What a collapse does to a field
At its defaults it draws what a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.
flux-freezing is one function in lib/figures/extremes.js —
radiation, cross-sections and self-gravity. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.
A pendulum with unequal steps
The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The potential −EJ cos φ of the junction against its phase, at EJ/EC = 50, with the lowest 4 levels of the circuit drawn across the well between their classical turning points, in units of the charging energy. The transitions are 18.94, 17.79, 16.50, each smaller than the one below it; a harmonic well of the same curvature would space them all at the square root of 8·EJ·EC, 20.00. The spacing shrinks because the cosine is flatter than a parabola away from its bottom, and the shrinking is what lets a microwave pulse tuned to the lowest transition leave the next one alone.
The levels that stop caring about stray charge
The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The lowest energy levels of a superconducting island joined to a reservoir through a Josephson junction, against the offset charge on the island in units of a Cooper pair's charge, measured from the lowest point of the ground level and in units of the first transition energy. With the Josephson energy 1 times the charging energy, the ground level wanders by 0.263 of a transition as the offset charge changes; with the Josephson energy 50 times the charging energy, the ground level wanders by 3.0e-8 of a transition as the offset charge changes. Every level repeats with a period of one pair, checked. On the left the levels are parabolas in the offset charge joined where the junction mixes them, and any stray charge moves the transition; on the right they are flat, and the circuit has almost forgotten which charge state it is near.
Exponentially deaf to charge
The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
How much each of the three lowest levels moves as the offset charge sweeps through half a pair, divided by the first transition energy, against the ratio of Josephson to charging energy, both on logarithmic axes. The curves are computed by diagonalising the circuit's Hamiltonian at the two extremes of the offset charge and are checked against the asymptotic result, which falls as the exponential of minus the square root of eight times EJ/EC. At EJ/EC = 50: level 0 3.0e-8, level 1 2.1e-6, level 2 6.5e-5. Each higher level is more sensitive than the one below, because it sits nearer the top of the cosine well where the phase is less confined. Between EJ/EC = 1 and 50 the first excited level becomes 5.4·10⁵ times less sensitive.
What the flatness costs
The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The difference between the second and first transition energies, as a fraction of the first, against the ratio of Josephson to charging energy. Zero would be a harmonic oscillator, whose transitions are all the same and whose two lowest levels cannot be addressed alone. The solid curve is computed by diagonalisation; the dashed curve is the result for a weakly anharmonic cosine well, the charging energy divided by the harmonic spacing less one charging energy, with a minus sign. At EJ/EC = 50 the relative anharmonicity is −6.1 per cent; at 1 it is +74.9 per cent. It falls only as one over the square root of the ratio, while the charge dispersion falls exponentially — the asymmetry the design exploits.
Where the design sits
The options are the ones The circuit that forgets its charge passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
For a qubit whose first transition is fixed at 5 GHz, the two quantities the ratio EJ/EC trades against each other, in hertz on a logarithmic axis: the size of the anharmonicity, which sets how fast the qubit can be driven without leaking into its third level, and the charge dispersion of the first excited level, which sets how much a stray charge moves its frequency. They are equal near EJ/EC = 5.4. At 50 the anharmonicity is 303 MHz and the dispersion 10.3 kHz, with a charging energy of 264 MHz. The notch near EJ/EC of 3 is where the anharmonicity passes through zero and changes sign. Beyond it the anharmonicity declines slowly and the dispersion by many decades.
What checks it
physicscheck asserts something about flux-freezing that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
The circuit that forgets its charge
A tiny superconducting island joined to its surroundings through a Josephson junction has discrete energy levels, and two of them make a quantum bit. The first such circuits were ruined by stray charges on nearby surfaces, which moved their levels and scrambled any superposition within a nanosecond. The cure was to make the junction's energy fifty times the charging energy. That makes the levels exponentially insensitive to charge while costing only a power-law loss in the unequal spacing that lets one transition be driven alone.
ElectromagnetismThe current a superconductor stops with its own field
A superconducting wire has no resistance, so it seems it should carry any current at all. It carries a definite amount and no more, and the amount is set not by the metal's ability to conduct but by the field the current itself makes: when that field at the wire's surface reaches the critical field, superconductivity fails there. The rule makes the largest current proportional to the wire's radius rather than its area, makes a thick wire worse per square millimetre than a thin one, and explains why the first superconducting magnet, wound in 1913, did not work.
