Four paths, one answer

At its defaults it draws four paths, one answer. The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

ampere-loop 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.

The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

The magnetic scalar potential along a path circling a wire carrying 10 amps, against the angle turned through, for 2 complete circuits. Away from the wire the magnetic field has no circulation round any small loop, so it is the gradient of something — and it is, except that the something does not come back to its own value. Each circuit lowers it by exactly the current, 10 amps, and a second circuit lowers it by 10 again. The potential is perfectly good locally and has no single value globally, and the amount by which it fails to close is the current threaded. So nothing has been lost in going from a circulation to a potential: Ampère's law has been rewritten as a statement about the shape of the region the potential lives in.

The circulation of H round 5 different closed paths, with the current fixed at 10 amps in each wire. The answer for each is the current multiplied by how many times the path winds round the wire — a whole number, with a sign — and nothing about the path's length, its shape or how far from the wire it runs appears anywhere. A path that goes round twice counts twice; one that goes round and comes back the same way counts nothing at all; and one enclosing two wires counts their sum, which is zero if they oppose. That is why the scalar potential's failure to close is a topological statement rather than a geometric one: it depends on what the path encircles and not on where it went.

Along the axis of a circular loop of current: the solid angle the loop subtends, divided by 4π, and the field that is minus the gradient of the potential built from it. The potential is the current times the solid angle over 4π and nothing else — which turns finding a magnetic field into a problem in geometry, with no integral over the wire anywhere in it. The solid angle runs from zero far on one side to 4π far on the other, passing through 2π at the plane of the loop, and it is that steady climb whose slope is the field. Crossing any surface the loop spans changes the solid angle by 4π at a stroke, so the potential jumps by exactly the current — which is the same multivaluedness the straight wire has, with the place it happens now a surface anybody may choose rather than a half-plane.

Along the axis of a circular loop of current: the solid angle the loop subtends, divided by 4π, and the field that is minus the gradient of the potential built from it. The potential is the current times the solid angle over 4π and nothing else — which turns finding a magnetic field into a problem in geometry, with no integral over the wire anywhere in it. The solid angle runs from zero far on one side to 4π far on the other, passing through 2π at the plane of the loop, and it is that steady climb whose slope is the field. Crossing any surface the loop spans changes the solid angle by 4π at a stroke, so the potential jumps by exactly the current — which is the same multivaluedness the straight wire has, with the place it happens now a surface anybody may choose rather than a half-plane.

The magnetic scalar potential along a path circling a wire carrying 5 amps, against the angle turned through, for 3 complete circuits. Away from the wire the magnetic field has no circulation round any small loop, so it is the gradient of something — and it is, except that the something does not come back to its own value. Each circuit lowers it by exactly the current, 5 amps, and a second circuit lowers it by 5 again. The potential is perfectly good locally and has no single value globally, and the amount by which it fails to close is the current threaded. So nothing has been lost in going from a circulation to a potential: Ampère's law has been rewritten as a statement about the shape of the region the potential lives in.

physicscheck asserts something about ampere-loop 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.