Theme

Approximations that lie

The small-angle assumption, the frictionless plane, the point mass — where each is fine, and precisely where it stops being.

The pendulum's phase portrait. Angle plotted against angular velocity. Closed loops are swinging back and forth; the open curves above and below are rotating all the way round; the dashed curve between them is the separatrix.

Mechanics

The pendulum, and the small lie that makes it simple

A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.

Trajectories at one speed and several angles. Projectile paths launched at the same speed and five different angles. The 45° launch travels furthest, and the 20° and 70° launches land in the same place.

Mechanics

The angle that throws furthest, and why nobody notices

Forty-five degrees is the answer, and the maximum is so flat that a throw ten degrees off loses almost nothing. Both halves of that are worth drawing.

A block on a 27° incline. Free-body diagram of a block resting on an inclined plane: weight straight down, resolved into a component pressing into the surface and one pulling along it, with friction opposing the slide.

Mechanics

The slope, and the two directions that make it easy

An inclined plane looks like a harder problem than a flat one. Split the weight into two components chosen to suit the slope and it becomes an easier one.

A converging lens making a real image. An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

Optics

What a lens is doing, and why three rays are enough

A lens bends every ray that reaches it. The construction uses three, because three are all that can be drawn without calculation — and any three that meet prove all the rest do.

A harmonic well. Potential energy against position, with a horizontal line at the total energy. The motion is confined to where the line lies above the curve, and the turning points are the intersections — computed by solving for them, not marked by hand.

Mechanics

The hill that gives it back, and the forces that do not

Potential energy turns a question about motion into a picture of a landscape. It works for gravity and springs, it fails for friction, and the difference between those two cases is the whole of what makes energy useful.

Velocity and acceleration in uniform circular motion. Velocity drawn tangent to a circular path and acceleration drawn toward its centre, at eight points around the circle. The speed never changes and the acceleration is never zero; the two vectors are perpendicular everywhere.

Mechanics

Turning is an acceleration, and constant speed does not help

An object going round a circle at unchanging speed is accelerating hard, all the time, toward a point it never reaches. The construction that shows this needs two arrows and no calculus.

The pendulum's period against its amplitude. The exact period of a simple pendulum divided by the small-angle period, plotted against amplitude. The small-angle formula is the horizontal line at one; the exact curve leaves it slowly and then climbs without limit as the amplitude approaches a half turn.

Mechanics

The period that depends on the swing, computed exactly

A pendulum's period is not independent of amplitude. The exact answer is an elliptic integral, it has no elementary form, and plotting it shows precisely where the famous approximation earns its keep and where it collapses.

A sphere does not have a focus. Parallel rays reflected off a spherical mirror. Rays striking further from the axis cross it nearer the mirror, so there is no single point where all of them meet — the blur is spherical aberration, and a perfect sphere has it inherently.

Optics

The mirror that cannot focus, and the shape that can

A perfect sphere does not bring parallel light to a point. The blur is not a manufacturing defect — it is what the shape does, and the shape is used anyway, for a reason worth knowing.

$An electron's wavelength against the voltage that accelerated it. The de Broglie wavelength of an electron after falling through a potential difference, in picometres. At 100 volts it is 122.6 picometres, at 400 volts it is 61.3 picometres, at 900 volts it is 40.9 picometres. The wavelength goes as the inverse square root of the voltage, so quadrupling the voltage halves it. The calculation is non-relativistic; at a kilovolt that costs a tenth of a per cent. The dashed line is the 215 picometre spacing between atomic planes in nickel, which is what makes an electron beam diffract off a crystal at all.$ Quantum

Everything has a wavelength, and almost nothing shows it

If light with a momentum can behave like a particle, a particle with a momentum can behave like a wave. The wavelength is Planck's constant over the momentum, which for anything larger than a molecule is a number too small to have consequences.

Where the electron actually is, by radius. The radial probability density of the hydrogen 1s, 2s, 2p states — the chance of finding the electron in a thin shell at each radius, in units of the Bohr radius. 1s is most likely at 1.00 Bohr radii and averages 1.50; 2s is most likely at 5.24 Bohr radii and averages 6.00; 2p is most likely at 4.00 Bohr radii and averages 5.00. Each curve integrates to one, and each has n − l − 1 radial nodes where the electron is never found.

Quantum

Where the electron probably is

The Bohr atom put the electron on a circle of definite radius. What replaced it keeps the radius as the most likely place to find the electron and gives up the circle, the speed and the trajectory entirely.

Where the particle is likely to be found. A particle confined between two walls one unit apart. States 1, 4, 16 are drawn, each riding on a line at its own energy — 1E₁, 16E₁, 256E₁ — because the energies go as n². The curves are |ψ|², the probability of finding the particle at each position. The dashed line on each is the classical answer: a ball bouncing between the walls at constant speed is equally likely to be anywhere, and the quantum density oscillates about it and converges onto it as n rises.

Quantum

Where the quantum picture hands back the old one

A confined particle's probability density oscillates violently at every quantum number, and never stops. What makes the classical answer come back is not that the oscillations die away — it is that nothing can resolve them.

Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

Fluids

The pressure that only knows depth

A litre of water and a swimming pool press equally hard on a floor at the same depth. Pressure in a still fluid is a scalar with no direction of its own, it depends on how far down and on nothing else, and the shape of the container falls out of the arithmetic entirely.

A heeled hull, and the couple it makes. A rectangular hull of beam 3 m heeled 18°, with the waterline solved so that it displaces the same volume it did upright. The centre of buoyancy has moved 0.244 m to the low side, and weight and buoyancy now act along two lines 0.059 m apart — a couple that turns the hull back upright.

Fluids

Why a ship comes back upright

Whether a floating body rights itself or rolls over is not decided by its weight, its density or how deep it sits. It is decided by the shape of the slice the water cuts through it, and the number that settles it can be worked out before the vessel is built.

Fluids

The fourth power in a pipe

Halve a pipe's radius and the flow through it falls to a sixteenth. The exponent is four rather than two, because narrowing a pipe both removes cross-section and slows what is left — and one law with that exponent in it governs a blood vessel, a hypodermic needle and a water main.

Fluids

The fluid that answers back

For water, stress is proportional to how fast it is sheared, and the constant of proportionality is its viscosity. For paint, blood, ketchup and cornflour in water, it is not — and once the proportionality goes, so does the idea that viscosity is a number a substance has.

Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

Astrophysics

Light has a pressure

Sunlight pushes on a square metre with about the weight of a grain of sand, which sounds like a curiosity until the object being pushed is small enough. The demonstration in every school cupboard turns the wrong way, and the reason it does is more interesting than the effect it is supposed to show.

Where a body can no longer be any shape it likes. The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary.

Astrophysics

The size at which a body becomes round

A mountain can be no taller than the height at which the rock beneath it begins to crush, and that height falls as the body gets bigger — so there is a size above which a mountain would have to be taller than the world it stands on. Above it, nothing can be any shape but a sphere.

What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 30.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 22.5 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 22.5 N at the right-hand edge of the axis.

Mechanics

The force that takes what it needs

Static friction has no value of its own. It supplies exactly what equilibrium demands and not a newton more, right up to the moment it cannot — which is the only instant in the whole business at which a coefficient of friction means anything at all.

Mechanics

Every minimum is a parabola

A pendulum, a bond between two atoms and a ship rolling in a swell obey the same equation, and the reason is not that they are alike. It is that a function with a minimum has no linear term there, so the first thing every potential well looks like is the same well.

Why a straight front stays straight. A plane wavefront, with 9 points on it treated as sources and a wavelet of radius vt drawn about each. The envelope of those circles — the curve touching all of them — is a second straight line, parallel to the first and displaced by exactly vt. That is the whole of straight-line propagation: nothing else has to be assumed, and in particular nothing has to be said about rays, which are afterwards defined as the normals to these fronts. The construction is drawn with the wavelets left in, because they are the part that does the work.

Waves

Every front is a source

Treat each point of a wavefront as though it were a little source of its own, and take the envelope of what they produce. That one rule gives straight-line propagation, reflection, Snell's law and diffraction — and its most famous failure is what told everyone what was missing from it.

Simultaneity at β = 0.866. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

Relativity

The pole that fits and does not fit

A twenty-metre ladder is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the ladder's own frame the barn is five metres long and there is plainly no room. Both accounts are correct, and the doors' closings are 57.8 nanoseconds apart in one of them.

