The law that is always true and rarely useful

The field wraps a current, and the standard demonstration of it is Ampère’s law: the circulation of $\mathbf{B}$ round any closed loop is $\mu_0$ times the current through it. Two lines of algebra then give the field of a straight wire, and the result is one of the first things anybody computes in the subject.

Almost every use of it is a use of the symmetry, and the law is doing the bookkeeping rather than the work. The way to see that is to take the symmetry away.

What the law actually says #

Ampère’s law with Maxwell’s correction is

$$\oint \mathbf{B}\cdot d\boldsymbol{\ell} = \mu_0\left(I_{\text{enc}} + \varepsilon_0\frac{d\Phi_E}{dt}\right)$$

and it is one equation for one number. It says what the sum of $B_\parallel$ round a loop is; it says nothing about how that sum is distributed round the loop, and it says nothing about $B_\perp$.

To get a field out of it, three things have to be supplied from outside:

The direction of the field, which must be known to be along the loop everywhere. The magnitude’s constancy along the loop, so that the integral is $B$ times the length. And a loop that can be chosen with those two properties.

All three come from symmetry. An infinite straight wire is invariant under translation along itself and rotation about itself, which forces the field to be azimuthal and to depend only on the radius, and a circle then works. A solenoid, a toroid and an infinite sheet each have a symmetry that supplies the same three things.

Take a wire of finite length and none of it survives. There is no translational symmetry, the field has a component along the wire’s direction nowhere but is not constant on any circle, and no loop of any shape makes the integral tractable.

What a finite wire gives instead #

The computation that does work is Biot and Savart’s: add up the contribution of each element of the current, which for a straight segment of length $L$ at perpendicular distance $d$ from its midpoint gives

$$B = \frac{\mu_0 I}{2\pi d}\cdot\frac{L/2}{\sqrt{(L/2)^2 + d^2}}$$

the second factor being the fraction of the infinite answer. It is 0.24 for a wire half as long as the distance, 0.45 at equal, 0.71 at twice, 0.93 at five times and 0.995 at twenty.

That is the practical content and it is reassuring: the infinite-wire formula is good to a few per cent for any laboratory wire more than a few times longer than the distance being measured at, which is nearly always. The failure is quick, though — half of the field is gone by the time the wire is as long as the distance — so a short lead near a sensitive probe is a real error rather than a theoretical one.

The field of a long straight wire is azimuthal and falls as the inverse of the distance, and the symmetry visible in that picture is what makes Ampère’s law a computation rather than a constraint. A finite wire has no such symmetry: its field still satisfies the law exactly, and there is no path on which the field is constant, so the circulation is known and the field is not.

Where the missing half goes #

The interesting failure is not the field but the circulation, because a finite segment carrying a steady current is not a legal steady-state problem at all.

Charge is arriving at one end and leaving the other, so charge is piling up. The current is not divergence-free, so “the current enclosed by a loop” is ambiguous: two surfaces bounded by the same loop can have different currents through them, which is exactly the contradiction that forced the displacement current into existence.

Compute the circulation of the finite segment’s own field round a circle of radius $d$ in its midplane. It comes out $\mu_0 I \cos\alpha$, where $\cos\alpha = (L/2)/\sqrt{(L/2)^2 + d^2}$ is the same factor as before — so for a short segment the circulation is a small fraction of $\mu_0 I$ rather than all of it.

The deficit is supplied by the electric flux. Each end of the segment is accumulating charge at rate $I$, and the flux of a point charge $q$ at distance $L/2$ on the axis through a disc of radius $d$ is $(q/2\varepsilon_0)(1-\cos\alpha)$. Two ends, differentiated, give a displacement term $\mu_0 I (1 - \cos\alpha)$.

The two sum to $\mu_0 I$. Exactly, at every length, and the figure integrates both numerically and finds them summing to one to better than a ten-thousandth of a per cent at all five ratios drawn.

So the law is not approximately true for an open circuit and exactly true for a closed one. It is exactly true always, and the two terms trade off against each other as the geometry changes. A closed circuit puts everything in the conduction term; an isolated segment puts most of it in the displacement term; a capacitor being charged puts all of it in the displacement term at a surface between the plates, which is the standard demonstration.

An electromotive force has two contributions, one from the field changing and one from the circuit moving, and the bookkeeping is the same kind as here: a law that is exactly true and that has to be split before it computes anything. The pattern recurs throughout electromagnetism — an integral statement is unconditionally correct and yields a number only when a symmetry or a decomposition is supplied from outside it.

Which loop, and the surface it bounds #

The same arithmetic settles the classic puzzle about which surface to use.

