The collection

Every essay — page 19

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Mechanics

Motion, force, and the quantities that refuse to change.

Long stretches of order, broken without warning. 1800 successive values of the logistic map at r = 3.828427 − 0.00002, a distance of 2.0 × 10⁻⁵ below the setting at which its stable three-cycle is born. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them the orbit bursts through the whole interval with no discernible pattern, and then, at an unpredictable moment, is captured into another calm. In a run of 400,000 iterates at this setting the calms last 173 iterates on average, and the channel the orbit creeps through — measured on the map's own third iterate — has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates. Nothing random is added: the sequence is the same every time it is computed from the same start.

The calm that is the ghost of a cycle

Just before a chaotic system settles into a stable cycle it does something stranger than either: it behaves perfectly periodically for long stretches, and then, at moments nothing in the record predicts, bursts into disorder and back. The calm is a cycle that does not exist yet, creeping through the narrow gap where it is about to be born, and how long each calm lasts is set by the square root of the distance to that birth.

5 figures · part 5 on Chaos

Waves

Oscillation, and everything that turns out to be an oscillation.

Two states at zero that only one side of the chain has. Every energy level of a chain of 20 cells, 40 sites, whose couplings alternate between v inside a cell and w = 1 between cells, against the ratio v/w from 0 to 2. The bulk levels fill two bands, bounded by the dashed lines ±|v − w| and ±(v + w), with a gap between them that closes at v = w. For v < w two levels sit at zero energy, in the middle of the gap, drawn in the warning colour: at v/w = 0.5 they are within a millionth of the coupling of zero. For v > w the gap is empty — at v/w = 1.5 the nearest level to zero is 0.527. Nothing about the chain's middle distinguishes the two sides of v = w; the difference is at its ends.

The end that knows how the middle was cut

A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.

5 figures · part 4 on Periodic media

A gap that every angle from air falls into. The stop bands of a stack of quarter-wave layers of index 4.6 and 1.6 — tellurium and polystyrene, the pair of the first such mirror — against frequency, in units of the design frequency, and the parallel index β = n₀ sin θ₀ the light brings along the layers; TE polarisation on the right, TM on the left. Shaded regions are gaps. Light from air can only have β between −1 and 1, the vertical lines; within those lines the gaps for both polarisations overlap between f = 0.848 and 1.321, the band marked across the figure, so every angle of incidence and both polarisations are reflected: a relative width of 43.6%. The TM gap narrows as β grows and closes at the internal Brewster index, 1.51, which light from air cannot reach.

The mirror that works from every direction

A stack of alternating transparent layers reflects nearly all the light of one colour arriving straight on, and less as the light tilts, because tilting moves the forbidden band. A structure that repeats in only one direction ought therefore to be a mirror for only a range of directions. It is not, if the layers differ enough: light arriving from air cannot bring enough sideways momentum to escape the forbidden band at any angle, for either polarisation, and a flat stack of plastic and tellurium reflects every angle over a band of frequencies almost half as wide as its centre.

5 figures · part 5 on Periodic media

One width that falls to nothing. The decay rates of the two modes of a pair of resonances that leak into one shared channel at rates 0.1 and 0.05 and are coupled to each other with strength 0.2, against the detuning of the first from the second. The two rates always add to 0.15, the trace of the leak matrix, to 10⁻¹². Where the resonances are far apart each mode keeps roughly its own resonance's leak. Near them the leaks interfere, and at a detuning of 0.1414 — κ(γ₁ − γ₂)/√(γ₁γ₂) — the slower mode's decay rate is zero to within 10⁻¹², and the faster carries all 0.15. That mode sits at a frequency where the channel is open and does not leak into it: the two routes by which it could escape cancel.

The resonance that refuses to leak

A resonance that sits at a frequency where waves can escape has a width, because it leaks. Put two such resonances into the same channel and let them talk to each other, and at one precise detuning one of their combinations stops leaking altogether — a mode with no width, surrounded by a continuum it could escape into and does not. It cannot be seen from outside, it traps whatever energy lands in it, and if a symmetry is what forbids the leak, it survives any change that keeps the symmetry.