A torque, and no force at all. A loop of 1 turn enclosing 20 cm² and carrying 10 A has a magnetic moment of 0.02 A m². In a uniform field of 0.05 T the torque on it is m B sin θ, drawn here against the angle between the moment and the field: zero when they are aligned, largest at 0.001 N m across, and zero again when they are opposed. The second curve is the energy, −m·B, whose minimum is the aligned position and whose maximum is the opposed one — which is why a compass needle settles one way round and not the other. The net force is zero at every angle on this axis, exactly and not approximately: the force is I dl × B summed round the loop, the sum of dl round any closed path is zero, and a constant B comes outside the sum. The inset shows the four forces on a rectangular loop; the pair across the axis is the couple, and the pair along it cancels.

Electromagnetism

The loop that behaves like a needle

Far enough away, a current going round in a circle is indistinguishable from a bar magnet, and one number describes both. That number tells a uniform field how to turn the loop and gives it no way to pull on it at all — which is why two magnets attract by the fourth power of the distance and not the second.

Waves

The correction that took a century

Newton derived the speed of sound in 1687 and got 290 metres a second against a measured 340. The arithmetic was right; the assumption was not. Heat cannot cross a wavelength in a period, so the compressions are adiabatic — and the factor that repairs the answer turns out to be a count of the ways a molecule can move.

The narrow packet is the one that spreads. Three packets on deep water, ω = √(gk), all built on the same 4 m carrier and differing only in bandwidth — 10%, 18%, 28% of the carrier wavenumber. Each curve is the width of the emitted envelope, measured as the second moment of its intensity about its own centroid in a frame moving at the group velocity, divided by that width at the start. The starting widths are 4.50 m, 2.50 m, 1.61 m, and the order of the curves is the reverse of the order of the widths: the shortest packet, which is the one with the widest spectrum, is the one that comes apart first. The dashed curves are √(1 + (t/τ)²) with τ = σ₀²/|d²ω/dk²| — computed from the dispersion relation, not fitted. They agree with the measured widths to 0.3% at 10% bandwidth, 3.3% at 18% bandwidth, 8.5% at 28% bandwidth, and that ordering is the second thing the figure says: the closed form keeps only the curvature of ω(k), so it is exact for a narrow spectrum and starts to fail for a wide one, by about as much as the cubic term is worth. A packet with no bandwidth would never spread at all, and would also never begin or end.

Waves

The packet that will not keep its shape

A group velocity is only the first thing a dispersion relation says. The second is that the packet spreads — at a rate fixed by its own bandwidth and by the curvature of ω(k) — so a short pulse comes apart quickly and a long one hardly at all. It is a trade quantum mechanics is usually given credit for, and classical waves make it too.

Cancelling a derivative, and what is left over. How far the focus of a 500 mm lens moves with wavelength, for a single crown element and for a cemented pair of crown and flint whose powers satisfy the achromatic condition. The singlet's focus runs over 18.1 mm across the visible — a fifth of a per cent of its focal length, and utterly ruinous at any useful aperture. The doublet's runs over 2.268 mm, some 8× less, and — this is the whole content of the figure — it is not flat. The condition sets the rate of change of power with wavelength to zero, so the curve is stationary rather than constant: it returns to the corrected focus at exactly 2 wavelengths — 486 nm and 656 nm, which are the two Fraunhofer lines the condition was written at — and departs from it everywhere else, most at 400 nm. That residual is the secondary spectrum, it has the same sign at both ends of the visible, and no pair of ordinary glasses removes it, because two conditions cannot be met with one free ratio.

Optics

Two glasses that cancel a derivative

A single lens focuses blue light closer than red, and the difference ruins the image. Cementing a second lens of another glass behind it fixes the fault at two wavelengths and at no others, because the condition sets a slope to zero rather than a value.

A peak that leaves before it should have arrived. Two pulses at the far face of a cell, both normalised to the peak the vacuum one reaches. One has crossed empty space; the other has crossed a medium with two gain lines either side of its carrier, whose group index there is -3.85 — negative, so the envelope's peak should emerge early, and it does. Measured off the two curves the advance is 616 in units where the carrier period is 2π, against 582 predicted from the group index alone; the difference is the higher-order dispersion the group index leaves out. The advance is 0.21 of the pulse's own duration, and the peak leaves the far face before the input peak has entered the near one. Nothing has outrun anything. The emergent pulse is a reshaped version of the input's leading edge, which arrived in plenty of time and already contained — for a smooth pulse — everything needed to reconstruct the rest; the medium amplifies it by 1.14× and delivers it early. Give the pulse a genuine front, a moment before which it is exactly zero, and that front travels at the speed of light in every medium there is.

Waves

The speed that carries no signal

In the right medium a pulse's peak emerges from the far side before it entered the near one. The measurement is real, it has been made, and nothing has outrun light — because the peak of a smooth pulse was never carrying any information in the first place.

Two costs, and the width that balances them. The energy of a particle in a harmonic well against how tightly its wavefunction is squeezed, in units of ħω and of the width that minimises the total. Two terms compete. Squeezing the particle into a smaller region raises its kinetic energy, because the uncertainty relation makes a narrow position spread a wide momentum spread and momentum is squared in the energy; that term rises as the inverse square of the width and goes to infinity as the particle is localised. Letting it spread out raises its potential energy, since the well gets steeper away from the bottom; that term rises as the square of the width. The sum has a minimum at a width of 1.0000 in these units, where the total is 0.5000 ħω and the two terms are equal at a quarter each. That number is exactly the true ground-state energy of a quantum harmonic oscillator, obtained here with nothing but the uncertainty relation and a minimisation. What the figure shows and the formula does not is why there is a floor at all: it is not that the particle happens to keep moving, but that every way of stopping it costs more than it saves.

Quantum

The motion that cannot be stopped

A particle in a well cannot sit at the bottom of it. Squeezing it into a smaller region costs kinetic energy faster than it saves potential energy, so there is a width that minimises the total — and the minimum is not zero. Helium never freezes because of it.

$Buoyancy that falls away as the body sinks. The net upward force on a body containing a little gas, against how deep it has been taken, for 3 gas fractions. The weight does not change with depth. The buoyancy does, because the gas obeys Boyle's law and the pressure rises by an atmosphere every ten metres, so a body that displaced its own weight at the surface displaces less at depth. Every curve therefore slopes downward, and that slope is the whole point: where a curve crosses zero the body is in equilibrium, and the crossing is always from above, which makes every one of these equilibria unstable. Push the body a little deeper and the force does not push back — it turns downward and grows. The crossings drawn are at 8.2 m, 10.0 m, 13.6 m, and a body sitting at one of them is balanced in the sense that a pencil is balanced on its point. This is why a diver at neutral buoyancy has to keep adjusting, why a submarine's depth is held by hydroplanes and not by ballast alone, and why a fish that loses the use of its swim bladder sinks rather than drifting.$ Fluids

The depth past which it must sink

A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.

Arms in: 4.33× the rate, and 4.33× the energy. A body of 1.2 kg m² carrying two 4 kg masses on arms, spinning freely at 60 revolutions a minute with the arms out at 0.75 m, as the arms are pulled in to 0.12 m. The axis runs right to left, in the direction the arms move. Angular momentum is flat — nothing exerts a torque about the axis, and pulling inward is a force along a radius, which has no moment about the centre. The rate rises as the inverse of the moment of inertia, by a factor of 4.33 here, and the kinetic energy L²/2I rises by exactly the same factor, which is where the usual account stops and where the question starts. The fourth curve is the work done by whoever pulled the arms in, integrated from the force needed to hold each mass on its circle. It lies on the energy curve, to 1.6e-7 joules. Nothing is unaccounted for and nothing is created: the energy is bought, at full price, by pulling against the force that would otherwise fling the arms out.

Mechanics

The quantity that survives a change of shape

A skater pulls her arms in and spins four times faster. Angular momentum is conserved, which is the usual explanation, and it accounts for only half of what happened — because the kinetic energy has gone up by the same factor, and something had to pay for it.

