The loop that behaves like a needle

A compass needle in the Earth’s field turns to point north and then stays where it is. It does not drift north. A fridge magnet, on the other hand, is pulled hard toward the fridge. Both are magnets in magnetic fields, and only one of them experiences a force — a difference that is not about the magnets at all.

One number for a loop of any shape #

Take a flat loop of wire carrying current $I$ and enclosing area $A$. Its magnetic moment is

$$\mathbf{m} = N I A \,\hat{\mathbf{n}},$$

with $N$ the number of turns and $\hat{\mathbf{n}}$ the normal fixed by the right-hand rule from the direction of the current. Three vectors have collapsed into one: the shape of the loop has gone, the material has gone, and what is left is an area times a current pointing along an axis.

The collapse is not an approximation of the near field, which depends on the shape in every detail. It is a statement about the far field, and it comes out of the multipole expansion: at distance $r$ much larger than the loop, the leading term of the vector potential depends on the loop only through $\oint \mathbf{r}' \times d\mathbf{l}'$, which is twice the enclosed area vector. Two loops of quite different outline but the same $IA$ produce the same field far away, to leading order, and there is no measurement at a distance that could tell them apart. It is the same collapse that lets a whole charge distribution be replaced by its total as soon as the enclosing surface is symmetric enough.

An electric dipole’s field lines are the same pattern, and that is the fact the word “dipole” is doing. Two systems with entirely different insides — a pair of separated charges and a circulating current — produce the identical field at a distance, so “dipole” names a far-field shape rather than a construction. It is a statement about what survives at range, and it deliberately says nothing about what is there.

How far away is far enough #

“Far enough” is doing work in every sentence above, and it is the kind of phrase this site is obliged to put a number on.

On the axis of a circular loop of radius $a$, the exact field is

$$B_{\text{exact}}(z) = \frac{\mu_0 I a^2}{2\,(a^2 + z^2)^{3/2}},$$

and the ideal point dipole of the same moment gives $B_{\text{dipole}}(z) = \mu_0 m / 2\pi z^3$ with $m = I\pi a^2$. Their ratio is $(1 + (a/z)^2)^{-3/2}$, which approaches one algebraically — as $1 - 3a^2/2z^2$ — rather than exponentially.

That slowness has a practical face. A Helmholtz pair, a loudspeaker motor and an MRI gradient coil are all specified by how much their field departs from an ideal over a working volume, and the departures are all this curve seen from a different angle.

It also settles a question of vocabulary that is otherwise a matter of taste. Calling something a dipole is not a claim about its size; it is a claim about the ratio of its size to the distance at which it is being described. The Earth is a dipole to a compass needle and is emphatically not one to a satellite in low orbit, whose altitude is a twentieth of the planet’s radius and which flies through a field with several higher terms in it. The same object is a dipole and not a dipole depending only on where the question is asked from.

Where the moment comes from #

The expression $m = IA$ says nothing about what is going round, and putting a particle into it turns the moment into a statement about angular momentum — which is the connection the rest of atomic magnetism is built on.

Take one charge $q$ of mass $M$ going round a circle of radius $r$ at speed $v$. It passes any point $v/2\pi r$ times a second, so the current is $qv/2\pi r$; the area is $\pi r^2$; and the moment is

$$m = \frac{qv}{2\pi r}\,\pi r^2 = \frac{qvr}{2} = \frac{q}{2M}\,L,$$

with $L = Mvr$ the angular momentum. Every geometric detail has cancelled. A moment and an angular momentum are the same quantity in different units, related by a ratio built from the charge and the mass alone.

That is where the natural unit comes from. Put in an electron’s charge and mass, and one unit of angular momentum — $\hbar$ — gives $e\hbar/2M_e = 9.274\times10^{-24}$ A m², the Bohr magneton, which is the number the alignment-energy comparison above uses. Nothing was measured to get it: it is two constants and a quantum of angular momentum.