5 figures · part 6 on Resonance $How close a network gets to a load that stores charge. The fraction of a wave's amplitude reflected from a resistance shunted by a capacitance, with RCωc = 2, against frequency in units of the band edge ωc, for the load alone and for networks of inductors and capacitors whose values, with an ideal transformer at the source, were optimised numerically to keep the reflection low across the band. The bare load holds it to 0.707 across the band; the 1-element network holds it to 0.392 across the band; the 2-element network holds it to 0.320 across the band; the 3-element network holds it to 0.289 across the band. The dashed line is Bode and Fano's floor, exp(−π/RCωc) = 0.208, which no network of any size can go beneath over the whole band; each added element brings the design closer to it, and each buys its flatter band with a reflection that climbs to total just beyond the band edge.$

The mismatch no network can remove

A quarter-wave layer or a taper can match a resistance to a resistance as well as anyone likes. Put a capacitance across the load and that stops being true for every network that could ever be built from lossless parts: Bode and Fano proved that the total amount of match available is fixed by the load's resistance and capacitance, so a network can only move it about — and a flat match across a band can never be better than e to the minus π over the load's time constant times the band.

4 figures · part 4 on Impedance $One slit, four distances, one multiplication. The intensity across the beam behind a slit 5 wavelengths wide, at distances of 0.5, 5, 25, 100 wavelengths, each computed by multiplying the slit's plane-wave spectrum by the phase each wave accumulates and transforming back — no approximation about angles. Close to the slit the pattern is the slit's own shape with ripples at its edges; further out the ripples move inwards and the beam develops a bright centre; far away it spreads into the diffraction pattern. The travelling part of the field keeps its power to 10⁻¹⁰, running it back 100 wavelengths recovers it to 5 × 10⁻¹⁴, and at 100 wavelengths the result matches a direct Fresnel integral to 2.75 per cent rms. Near field and far field are not two theories; they are one multiplication at different distances.$

The fan of plane waves inside every beam

Huygens added up wavelets from every point of a front. The same content can be written as a sum over plane waves travelling in every direction, and then propagation stops being an integral and becomes a multiplication: each plane wave picks up a phase in proportion to the distance. One square root in that phase holds all of diffraction, near field and far field alike — and when the square root turns imaginary, it holds the reason no instrument a wavelength away can see detail finer than half a wavelength.

4 figures · part 5 on Huygens

Optics

Light, and the small number of rules it obeys.

Light that walks through a cloud. Seven photons entering a slab 8 scattering mean free paths thick from above, straight down, and followed until they leave, with every scattering equally likely to send them in any direction; their paths are projected onto the page. Of 20000 photons followed the same way, 82.5 per cent come back out of the top and 17.5 per cent out of the bottom. Diffusion theory predicts 18.2 per cent through the bottom. Only 10 photons cross without scattering at all, where e⁻⁸ predicts 6.7 — within the counting error of so few: nearly all of what is transmitted has walked, and the direction it arrived from is forgotten on the way.

The cloud light has to walk through

A beam crossing thirty scattering lengths of anything should keep e⁻³⁰ of itself — a ten-millionth of a millionth. A cloud thirty scattering lengths thick lets through nearly a third of the sunlight falling on it. The light has not crossed; it has walked, one scattering at a time, and a walk through a slab obeys a law with the shape of Ohm's rather than of an exponential.

5 figures · part 5 on Scattering

Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength.

The reflection that needs no surface

Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

5 figures · part 5 on Refraction

Electromagnetism

Charge, field, and the lines drawn between them.

$The potential at a point is where its walkers end up. A square divided into a 24-step grid, with its top edge held at a potential of 1 and the other three edges at 0. From the probe point (0.29, 0.71), 4000 random walkers each step to one of their four neighbours with equal chance until they touch an edge; six of them are drawn, each ending with a dot on the edge it reached. The fraction that end on the held edge is 0.405 ± 0.008, one standard error, after an average of 122 steps. Solving Laplace's equation on the same grid by repeatedly replacing every value with the average of its four neighbours gives 0.408, and the series solution for the continuous square gives 0.408. The walkers were never told the equation: a value that is the average of its neighbours and a probability of ending somewhere are the same arithmetic.$

The potential is where the wanderers stop

Start a random walker at a point between charged conductors and let it wander until it touches one of them. The average potential of the surfaces the walkers touch is the potential at the starting point — exactly, with no equation solved — and the charge a conductor keeps at each place on its surface is the chance that a walker arriving from far away touches it there first.