A loop bounds infinitely many surfaces. For the capacitor, a flat one between the plates has no conduction current through it and full displacement current; one bulging round to cut the lead has full conduction current and none of the displacement. The two agree because charge is conserved, which is the same statement as the divergence of the total current being zero — and it is why Maxwell’s correction had to be exactly the term it is rather than something similar.

The general rule is worth stating, because it turns a puzzle into a routine: the total current, conduction plus displacement, is divergence-free, so its flux through any surface bounded by a given loop is the same. Once that is known there is nothing left to be surprised by, and the choice of surface becomes purely a matter of which one is easier to integrate over.

The three cases where an integral law is a computation #

It is worth stating the criterion positively, because it is short and it decides the matter in every case.

An integral law computes a field when a loop or surface can be found on which the field’s direction is known and its magnitude is constant. Both conditions come from a symmetry of the source that the loop or surface respects.

There are exactly three symmetries in three dimensions that deliver this: planar (invariance under translation in two directions), cylindrical (translation along an axis plus rotation about it) and spherical (rotation about a point). Every textbook application of Gauss’s law or Ampère’s law is one of those three, and there are no others, because there are no other continuous symmetry groups that leave a point’s neighbourhood with enough structure.

That is why the standard problems are a short list — the infinite sheet, the infinite line, the infinite cylinder, the sphere, the solenoid, the toroid — and why the list has not grown since the nineteenth century. It is not that nobody has been clever enough; it is that the geometry is exhausted.

Everything outside the list is done by superposition or by solving a differential equation with boundary conditions, and the integral law is then used as a check rather than a method — which is a perfectly good use for it, and one this collection relies on repeatedly.

The counting argument in its electrostatic form works on a surface chosen so that the field is radial and constant everywhere on it. Take that choice away and the law is still true and no longer useful, which is exactly the situation with Ampère’s — so the two laws share a limitation as well as a form, and in both cases the limitation is that the integral averages away the information one wants.

Where the symmetry genuinely does the work #

Three standard cases, and it is worth being explicit about which symmetry each one leans on.

The solenoid. Translational invariance along the axis and rotational invariance about it force the field to be axial and independent of position along the winding, so a rectangular loop with one side inside and one far outside gives $B = \mu_0 n I$. The argument is exact for an infinite winding and, as the second rung of this ladder shows, the “exactly zero” outside becomes a bar magnet’s dipole field the moment the winding is finite.

The toroid. Rotational symmetry about the axis, with no ends at all, gives $B = \mu_0 N I/2\pi r$ inside and zero outside — and here the zero really is zero, because the toroid closes on itself and there is no end for flux to leave from. That is why a toroidal inductor radiates far less than a solenoidal one of the same inductance.

The coaxial cable. Cylindrical symmetry with the return current inside the same structure, so beyond the outer conductor the enclosed current is zero and the external field vanishes exactly. The whole point of coaxial construction is to arrange a symmetry that makes Ampère’s law give zero outside.

Each of these is a case where somebody arranged the geometry so the law would be usable, and that is the honest description of how the law is used: not as a tool for finding fields but as a constraint that a symmetric arrangement makes into one.

Summing the contribution of every turn gives a solenoid’s axial field without any symmetry argument, and near the middle of a long winding the two agree. That is where the symmetry genuinely does the work: the infinite solenoid has a translational symmetry that a real one approximates, and the approximation is good in the middle and bad at the ends — which is exactly where the symmetry argument stops applying.

The law as an instrument #

There is one use of Ampère’s law in which giving a single number about a loop is not a limitation but the entire point, and it is worth a section because it inverts the essay’s complaint.

Wind a long flexible coil of uniform turn density, bring its two ends together round a conductor, and measure the voltage it produces. The coil is a discrete approximation to $\oint \mathbf{B}\cdot d\boldsymbol{\ell}$ — each turn samples the component of $\mathbf{B}$ along the path at its own position, and the winding sums them. The output is therefore proportional to $\mu_0 \,\mathrm{d}I/\mathrm{d}t$, and integrating it gives the current.

That is a Rogowski coil, and its virtues are exactly the law’s properties read as specifications. The reading does not depend on where inside the loop the conductor sits, because the law refers only to the enclosed current. It does not depend on the conductor’s shape or on how many conductors there are — it returns their sum. And a current outside the loop contributes nothing, however large and however close, so a coil round one busbar is immune to the one beside it.

Two details of the construction are the law being taken seriously. The winding is returned along its own axis so that the whole assembly encloses no net flux as a single turn, which would otherwise add a spurious response to any field threading the loop. And the coil has no iron in it, so nothing saturates — which is why the same instrument measures a hundred amperes of mains and a hundred kiloamperes of lightning with the same calibration.

The clamp meter in an electrician’s bag is the iron-cored version of the same idea, and it works for the same reason. Neither instrument computes a field anywhere.