A 120 g top at 3000 rpm, precessing once every 1.92 s. A disc of radius 30 mm spinning at 3000 revolutions a minute on a shaft 45 mm long, tilted 30° from the vertical. The weight acts at the centre of mass and the pivot holds the bottom, so the torque about the pivot is horizontal and at right angles to the plane containing the axis and the vertical. Angular momentum points along the axis; a torque at right angles to a vector turns it without changing its length, so the axis sweeps round the dashed circle instead of falling. The precession rate is Mgl divided by I₃ω₃ to leading order, which is 3.269 radians a second here, or one turn every 1.92 seconds — slower the faster it spins.

Mechanics

The push that comes out sideways

Push down on a spinning wheel's axle and it swings horizontally. Nothing about that is mysterious once angular momentum is a vector — but the steady precession every demonstration shows is a solution nobody's initial conditions select, and a top released from rest does something else first.

The rotation two boosts leave behind. The angle through which a frame's axes are turned after two boosts of equal size, against the angle between the two boosts, for 4 speeds. Two boosts in the same direction compose to a boost and nothing else, which is the zero at the left; two in different directions do not. What is left over is a rotation, and it is not small at large speeds: at β = 0.3 it peaks at 2.7° when the boosts are 91° apart, at β = 0.6 it peaks at 12.8° when the boosts are 96° apart, at β = 0.85 it peaks at 36.1° when the boosts are 108° apart, at β = 0.95 it peaks at 63.2° when the boosts are 122° apart. Each curve here is computed by multiplying the two boost matrices and pulling the rotation out of the product, not by evaluating a formula; the closed form for perpendicular boosts is used to check the extraction and appears nowhere in the drawing. The consequence is that the Lorentz boosts do not form a group by themselves — compose two and you leave the set — and that an object carried round a closed path in velocity space comes back turned.

Relativity

The turn that two pushes leave behind

Two boosts in different directions do not compose to a boost. The product carries a rotation, so a frame carried once round a closed path comes back turned — and the size of that turn was the factor of two standing between the calculated and the measured splitting of a spectral line.

Thermodynamics

Why heating a perfect spring changes nothing

A harmonic solid vibrates harder when heated and does not get any longer. Thermal expansion lives entirely in the term that the harmonic approximation throws away — and so does the fact that a solid conducts heat at a finite rate, which is the same discarded term doing a second job nobody would have connected to the first.

Pushing one way and going another. The angle between an applied force and the acceleration it produces, against the angle between the force and the body's velocity, at four speeds. At 0° and 90° the two are parallel, because those are the two directions the γ³ and γ divisors do not mix. Everywhere between, they are not: at 0.99c the worst case is 73.9°, reached with the force at 8.0°. A body under a steady sideways-ish push does not travel along it.

Relativity

The push that does not point where the body goes

Newton's second law survives relativity in the form F = dp/dt and in no other. Written as F = ma it fails outright, and not merely by a factor — at high speed a body pushed at forty-five degrees accelerates at eighty, because the same force is divided by γ³ along the motion and by γ across it.

The same block, one of them with no upthrust at all. Two identical blocks 0.8 m tall with their tops 1.2 m under the surface, drawn with the pressure on every wetted face at its true relative size. On the right the block is clear of the floor and the pressure on its underside exceeds that on its top by 7.8 kPa, which is ρgh and is exactly Archimedes' 7.8 kPa. On the left the bedding is perfect and there is no water under it, so nothing pushes up: the resultant is 11.8 kPa downward and the block presses on the floor with more than its own weight. Buoyancy is not something the fluid has. It is what the bottom face is doing, and a face the fluid cannot reach does nothing.

Fluids

The block the water does not lift

A block bedded flat on the bottom of a tank, with no water underneath it, feels no upthrust at all. It is fully submerged, Archimedes' principle is not suspended, and it presses on the floor with more than its own weight — because buoyancy is not something a fluid has, it is what the bottom face is doing, and a face the water cannot reach does nothing.

Two rockets that keep their distance, and the string that does not. Two rockets 0.5 unit apart in the laboratory, given identical acceleration programmes there, drawn in units where the light speed is one and c²/a is one. Their laboratory separation is constant for ever — the two worldlines are the same curve shifted sideways, and every horizontal line meets them 0.5 apart. The slanted lines are the rockets' own lines of simultaneity, and the distance between the worldlines measured along those is what a string tied between them has to span: at 0.3c it is 0.512, a stretch of 2 per cent; at 0.6c it is 0.557, a stretch of 11 per cent; at 0.8c it is 0.631, a stretch of 26 per cent; at 0.9c it is 0.710, a stretch of 42 per cent. The γL that is always quoted — 0.524, 0.625, 0.833, 1.147 here — is the limit of that measurement for a vanishing gap, and at a gap of 0.5 in these units it overstates the stretch by up to 38.1 per cent; shrinking the gap a hundredfold brings the two within 0.46 per cent. Either way the string is stretched and breaks, while the gap in the laboratory never changes by a millimetre. Length contraction is not something that happens to a rod. It is a statement about which events count as simultaneous, and a rod that is not allowed to contract is a rod that is being pulled apart.

Relativity

The string that breaks between two rockets

Two rockets a metre apart, given identical acceleration programmes, stay a metre apart in the laboratory for ever. A string tied between them breaks anyway. Nothing pulls on it, nothing in the laboratory moves relative to anything else, and the string is stretched — because the distance it has to span is measured on the rockets' slices of simultaneity and not on the laboratory's.

Thrust against the air it is supposed to be pushing on. The thrust of one F-1 engine against ambient pressure, in atmospheres. It is 6.78 meganewtons at sea level and 7.77 in vacuum — 14.6 per cent more with the air taken away. The line falls at exactly 9.787 newtons per pascal, which is the nozzle's exit area, because the term is (p_e − p_a)·A_e: the ambient pressure pushes on the exit plane from outside and there is nothing to push back on it. The account in which the exhaust shoves against the atmosphere makes the opposite prediction — thrust falling with the pressure and vanishing in vacuum — and it is not a small disagreement about a coefficient. It has the sign wrong. A rocket works better in vacuum than in air, and every engine ever fired has said so.

Mechanics

The push that needs nothing to push against

A rocket engine produces more thrust in vacuum than at sea level — fifteen per cent more, for the engine drawn here, and the extra is exactly the ambient pressure times the nozzle's exit area. The account in which the exhaust shoves against the atmosphere does not merely overstate a coefficient. It has the sign wrong, and every engine ever fired has said so.

Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

Quantum

The exponential that is only true in the middle

A decay law is not an assumption about nuclei; it is the Fourier transform of an energy distribution. Do that transform honestly and the exponential fails at both ends — flat at the start, because the state is normalisable, and an inverse square at the end, because no system has states of arbitrarily negative energy. Neither departure is a correction that could be made small.

Every line that goes in has to come out. Field lines of a 5:1 solenoid in the plane through its axis, each traced by stepping along the local direction of a field summed turn by turn from Biot–Savart. Inside the winding they are parallel and evenly spaced, which is the picture the textbook argument is about. Outside they are not absent: they are spread over the whole of the rest of space, which is why the field there is small — 1.60e-2 of the centre value at 2 radii off the axis — and why it cannot be zero. A line has no end, so every one of the lines through the bore returns outside, and a field with no outside would be a field whose lines stop.

Electromagnetism

The field outside the solenoid, which is not zero

Ampère's law says the field outside a solenoid vanishes, and every step of that argument is exact — for a winding of infinite length. A real one is a bar magnet seen from outside, its external field falls as the inverse square of its length rather than to nothing, and the "exactly zero" that makes the derivation so satisfying is the one part of it a laboratory cannot have.

An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

Waves

The distance that takes the treble out

Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles.

What a scale reads while a chain falls onto it. The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.

Mechanics

The pile that lands heavier than it weighs

Drop a chain onto a scale and the reading is three times the weight of the part that has landed — not approximately, exactly, all the way through the fall. The extra two parts are the force needed to stop links that are still arriving, and the same arithmetic run backwards says that picking a chain up wastes exactly half the energy it takes to get it moving.