5 figures · part 3 on Potential

The interaction that orders a magnet is not the magnetic one. For six ordered magnets, two temperatures on a logarithmic scale: the energy of the magnetic interaction between two neighbouring moments, expressed as a temperature, and the temperature at which the material actually orders. iron orders at 1043 K against a dipolar scale of 0.201 K, a factor of 5182; cobalt orders at 1394 K against a dipolar scale of 0.117 K, a factor of 11961; nickel orders at 627 K against a dipolar scale of 0.015 K, a factor of 42312; gadolinium orders at 293 K against a dipolar scale of 0.790 K, a factor of 371; europium oxide orders at 69 K against a dipolar scale of 0.615 K, a factor of 112; lithium holmium fluoride orders at 1.53 K against a dipolar scale of 1.230 K, a factor of 1.24. The five ferromagnets order between a hundred and forty thousand times above the only interaction their moments have with each other, so whatever aligns them is not magnetism. The sixth is the control: lithium holmium fluoride is a magnet whose ordering really is dipolar, and its two temperatures agree.

What holds a magnet together is not magnetism

Two neighbouring moments in iron interact magnetically with an energy worth a fifth of a kelvin, and iron keeps its order to 1,043 kelvin. Whatever aligns them is five thousand times stronger than the only force they exert on one another — and it is electrostatic, with the exclusion principle deciding which of two spatial arrangements two electrons may use.

6 figures · part 4 on Magnetisation

Thermodynamics

Heat, disorder, and the one law with a direction in it.

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.

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.

5 figures · part 6 on Heat engines $Two entropies that agree until half filling. The entropy per unit of 100 two-level units against the fraction excited, by Boltzmann's definition, the logarithm of the number of arrangements at that energy, and by Gibbs's, the logarithm of the number at or below it. Below half filling they nearly coincide: at a quarter excited they are 0.538 and 0.542 per unit, and both approach the dashed large-system curve. Boltzmann's entropy then turns over and falls back to zero when every unit is excited; at three-quarters it is 0.538. Gibbs's cannot fall, because a running total cannot, and it levels off at ln 2 = 0.693, reaching 0.693 at three-quarters. The slope of each is one over its temperature.$

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.

5 figures · part 6 on Third law $Nitrogen that runs the wrong way. The nitrogen mole fraction in each of two bulbs joined by a capillary, as in Duncan and Toor's experiment: one bulb starts with 0.50086 nitrogen and the rest carbon dioxide, the other with 0.49879 nitrogen and the rest hydrogen, at 35 °C. Solid curves: the capillary solved at each instant from the Maxwell–Stefan equations with the three pairs' diffusivities, 83.8, 68.0 and 16.8 mm²/s. Dashed: Fick's law for nitrogen alone, which can only let the two start values relax together. At the start the nitrogen gradient is 0.00207, yet nitrogen flows at 309 times the rate that gradient would drive. From 0.1 to 6.5 hours it flows from the bulb with less nitrogen into the bulb with more, opening a difference of 0.1468 at 6.5 hours; at 6.6 hours its flux passes through zero with a difference of 0.1468 still in place. Each gas is conserved to a part in 10⁹. The carbon dioxide moving out of the first bulb drags nitrogen with it, because the nitrogen–carbon dioxide pair has by far the smallest diffusivity and so the strongest friction.$

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.

4 figures · part 7 on Diffusion

Relativity

Space and time, drawn on the same axes.

