Illustrated Physics
physics, drawn
Concept

Ampere law — where it appears

The statement that the magnetic field's circulation round any closed path counts the current threaded through it and nothing else about the path. In matter there are two versions, differing in whether the atomic currents bound inside the material are allowed to be counted.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

How much of the answer a finite wire gives back. The field beside a straight segment of wire, divided by what the infinite-wire formula would give, against the length of the segment in units of the distance to the field point, on a logarithmic horizontal axis. The field is integrated element by element along the segment rather than evaluated from a closed form. A wire ten times as long as the distance already gives 92.8 per cent of the infinite answer, and one as long as the distance gives 45 — which is the practical content, and the reason the infinite-wire result is used for laboratory wires without apology. The second column is the part that matters for the law rather than for the number. The circulation of this field round a circle of radius d is not μ₀I; it falls short by exactly the fraction the segment fails to subtend. What makes up the difference is the displacement current of the charge piling up at the segment's two ends, and the two terms, both integrated here, sum to μ₀I to within 2.2e-8 per cent at every length. So Ampère's law is not approximately true for an open circuit and exactly true for a closed one — it is exactly true always, and it is a computation only when symmetry supplies the direction and the magnitude along the loop. Symmetry is doing the work; the law is doing the bookkeeping.

The law that is always true and rarely useful

Ampère's law holds for every loop and every current. Apply it to a wire of finite length and it gives an answer that is wrong by half — until the displacement current of the charge piling up at the wire's two ends is put back in, whereupon the two terms sum to μ₀I exactly at every length. The law was never approximate. It was never a computation either.

electromagnetism · Ampere law Two fields of the same magnet, and inside they point opposite ways. A uniformly magnetised sphere, with the field lines of B on the left and of H on the right, both computed from the exact solution — uniform inside, a dipole outside. Outside the sphere the two pictures are identical up to a constant, because there B is μ₀ times H and nothing else. Inside they are opposite: B is 0.67 tesla pointing along the magnetisation and H is 267 kiloamps a metre pointing against it. The B lines close on themselves and never end; the H lines begin on the top face and end on the bottom, which is what a field with sources looks like. Nothing about the magnet changed between the two panels — only which currents the circulation is allowed to count.

The field that points against the magnet it is in

There are two magnetic fields in use and the difference between them is which currents a loop is allowed to count. The consequence nobody expects on being told the definitions: inside a permanent magnet H points the other way from B. It has to — a loop inside the magnet threads no wire, so its H circulation is zero, and the only arrangement left has H running backwards.

electromagnetism · Ampere law A potential that is lower every time round. 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.

A potential that does not come back to itself

Where no current flows, the magnetic field has no circulation round any small loop, so it is the gradient of something and a magnetic problem becomes an electrostatic one. The catch is not that the potential fails to exist. It is that walking once round a wire lowers it by the current, and walking round again lowers it by the current again — so Ampère's law survives the translation as a statement about what the path encircles rather than about where it went.

electromagnetism · Ampere law Two pieces of current that one law makes push each other unequally. Two short elements of current a unit distance apart: element 1 pointing along the line joining them, element 2 at 45° to it. Left, the forces by the law that follows from Biot and Savart (Grassmann's): element 1 pushes nothing on element 2, because 1 points straight at it, while element 2 pushes on element 1 with a force of 0.71 units at right angles to it; the pair would feel a net 0.71 units. Right, the forces by Ampère's own law of 1826: 0.71 units each, along the line joining the elements, equal and opposite. Both laws give the same force between any two complete circuits, so no experiment on closed circuits can say which is right.

The force between pieces of a current

Ampère spent six years finding the force one short piece of current exerts on another, and the law he found is not the one used today. The modern law, which follows from the Biot–Savart field, disagrees with his for almost every pair of pieces — his obeys Newton's third law and the modern one does not. Both give exactly the same force between any two complete circuits, which is the only thing that has ever been measured. The force between two pieces of current is a quantity with two correct values, because a piece of current on its own does not exist.

electromagnetism · Ampere law A current limited by the wire's circumference. The largest supercurrent a wire can carry before the field of its own current destroys superconductivity at its surface, Ic = 2πaHc, against the wire's diameter, for lead, mercury and tin at 1.5 K, on logarithmic axes; dashed, a niobium–titanium wire carrying 3,000 A/mm² in a field of 5 tesla, which grows with the cross-section rather than the circumference. A lead wire 1 mm across carries 192 A — 244 A/mm² — and one 1 cm across only ten times as much, at a tenth of the density. Below 81 micrometres lead carries more than niobium–titanium; above it niobium–titanium wins by a factor that grows with the wire, and it does so in a field of 5 tesla, where lead cannot superconduct at all.

The current a superconductor stops with its own field

A superconducting wire has no resistance, so it seems it should carry any current at all. It carries a definite amount and no more, and the amount is set not by the metal's ability to conduct but by the field the current itself makes: when that field at the wire's surface reaches the critical field, superconductivity fails there. The rule makes the largest current proportional to the wire's radius rather than its area, makes a thick wire worse per square millimetre than a thin one, and explains why the first superconducting magnet, wound in 1913, did not work.

electromagnetism · Superconductivity

Named alongside it

The objects these essays reach for when they reach for this one.

Magnetic fieldBoundary conditionsCirculationAmperes force lawBiot–SavartBiot savart lawBound chargeConstitutive lawCritical currentCritical fieldCurrent elementDemagnetising field

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