Abbe's ratio, measured on the traced rays. The quantity h divided by the sine of the angle at which the ray converges on the focus, in units of the paraxial focal length, against how far up the aperture the ray entered. Abbe's sine condition says that a system already free of spherical aberration images a small region round the axis faithfully only if this ratio is the same for every ray. A horizontal line means the condition is met. The parabola departs by 12.96 per cent across the aperture; The sphere departs by 7.18 per cent across the aperture. The paraboloid is the interesting case, because it is exactly stigmatic on axis — every ray from infinity crosses at one point, which is the definition of the shape — and it still fails this test. Perfection at one point buys nothing at the next one along. What the departure predicts is coma, a blur that grows linearly with the distance off axis and quadratically with the aperture, and the offaxis figure measures exactly that blur on the same surfaces. The condition is not a design rule invented for telescopes: it follows from requiring that the same optical path length join object and image for every route, and any instrument that images a field rather than a point has to meet it.

Optics

The condition a lens must meet

A paraboloid brings every parallel ray to exactly one point. Move the source a fifth of a degree off axis and the image is a fan rather than a point, and the reason is a condition Abbe wrote down that has nothing to do with the axis — perfection at one point buys nothing at the next one along.

Four product states, and the four combinations that have a total spin. The four ways two spin-halves can be arranged, on the left, and the four combinations of them that are eigenstates of the total spin, on the right. Two of the products — both up and both down — are already eigenstates. The other two are not: one spin up and the other down does not specify a total spin, because it does not say which spin is which, and the states that do are the sum and the difference. The eigenvalues printed beside them are computed by applying S² as a matrix in the product basis and reading the result off, then solving s(s + 1) for s: |↑↑⟩ gives 2ħ², so s = 1; (|↑↓⟩ + |↓↑⟩)/√2 gives 2ħ², so s = 1; |↓↓⟩ gives 2ħ², so s = 1; (|↑↓⟩ − |↓↑⟩)/√2 gives 0ħ², so s = 0. The column on the far right is the eigenvalue of the operator that swaps the two particles: the three states with s = 1 come back unchanged and the one with s = 0 comes back with a minus sign. That sign is the whole of the difference. It is why the three are called a triplet and the one a singlet, why they behave differently in a magnetic field, and — through the requirement that the total state of two electrons be antisymmetric — why the two families occupy space differently before any force between them has been mentioned.

Quantum

Four states, and one of them is odd

Two spin-halves make four states, and they split three and one rather than into four of a kind. Three come back unchanged when the two particles are swapped and one comes back with a minus sign — and that single sign decides how far apart two electrons sit before any force between them has been mentioned, and why hydrogen gas is two gases that do not interconvert.

The same launches with drag 2.4 times the weight. Five launches at the same speed and at 20, 32, 45, 60, 70 degrees, drawn twice: in vacuum, where the arcs are symmetric parabolas, and with quadratic drag whose force at launch is 2.4 times the projectile's weight. Nothing about the drag figure is a parabola. Each path rises at nearly the vacuum angle, loses horizontal speed that nothing restores, and comes down far more steeply than it went up — the 32° launch leaves at 32° and arrives at 50°. The best of these angles in vacuum is 45° and in air is 32°, and the best range has fallen by 60 per cent. The asymmetry is the whole of the difference: drag removes speed in proportion to speed squared, so it takes most from the fast early part of the flight, and the descent happens at a speed the drag has already limited.

Mechanics

The angle that drag moves

Forty-five degrees is the answer in vacuum and almost nowhere else. Add one velocity-dependent force and the two equations of motion lock together, the closed form disappears, and the best launch angle falls — to thirty-eight degrees for a golf ball's drag and to twenty-nine for a shuttlecock. What moves it is not the loss but the asymmetry.

The potential a shaken pivot creates. The effective potential of a pendulum whose pivot is shaken vertically, against the angle from hanging, for shaking rates of 8, 14, 20, 30 times the pendulum's own frequency at an amplitude of 0.12 of its length. The shaking averages to no force at all — it is up as often as down — and yet it adds a term to the potential, because the pendulum's position is correlated with the phase of the shake rather than independent of it. The added term is proportional to sin²θ, so it is largest sideways and zero at both the hanging and the inverted positions, and it turns the maximum at 180° into a minimum once 14× and 20× and 30× the natural frequency is reached. The criterion is (aΩ)² > 2gL: the shake speed must beat the speed a fall through the pendulum's own length would give. Upside down then becomes a stable equilibrium, with a restoring force and a period of its own.

Mechanics

Held up by a force that averages to nothing

Shake a pendulum's pivot up and down fast enough and the pendulum will stand upside down, balanced, and push back if it is nudged. The shaking supplies no average force at all — it is up as often as it is down — and the reason it nevertheless holds is that the pendulum's position and the phase of the shake are not independent.

The action along a family of paths. On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve.

Mechanics

Least action, except that it is not least

Mechanics can be stated twice over. Once as a rule about every instant — force equals mass times acceleration — and once as a rule about the whole path at once, which says that one number computed along it is stationary. The two pick out the same trajectory, and the second name for it is wrong — past a certain duration the real path has more action than its neighbours, not less.

$A caustic, by rays and by waves. The brightness across a fold caustic, computed two ways. Geometric optics gives the rising curve: on the illuminated side two rays arrive at every point and the intensity goes as the inverse square root of the distance from the caustic, so it becomes infinite exactly at it; on the other side no ray arrives at all and the intensity is zero. The wave answer is the squared Airy function, and it disagrees in three ways that are all observable. It is finite, peaking at 1.0188 in the scaled variable rather than at the caustic itself, so the brightest line is displaced onto the bright side. It oscillates, with maxima at -1.02, -3.25, -4.82, -6.16 — those are the supernumerary fringes, and they are not interference between two separate objects but between the two rays the caustic joins. And it leaks: on the dark side, where geometry forbids any light, the Airy function decays exponentially rather than stopping, which is the same mathematics as tunnelling and is why the edge of a shadow is soft before diffraction from any aperture is considered. The two curves agree far from the caustic, which is where the ray picture is a good approximation and where they have been matched here.$ Optics

The fringes below the rainbow

Geometric optics puts the whole rainbow at one angle and predicts an infinite brightness there. What is seen instead is a peak displaced inside that angle, followed by a train of pink and green arcs — and their spacing is a measurement of the raindrops, because a caustic's structure is set by the wavelength to the two-thirds power over the drop radius to the two-thirds.

Thermodynamics

The barrier a new phase has to climb

Water vapour three times supersaturated is thermodynamically desperate to condense and will sit there indefinitely if it is clean enough. The obstacle is that a droplet has to start small, and a small droplet is nearly all surface — so the first nanometre of every phase transition costs energy rather than releasing it, and what decides whether anything happens is the height of that cost divided by kT.

Relativity

The speed that cannot be measured one way

Every measurement of the speed of light ever made has sent it out and brought it back. Measuring it one way needs two clocks that agree, and making two distant clocks agree needs a rule about when the far one should read what — which is a choice, not a discovery. The constancy of c is a fact about round trips; its isotropy is a convention, chosen because it makes the equations simple.

A sine on its way to a vertical face. One period of a sine, followed by the equation whose only nonlinearity is that the local speed depends on the local height. The profiles are at σ = t/t_b of 0, 0.3, 0.6, 0.9, 0.995, each obtained by solving the implicit relation u = u₀(x − (c₀+βu)t) for u at every point by bisection — and checked against the partial differential equation itself, which it satisfies to 2.6e-7. The crest travels faster than the trough, so the descending front leans forward and the ascending one leans back; the wave stays exactly as tall as it started and exactly as long, and only its shape changes. The steepest gradient grows as one over (1 − σ) — measured here as 9.83 times its initial value at σ = 0.9, against ten — so it is infinite at σ = 1 and the curve has a vertical tangent. The picture cannot be drawn past that point, which is not a failure of the drawing: the solution genuinely becomes three-valued, and what actually happens is a jump whose width is set by the dissipation this equation does not contain.

Waves

The front that steepens until it cannot

In a linear medium every wave keeps its shape, because every part of it travels at the same speed. Let the speed depend on the height by even a little and the crest overtakes the trough, the front leans forward, and after a time that can be written down the wave demands two values at one place — which is where the description ends and a shock begins.

