The two coils one radius apart

The field that wraps a current found that Ampère’s law gives the magnetic field of a straight wire or a long solenoid in one line, and that for nearly any other shape it gives nothing useful, so the field of a single circular loop has to be built up piece by piece from Biot and Savart’s law. On the loop’s axis the sum can be done in closed form. At a height $z$ above the centre of a loop of radius $a$ carrying a current $I$,

$$B_z = \frac{\mu_0 I a^2}{2\left(a^2 + z^2\right)^{3/2}},$$

strongest at the centre and falling away on either side, and the loop that behaves like a needle found that far away it falls as the inverse cube, like the field of a bar magnet.

For most purposes the shape of that curve is a nuisance. An experiment that needs a known, steady magnetic field across a sample — a calibration of a magnetometer, a measurement of an electron’s charge-to-mass ratio, the cancellation of the Earth’s field round a sensitive instrument — wants the field to be the same everywhere the sample is, and a single loop gives a field that varies from its first millimetre outward. The need is old. A charged particle crossing a uniform field moves on a circle whose radius is its momentum divided by its charge times the field, as the force that does no work found, and a measurement of that radius is a measurement of the charge-to-mass ratio only if the field is the same all round the circle. In a field that varies, the particle’s orbit drifts sideways along the gradient, at a speed the drift that does not care what the charge is computed, and the circle does not close. The fix Hermann von Helmholtz described is to add a second loop on the same axis. Where the second goes is the whole content of this essay.

A peak, a dip, and a flat place #

Put two identical loops on one axis, a distance $d$ apart, carrying the same current in the same sense. On the axis their fields add, and the total is the sum of two copies of the single-loop curve, one centred at $-d/2$ and one at $+d/2$.

If the loops are close together their two peaks merge into one, and the total is a single hump: largest at the midpoint, smaller on either side. If they are far apart the two peaks stand separately and the midpoint lies in a valley between them. Somewhere in between, the hump turns into the valley, and at that spacing the curve is flat on top: neither a maximum nor a minimum at the centre, but a point where the curvature is zero.

Finding that spacing is a calculus exercise. The field is symmetric about the midpoint, so its first derivative there is zero for any spacing, and its third and every odd derivative too. The second derivative is what changes sign between hump and valley, and setting it to zero for the sum of two single-loop curves gives

$$d = a.$$

There is a way to see the answer without the calculus. A single loop’s axial field, the hump of the formula above, bends downward near its peak and upward far out, and between those it has a point of inflection, where its curvature is zero. Differentiate twice and the inflection sits at exactly half a radius from the loop’s plane. Put two loops a radius apart and the midpoint is half a radius from each, so it sits at the inflection point of both curves at once: each contributes no curvature there, and their slopes cancel by symmetry. The flat top is not a balance between a hump and a valley; it is two curves each caught at the one place where they are straight.

The loops must be exactly one radius apart. That is a Helmholtz pair, and the figure shows what it buys: within 0.17 of a radius of the centre, along the axis, the field departs from its central value by less than a part in a thousand. A pair of coils half a metre across holds the field constant to that precision over a span of eight centimetres.

Each coil cancels one more term #

The reason the flat region is so flat is that the first non-zero term in the field’s expansion about the centre has been pushed up a power. Expand the axial field as a series in distance $z$ from the midpoint:

$$B_z(z) = B_0 + \tfrac{1}{2}B''z^2 + \tfrac{1}{24}B''''z^4 + \dots$$

For one loop, $B''$ is not zero, and the departure from uniform grows as $z^2$. For a Helmholtz pair the second derivative has been cancelled and the departure starts at $z^4$.

On logarithmic axes a power law is a straight line, and its slope is the power. The single loop’s line has slope two, the Helmholtz pair’s slope four. Close to the centre the difference is enormous: a hundredth of a radius away, one loop is already off by a part in ten thousand and a Helmholtz pair by a part in a hundred million.