The rectangle a spin tilts. The laboratory energy of the pion from a tau into a pion and a neutrino moving at 0.8 of the speed of light, for samples of 60,000 decays whose parents have αP = +1, 0, −1 along their line of flight. Every sample fills the same interval, 313 to 2667 MeV: the edges are set by the masses and the speed, and they do not move. Inside that interval an unpolarised sample is flat and a polarised one is tilted, its height at each energy 1 + αP times the cosine of the rest-frame angle that energy corresponds to. Reading the slope back from each sample gives +0.993, −0.010, −1.006, each with a standard error near 0.006. A parent spinning along its flight throws the pion forwards in its own frame, and the boost turns forwards into more energetic, so the tilt of an energy spectrum measures a polarisation without any angle being measured at all.

The slope a spin leaves in a spectrum

An unpolarised parent decaying in flight gives its products a rectangle of energies. Give the parent a spin along its line of flight and the rectangle tilts, while its two edges stay exactly where they were. The tilt is the polarisation, it can be read without ever seeing which way the parent was going — and which variable it is read from decides how many decays the reading costs.

5 figures · part 6 on Relativistic dynamics $Resonances drawn as bands. 60,000 decays of a D⁰ decaying to K⁻π⁺π⁰, accepted from 1,287,365 flat ones in proportion to the square of an amplitude built from 3 short-lived intermediate states, each decaying to two of the three products. A ρ⁺(770) in m²(π⁺π⁰), 67.0 per cent of the rate on its own; a K⁻(892) in m²(K⁻π⁰), 25.5 per cent of the rate on its own; a K⁰(892) in m²(K⁻π⁺), 33.2 per cent of the rate on its own. Each appears as a band at its own mass squared — vertical, horizontal or diagonal according to which pair it decays to — holding 51 per cent, 19 per cent, 23 per cent of the decays within one width of its mass, where phase space alone would put 34, 10, 9. The separate fractions add to 126 per cent, not 100, because the amplitudes interfere where the bands overlap. The magnitudes and phases are a model chosen to make all three visible, not a fit to data.$

The plane in which three bodies are flat

A particle breaking into two gives each product a fixed energy; one breaking into three gives none of them one. What it gives instead is a plane of two invariant masses in which a decay with no forces spreads perfectly evenly inside a curved boundary — so every band, dark stripe and bright crossing a real decay draws there is a force, its spin, or a phase between two routes to the same three particles.

5 figures · part 7 on Relativistic dynamics

What a beam's energy buys, three ways. The speed reached against the energy intercepted, measured in the body's own rest energy, for a perfect mirror pushed by a beam, a perfect absorber pushed by the same beam, and a photon rocket that carries the same energy as fuel and throws it out behind. The mirror's curve is γ(1 + β) = 1 + 2E/mc², a rapidity of ln(1 + 2E/mc²); the dots integrate the reflected beam's force, (2P/c)(1 − β)/(1 + β), and agree with it to 10⁻¹⁵. To reach 0.2c the mirror needs 0.1124 of its rest energy, the absorber 0.2500 and the photon rocket 0.2247 — for a 1 g sail, 10.1 terajoules against 20.2. The rocket's rapidity is ln(1 + E/mc²), so the mirror is the rocket with its fuel left at home and each joule used twice, once arriving and once leaving. The absorber does worst, because the energy it keeps becomes rest mass it then has to carry.

The rocket that leaves its fuel at home

A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.

5 figures · part 6 on Mass-energy

Quantum

Where the continuous picture runs out, and what replaces it.

The field an atom calls strong

A magnetic field splits sodium's two yellow lines into ten, at spacings set by Landé's factors — until the field grows past the one the electron already feels from its own motion, when the ten reorganise into the three a theory without spin predicts. Nothing about the magnet decides which pattern appears. The atom does: the same 45 tesla is a strong field for hydrogen, a middling one for sodium and a weak one for caesium.

5 figures · part 3 on Atomic spectra

The floor a state's survival cannot go below. The probability that a quantum state is still found in its initial state, against time measured as its energy spread times time over ħ, for four states with the same spread. The shaded region under cos²(ΔE t/ħ) is forbidden by Mandelstam and Tamm's theorem, and every curve — each a sum of phases over the state's energies — stays out of it. Two equally weighted levels run along its edge and reach an orthogonal state at exactly π/2, the fastest any state with this spread can. The same two levels driven off resonance, with the same spread, never get further than a survival of 0.500. Three equally spaced levels become orthogonal only at 1.7101, 1.0887 times the limit, and a coherent state never does, bottoming out at 0.0183. A spread of energy is permission to change, not an obligation.