Throttling a gas, and the curve that says which way it goes. Curves of constant enthalpy for a van der Waals gas, in temperature and pressure both measured against the critical values. A gas pushed slowly through a plug or a valve keeps its enthalpy, so it moves along one of these curves — from right to left, since the pressure falls. Where a curve slopes upward to the right the gas cools as it expands; where it slopes downward it warms. An ideal gas would give horizontal lines and no change at all, because its enthalpy depends on the temperature alone; every curve here is bent, and the bending is the attraction between molecules and the room they take up, fighting. The dashed line through the tops of the curves is the inversion curve, and the maxima were found on the drawn points rather than put there — they lie on the closed form to 0.65 per cent. Which side of it a gas starts on decides the sign of the effect, and that is the whole of why air can be liquefied by throttling at room temperature and hydrogen cannot: hydrogen has to be pre-cooled below its own inversion temperature first, which is why Dewar needed liquid air before he could get liquid hydrogen, and why Onnes needed liquid hydrogen before he could get helium.

Thermodynamics

The gas that cools by being let go

Push a gas through a plug and its temperature changes, although no work is done on anything and no heat goes anywhere. Which way it changes depends on where it starts: inside a dome in the pressure–temperature plane it cools, outside it warms, and hydrogen at room temperature is outside — which is why liquid hydrogen needed liquid air first.

Four beads, four heights, one arrival time. One arch of a cycloid of radius 1, drawn with four beads on it at heights 0.061, 0.235, 0.592, 2.000 — a range of 33.0 to one. Each bead's time to slide, from rest and without friction, to the bottom of the arch is computed as a quadrature of ds/v along the curve as drawn, and the four answers are 1.003205 s, 1.003205 s, 1.003205 s, 1.003205 s: identical to 6.9e-13 of themselves. The closed form for this curve is π√(a/g) = 1.003205 s, which the quadrature reproduces without being told it. A bead let go at the cusp travels 5.7 times as far as the lowest one and arrives with it, because the extra distance is exactly paid for by the extra speed the extra height buys.

Mechanics

The curve that does not ask where it started

A pendulum's period depends on how far it swings, and the dependence is small but never zero. There is exactly one curve for which it is zero, and the reason has nothing to do with pendulums: on that curve the height above the bottom is proportional to the square of the distance travelled along it, which makes the motion harmonic by construction rather than by approximation.

How fast something looks as it moves across the sky. The apparent transverse speed of a source, in units of the speed of light, against the angle between its motion and the line of sight, at β = 0.8, β = 0.95, β = 0.99. Every curve rises above one over a range of angles, reaching 1.33 at 36.8°, 3.04 at 18.4°, 7.02 at 8.1° — and those maxima are γβ at arccos β, found by searching the drawn curves rather than put into them. Nothing is moving faster than light. What has happened is that the source has come closer between the two observations, so the second flash had less far to travel and arrived sooner than it would have done; dividing the transverse distance by the interval between arrivals therefore gives too large a speed. Below β = 1/√2 no angle produces the illusion at all, so seeing it is a measurement: it puts a floor under the speed and a ceiling on the angle at once.

Relativity

The motion that measures faster than light

Take two photographs of a jet a year apart, measure how far a blob moved across the sky, divide by a year, and the answer can be seven times the speed of light. Nothing has broken. The blob came closer between the two pictures, so the second flash had less far to travel and arrived early, and the interval between arrivals is not the interval between departures.

An average that follows Newton's law exactly. The centre of a wavepacket in a harmonic well, and the classical orbit started from the same place, drawn on top of each other. They agree to 5.5e-6 over 2.2 periods, which is the split-step integrator's own error and not a physical gap. This is exact and it is exact for every state of a harmonic oscillator, however wide, however lumpy, however far from classical: the theorem needs ⟨−V′(x)⟩ = −V′(⟨x⟩), which for a linear restoring force is true term by term. It is worth being suspicious of how strong that looks. The harmonic oscillator is the one potential where the average force over a state and the force at the state's centre cannot differ, so it is the worst possible example from which to conclude that quantum averages follow classical paths.

Quantum

The average that obeys Newton

Ehrenfest's theorem says the centre of a wavepacket moves according to the average force over the state. That is exact, it is often quoted as the reason classical mechanics survives, and the two statements are not the same — because the average of a force is the force at the average only when the force is linear, which is to say almost never.

The same top, let go four ways. The path traced by the top of the axis, seen from directly above, over 1.2 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 46.8° from the vertical. Released from rest the axis falls, and the fall is what generates the sideways motion: the path comes to a cusp each time it returns to the starting tilt, because at that instant the precession rate is momentarily zero. Launched at exactly the steady rate the path is a circle and the nutation is absent. Launched slower it waves; launched faster it loops: at 0× the steady rate the path comes to cusps, at 0.45× the steady rate the path waves, at 1× the steady rate the path stays a circle, at 1.9× the steady rate the path waves. Every one of these is the same equation with the same top and the same spin.

Mechanics

The top that nods before it settles

A spinning top let go from rest does not begin to precess. It falls, catches itself, and comes back up, over and over, at a frequency that has nothing to do with gravity — and the steady precession every textbook draws is what is left after friction has removed the nod.

A drive at one frequency, and what comes back at three times it. The Fourier components of the steady motion of an oscillator driven at a single frequency ω = 1.35, for drive strengths of 0.02, 0.05, 0.1. The equation is a harmonic oscillator with a cubic term added, and the components are projected out of the integrated motion rather than assumed. A linear oscillator answers only in the first column. This one answers in the third as well, because x³ of a cosine contains a cosine of three times the angle. The third harmonic grows as the drive to the power 3.00 where the fundamental grows as the power 1.00, so it is negligible at small drive and not at large — which is why nonlinearity in an instrument is a specification rather than a yes or no.

Mechanics

The oscillator that answers at three times the question

Push a spring hard enough that the parabola stops being the whole story, and three things happen that a linear oscillator cannot do: it emits frequencies nobody supplied, its resonance leans over, and its amplitude at one drive frequency depends on where the drive has been.

Image distance against object distance. Image distance in focal lengths against object distance in focal lengths. At exactly one focal length the image runs off to infinity; inside it the image distance goes negative, which means virtual.

Optics

The focus that is a slab, not a plane

A lens images one plane and no other, which would make every photograph and every micrograph almost entirely out of focus. What rescues them is a tolerance — and there are two of them, one from rays and one from waves, which give different answers and stop being interchangeable exactly where microscopes work.

Three ways for a wavelet to be strong, and what each leaves behind. On the left, the strength of a secondary wavelet against the angle from the forward direction, for three candidate rules. Huygens' construction as stated has no such rule: a wavelet is spherical and equally strong in every direction. On the right, what each predicts when the wavelets over a whole plane are added up, on the axis, in front of the plane and behind it. All three reproduce the incident wave in front, which is the part of the construction that has always worked. Only the rule that falls to exactly nothing at a hundred and eighty degrees leaves nothing behind, and that rule is not a repair invented for the purpose — it comes out of solving the wave equation.

Waves

The backward wave Huygens had to remove

Every point of a wavefront is a source of a spherical wavelet, and a spherical wavelet goes in every direction — so the construction predicts a wave travelling backwards as well as forwards. Nothing of the kind exists, and the repair is a factor that Huygens' geometry has no room for.

Melting curves, and the one that leans the wrong way. Melting temperature against pressure for water, benzene, naphthalene, each measured from its own melting point at one atmosphere, with pressure in bars. The slope of every coexistence line is the latent heat divided by the temperature and the change in volume, and the latent heat of melting is positive for everything — so the sign of the slope is the sign of the volume change, and nothing else. Almost everything expands on melting and its line leans forwards. Water's solid is less dense than its liquid, so its line leans backwards at 135 bars a kelvin: pressing on ice at just below zero melts it, and it takes 135 atmospheres to gain a single degree. The anomaly is not in the thermodynamics; it is in the fact that ice floats.

Thermodynamics

The melting curve that leans the wrong way

The slope of any coexistence line is the latent heat divided by the temperature and the change in volume. Latent heat is always positive, so the sign of the slope is the sign of the volume change — and for water the volume change is negative, which is the whole of why ice floats and why the melting curve leans backwards.

The pair potential, and the two things it does to a gas. The Lennard-Jones potential between two molecules, in units of its own depth and range, with the Mayer function it produces at 1, 3, 8 times the well depth in temperature. The virial coefficient is minus the integral of that function over volume, so the two parts of the potential contribute with opposite signs: the steep repulsive core makes the function minus one there, giving a positive contribution — molecules take up room — and the attractive well makes it positive, giving a negative one. At low temperature the attraction dominates and a gas is easier to compress than an ideal one; at high temperature the core dominates and it is harder. Between them is one temperature at which they cancel.