The number the loop returns is a linking number #

What the enclosed current means deserves saying precisely, because it is a topological quantity rather than a geometric one and the difference has consequences an instrument can feel.

The right-hand side counts each circuit according to how many times the loop links it. Wind the measuring path twice about a wire and it reads twice the current. Take a path that runs alongside the wire, at any distance and for any length, and returns without going round it, and it reads exactly zero — not approximately zero, and not small because the field is weak. The field along such a path is substantial and its circulation cancels.

So the quantity is insensitive to everything except a whole number. Deform the loop however violently, stretch it, knot it, drag it across the room; as long as it does not cut through a conductor, the reading does not change at all. It changes discontinuously, by exactly one unit of $\mu_0 I$, at the moment the loop is pulled across the wire.

That is what a clamp meter’s jaws are for, and why a reading taken with them not quite shut is not merely degraded. An open clamp is a path that does not close, and the law makes no statement whatever about an unclosed path — there is no partial linking, and the number the instrument returns in that condition is not a fraction of the current but a measurement of nothing in particular.

The general lesson #

Ampère’s law belongs to a family — Gauss’s law for electricity, Gauss’s law for magnetism, the circulation theorems — all of which share the same character: an integral statement that is exactly true and computes a field only when a symmetry supplies the missing information.

Counting what comes out of a surface gives the field of a point charge because the field must be radial and constant on a sphere; it gives nothing at all for two charges. The pattern is identical, and recognising it saves a great deal of time, because the question of whether the integral form applies reduces to whether there is a surface or loop on which the integrand is constant and that question is usually answerable at a glance.

When the answer is no, the route is superposition: add up the contributions of the pieces, which is what Biot and Savart’s law and Coulomb’s law are for, and which is what the hero figure does twenty thousand elements at a time.

The history, which reads the other way round #

The order in which these ideas arrived is worth knowing, because it explains why the law bears Ampère’s name while looking nothing like what he did.

Ampère’s own work, from 1820 onward, was a force law between current elements — an inverse-square expression giving the force one small piece of current exerts on another, obtained from a series of null experiments of great ingenuity. He never wrote the circulation law, and the field concept he would have needed for it did not yet exist.

The integral law is Maxwell’s reformulation, and Maxwell’s own reason for it was that it made the theory local: a statement about what the field does in the neighbourhood of a point rather than about action at a distance between distant elements. That reformulation is the whole reason the field survived as a concept, and the difference between a force law and a field law is exactly the difference between the two descriptions of this same phenomenon.

The correction that makes the law universally true is Maxwell’s too, from 1861, and it was introduced on precisely the grounds this essay uses: that without it the law is inconsistent for a current that is not closed. He had a mechanical model behind it which nobody now believes; the term is right anyway, which is a recurring and slightly uncomfortable pattern in the history of the subject.

The irony worth recording is that the law now used to compute the field of a wire is not the one Ampère wrote, and the law Ampère wrote — the force between elements — is not used at all, because the force between two isolated current elements is not a measurable thing. Only closed circuits exist, and for closed circuits several inequivalent element-level force laws give identical answers.

What the picture cannot show #

A steady current in an isolated segment is not physical. Charge really would pile up, the potentials would rise without limit, and the current would stop. What the calculation describes is an instant of a real situation — one leg of a circuit, or the moment during charging of a capacitor — and treating it as a standing state is a decomposition rather than a scenario.

The circulation is computed in the segment’s midplane. For a loop elsewhere the geometry is worse and the split between conduction and displacement is different, though the sum is the same. The midplane is chosen because the field there is azimuthal by symmetry and the integral is doable.

Superposition is being used and is not free. Adding the fields of current elements assumes the elements do not affect one another, which is exact in vacuum and is exactly what fails inside magnetic material — where the medium’s response to the field of one element changes what the next element produces.

And the whole calculation is magnetostatic in a situation that is not static. Charge accumulating at the ends produces a growing electric field, that field’s own changes produce further magnetic field, and the honest treatment is the full retarded solution. The corrections are small when the geometry is small compared with a wavelength, which for a laboratory wire at any ordinary frequency it is.

The ladder from here #

Later rungs on this anchor: the vector potential as the general solution, where the symmetry requirement disappears and is replaced by an integral anybody can do; the magnetic scalar potential and the regions where it is single-valued; Ampère’s law in matter, with $\mathbf{H}$ and the bound currents, and the reason the two versions of the law are so often confused; the Biot–Savart law derived from the retarded potentials rather than postulated; and the surface-current formulation, where a problem is solved by finding the currents that make the boundary conditions hold.

The neighbouring ladders are the field wrapping a current, which is where the law is set up and its symmetric cases worked, and the displacement current, without which the arithmetic in this essay would not close. The solenoid’s exterior is the same lesson about finite length applied to the other symmetric case.