AstrophysicsThe dynamo that reverses with nothing pushing it
A copper disc spinning in a magnetic field drives a current from its axle to its rim. Wire that current through a coil wound round the disc and the coil supplies the field the disc needs — and above one definite speed, a resistance divided by an inductance, the arrangement makes its own field out of any stray one, with nothing else magnetic anywhere. It cannot choose which way the field points; the stray field chooses. Couple two such discs, each feeding the other's coil, and with the motors' torque held perfectly steady the field flips direction again and again at intervals that follow no pattern. The simplest dynamo that works is also the simplest one that reverses.
ElectromagnetismThe field a spinning superconductor makes of itself
Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched.
ElectromagnetismThe field a superconductor keeps after all
A superconductor is famous for throwing magnetic field out, and the experiment that showed it is the one that made superconductivity a state of matter. Most superconductors that are actually used do something closer to the opposite: they let the field in as threads, grip the threads where they are, and keep them when the field is taken away. The result is a slope of field inside the material that is as steep as the grip allows and no steeper, which is all that is needed to explain why such a superconductor has a hysteresis loop, heats up in an alternating field, can hang a magnet underneath itself, and can be made into a magnet stronger than any of iron.
AstrophysicsThe field no symmetric flow can keep
The Earth's core would forget its magnetic field in about forty thousand years, and the field is at least three and a half thousand million years old, so something must be making it again all the time. The obvious candidate is the flow of liquid iron. In 1934 Thomas Cowling proved that the simplest version cannot work: no motion of a conductor, however fast or however arranged, can keep up a field that is symmetric about an axis. The reason is a single ring inside the field where the current it needs cannot be driven.
AstrophysicsThe field that cannot get out
Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.
ElectromagnetismThe field that is pushed out
A perfect conductor keeps whatever field was inside it when its resistance vanished. A superconductor expels the field either way, and the difference between remembering and expelling is the experiment that showed superconductivity is a state of matter rather than a very good conductor.
AstrophysicsThe same force whichever way the surface faces
A magnetic field's stress is usually described as a pressure across the lines with a tension along them, as though it were two effects. It is one. The force per unit area is B²/2μ₀ on every surface however it is oriented, and only the direction changes — the surface normal reflected in the field, so that turning the surface one way turns the force the other. At 45° neither name applies.
ElectromagnetismThe stripes a superconductor makes of a field
A superconductor expels a magnetic field, but only if the field it pushes aside stays below the critical field everywhere at its surface — and for anything but a long rod held along the field, the field it pushes aside piles up at its edges. A flat plate laid across a field cannot expel any of it. It splits instead into interleaved bands of normal metal carrying exactly the critical field and superconducting metal carrying none, in the proportion flux demands, and the bands have a definite width. Walls between the two cost energy and favour few, broad bands; the field fanning out of the surface favours many narrow ones. The compromise is the geometric mean of the plate's thickness and a microscopic length.
AstrophysicsThe twist a bottle of plasma has to have
A straight magnetic bottle leaks out of its ends, so the obvious repair is to bend it round into a ring. A ring of field with no twist in it loses its plasma in about three millionths of a second. Adding a twist repairs that by making a drift average to nothing, and adding too much makes the whole column kink like an overwound rubber band. A tokamak is built in the narrow band between those two failures.
AstrophysicsThe twist that outlives the turbulence
A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.
ElectromagnetismThe two in the flux quantum
A superconducting ring cannot hold whatever flux is applied to it. It holds a whole number of quanta and drives a current to make up the difference, and the size of that quantum is Planck's constant divided by twice the electron's charge. The factor of two was measured in 1961, four years after somebody predicted that the carriers are pairs — by an experiment in which no charge is measured at all.
ElectromagnetismThe voltage that is a frequency
Two superconductors separated by a barrier a nanometre thick carry a current with no voltage at all, set by the difference of their quantum phases. Push harder and a voltage appears — and a voltage makes that phase difference run, so the current oscillates at 483.6 gigahertz for every millivolt. Shine microwaves on the junction and the voltage locks to exact multiples of the frequency divided by a ratio of fundamental constants, with nothing about the junction in it. That is why a volt is now counted in cycles.
AstrophysicsThe wave that does not know what the gas is made of
Give the magnetic tension of a bent field line an inertia and it becomes a string. The wave that runs along it goes at the same speed at every wavelength, compresses nothing anywhere, and has a speed containing no temperature, no pressure and no sound speed — it is the only wave in classical physics that is indifferent to what the medium is. Its energy travels along the field whatever direction the wave was sent in.
ElectromagnetismTwo lengths, and which one is longer
A superconductor has a depth to which a field leaks in and a distance over which superconductivity itself can be built up. Which of the two is larger decides the sign of the energy of a boundary — and therefore whether the material keeps every field out or fills itself with a lattice of holes.