Thermodynamics

The first correction to the gas law

An ideal gas has no forces between its molecules. The first correction to what it does is computable from those forces alone — one integral over the pair potential — and its sign flips at a temperature where a real gas obeys the ideal law without being ideal at all.

A bundle that swings and never spreads. A Gaussian of the ground state's own width, released at rest from x = 3 in a harmonic well and propagated on a grid by split-step Fourier, drawn at 0 of a period, 0.25 of a period, 0.5 of a period. The packet slides from side to side and its shape does not change: over 2.2 full periods the width moves by 6.6e-5 per cent, and its centre tracks x₀cos t to 8.6e-6. Every other initial width breathes. This one is the displaced ground state, and it is the closest a quantum state comes to being a classical oscillator — a definite thing at a definite place, moving on the classical trajectory, staying the size it was.

Quantum

The state that swings like a pendulum

Most quantum states of an oscillator look nothing like a swinging weight. One family does: it follows the classical trajectory exactly, never spreads, and sits at the uncertainty minimum for ever — and it is the state a laser and a driven circuit actually produce.

Where the second tick actually goes. A spacetime diagram with a second observer's axes at β = 0.6. The hyperbolae are the sets of events one second and one metre from the origin — invariantly, by the interval — and every observer's unit tick is where their own axis crosses them. That is checked here rather than drawn by eye. The moving observer's one-second mark sits 1.458 times further from the origin on the page than the stationary one's, so a ruler laid on this picture reads the two frames on different scales. The picture is not distorted; the page is Euclidean and spacetime is not. The Lorentz factor here is 1.2500.

Relativity

The diagram a ruler cannot read

A spacetime diagram is drawn on flat paper, and the geometry it depicts is not flat. The tick marking one second on a moving observer's axis sits further from the origin than the stationary observer's, by an amount that is not the Lorentz factor and means nothing at all.

$The fraction of the pressure a membrane can hold. The osmotic pressure actually developed across a membrane against a 300 mol/m³ solution at 298 K, as the reflection coefficient runs from a membrane the solute passes freely to one it cannot pass at all. The line is straight with the van 't Hoff pressure 743.69 kPa as its slope — checked against the drawn line — because the coefficient enters as a simple factor. At σ = 1 the pressure is 743.69 kPa; At σ = 0.6 the pressure is 446.21 kPa; At σ = 0.2 the pressure is 148.74 kPa. Van 't Hoff's law is the ceiling rather than the answer, and a membrane's coefficient against a given solute is as much a property of the pair as the concentration is of the solution.$ Fluids

The membrane that almost holds

Van 't Hoff's law gives the osmotic pressure a perfectly selective membrane would develop, and no membrane is. What a real one develops is a fraction of it — a number between zero and one that belongs to the membrane and the solute together, and that decides whether a solution is isotonic in effect or only on paper.

$Where a small weight is felt, and where it is not. The fractional change in the frequency of harmonic 3 of a stretched string when a point mass of 0.06 of the string's own mass is placed at each position along it. The solid curve is the exact answer, found by solving the string's frequency equation for the loaded string at each position; the dashed curve is the first-order prediction, minus the mass fraction times the square of the mode shape. The two agree to 10.4 per cent at the antinode, where the shift is largest. Every node is a place where the exact shift is zero to better than a part in a thousand million, and that is not an approximation: a mass at a node is never moved by the mode, so it takes no part in the motion and cannot change its rate. Between the nodes the shift follows the square of the displacement, which is the square of the amplitude — the mass is felt in proportion to the kinetic energy the mode was already keeping there.$ Waves

The dent that raises the note

Push a wall of a resonator inwards and the pitch goes up or down depending entirely on where the wall is pushed. A mass added at a node changes nothing at all; the same mass at an antinode changes as much as it can. One rule covers a loaded string, a tuned microwave cavity and a bead drawn through a resonator to read out its field.

Three charges, three orbits, one drift. Three particles released at rest in crossed fields — 1000 V/m across 0.1 T — with their paths integrated by a scheme that rotates the velocity rather than adding to it, so the magnetic part changes no speeds. The three loops have wildly different sizes and periods: the the electron turns at 2799.25 MHz, the proton turns at 1.52 MHz, the α particle turns at 0.76 MHz. Their guiding centres all creep along the same line at the same rate, measured here from the orbits at -1.000e+4, -1.000e+4 and -1.000e+4 m/s against −E/B = -1.000e+4 m/s, a spread of -0.00 per cent. Neither the charge nor the mass nor the sign appears in the answer. A plasma in crossed fields therefore moves bodily and carries no current from this drift at all, which is the opposite of what an intuition built on ions being heavier than electrons expects.

Electromagnetism

The drift that does not care what the charge is

A charge in a uniform magnetic field goes round in a circle and arrives nowhere. Add anything at all — an electric field, gravity, a gradient in the magnetic field itself — and the circle's centre creeps sideways, at right angles to both. One of those drifts is the same for every particle regardless of charge, sign or mass; the others are not, and the difference decides what a plasma does.

Force against slip, and the knee between them. The force a tyre delivers against how much faster its tread is going than the road, for a patch 120 mm long under 4000 N with a friction coefficient of 1. The curve is the integral over the bristles, and the dashed line is the cubic the brush model gives in closed form; they agree to 0.00 per cent of the sliding force. The first slope is 80 kN per unit slip, and it belongs entirely to the elasticity of the rubber — at vanishing slip nothing is sliding, so no friction coefficient can appear in it. Full sliding is reached at 15.0 per cent slip and not before. Everything a driver calls grip lives on the rising part of this curve, at a few per cent of slip, where the patch is partly stuck and partly sliding — and the quantity that decides handling in that region is the slope rather than the friction coefficient at the top.

Mechanics

The grip that needs a little slipping

A wheel that transmits any force at all is not rolling. Part of its contact patch is stuck to the road and part is already sliding, and the force it delivers is a measure of how much has given up. The useful part of the curve is a few per cent of slip, the peak is not the end of it, and everything past the peak is unstable.

Thermodynamics

The energy that refuses to be shared

Put all the energy of a chain of masses into its longest mode and add a few per cent of nonlinearity, and equipartition says it should spread out among all thirty-two modes and stay there. It does not. It leaks into three or four neighbours and then comes back — almost exactly — and goes on doing so, and the calculation that found this was expected to be a demonstration that it would not happen.

$The fringes a flow of water moves. The interference fringe shift against the speed of the water, for two tubes 1.5 m long, an index of 1.333, and light of 526 nm — Fizeau's apparatus. The beam is split, each half goes with the flow in one tube and against it in the other, and the two are recombined; the shift is the difference in transit time counted in wavelengths. Three predictions are drawn and they are not close together. If the water did not affect the light at all the shift would be zero, flat along the bottom. If the water carried the light with it completely the shift would be the steepest line. Fresnel's partial drag is the middle one, and at 7 m/s it gives 0.207 of a fringe — which is what was measured, to the accuracy of an eye reading a fringe pattern in 1851. The experiment therefore did not merely detect an effect; it chose between three quantitative possibilities that differ by factors of two, which is why a fraction of a fringe settled something.$ Relativity

The drag that was only an addition

Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.

$A hundred thousand taps, and still not finished. The packing fraction of a column of grains against the number of taps it has been given, on a logarithmic horizontal scale, for several tap intensities. The grains start where pouring leaves them, around 0.55, and climb towards something near 0.64. At an intensity of 1.2 the packing reaches 0.6318 after a hundred thousand taps; At an intensity of 2 the packing reaches 0.6327 after a hundred thousand taps; At an intensity of 3 the packing reaches 0.6338 after a hundred thousand taps, which is still 0.0097 short of the asymptote. The shape is what matters. On a logarithmic axis the curve is close to a straight line over four decades, which means the packing improves by about the same amount for each factor of ten in the number of taps — not for each additional thousand. Going from a hundred taps to a thousand buys as much as going from a thousand to ten thousand. An exponential relaxation is over after a few time constants and this is not one. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure.$ Fluids

The pile that is never finished settling

Tap a jar of grains and it settles. Keep tapping and it goes on settling — logarithmically, so that each factor of ten in the number of taps buys the same small improvement as the last. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure, and the asymptote everyone quotes is a fitted number rather than a measured one.

