefficient decides only where the curve stops rising. Two tyres with the same coefficient and different tread stiffness behave completely differently at the slips a driver uses.

Tested in The grip that needs a little slipping · the friction reading path

“The intermediate-axis theorem says which spins are stable, so a body may spin stably about either of its extreme axes.”

The theorem is a statement about a perfectly rigid body, where both extreme axes give oscillatory linearised motion and the middle one does not. It says nothing about dissipation. Adding any internal energy sink leaves exactly one stable spin, and the figures here integrate the flip from a spin the rigid theorem calls stable.

Tested in The axis a leak of energy chooses · the rotation reading path

“Stopping down fixes soft corners.”

It divides the blur by the f-number and does nothing to the defocus that causes it, which is checked here across three apertures. Going from f/2.8 to f/16 improves the corner by a factor of six and leaves it hundreds of micrometres out of focus. On the figures here the corner at f/16 is still blurred over more than a dozen sensor pixels.

Tested in The flat scene that comes back curved · the imaging reading path

“A polarising filter darkens the sky.”

It darkens the band ninety degrees from the sun and does almost nothing towards or away from it, where the light is unpolarised and any filter can only halve it. The transmission curves computed here converge at both ends and separate by a factor of several only near a right angle. On a wide-angle lens the result is a dark stripe across the frame with pale sky either side.

Tested in The pattern the sky is written in · the polarisation reading path

“Making the pawl spring stiffer, or the notch deeper, would let it work.”

Both slow the device down exponentially and neither changes the balance, because the same energy appears in both rates. The net rate is computed here for notches from two to twenty thermal energies deep and is zero at zero load in every case. A rectifier cannot be improved into working; it has to be given a second temperature.

Tested in The engine a fluctuation cannot run · the heat engines reading path

“Any nonlinearity couples the modes, so a nonlinear system reaches equipartition.”

Coupling is necessary and not sufficient. The chain integrated here has coupled modes throughout — energy visibly moves between them — and returns to its starting distribution again and again for the whole run. What decides whether the sharing sticks is the energy density, not the presence of a nonlinear term.

Tested in The energy that refuses to be shared · the equipartition reading path

“An atmosphere warms the surface by trapping heat, like a blanket.”

A blanket works by suppressing convection and conduction; the atmosphere is where the convection happens. The mechanism computed here is different: absorbing gas raises the level from which the planet's radiation finally escapes, that level is colder because temperature falls with height, and the whole profile must shift to restore the balance. The lapse rate doing the shifting is set by convection, not by radiation.

Tested in The height a planet is seen from · the blackbody reading path

“The area theorem is a consequence of general relativity, so it holds whenever general relativity does.”

Its proof requires an energy condition — no observer measures a negative energy density — which ordinary matter satisfies and Hawking radiation does not. An evaporating hole's area falls to zero, computed here, without any failure of the field equations. What survives is the generalised statement including the entropy outside the horizon.

Tested in The area that is not allowed to shrink · the horizons reading path

“A clock measures the strength of gravity where it is.”

It measures the potential, which is a different quantity: the figure here shows gravity varying by half a per cent along sea level while a clock carried along the same surface reads one rate throughout, because sea level is by definition a surface of constant potential. A gravimeter reads the gradient and a clock reads the thing being differentiated.

Tested in The clock that measures a height · the gravitational redshift reading path

“Dropping two objects and seeing them land together tests the equivalence principle.”

It tests it to about a part in a hundred, which was settled before the principle was stated. Every advance since has come from a null instrument that measures a difference directly rather than two large quantities that are then subtracted, and from modulating the signal at a frequency where the apparatus is quiet.

Tested in The fall that does not depend on what is falling · the equivalence principle reading path

“The experiment distinguishes special relativity from its predecessors.”

It distinguishes partial drag from no drag and from complete drag, which differ by factors of two, and on that it was decisive. The difference between Fresnel's first-order coefficient and the exact composition is second order in the water's speed — a part in 10^17 here, computed in the figures — and no nineteenth-century measurement could see it.

Tested in The drag that was only an addition · the velocity addition reading path

“The lab energy spectrum of a decay product carries information about the decay dynamics.”