The same trick can be played again. James Clerk Maxwell, in his Treatise of 1873, added a third coil and adjusted the sizes and currents so that the fourth derivative also cancelled: a central loop of radius $a$ and two smaller loops of radius $\sqrt{4/7}\,a$ at distances $\sqrt{3/7}\,a$ either side, carrying $49/64$ of the central current. The measured slope rises to six, and a hundredth of a radius from the centre the departure is a part in a million million. Each additional coil buys one more term of the series. The series is the inside counterpart of the multipole expansion the shape that takes five numbers used for the field far outside a distribution: there, each term falls off one power faster than the last and the distribution’s shape enters term by term; here, each term grows one power faster from the centre, and the coil designer’s job is to set the low terms to zero so that only the fast-growing high ones remain, too small to matter near the middle. Arrangements of four and six coils, and of continuous windings shaped to cancel many terms at once, are what the magnets of nuclear magnetic resonance spectrometers use to reach a part in a billion across a sample.

Uniform sideways, without being asked #

All of that is a statement about the axis. But the sample in a real experiment has width, and nothing in the condition $B'' = 0$ mentions the field off the axis. It is natural to expect that the region of uniform field is a thin needle along the axis, and that making it wide would require a separate set of conditions. It is not, and it does not.

The map is computed from the exact field of each loop, on and off the axis, which involves the complete elliptic integrals and has no simpler form. It shows the region of uniform field as a squat, rounded volume, slightly wider than it is long. The contour at a part in a thousand extends 0.17 radii along the axis and 0.22 radii out from it. The contour at a part in ten thousand is the same shape at about half the size.

The width comes free because of where the field lives. Between the coils there are no currents, so the magnetic field there has no curl and no divergence, and each of its components obeys Laplace’s equation,

$$\frac{\partial^2 B_z}{\partial r^2} + \frac{1}{r}\frac{\partial B_z}{\partial r} + \frac{\partial^2 B_z}{\partial z^2} = 0.$$

At the centre, by symmetry, $B_z$ depends on $r$ only through $r^2$, and the radial part of the equation contributes twice its curvature in $r$. So the equation says that the curvature along the axis and the curvature across it are tied together: $2\,\partial^2 B_z/\partial r^2 = -\partial^2 B_z/\partial z^2$. Make one vanish and the other vanishes with it.

The same thing happens at every order. A function that obeys Laplace’s equation and is symmetric about an axis is entirely determined, near that axis, by its values on the axis: each term $z^{2n}$ in the axial series comes with a fixed set of terms in $r$ that the equation will not let it shed. For the fourth-order term the full form is

$$z^4 - 3z^2 r^2 + \tfrac{3}{8} r^4,$$

so on the midplane, where $z = 0$, the sideways departure is three-eighths of the axial departure at the same distance.

The exact sideways curve follows three-eighths of the axial curve closely until higher-order terms take over, near a third of a radius out. That is why the uniform region is wider than it is long: along the axis the departure is the full fourth-order coefficient, sideways it is three-eighths of it, and the contours bulge outward accordingly.

This is the same property of Laplace’s equation that the potential is where the wanderers stop found from a different direction: a solution’s value at any point is the average of its values on any sphere around that point. A harmonic function can have no local maximum or minimum, and the flat place at the centre of a Helmholtz pair is the closest it can come — a saddle so shallow that its first four derivatives in every direction are zero. It is the same reason the window a spinning magnet floats in found no static arrangement of magnets able to hold another, fixed magnet in a stable equilibrium: that magnet’s energy is a sum of field components, each harmonic, so it can have no minimum where there are no sources, and Earnshaw’s theorem is the statement of it.

Opposite currents: a zero with a slope #

Reverse the current in one of the two loops and the fields subtract instead of adding. At the midpoint they cancel exactly, and on either side the field grows, pointing outward along the axis on one side and inward on the other. The arrangement is an anti-Helmholtz pair, and it makes the simplest possible magnetic field with a zero in it.

A zero of this kind is a different object from the zeros the zeros a handful of charges can make found between point charges, which were saddles of the potential with the field pointing in and out along different axes. Here too the field points inward along the axis and outward across it, because its divergence must vanish; what makes the zero useful is not its shape but that the field’s strength rises from it in every direction. Near the zero the field grows linearly in distance, and what matters now is the uniformity of the slope rather than of the field. By symmetry the even derivatives of the axial field vanish at the centre; the third derivative is what bends the slope, and setting it to zero gives a different spacing:

$$d = \sqrt{3}\,a.$$

At that spacing the gradient is constant to fifth order along the axis, and by the same harmonic argument it is uniform sideways too, at half the strength — the field’s divergence must vanish, so a gradient of $G$ along the axis forces gradients of $-G/2$ in each sideways direction.