Pushed one way, travelling another. A parcel released from rest under a steady pressure-gradient acceleration of 2.0e-4 metres per second squared pointing east, integrated for 36 hours at 60°, 30°, 10°. Without rotation it would accelerate east indefinitely. With it, the motion is an inertial circle about a mean velocity at right angles to the push, and the mean over a whole number of inertial periods is measured off each path and matches the geostrophic value a/f to two per cent. At high latitude the loops are tight and the drift is almost purely along the isobars; near the equator the parcel travels a long way down the gradient before the rotation has had time to turn it, which is why geostrophic balance is a high-latitude statement and the tropics need a different set of approximations.

Mechanics

The ratio that decides whether the planet is turning

Whether the rotating terms matter is not a question about size. It is one dimensionless ratio, U over fL, and it runs from ten thousand in a teacup to a millionth in the Earth's core. Where it is small the pressure gradient stops accelerating the fluid and starts balancing a force on fluid already moving across it — so the flow runs along the pressure contours instead of down them, and a weather map is a streamline plot.

$A wave with two directions in it. Light at 60° entering silver, 550 nm, whose index is 0.055 + 3.32i. Phase matching along the boundary fixes the transmitted wave's tangential wavenumber and leaves the normal one to the medium, which supplies a complex answer. The real part decides where the phase goes and the imaginary part where the amplitude does, and they are not the same direction: the surfaces of constant amplitude are parallel to the interface, because the decay is entirely into the metal, while the surfaces of constant phase are tilted by 86.5° from the normal. There is therefore no single refracted angle to quote. The familiar picture, in which one set of parallel planes carries both, requires the absorption to be exactly zero.$ Optics

The angle that is two angles

Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

A swing that dies away and comes back. The envelope of the mean position of a coherent state with 9 quanta on average, in an oscillator whose levels carry a small quadratic term, Eₙ = n + n²/240, over one revival time of 240 oscillator periods. The swing collapses within about 9.1 periods, when the packet has spread round its orbit, and it stays at zero for most of the run. It returns whole at half the revival time, on the opposite side, and whole again at the full revival time; the envelope is summed from the state's energy components and checked against α·exp(−2n̄ sin²χt) to a part in a hundred million. The dashed curve is a classical ensemble started from the same distribution, each member orbiting at the frequency its own energy gives: it collapses in the same way and never returns, its swing at the half and full revival times 0.1% and 0.1% of the start.

Quantum

The return a classical cloud never makes

Put a swinging quantum packet in a well whose frequency depends a little on the amplitude and it spreads round its orbit until its swing has vanished. That part is not quantum at all: a cloud of classical oscillators does exactly the same. What no classical cloud can do is come back — and the quantum packet reassembles whole, on schedule, splitting into copies on the way, because its energies are discrete.

A quasi-probability that goes negative. The Wigner function of the oscillator's state at a quarter of its revival time, with 4 quanta on average: two copies of the packet, at x = ±2.83, in equal superposition. It is computed from the wavefunction by the Wigner transform on a lattice, and it integrates to one. The two copies are the two positive blobs. Between them lies a pattern of stripes with no classical counterpart, running from 0.289 down to −0.289, where the shaded warm regions and their outlines mark negative values. A negative probability is not a probability, so this distribution cannot describe a cloud of classical particles. The stripes are also taller than the blobs, so the interference carries more structure than the copies themselves: the highest stripe reaches 0.289 and the centre of a blob 0.159.

Quantum

The probability that goes below zero

Classical mechanics describes an uncertain state as a cloud of points in the plane of position and momentum. Quantum mechanics has an exact counterpart, the Wigner function, whose shadows are the true position and momentum distributions — and which goes negative. It goes negative for a single photon, for every superposition of two packets, and for every pure state that is not a Gaussian. Where it is negative no classical cloud can imitate the state, and losing energy to the surroundings erases the negative regions first.

Grip, then power, then air. The force a 1500 kg car can put through its driven wheels against road speed, with 100 kW at the wheels and a tyre friction coefficient of 0.9. The grip allows 13.2 kN at any speed; the engine allows its power divided by the speed, a hyperbola; the car gets whichever is smaller, drawn solid. The two are equal at 7.6 m/s, 27 km/h: below it the car is limited by friction and extra power would change nothing, above it by power and better tyres would change nothing. The rising curve is the resistance, rolling plus air, which grows as the square of the speed; it meets the drive at 61.2 m/s, 220 km/h, the top speed, solved for and checked there.

Mechanics

The speed at which grip hands over to power

A car's specification lists its power, and power does not limit how hard a car can push. It limits how hard it can push at a given speed, and at low speed that limit is higher than anything the tyres can transmit. So every car leaves the line as a friction problem and becomes a power problem a second later, at a crossover speed that decides which upgrade would make it faster — and at the top of its speed range a third limit, the cube of the speed, takes over from both.

The rectangle every cycle is equal to. An ideal Otto cycle for air on a temperature–entropy diagram: compression ratio 9, intake at 300 K, 1400 kJ/kg added at constant volume. Compression takes the charge to 722 K, combustion to 2672 K, expansion back to 1110 K, and the exhaust cools at constant volume. The shaded loop is the work; the region under the lower curve is the heat rejected. Heat enters over a range of temperatures, and its entropy-weighted mean — heat divided by the entropy it brings — is 1491 K; the heat leaves at a mean of 619 K. The dashed rectangle between those two temperatures has exactly the loop's area, and 1 − 619/1491 = 58.5%, which is the Otto efficiency, checked to rounding. A Carnot engine between the coldest and hottest points of the same cycle would reach 88.8%.

Thermodynamics

The temperature an engine really takes its heat at

Carnot's ceiling is set by two temperatures, and no engine that burns fuel takes its heat in at one temperature or gives it out at another. It takes heat over a range, from the moment combustion starts to the moment it ends. For any reversible cycle there is an exact replacement for Carnot's two numbers: the average temperature at which heat arrives and the average at which it leaves, each weighted by the entropy the heat carries. The gap between a real cycle and Carnot is a gap between those averages and the extremes.

Two bodies an engine draws together. Two equal bodies of 4.186 kJ/K — a kilogram of water each — one at 90.0 °C and one at 10.0 °C, against the heat drawn from the hot one. The solid curves are the best possible engine running between them, a reversible one, which leaves the product of the two temperatures unchanged and brings both to the geometric mean, 320.7 K (47.5 °C). It draws 177.8 kJ from the hot body and delivers 20.8 kJ of work, C(√T₁ − √T₂)², checked against the heat balance. The dashed lines are the same bodies simply touching: they meet at the arithmetic mean, 323.1 K, having exchanged 167.4 kJ and delivered nothing. The 2.5 K between the two endpoints is the work, left behind as heat.

Thermodynamics

The work left in two buckets of water

Carnot's ceiling assumes reservoirs so large that taking heat from one and giving it to the other changes neither temperature. Two buckets of water are not reservoirs. Run the best possible engine between a hot one and a cold one and both temperatures move, the efficiency available shrinks as they do, and the engine stops when they meet — at the geometric mean of the starting temperatures, not the ordinary one. The work it delivered is exactly the difference between those two meeting points, and it is far less than the starting temperatures promise.

$Two entropies that agree until half filling. The entropy per unit of 100 two-level units against the fraction excited, by Boltzmann's definition, the logarithm of the number of arrangements at that energy, and by Gibbs's, the logarithm of the number at or below it. Below half filling they nearly coincide: at a quarter excited they are 0.538 and 0.542 per unit, and both approach the dashed large-system curve. Boltzmann's entropy then turns over and falls back to zero when every unit is excited; at three-quarters it is 0.538. Gibbs's cannot fall, because a running total cannot, and it levels off at ln 2 = 0.693, reaching 0.693 at three-quarters. The slope of each is one over its temperature.$ Thermodynamics

The count that decides which entropy is right

There are two ways to count the states of an isolated system: the states at its energy, which is Boltzmann's entropy, and the states at or below it, which is Gibbs's. For large systems in ordinary conditions they agree to the last measurable digit. For a system whose energy has a ceiling, past the halfway point, one gives negative temperatures and the other forbids them. Definitions cannot settle which is right, but a temperature is for something — saying which way heat will flow — and putting two such systems in contact lets the count of states answer.

