A potential that does not come back to itself

The two magnetic fields and which currents each counts establishes the field whose circulation counts only free current. Take that seriously in a region where there is no free current at all, and the circulation of $\mathbf{H}$ round every closed path in that region is zero.

A field whose circulation vanishes round every path is the gradient of a scalar — the same argument that replaced three numbers per point by one for the electric field, and lost nothing in doing it. So in any current-free region

$$\mathbf{H} = -\nabla\psi,$$

and since $\mathbf{B}$ has no divergence, $\psi$ satisfies Laplace’s equation wherever the material is uniform. The whole apparatus of electrostatics — one number for every point, uniqueness given boundary values, separation of variables, every numerical method written for Laplace — becomes available for magnetic problems.

That is an enormous economy and it is the reason magnetostatic solvers use it. One unknown per point rather than three.

And it does not quite work.

The obstruction is global, and that is the whole difference #

It is worth being precise about what fails, because the natural assumption is wrong.

A field is the gradient of something locally whenever its circulation round every small loop vanishes — that is a statement about derivatives at a point, and $\mathbf{H}$ satisfies it everywhere off the wire. So a potential exists in any small ball. Nothing is wrong with the field.

What can fail is patching those local potentials together into one function on the whole region, and whether it fails depends on the region’s shape. In a region with no holes in it, the local potentials always agree and a single-valued potential exists. In a region that has a hole — and the complement of a wire is exactly a region with a hole through it — they need not.

The wire’s exterior has a hole. Following the potential round a circuit, each local patch agrees with its neighbour, and after a complete circuit the value has drifted. It drifts by

$$\Delta\psi = -\oint \nabla\psi\cdot d\boldsymbol{\ell} = \oint \mathbf{H}\cdot d\boldsymbol{\ell} = I.$$

So the amount by which the potential fails to close is the current threaded, exactly. Ampère’s law has not been lost in going to a potential; it has been converted into a statement about the region’s topology.

That is a better trade than it sounds. A circulation is something to evaluate; a winding number is something to count.

The potential of a real circuit is a solid angle #

For a closed loop of current — which is what any real circuit is — the potential has a form that is almost embarrassingly simple:

$$\psi = -\frac{I\Omega}{4\pi},$$

where $\Omega$ is the solid angle the loop subtends at the point in question.

That is the whole answer. To find the magnetic field of an arbitrary circuit at an arbitrary point, work out what fraction of the sky the circuit takes up as seen from there, multiply by the current, and take the gradient. There is no integral along the wire, no cross product, and no Biot–Savart law anywhere in it.

The check in that figure matters. Taking minus the gradient of the solid-angle potential on the axis of a circular loop must give $I a^2/2(z^2+a^2)^{3/2}$, which is what the Biot–Savart integral gives, and it does. The two constructions are the same field.

The multivaluedness reappears in its most concrete form. Walk from far above the loop, down through the middle of it, and away below. The solid angle rises steadily from zero, and at the instant of passing through the loop’s own disc it jumps by $4\pi$ — because the loop has gone from filling part of the sky to filling the complementary part. So the potential jumps by exactly $I$, and the place it jumps is a surface spanning the loop.

Which surface is entirely up to whoever is doing the calculation. A flat disc, a hemisphere, a bag of any shape with the loop as its rim: all give the same field, and all put the jump somewhere different. The physics is on the rim; the surface is a choice.

Where the solid angle comes from #

The solid-angle result looks like a coincidence and is not, and the derivation is short enough to give.

Take any surface with the current loop as its rim, and cover it with magnetic dipoles pointing along its normal, at a density chosen so that the dipole moment per unit area is $I$. Such a sheet is called a double layer, and its magnetic field is identical to the loop’s.

The reason is one line of vector calculus: a small element of the sheet, of area $dA$, is a dipole of moment $I\,dA$, which is the same thing as a tiny current loop of current $I$ round its edge. Lay those tiny loops edge to edge over the surface and every internal edge is traversed twice in opposite directions and cancels. What is left circulating is the rim, which is the original loop.

So the loop and the sheet produce the same field, for the same reason a magnetised block is equivalent to a surface current — which is exactly the substitution the bound-current substitution makes, run backwards.