For a two-body decay that is isotropic in the rest frame the spectrum is exactly rectangular, whatever the interaction responsible. All the dynamics can do is make the rest-frame distribution anisotropic, and even then the edges are pure kinematics. The two edge positions give the parent's speed and the rest-frame momentum, and nothing else in the spectrum says anything.

Tested in The cone a decay cannot leave · the relativistic dynamics reading path

“Special relativity rests on the constancy of the speed of light.”

It can be derived from homogeneity, isotropy, the group property and the relativity principle alone, which leave a one-parameter family with an undetermined constant. The light postulate is one way of fixing that constant and is not the only one — any measurement of the composition of two speeds fixes it too, which is what the last figure shows.

Tested in The transformation that never mentions light · the spacetime diagram reading path

“Osmotic pressure counts the particles a membrane holds back.”

It counts the excess of particles on one side over the other, and a charged macromolecule brings counterions with it that are free to cross and stay anyway. The figures solve for that partition: at physiological salt a gel with 200 mM of fixed charge holds about 60 mM more mobile ions than its bath, and almost all of the pressure is theirs rather than the polymer's.

Tested in The swelling a membrane cannot stop · the osmosis reading path

“Surface tension is what holds a soap film together.”

Surface tension acts along the film and holds it taut; it does nothing to stop the two surfaces from approaching each other. What resists thinning is a pressure between the surfaces that has no analogue at larger separations — electrostatic repulsion at tens of nanometres and contact at a few — and the figures compute where it balances the suction pulling the water out.

Tested in The film that goes black before it bursts · the surface tension reading path

“The densest random packing of spheres is 0.64, so a tapped column reaches 0.64.”

The published number is the asymptote of a fitted curve, and no experiment has reached it. The hundred-thousand-tap experiments end measurably short of it. Three different quantities get the same name — a fitted asymptote, a value reached after a stated number of taps, and a geometric limit — and they are not the same number.

Tested in The pile that is never finished settling · the granular matter reading path

“Sunspots are dark because the magnetic field is strong there.”

The field strength does not enter the brightness. What the field does is take over some of the pressure, so the gas beneath it is thinner and convection is suppressed; the surface then cools to about 4,100 K against 5,780 K, and the Stefan–Boltzmann fourth power turns that into a brightness ratio of 0.253 against a measured fifth. The field is two steps upstream of the darkness, and the pressure balance is the first of them.

Tested in The same force whichever way the surface faces · the flux freezing reading path

“Advanced solutions are unphysical because they would let effects precede causes.”

A converging wave is not an effect preceding a cause; it is a field specified on a surface at an earlier time. Time-reversal mirrors build them routinely — record a wave on a surrounding surface, replay it reversed, and a wave converges onto where the source was. The figure computes the focus that a spherical cap produces and finds it narrows as the cap approaches a full sphere, which is exactly what surrounding the target means. What is expensive is the surface, not the causality.

Tested in The solution that is thrown away · the retardation reading path

“A wave focused onto a line has a large amplitude there.”

It has a small wavelength there, which is not the same thing. The contraction squeezes the wavelength, so the shear rises even where the energy does not, and viscosity acts on the shear. The attractor is a place where energy is destroyed rather than concentrated — which is why it matters for mixing an ocean and not for storing energy in one.

Tested in The reflection that changes the wavelength · the stratification reading path

“Geostrophic balance describes large-scale atmospheric flow.”

It describes large-scale flow away from the equator. The geostrophic wind is the pressure gradient divided by ρf, and f vanishes at the equator, so the expression runs away below about fifteen degrees of latitude — the same gradient that implies 32 metres a second at 45° implies over a hundred at eight. Tropical meteorology uses different balances for that reason, and the failure is visible in the formula rather than discovered afterwards.

Tested in The ratio that decides whether the planet is turning · the circular motion reading path

“Light entering a metal refracts almost along the normal, because the index is large.”

That holds when the real part of the transmitted normal wavenumber is large, which is a far-infrared statement. In copper at ten micrometres the phase fronts do sit within five degrees of the normal. In silver at 550 nanometres the real part is 0.053, so at sixty degrees of incidence the fronts sit at 86 degrees — nearly along the surface. Both are computed from measured optical constants, and they are opposite.