$Nitrogen that runs the wrong way. The nitrogen mole fraction in each of two bulbs joined by a capillary, as in Duncan and Toor's experiment: one bulb starts with 0.50086 nitrogen and the rest carbon dioxide, the other with 0.49879 nitrogen and the rest hydrogen, at 35 °C. Solid curves: the capillary solved at each instant from the Maxwell–Stefan equations with the three pairs' diffusivities, 83.8, 68.0 and 16.8 mm²/s. Dashed: Fick's law for nitrogen alone, which can only let the two start values relax together. At the start the nitrogen gradient is 0.00207, yet nitrogen flows at 309 times the rate that gradient would drive. From 0.1 to 6.5 hours it flows from the bulb with less nitrogen into the bulb with more, opening a difference of 0.1468 at 6.5 hours; at 6.6 hours its flux passes through zero with a difference of 0.1468 still in place. Each gas is conserved to a part in 10⁹. The carbon dioxide moving out of the first bulb drags nitrogen with it, because the nitrogen–carbon dioxide pair has by far the smallest diffusivity and so the strongest friction.$ Thermodynamics

The gas that flows towards more of itself

Fick's law says a substance diffuses from where there is more of it to where there is less. In a mixture of three gases, nitrogen can do the opposite for hours — flowing into the bulb that already holds more nitrogen, and then stopping while a difference remains — and in a welded bar of steel, carbon crosses into the side that is already richer. Nothing is wrong with the second law. Diffusion flattens chemical potential, and with more than two components, or a second element changing it, that is not the same as flattening concentration.

The one prediction the Planck scale makes. The energy density the vacuum should have, from summing the zero-point energy of a field's modes up to a cutoff, against where that cutoff is put — thirty decades of cutoff energy and a hundred and thirty of density, both logarithmic. The horizontal line is what is measured: 5.34e-10 joules per cubic metre, the dark energy that accounts for sixty-nine per cent of the universe. Cutting the sum off at the Planck energy — which is where dimensional analysis says every description available runs out — overshoots it by 10^121. That is the largest disagreement between an estimate and a measurement anywhere in physics, and the slope of the line is why it cannot be argued away: the density goes as the fourth power of the cutoff, so cutting off at the electroweak scale still overshoots by 10^54 and cutting off at one electronvolt — below which no physics is in doubt at all — still overshoots by 10^8. The cutoff that would give the right answer is 8.0e-3 electronvolts, which is a wavelength of about a tenth of a millimetre and corresponds to no known physics whatever.

Astrophysics

The estimate that misses by a hundred and twenty

Every argument about the Planck scale is an argument about consistency rather than about data, with one exception. The zero-point energy of the quantum fields gravitates, dimensional analysis at the Planck cutoff says how much, and what is measured is 10¹²¹ times smaller. It is the largest disagreement between an estimate and a measurement anywhere in physics, and lowering the cutoff does not rescue it.

The edge that is a straight line, not a step. The absorption of gallium arsenide near its own band gap of 1.424 electronvolts, on a logarithmic axis, at 4 temperatures. Below the gap the absorption does not stop; it falls exponentially, along a straight line whose slope is an energy, and the straightness holds over several decades. Raising the temperature makes the line shallower — the tail reaches further below the gap — and the slope runs from 5.6 millielectronvolts at 10 kelvin to 7.5 at 300, a factor of 1.33. The lines pivot about a point just above the gap rather than rotating about nothing, which is what makes the slope a single number worth quoting. The band gap's own shift with temperature has been removed here, so that the fan is the tail's doing and not the gap's.

Waves

Below the gap, where there is nothing to absorb

A semiconductor is supposed to be transparent below its band gap, and it is not. The absorption falls exponentially instead, over seven decades, along a straight line whose slope is an energy of a few millielectronvolts — and the description that produces that line has no states in the gap at all. What the slope measures is how much the gap itself is moving about.

The four vortices a standing wave leaves behind. Streamlines of the steady flow that a standing sound wave sets up in a channel, over half an acoustic wavelength, with the horizontal axis in units of the wave's own phase and the vertical axis scaled to the channel. The sound itself is a back-and-forth motion that averages to nothing; this is what does not average to nothing. Four closed cells fill each wavelength, two above the centreline and two below, turning in opposite senses, with the fluid moving along the walls toward the velocity nodes and back along the centre. The boundary layer that generates all of it is 69 micrometres thick, which is 0.7 per cent of the channel and is thinner than the width of a line in this drawing. The cells are not in the layer; they fill the channel.

Fluids

The drift a sound leaves behind

A sound wave moves fluid back and forth and puts it back where it started. Over many cycles it does not: a steady circulation appears, four cells to a wavelength, driven entirely from inside a boundary layer seventy micrometres thick. Its speed contains the sound speed and the amplitude, and it contains no viscosity at all — so making the fluid thinner does not make the drift weaker.

Electromagnetism

The field that points against the magnet it is in

There are two magnetic fields in use and the difference between them is which currents a loop is allowed to count. The consequence nobody expects on being told the definitions: inside a permanent magnet H points the other way from B. It has to — a loop inside the magnet threads no wire, so its H circulation is zero, and the only arrangement left has H running backwards.

The floor does no work and the jumper leaves the ground. A 70-kilogram person pushing off the floor: the floor's force in units of body weight against time, with the centre of mass's height and speed drawn on the same axis, each scaled. The force reaches 2.6 body weights, the contact lasts 260 milliseconds, and the take-off speed that comes out of integrating it is 1.67 metres a second — a jump of 14 centimetres. Integrating the floor's force over the centre of mass's rise gives 247 joules. The work the floor does is zero, because the patch of floor under the foot never moves and work is a force times the displacement of its own point of application. Both numbers are correct and they are answers to different questions: the first is what Newton's second law integrated over the centre of mass gives, and the second is what crosses the boundary between the floor and the person, which is nothing.

Mechanics

The floor that does no work

A jumper leaves the ground with three hundred joules of kinetic energy, supplied by a floor that does exactly zero work — because work is a force times the displacement of its own point of application, and the patch of floor under the foot never moves. Newton's second law integrated over the centre of mass gives the right kinetic energy and is not the work-energy theorem, and telling the two apart is what the first law of thermodynamics is for.

Mean field always pushes; correlations pull. The pressure between two planes of equal charge with only their own counterions between them, against their separation, in units of the Gouy–Chapman length μ for the separation and 2πℓ_Bσ²kT for the pressure. The upper curve is the Poisson–Boltzmann result, solved from k·tan(kd/2) = 1: it is the density of counterions at the midplane and is positive at every separation, falling from the ideal-gas 2/d at contact to π²/d² far apart — 1.71 at 1μ, 0.290 at 4μ. The lower curve is the strong-coupling limit, 2/d − 1, which the same ions reach when their valence and the surface charge are high. It crosses zero at d = 2μ and is negative beyond, tending to −1: the two like-charged planes attract, and the separation 2μ is where they come to rest.

Fluids

The like charges that pull together

Two surfaces carrying the same charge, with nothing between them but the ions that neutralise them, ought to repel, and the standard mean-field theory proves that they always do. With calcium or spermine as the counterions they attract, and come to rest a fraction of a nanometre apart. The mean field misses it because it averages the ions into a smooth cloud, and multivalent ions are too strongly repelled by each other to form one. Each keeps a patch of surface to itself, and the pressure between the plates becomes a single ion's business.

Circles that go nowhere, and a current across the line. Gyrating ions in a uniform magnetic field pointing out of the page, with 60 guiding centres drawn from a density that falls by a factor of e every 3 gyroradii to the right, and Maxwellian speeds. Every ion goes round clockwise and none of the circles moves. Of the circles that cross the dashed vertical line, those centred to its left cross it moving down and those centred to its right cross it moving up; in this sample 11 cross moving down and 7 moving up, a count a sample this small could turn either way. Because there are more circles on the left, the ions at the line move downwards on average over every speed and phase, at exactly the thermal speed squared over the gyrofrequency times L, 0.33 thermal speeds, computed by averaging over speeds and phases and checked against that value. It is a current, carried by circles whose centres are still.

Electromagnetism

The current no particle carries

A magnetised plasma holds its own pressure against the field only if a current flows across the pressure gradient, and the fluid equations say exactly how much. Follow the particles in a uniform field and none of them is going anywhere; every guiding centre is still. The current is real all the same. It is made of circles that are more crowded on one side of a line than the other, and when the field is not uniform, the drifts that do move the guiding centres flow the wrong way.

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