A conductor in a field, with the surface charge solved for

At its defaults it draws a conductor in a field, with the surface charge solved for. Field lines approaching an isolated conducting cylinder. The surface charge was found by requiring the conductor to be an equipotential, and the lines then end on that charge, meeting the surface at right angles and leaving the interior empty.

conductor-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.

Field lines approaching an isolated conducting cylinder. The surface charge was found by requiring the conductor to be an equipotential, and the lines then end on that charge, meeting the surface at right angles and leaving the interior empty.

The surface charge on three conducting spheroids of the same length, a sphere and spheroids 3 and 10 times as long as they are wide, drawn as a band whose thickness is proportional to the charge per unit area at each point. On the sphere it is uniform. On an ellipsoid the density is exactly proportional to the fourth root of the surface's Gaussian curvature, and so it is 3 and 10 times as dense at the tips as at the waist — not the 9 and 100 times by which the tips' curvature exceeds the waist's, and not the ratio of their mean curvatures either. The charge does crowd to the sharp ends, but by how much is set by the whole shape, not by any property of the surface at the point.

The surface charge density along a conducting spheroid 5 times as long as it is wide, from the waist to the tip, relative to its value at the waist, beside three measures of how sharply the surface curves there, each relative to the waist: the Gaussian curvature's fourth root, the mean curvature, and the Gaussian curvature's square root. The charge follows the fourth root of K exactly, the two curves lying on one another, and reaches 5.0 times the waist value at the tip; the mean curvature rises 9.6 times, and the square root of K 25 times, off the top of the plot. The fourth root is a fact about ellipsoids, derived from their potential, and not a law: two spheres joined by a wire share charge as the square root of curvature instead.

How many times denser the charge is at a conductor's sharpest point than at its bluntest, against how many times more curved the sharp point is (Gaussian curvature), on logarithmic axes, for two families: spheroids from a sphere to a needle 100 times as long as it is wide, and pairs of spheres of radius ratio 1 to 100 joined by a long wire. The spheroids lie on a line of slope exactly 1/4; the joined spheres on a line of slope 1/2, because each sphere's charge is proportional to its radius and is spread over an area proportional to the radius squared. At the same contrast in curvature, ten thousand, the spheroid's tip carries ten times the waist's density and the small sphere a hundred times the large one's. A rule that gave σ from the local curvature would put both families on one line.

The electric field at the tip of a grounded conducting spheroid standing along a uniform field, as a multiple of that field, against how many times longer the spheroid is than it is wide, on logarithmic axes; with the slender-body approximation (a/b)²/(ln(2a/b) − 1) dashed. A sphere triples the field at its poles. A spheroid ten times as long as it is wide multiplies it 49 times, and one 100 times as long 2326 times — so under a storm cloud's field of a few tens of kilovolts a metre, a slender conductor's tip reaches the three megavolts a metre at which air breaks down long before anything blunt does. The enhancement grows nearly as the square of the slenderness, but it depends on the length as well as the tip: a needle's tip is sharp in proportion to its whole length, not to its point alone.

The charge per unit area on the face of a conducting oblate spheroid, against distance from the centre in units of its radius, relative to its value at the centre, for thicknesses of a half, a tenth and a hundredth of the radius, flattening towards a disc; the disc's own distribution, the inverse square root of one minus the square of the distance from the centre, dashed. At nine-tenths of the way to the edge the density is 1.60, 2.25 and 2.29 times its central value; for the disc it is 2.29, and it diverges at the rim as one over the square root of the distance from it. Half the charge on a disc lies in the outer 13% of its radius, a quarter of its area. The rim is where any edge in a real capacitor puts its charge, and why every high-voltage electrode has its edges rolled.

physicscheck asserts something about conductor-field 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.