True where it was derived, used where it was not — part 1 of 5

“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