Tested in The angle that is two angles · the refraction reading path

“A negative-index slab is a perfect lens, so it has unlimited resolution.”

It has unlimited resolution only at exactly ε = μ = −1 with no loss. The transfer function computed here from the slab's own Fresnel coefficients with complex ε and μ falls through a half at transverse wavenumbers of 18, 25 and 33 times k₀ for losses of 10⁻², 10⁻³ and 10⁻⁴. Resolution is a logarithm of the loss divided by the thickness, so it is finite for every real material and improves only very slowly as materials improve.

Tested in The ray on the wrong side of the normal · the refraction reading path

“Radioactive equilibrium means the amount of each isotope is constant.”

Every amount is falling, at the parent's rate. Secular equilibrium is not a steady state; it is a state in which every member's decline is slaved to one exponential, the parent's, and it looks constant only because that exponential is slow. Over four and a half billion years the whole chain halves together.

Tested in The chain that runs at its slowest member's rate · the decay reading path

“Chemical effects on decay rates are a per cent, so they are a per cent for any nuclide.”

They are a per cent for beryllium and parts per million for anything heavy. The density of a 1s electron at the nucleus goes as Z³, so the two innermost electrons dominate more and more completely as the element gets heavier and the share belonging to the valence electrons — the ones chemistry rearranges — falls as Z⁻³. The exponent is measured off the curve here.

Tested in The half-life that chemistry can change · the decay reading path

“Cooling slowly enough would reach the Kauzmann temperature and settle the question.”

The glass transition moves down by about three and a half kelvin per decade of cooling rate, and the crossing is thirty kelvin below the ordinary transition. Reaching it needs of order 10⁴¹ seconds per kelvin, computed here from the same relaxation law that locates the transition — some 10³³ years. The experiment that would decide is not slow, it is impossible.

Tested in The entropy that depends on how fast it was cooled · the third law reading path

“Residual entropy is a violation of the third law that has to be explained away.”

It is what the law predicts for a degenerate ground state, and the law's statistical form contains it. What would be a violation is a system in a unique ground state with non-zero entropy, and none has been found. The cases that look like violations divide into degeneracies — where the count works — and systems that never reached their ground state, where the count is an upper bound the measurement falls below.

Tested in A law about spectra, not about heat · the third law reading path

“Lead is the best shielding material at every photon energy.”

It is spectacularly good below its K edge and much less special above a megaelectronvolt. At 100 keV lead's half-value layer is 0.011 cm against water's 4.06, a ratio of 369; at 1 MeV the same ratio is 11.4, because absorption there is Compton scattering off electrons and every material has about the same number of them per gram. A megaelectronvolt shield is chosen by mass per unit area, not by atomic number.

Tested in The steps in an absorption curve · the attenuation reading path

“Brighter beams cool an atom further.”

The limit contains no intensity at all once the beams are weak, and above that it gets worse. The coldest temperature computed for sodium is 247 µK at a tenth of saturation per beam and 332 µK at full saturation, because the heating follows the total scattering rate while the friction saturates first.

Tested in The friction made of light · the radiation pressure reading path

“Colder atoms need beams tuned closer to resonance, where the cooling force is strongest.”

That is the Doppler mechanism's trend, and the mechanism that actually sets the temperature runs the other way. Its temperature is a straight line in the light-shift well depth, and at fixed intensity the depth falls as one over the detuning: for caesium at five linewidths the Doppler theory predicts 664 µK and the polarisation-gradient model 10.8 µK, and the second gets colder still as the beams are tuned further away.

Tested in The limit that belonged to a simpler atom · the radiation pressure reading path

“Forty-five degrees gives the longest throw.”

It gives the longest throw onto ground at the height of the hand, and nowhere else. Along a 20° hillside the best launch is 55° and 45° lands at 91 per cent of the best; along a 30° downward slope the best is 30° and the reach is twice the level range. Every one of these was found by searching the launches and agrees with 45° plus half the slope to a millionth of a degree.

Tested in One curve answers every slope · the projectile reading path

“Athletes who launch well below 45° are using a poor technique.”