The fastest a state can stop being itself

Time has no operator, so the energy–time relation cannot be the commutator inequality it resembles. What stands in its place is sharper: a state whose energy is spread by ΔE cannot become a different, orthogonal state in less than πħ/2ΔE, and cannot do it faster than its mean energy above the ground state allows either. Two equally weighted levels reach both limits exactly. Nothing else does.

5 figures · part 4 on Uncertainty

Fluids

Matter that will not hold a shape, and the forces that act in it anyway.

Above a half, the chains stop letting go. The extensional viscosity of a dilute polymer solution, as a multiple of its zero-shear viscosity, against the accumulated stretch (Hencky strain, the stretch rate times the time), on a logarithmic axis, for stretch rates whose product with the relaxation time is 0.1, 0.4, 0.6, 1. The polymer carries 90 per cent of the viscosity. Dashed curves are the Oldroyd-B dumbbell, integrated from its conformation equation and matching its closed form; solid curves are the same chains with a finite length, FENE-P with L² = 400. At 0.1 the Oldroyd-B viscosity reaches 3.37 and the finite chain 3.37 by a strain of 6; at 0.4 the Oldroyd-B viscosity reaches 9.49 and the finite chain 9.25 by a strain of 6; at 0.6 the Oldroyd-B viscosity reaches 58 and the finite chain 49 by a strain of 6; at 1 the Oldroyd-B viscosity reaches 725 and the finite chain 329 by a strain of 6. Below one half the viscosity settles at a few times the shear value. Above it the Oldroyd-B chain stretches without limit and its resistance grows exponentially, while the finite chain grows until it is nearly fully extended and then stops, hundreds of times higher than it began.

The stretch a chain cannot outrun

A Newtonian liquid pulled into a thread resists exactly three times as hard as it resists being sheared, whatever it is made of. A polymer solution resists three times as hard only until the stretch rate passes one over twice its relaxation time. Past that, its chains can no longer recoil as fast as they are pulled apart, and the same liquid that is barely thicker than water in a stirred beaker becomes hundreds of times stiffer in a thread.

5 figures · part 5 on Rheology $Most of a chain's counterions never leave it. The fraction of a charged rod's counterions lying within a distance r of it, against r in rod radii on a logarithmic axis, for a charge parameter ξ = 4.2 — the Bjerrum length of water, 0.7135 nm, over a charge spacing of 0.17 nm, which is DNA's. Each curve is the Poisson–Boltzmann solution for the rod at the centre of a cell of radius 10², 10⁴, 10⁶ rod radii, with its counterions checked to neutralise it exactly. Diluting the solution widens the cell by four decades at a time, and a counterion free to go anywhere in the cell ought to spread with it; instead each curve keeps a plateau near the rod whose height does not change. At the inflection of every curve the enclosed fraction is 0.762, which is Manning's 1 − 1/ξ, and the plateau sits there: 76 per cent of the counterions stay bound to the chain however dilute the solution, and only 24 per cent spread through it.$

The counterions that never leave the chain

Dilute a solution of DNA a million times and its counterions ought to scatter through the whole volume. Three quarters of them do not. A line of charges closer together than the Bjerrum length — 0.71 nm in water — holds on to its counterions however much room they are given, until the chain's charge is cut back to one per Bjerrum length, and every osmotic pressure, swelling gel and packed virus built from such chains is set by that length rather than by the chemistry.

5 figures · part 5 on Osmosis

The latitude past which a tide cannot split

The ocean's internal tide carries about a terawatt, and somewhere it has to be broken into waves small enough to mix the water. One of the ways it breaks is by pumping waves at half its own frequency, the way a child on a swing pumps at twice the swing's. Those half-frequency waves cannot exist where the planet's rotation forbids oscillations that slow — poleward of 28.8° for the semidiurnal tide — so the route has an edge on the map, fixed by the Moon's period and the Earth's spin.

