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.

A charge of 1 nC held 20 mm above an earthed conducting plane. The lines are traced through the field of the real charge plus an equal and opposite one at the mirror position, and then cut at the plane, because below it there is metal and no field whatever. Nothing in the tracing knows about the surface: each line follows the local field direction and stops where it arrives. That every one of them arrives perpendicular — the worst departure among the 9 drawn is 2.2° away from square — is the boundary condition showing itself rather than a rule imposed on the drawing. The image charge is drawn faint because it is not there: it is a way of writing a function that happens to satisfy the equation and the boundary values, which by the uniqueness theorem makes it the field and not a model of the field.

Two curves against distance from the foot of the perpendicular, both in units of the charge's height, for 1 nC at 20 mm. The first is the induced surface charge density σ = −qd divided by 2π(r² + d²) to the three-halves power, at its largest directly underneath (3.979·10⁻⁷ C/m² there) and falling as the inverse cube far away. The second is how much charge lies inside a circle of that radius, which is the first integrated over the surface: it passes half the total at r = √3 d = 34.6 mm and tends to exactly −q, reaching 99.9999% of it by the edge of the arithmetic. That is the sense in which the image charge is real. It is not a charge at a point below the plane; it is this, spread over the surface, and it adds up to the same.

For 1 nC above an earthed plane, on logarithmic axes: the attraction, which is exactly the Coulomb force between the charge and its image a distance 2d apart and so goes as 1/d² — 5.617·10⁻⁶ N at 20 mm. Below it are two energies. The upper one is what the two-charge picture would give if the image were a real charge; the lower is the work actually done bringing the charge in from far away, which is the integral of the force and comes to exactly half as much, a factor of 2. The difference is the whole content of the image being a fiction: a real partner would stay put, and the induced charge moves as the charge approaches, so half the work goes into rearranging it.

Two curves against distance from the foot of the perpendicular, both in units of the charge's height, for 2.5 nC at 8 mm. The first is the induced surface charge density σ = −qd divided by 2π(r² + d²) to the three-halves power, at its largest directly underneath (6.217·10⁻⁶ C/m² there) and falling as the inverse cube far away. The second is how much charge lies inside a circle of that radius, which is the first integrated over the surface: it passes half the total at r = √3 d = 13.9 mm and tends to exactly −q, reaching 99.9999% of it by the edge of the arithmetic. That is the sense in which the image charge is real. It is not a charge at a point below the plane; it is this, spread over the surface, and it adds up to the same.

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.

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.