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.
The potential a stream of charge bends
The options are the ones The current a vacuum will not carry 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 and electric field across a vacuum gap of width d with a voltage V across it, drawn in units of V and V/d. With no charge in the gap the potential is a straight line and the field uniform. With a stream of charge crossing, the charge's own field pushes the potential curve down, and the current can rise only until the field at the emitter falls to zero: past that the next particle would be pushed back. At that limit the potential goes as the four-thirds power of the distance and the field as the cube root, so the emitter sees no field at all and the far side sees four-thirds of what it saw before. Child and Langmuir found the same curve independently, in 1911 and 1913, and it fixes the current completely.
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 current a vacuum will not carry
A vacuum has no resistance, and yet a gap of empty space will carry only so much current for a given voltage, however freely its cathode offers electrons. The charge already crossing cancels the field that would pull the next one off. Gauss's law gives the result in a few lines, and it says something odd about ion engines: their thrust per square metre does not depend on what the ion is, and it is exactly the electric stress on their grid.
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 imbalance of charge that would outweigh gravity
Gravity and electricity both fall as the inverse square, so Gauss's law treats them identically: the field of any round body counts what is inside it. If every nucleon in ordinary matter carried a tiny unbalanced charge, the body's charge would follow its mass everywhere, and its electric repulsion would be a fixed fraction of its gravity at every size from a grain of sand to the universe. Below about one part in a billion billion of an electron's charge, gravity always wins; above it, repulsion always does. In 1959 two astronomers proposed that the universe expands because the true imbalance is just above that line.
ElectromagnetismThe mass that would make Gauss's law leak
Gauss's law says the flux out of any closed surface counts the charge inside, whatever the surface's size. If the photon had even a tiny mass, that would stop being true: the field of a charge would die off exponentially beyond a length ħ/mc, the flux through a large sphere would fall short of the charge, and the inside of a charged shell would no longer be field-free. There are two different ways Coulomb's law could fail, and they are caught by opposite strategies. A wrong exponent has no length in it and shows equally in any apparatus; a mass has a length, and shows as the square of the apparatus's size. That is why the tightest limit on the photon's mass comes not from a laboratory but from the magnetic field the solar wind carries past Pluto.
ElectromagnetismThe number of dimensions a planet can orbit in
Gauss's law says that a point source's field is its strength spread over the surface around it, and the size of that surface depends on how many dimensions space has. In three it grows as the square of the distance, and the field falls as the inverse square. Count differently and the field falls differently — and above three dimensions it falls so steeply that no planet could hold a stable orbit and no atom could hold its electron. The inverse-square law is not a fact about gravity or electricity. It is a fact about space, and it is one of the conditions for there being anything in space to orbit.
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.