5 figures · part 5 on Stratification

Astrophysics

Gravity read as geometry, and the laws carried where no laboratory can follow.

Radiated while the push holds, paid for when it stops. The power a charge radiates, the power the radiation reaction force takes from its motion, and the rate of change of the Schott term mτ a·v, through a push that rises over the first 20 per cent of its duration, holds steady, and falls away over the last 20, in units of mτa₀² where a₀ is the steady acceleration. Radiated power is a², the reaction force's take is −ȧv, and the Schott rate is found by differencing a·v along the trajectory; at every instant the first equals the sum of the other two. While the push is steady the reaction force is exactly zero and the charge still radiates at the full rate, all of it drawn from the Schott term. When the push stops, ȧ is large and negative while the charge is moving fast, and the reaction force takes 0.775 units, more than the 0.750 radiated over the whole push. The difference is what it handed back while the push was starting: then the charge is still slow, the reaction force points along the rising acceleration, and it does 0.025 units of work on the charge instead of taking any. The totals agree to a part in a hundred thousand.

The bill that arrives when the pushing stops

A charge accelerating steadily radiates at the full Larmor rate while the radiation reaction force on it is exactly zero, so for as long as the push holds, nothing about the charge's motion pays a single watt. The energy is lent by the field that travels with the charge, the loan is called the Schott term, and it is repaid the moment the acceleration changes.

5 figures · part 4 on Radiating charge

The month a binding energy would bend. The Moon's orbit seen with the Sun held off to the right, drawn as it would be if the Earth's binding energy fell towards the Sun more weakly than the rest of it. The Earth is bound by 4.5 × 10⁻¹⁰ of its mass-energy and the Moon by 1.9 × 10⁻¹¹, so with the Sun pulling at 5.93 mm/s² the Moon is pushed sunward relative to the Earth by 2.6 × 10⁻¹² m/s² for every unit of η. That push turns once a synodic month relative to the orbit, 29.53 days, and Hill's equations about a circular orbit — integrated from the forced solution and held on it to 4 × 10⁻¹¹ over twelve months — give a radial displacement of 8.0 m times η times the cosine of the lunar phase: outward at new moon, inward at full. The complete lunar theory, with the Sun's tide on the orbit included, gives 13.1 m. The displacement is drawn about 7 × 10⁶ times larger than it would be at η = 1.

The binding energy that has to fall too

Every laboratory test of the equivalence principle compares bodies whose own gravity is a part in 10²⁵ of their mass, so none of them can ask whether gravitational binding energy falls like everything else. The Earth is bound by five parts in ten billion and the Moon by twenty times less, and if that difference fell differently the Moon's orbit would lean towards the Sun once a month — by a distance lasers have been measuring since 1969.

5 figures · part 4 on Equivalence principle

The electrode that walks itself negative. An electrode driven through a blocking capacitor by a symmetric waveform of amplitude 20 kTe, in an argon-like plasma whose electrons are collected 108 times as readily as its ions at saturation, starting uncharged. Potentials are relative to the plasma, in electron temperatures. On the first positive half-cycle the electrode draws a flood of electrons and nothing on the negative half-cycle can return the charge, because the ions arrive at a fixed, small rate; the electrode walks down, and within 6 cycles its mean potential is within half an electron temperature of the self-bias. That bias, found by integrating the charging, is −22.27 kTe, and the charge balance with a Bessel function gives −22.27. The floating potential of the same wall undriven is −4.68. At 3 eV, a 60 V drive makes a 67 V bias with no direct current supplied anywhere.

The bias no battery supplies

Drive an electrode in a plasma through a capacitor with a perfectly symmetric waveform and it charges to a steady negative voltage almost as large as the waveform's own amplitude. No direct current flows anywhere and no battery is connected. The plasma's sheath lets electrons in far more easily than it lets them out, and every semiconductor wafer etched in the last forty years was held at a voltage made that way.

5 figures · part 6 on Plasma oscillation

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