Forty-five degrees is the best throw for a set of launches that is a circle centred on the origin, and no body produces that set. A thrower whose speed falls by 30 per cent per radian of elevation does best at 34.9°, and a long jumper whose horizontal speed comes from a 9.5 m/s run-up and whose vertical speed is a 3.2 m/s push does best at 18.6°. Both come from the same tangency as 45°, applied to a different shape.

Tested in The best throw is a tangency · the projectile reading path

“The water held by a porous material is fixed by the humidity of the air around it.”

It depends on which way the humidity last moved, even in perfectly smooth, perfectly wetting pores. A pore fills through a cylindrical film with one curvature and empties through a hemispherical meniscus with two, so it fills at a higher humidity than it empties at. For pores spread about 4 nm, half the volume is full at 87.7 per cent on the way up and still full at 76.9 per cent on the way down.

Tested in The pore that fills from dry air · the capillarity reading path

“Electrowetting works by lowering the surface tension of the liquid.”

The liquid's own surface tension is unchanged, and so is the angle at the very edge of the contact line. What the voltage adds is capacitor energy under the wetted area, and the force it produces is concentrated in the fringing field within about a coating thickness of the contact line — the same force that pulls a dielectric slab into a charged capacitor. At 110 V on a 1 µm coating that pull reaches γ(1 − cos θ₀) = 104 mN/m, which is what complete wetting would need.

Tested in The angle a voltage can set · the capillarity reading path

“A high-index covering on a chip lets more light out than the invariant allows.”

It lets more light across the chip's face — 10.5 per cent into epoxy against 4.35 into air for a gallium nitride chip — because the critical angle depends on the ratio of the two indices. The light is then inside the covering at that material's radiance and must still leave it. The covering is shaped as a dome much larger than the chip so that every ray meets its surface near the normal, and the étendue crossing each surface in turn is conserved throughout.

Tested in The cone light has to find to get out · the etendue reading path

“Twice as much fibre gives twice as much polarisation delay.”

The sections' delays add as the steps of a random walk in three dimensions, because each section rotates the polarisation it receives before adding its own. The ensemble delay grows as the length to the power 0.50: 5.18 ps at 100 km for a coefficient of 0.5 ps/√km, where the same sections laid with their axes aligned would have given 172 ps.

Tested in The delay that is a random variable · the dispersion reading path

“Inside a planet, gravity falls in proportion to the distance from the centre.”

That is true only of a planet of uniform density. The local form of Gauss's law gives dg/dr = 4πG(ρ − ⅔ρ̄), so gravity decreases going down only where the local density exceeds two thirds of the mean density enclosed. The Earth's mantle is lighter than that everywhere, and its core heavier, and the figures check the formula against the slope of the computed profile to half a per cent.

Tested in The pull that grows on the way down · the Gauss's law reading path

“The voltage produced by a Josephson junction depends on how well it is made.”

The mean voltage between steps does, through the critical current and the resistance. The voltage of a microwave-induced step does not: it is n times the drive frequency times h/2e, and the integrated steps sit at 0.60, 1.20 and 1.80 in units chosen so that the step voltage contains only the drive frequency. Junctions of different superconductors have been compared at the level of parts in 10¹⁶ and give the same steps.

Tested in The voltage that is a frequency · the superconductivity reading path

“The spreading of a wave packet in a potential well is a quantum effect.”

Spreading in an anharmonic well is what any cloud of oscillators with slightly different frequencies does. Through the collapse the quantum mean position and that of a classical ensemble with the same starting spread agree to a few per cent of the swing. The quantum signature is the return, which requires the energies to be discrete.

Tested in The return a classical cloud never makes · the correspondence reading path

“A negative Wigner function is the mark of a large or exotic superposition, such as a cat state.”

Every state of definite energy except the ground state goes negative, and each takes the value ±1/π at the origin — the largest magnitude a Wigner function can have. A single quantum already has a negative volume of 4/√e − 2 = 0.425, two-thirds of the 0.624 a superposition of two packets carrying six quanta reaches. Among pure states only the Gaussian ones never go negative.

Tested in The probability that goes below zero · the correspondence reading path

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“The diffraction limit is a limitation of lenses.”

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

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

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

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

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

“Coercivity is a property of a magnetic material.”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Tested in How big now is · the simultaneity reading path

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“Energy is always conserved.”

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

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

“A larger telescope has fainter diffraction rings.”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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