The same field, sampled as arrows
At its defaults it draws the same field, sampled as arrows. The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.
vector-field is one function in lib/figures/fields.js —
charge, current, flux and the lines drawn between them. 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.
The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.
The field of a dipole
The options are the ones Field lines are a choice, not a discovery passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Field lines traced from a positive charge toward a negative one. Every line does eventually close on the negative charge, but the outer ones loop far outside any frame, so this picture is a crop rather than the whole field.
The same field, sampled as arrows
The options are the ones Field lines are a choice, not a discovery 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 field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.
The field of a single charge
The options are the ones Field lines are a choice, not a discovery passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Field lines radiating from one positive charge, straight and evenly spread.
The field of two like charges
The options are the ones Field lines are a choice, not a discovery passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Field lines from two positive charges. None connects them; between them is a point where the field vanishes entirely.
Every stationary point has a way out
The options are the ones Nothing can be held still by a static field 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 along four lines through the most symmetric point of a symmetric arrangement of 4 equal charges at the corners of a square — the one place a trap might be expected. In the plane of the square the potential rises in every direction; out of the plane it falls. The point is stationary and it is a saddle, which is what Laplace's equation forces: the three second derivatives must sum to zero — computed here as 1.4e-5 against the individual values of order 2e+0 — so if two of them are positive the third must be negative. A particle released here rolls away along the direction that falls. No amount of ingenuity in placing the charges changes this, because the constraint is on the equation rather than on the arrangement, and it is why every real trap for a charged particle either uses time-varying fields, or a magnetic field with a velocity, or a material with a negative response.
What checks it
physicscheck does not draw vector-field at all.
Its figures are checked for overflow, contrast, legibility and markup by the shared gates and
for nothing about the physics — a debt recorded in _plan/PHASE_PLAN.md since the
scale phase, published here so it is visible from the site rather than only from
the gate.
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.
Field lines are a choice, not a discovery
Nothing in space is arranged in lines. The lines are a drawing convention — and an unusually good one, because three separate facts about the field survive the translation.
ElectromagnetismNothing can be held still by a static field
However many charges are arranged, however cleverly, a charge placed among them has somewhere to fall. The reason is one line of arithmetic — the potential in empty space satisfies Laplace's equation, and a solution of that equation has no interior maximum or minimum — and the consequence is that every real trap for a charged particle works by breaking one of the assumptions rather than by being cleverer.
ElectromagnetismOne number for every point, and nothing at all is lost
The electric field is three numbers at every point of space. Replacing it with one number loses nothing — and the reason it loses nothing is the same reason a hill can be drawn as a contour map.
ElectromagnetismThe atom whose nucleus is a reflection
An electron set down on liquid helium cannot go in, because the helium atoms' closed shells repel it, and it cannot go far, because the liquid polarises and pulls it back with the field of an image charge. The image is weak, three-hundredths of the electron's charge. So the electron sits in a one-dimensional hydrogen atom whose nucleus is its own reflection: its levels follow hydrogen's 1/n² series at 0.66 milli-electronvolts instead of 13.6 electronvolts, and it hovers eleven nanometres above the surface. Microwaves at 126 gigahertz lift it to the next level, and a field pressing down on it tunes the line.
ElectromagnetismThe attraction that needs no charge
Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.
ElectromagnetismThe field a ring of wires lets through
A cage made of wires is supposed to work almost as well as a solid box, because the field that leaks through the gaps dies away exponentially within a gap's width. The exponential is real and it is the wrong part of the story. It describes the ripple the separate wires make; underneath it the average field goes straight through, at a level that falls only in proportion to the spacing and depends on the wires' thickness through a logarithm. A cage of twelve thin wires lets a quarter of the field in.
ElectromagnetismThe field before the lines were drawn on it
A field is a vector attached to every point of space. Drawing it as arrows on a grid is honest and ugly; drawing it as lines is beautiful and throws information away.
ElectromagnetismThe field that keeps Greenwich time
Over a calm ocean, a thousand kilometres from any storm, the air carries a downward electric field of about a hundred volts per metre, and it rises and falls by a sixth once a day. It peaks at the same instant on every ship on every ocean — near seven in the evening in Greenwich, whatever the local clock says. Nothing local can do that. The field is the far end of a circuit whose upper plate is a single conductor round the whole planet, and a conductor has one potential, so every patch of fair-weather sky is reading the sum of all the world's thunderstorms at once.
ElectromagnetismThe path a charge takes is not its field line
A field line is drawn by stepping along the direction of the force, and it is natural to read it as the route a charge would take. It is not. A field line is the curve whose direction is the force; a charge's path is the curve whose bending is the force, and the two agree only where the charge starts from rest. A charge released beside a dipole sets off along its line and then swings wide of it, carried by the speed it has gained; a charge launched round a point charge circles it on an ellipse that crosses every line. Only two things make a charge keep to its line: enough friction that it never has a speed of its own, or a magnetic field.
ElectromagnetismThe 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.
ElectromagnetismThe shape that takes five numbers
A charge distribution's net charge is one number and its dipole three. The next term, the quadrupole, is where the distribution's shape first reaches the far field, and it is a table of nine numbers of which only five survive: two for the shape and three for which way it points. Its field vanishes on a cone at 54.74° — the same angle at which a spinning sample makes a nuclear magnetic resonance spectrum sharp.
ElectromagnetismThe voltage that bends an electron like glass
An electron that has fallen through a potential difference V moves at a speed proportional to √V. Crossing a surface of constant potential, it keeps the part of its momentum that runs along the surface and changes the part across it, and the two rules together are Snell's law with √V in place of the refractive index. Every arrangement of charged electrodes is therefore a piece of glass for electrons. Three metal tubes make a lens that focuses whichever way its middle tube is charged — and, by a theorem proved in 1936, no lens built from round electrodes can be free of spherical aberration. That theorem kept electron microscopes some fifty times coarser than their wavelength for sixty years.
ElectromagnetismThe water drops that charge themselves to a spark
Two streams of water drip from one earthed pipe through two metal rings into two metal cans, and each ring is wired to the opposite can. Nothing is charged to begin with, nothing is rubbed, nothing is plugged in. Within a minute a spark cracks between the cans at ten thousand volts. A stray volt between them makes the drops forming in one ring pick up a slight charge, which they carry to the can that makes the other ring's drops pick up the opposite charge — a loop in which every drop strengthens the imbalance that charged it. Kelvin built it in 1867; it is a dynamo made of water and gravity.
ElectromagnetismThe zeros a handful of charges can make
Between two equal charges there is a point where their fields cancel. With three there can be two such points, or one, or four. Every such point is a saddle, never a place where a charge could rest. In two dimensions the count is fixed by algebra: n equal line charges make exactly n − 1 zeros, the roots of a polynomial's derivative. In three dimensions nobody knows the answer. In 1873 Maxwell conjectured that n point charges can make at most (n − 1)² zeros. Three charges on a triangle make exactly four, and the conjecture has not been proved even for three charges.
ElectromagnetismTwo dipoles that agree everywhere but inside
A pair of opposite charges and a small loop of current make the same field at any distance worth measuring. Between the charges that field points one way, and through the loop it points the other. Shrink both to a point and the disagreement does not vanish: it becomes a term concentrated at the origin, and hydrogen's 21-centimetre line depends on it. The level order of the hydrogen atom says which kind of dipole a proton is.