Then the potential of a dipole is its moment times the cosine of the angle over the distance squared — the standard dipole result, which is the first term in the tower of falloffs after the monopole has vanished — and integrating that over the sheet gives

$$\psi = -\frac{I}{4\pi}\int \frac{\hat{n}\cdot\hat{r}}{r^2}\,dA.$$

The integrand is the definition of an element of solid angle. So the integral is the solid angle the surface subtends, which for a closed rim is the solid angle the rim subtends, and the result follows.

It also explains the jump, and explains why the jump is where it is. Crossing a double layer changes the potential by the layer’s strength, which here is the current — and the layer sits on whichever surface was chosen. Different surfaces are different sheets producing the same field, which is the arbitrariness already noted, now with a mechanism attached.

The pole picture, rebuilt as arithmetic #

The scalar potential also gives back the nineteenth-century picture of magnets as things with poles, in a form that is a calculation rather than a metaphor.

Inside magnetised matter with no free current, $\mathbf{H} = -\nabla\psi$ still holds, and the divergence of $\mathbf{H}$ is $-\nabla\cdot\mathbf{M}$, since the divergence of $\mathbf{B}$ is zero. So $\psi$ satisfies Poisson’s equation with a source density

$$\rho_m = -\nabla\cdot\mathbf{M},$$

plus a surface density $\sigma_m = \mathbf{M}\cdot\hat{n}$ wherever the magnetisation stops abruptly. Those are the magnetic charges, they are computed rather than postulated, and every method for counting what comes out of a closed surface applies to them unchanged.

Take the uniformly magnetised sphere. Its magnetisation is constant, so its divergence is zero and there is no volume charge at all; and its surface charge is $M\cos\theta$, the component of $\mathbf{M}$ along the outward normal. A sphere carrying a surface density proportional to the cosine of the polar angle is one of the handful of exactly solvable problems in electrostatics, and its interior field is uniform and equal to minus a third of the surface density.

So $\mathbf{H}$ inside is $-M/3$, pointing against the magnetisation — which is the number the previous essay draws, arrived at here by solving an electrostatics problem with no magnetism in it.

That is the payoff of the translation. A magnetostatics problem with no free current in it is an electrostatics problem, the poles are where the magnetisation has a divergence, and nothing about the field has been approximated on the way.

One consequence comes free and is worth stating. Laplace’s equation has no interior maximum or minimum, so a magnetic scalar potential cannot have one either — and a static field can hold nothing still, for magnets exactly as for charges. Every stable magnetic levitation there is works by breaking one of the assumptions: by using a diamagnet, whose response reverses the sign; by spinning; by superconducting; or by feedback.

Why anybody would use it #

The economy is real and it is the reason this is not a curiosity.

A three-dimensional magnetostatic problem solved for the vector potential carries three unknowns at every point, plus a gauge condition to fix the arbitrariness — since a vector potential is not unique and something has to decide which of the many is computed. The same problem solved for a scalar potential carries one unknown per point and no gauge condition at all.

A factor of three in unknowns is a factor of nine or twenty-seven in the cost of solving the resulting linear system, depending on the method. For a magnet design of a few million nodes that is the difference between a calculation somebody runs and one they do not.

The standard arrangement is therefore to split the problem. Regions containing current — the windings — are solved with a vector potential or with a direct Biot–Savart integral, since the coil’s own geometry is known and simple. Regions containing no current — the iron, the air gap, the space around the machine — are solved with a scalar potential. The two are matched at the interfaces.

The trouble the split creates is the multivaluedness, and it is handled exactly as the figures suggest: by choosing surfaces that span the current-carrying loops, declaring them cuts, and requiring the potential to jump by the enclosed current across each one. Finding those surfaces automatically for a complicated winding is a genuinely difficult problem in computational topology and is a real cost of the method.

The planet’s field is stored as one of these #

The largest routine use of the magnetic scalar potential is not in a solver at all. It is how the Earth’s magnetic field is written down.

At the surface of the Earth and for some distance above it there is very little electric current — the atmosphere is a poor conductor below about eighty kilometres — so in that shell the field is curl-free and a scalar potential exists. Expanding it in spherical harmonics turns the whole geomagnetic field into a list of coefficients, and that list is the standard model: a few hundred numbers, revised every five years, from which the field at any point of the Earth’s surface is evaluated.