An atom with a magnetic moment has an energy that depends on the strength of the field it sits in. If the energy rises with field strength, the atom is pushed towards weaker field, and near a field zero that means it is pushed towards the zero from every direction. This is the magnetic trap of cold-atom physics, and with a beam of laser light added it is the magneto-optical trap that cools and holds atoms in laboratories everywhere: the light does the cooling of the friction made of light, and the anti-Helmholtz field tells the light which way each atom is displaced. It does not violate Earnshaw’s theorem, because the trapped atom is not a fixed magnet: its moment follows the field’s direction, so its energy depends on the field’s strength rather than on any one component. The strength of a static field in empty space can have no maximum — an atom drawn towards strong field can never be held — but it can have a minimum, and a zero is the simplest one there is.

The same pair, run as a gradient coil, is how a magnetic resonance scanner knows where a signal came from. The image made of frequencies found that the nuclei in a uniform field all precess at one frequency, and that a field gradient added on top makes the frequency a label of position. A uniform gradient makes the label linear, so the image is not distorted, and the coils that produce it are elaborations of the anti-Helmholtz pair.

What the coils cannot do #

The uniform region is small. A part in a thousand over a third of the coils’ diameter is excellent for calibrating a probe and inadequate for a sample the size of the coils. Uniformity over most of the interior needs many more coils, or a long solenoid, whose field is uniform over most of its length because its windings extend far beyond the sample in both directions, and whose inside is shielded by the field outside the solenoid being almost nothing.

The coils are not loops. Real coils have a cross-section — many turns spread over a rectangle of winding — and the ideal spacing for a coil of finite cross-section is not exactly one radius. The correction depends on the winding’s shape and is of order its width over its radius squared; a pair wound carelessly gives back a second-order term the geometry was supposed to cancel.

Nothing nearby may be magnetic. The calculation assumes empty space round the coils. A steel bench, a building’s reinforcing bars or the Earth’s own field adds a field of its own, and for the Earth’s that is about fifty microtesla — comparable to the field a teaching Helmholtz pair makes, and enough to spoil any measurement that needs the pair’s field to be the only one present. Cancelling it is one of the commonest reasons a laboratory builds a Helmholtz pair at all, usually three of them, one for each direction.

The field is static. Everything here uses the steady-current field. Switched quickly, a pair’s field takes time to settle, as any field does when its current changes; at radio frequencies the loops couple to each other and to their surroundings, and the uniform region shrinks.

Still open: how uniform a field can be made #

The best magnets in nuclear magnetic resonance reach uniformity of a part in a billion over a sample a few millimetres across, and they get there by a long sequence of corrections. A superconducting main coil, wound to cancel as many terms of its series as its geometry allows, is followed by a set of superconducting “shim” coils, each designed to produce one term of the expansion of a harmonic field — one $z^2$-like term, one $xy$-like term, and so on — whose currents are adjusted to cancel whatever the main coil left behind. Then come room-temperature shims for the same purpose, adjusted with a sample in place, because the sample itself, being slightly magnetic, distorts the field it is measured in.

Every one of those corrections relies on the fact used above: a field in empty space is harmonic, so its imperfections can be written as a sum of harmonic terms, each with its own shape, and cancelled one at a time. What limits the final uniformity is not the theory but the materials — the magnetic properties of the sample tube, the probe, the solvent, and the slow drift of the persistent current in the main coil — and the number of shim terms that can be independently controlled, which in the largest systems is several dozen.

Whether that sequence can be pushed to a part in ten billion over a larger volume, which would sharpen the spectra of complex molecules beyond what any present magnet gives, is a question about engineering rather than physics. The physics answer to how uniform a field can be is simple: as uniform as the number of independent coils allows, since each removes one more term of a series whose structure Laplace fixed. The first two coils, one radius apart, already remove the term that matters most.