The same law, three shapes of source
At its defaults it draws the same law, three shapes of source. Field strength against distance on logarithmic axes, for a point, a long line and a wide plane carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law.
falloff-compare 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.
Field strength against distance on logarithmic axes, for a point, a long line and a wide plane carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law.
Why the field falls off as the square
The options are the ones Counting what comes out, and never looking inside 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 same number of field lines crossing shells at one, two and three times the distance. The shell's area grows as the square of the radius, so the lines per unit area falls as its inverse.
Why the field falls off as the square
The options are the ones Counting what comes out, and never looking inside 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 same number of field lines crossing shells at one, two and three times the distance. The shell's area grows as the square of the radius, so the lines per unit area falls as its inverse.
Why the field falls off as the square
The options are the ones How a wave thins out 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 same number of field lines crossing shells at one, two and three times the distance. The shell's area grows as the square of the radius, so the lines per unit area falls as its inverse.
The same law, four shapes of source
The options are the ones The attraction that needs no 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.
Field strength against distance on logarithmic axes, for a point, a long line, a wide plane and a dipole carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law. The steeper cases are not other laws. A dipole is two opposed charges whose fields nearly cancel, which costs one power of the distance.
Three bodies of one mass, one field outside, three fields inside
The options are the ones The field outside that cannot find the core 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 gravitational field against distance from the centre, both in units of the body's surface values, for 3 spherical bodies of the same mass and radius: a uniform ball; a dense core under a light mantle; a hollow shell. Outside the surface the three curves are one curve — checked at 1.7 radii by adding up the pull of every mass element in each body, which agrees with the pull of a point of the same mass to better than a part in five hundred. Inside they part: the uniform ball's field falls in a straight line, the layered body's rises to 1.24 times the surface value at the top of its core, and the hollow shell's is zero throughout its cavity. Their moment-of-inertia factors are 0.400 (uniform ball), 0.330 (dense core under a light mantle), 0.551 (hollow shell) — a number that no measurement of the field outside can supply.
What checks it
physicscheck asserts something about falloff-compare 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.
Counting what comes out, and never looking inside
Draw any closed surface. The field crossing it depends only on the charge enclosed — not on where that charge sits, not on its shape, not on anything outside.
WavesHow a wave thins out
A wave gets weaker with distance for two quite different reasons, and only one of them is a loss. Geometry alone fixes the first exactly — three exponents for three dimensions, with nothing about the medium in them — and whatever is left over is the medium eating the wave.
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 outside that cannot find the core
Gauss's law gives the field outside a body from what it encloses, and read backwards it is a limit. A uniform ball, a planet with an iron core and a hollow shell of the same mass have identical gravity everywhere outside. The field fixes a list of numbers — the mass, the flattening, higher moments — and leaves free everything else, including the moment of inertia; the Earth's core was weighed by its wobble, not by its pull.
ElectromagnetismThe pull that grows on the way down
Inside a uniform ball, gravity falls in a straight line from the surface to nothing at the centre, and that is the answer usually given for the Earth. It is wrong for almost three thousand kilometres. Going down through the mantle, gravity rises, reaching nearly nine per cent above its surface value where the core begins — because Gauss's law counts only the mass inside, and the local form of the law says gravity grows inward wherever the rock is lighter than two thirds of the average beneath it.
ElectromagnetismThe shape decides the falloff, and the force law never changes
A point charge gives an inverse square, a line gives an inverse, a plane gives a constant. All three come from the same law, and the exponent belongs to the geometry of the source rather than to the physics.