What makes the expansion more than a convenient parameterisation is that it separates two things no measurement can separate directly. A harmonic of degree $n$ from a source inside the shell falls as $r^{-(n+1)}$; the same harmonic from a source outside it grows as $r^{n}$. Measure the field on a surface and the two contribute together and cannot be told apart. Measure it on a surface and also know how it varies with height — or measure on two surfaces — and the separation is exact.

So a single scalar potential, fitted to surface and satellite data, splits the planet’s field into the part generated by the core beneath and the part generated by currents in the ionosphere and magnetosphere above. The core’s contribution is about 98 per cent of it, the external part is what varies on the timescale of a day and of a solar storm, and the two are distinguished by a power of $r$ and by nothing else.

The same argument was Gauss’s, in 1839, and it was the first proof that the Earth’s field originates inside the Earth. The instrument was a network of observatories; the method was that the exponents differ; and the conclusion was not available by any amount of measuring in one place.

The potential earns its keep there for a reason worth generalising. A field written as a gradient of one function has its structure exposed as a list of coefficients, and the coefficients are what carry the physics — where the sources are, how deep, how fast they change. The vector field itself is three numbers per point and says none of that until it is analysed.

Only where no current runs, and only in a uniform medium #

The potential exists only where there is no current. Inside a wire, inside a plasma, inside a conductor carrying eddy currents, the circulation round a small loop is not zero and no scalar potential is available. Everything here applies to the empty space and the current-free iron around a circuit, and stops abruptly at the copper.

Laplace’s equation requires a uniform medium. Where the permeability varies the equation for $\psi$ picks up a term in the gradient of $\mu$, and where the material is nonlinear — which any working iron is — the equation is nonlinear too. What remains true is the existence of the potential, which depends only on the absence of free current.

And the “total” scalar potential is numerically delicate. In a region where the field of the coil and the field of the magnetised iron nearly cancel — which is the normal situation inside a good core — the potential is a small difference of two large numbers, and a straightforward solver loses most of its significant figures. The standard repair splits the potential into a part computed directly from the coils and a reduced part solved for, which keeps the cancellation out of the numerics. That repair is older than most of the solvers using it and is still where the method goes wrong when it does.

The cut is a surface and these are slices through it #

They cannot show the cut. A surface spanning the wire, across which the potential jumps, is a two-dimensional object in three dimensions and the figures here are one-dimensional slices through the problem — an angle, a distance along an axis. The sawtooth’s vertical drop is where a real calculation would have a surface, and the surface’s position is arbitrary while the drop across it is not.

Nor can they show that the choice of cut is invisible in the answer. Two engineers choosing two different spanning surfaces compute different potentials everywhere near their respective cuts and identical fields at every point — which is the same kind of statement as a gauge choice and has the same resolution: the potential is a device, the field is what is measured, and the arbitrariness cancels before anything observable is reached.

And they cannot show why the winding number is the only thing that survives. That fact has a proof and the proof is not a picture: it is that two paths which can be deformed into one another without crossing a wire give the same circulation, since the difference between them bounds a surface threading no current, which is the argument four different loops round one wire make. The whole content of the figures is one corollary of a deformation argument.

Still open: how to find the cuts automatically #

The practical obstacle to using scalar potentials in a general solver is not the physics but the geometry. Given an arbitrary arrangement of conductors meshed as a few million tetrahedra, find a set of surfaces in the mesh that make the current-free region simply connected — and do it fast, robustly, and without human intervention.

The problem is a computation of the mesh’s homology, algorithms for it have been developed since the 1980s, and they work. What is not settled is doing it at scale on the meshes industrial solvers actually use: the surfaces produced can be enormous and badly shaped, which degrades the conditioning of the linear system that follows, and choosing among the many valid cuts the one that is numerically best is an optimisation nobody has a good criterion for. The compromise in practice is often to give up and use a vector potential in the regions where the topology is awkward.

The habit worth carrying away is about obstructions. A condition that holds at every point need not hold on every path, and when it fails the failure is a property of the region rather than of the field. The magnetic field off a wire is curl-free everywhere and is still not a gradient of anything single-valued, because the region has a hole in it — and the size of the failure, once measured, turns out to be the physical quantity the local statement had thrown away.