And it is where the picture visibly runs out. An electron’s spin is half a unit of angular momentum and produces a moment of one whole Bohr magneton rather than a half — the ratio is twice what any circulating current can give. No arrangement of charge going round in circles produces that factor, which is the first hard evidence that spin is not a rotation of anything.

The experiment that made the connection physical rather than formal is worth recording. Suspend an iron cylinder on a fine fibre and magnetise it: aligning the moments inside must align their angular momenta too, and since the cylinder was not rotating before, it has to rotate the other way to keep the total at zero. It does, by a tiny and measurable amount, and the size of the rotation returns the ratio of moment to angular momentum for whatever is doing the magnetising. The answer came back at roughly twice the orbital value — the same factor of two, arrived at by hanging a lump of iron on a thread.

A torque, and no force whatever #

The force on a current-carrying element is $I\,d\mathbf{l} \times \mathbf{B}$. Summing round a closed loop in a uniform field,

$$\mathbf{F} = I \oint d\mathbf{l} \times \mathbf{B} = I\left(\oint d\mathbf{l}\right) \times \mathbf{B} = \mathbf{0},$$

because $\oint d\mathbf{l}$ is the sum of the displacement vectors round a closed path, and a closed path returns to where it started. The result is exact, holds for any shape of loop, and does not depend on the angle. A uniform field cannot pull on a magnet.

The moments do not cancel, because the forces do not share a line of action. For a rectangular loop with two sides across the field and two along it, the two along contribute nothing and the two across carry equal and opposite forces separated by the width — a couple, of magnitude

$$\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B},$$

whose size is $mB\sin\theta$ and whose associated energy is $U = -\mathbf{m}\cdot\mathbf{B}$.

A couple in its plainest form is two equal and opposite forces whose lines do not coincide, so their moments add instead of cancelling. Its magnitude depends on the separation between the lines and not on where the origin is put — which is why a loop in a uniform field feels a torque with no net force, and why that torque is the same about every point.

So a compass needle turns and stays put, which is the observation the essay opened with. The Earth’s field is very nearly uniform over the size of a needle — its gradient is about 3 × 10⁻¹¹ T per metre, so across 30 mm the field changes by one part in ten million — and the residual force is far below the pivot’s friction.

Where a force comes from, then #

A magnet is pulled to a fridge because the fridge’s field is not uniform. Expanding to the next order, the net force on a dipole is

$$\mathbf{F} = \nabla(\mathbf{m}\cdot\mathbf{B}),$$

which is a gradient, and therefore zero whenever $\mathbf{B}$ is the same everywhere. Magnetic attraction is a second-order effect in a sense that can be stated precisely: it needs the field to vary over the size of the object, and it vanishes to the extent that it does not.

A Stern–Gerlach analyser deflects magnetic moments by putting them through a deliberately non-uniform field, and the whole apparatus is built on the fact that a uniform one would do nothing at all. That is where a force comes from: not from the field but from its gradient, and an instrument that wants to move moments rather than turn them has to be designed around producing one.

Now take the second dipole to be another magnet. Its field falls as $1/r^3$, so its gradient falls as $1/r^4$, and the force between two aligned dipoles is

$$F = \frac{3\mu_0}{2\pi}\frac{m_1 m_2}{r^4}.$$

A point source gives an exponent of $-2$ and a dipole gives $-3$, because two nearly cancelling contributions leave only their difference — and a difference falls one power faster than either term. That is the arithmetic behind “how far away is far enough”: the dipole term outlives nothing and is outlived by nothing, and the correction beyond it falls faster still.

A fourth power is a startling rate of decline and it is the everyday experience of magnets: at twice the distance the pull is a sixteenth. Compare the inverse square of a point charge, where twice the distance costs only a quarter, and the difference in how the two forces feel is entirely this exponent. It is also why magnetic attraction feels like a threshold rather than a field — a magnet does nothing at all until it is close, and then it snaps.

The energy, and why hardly any moments are aligned #

The torque integrates to an energy, $U = -\mathbf{m}\cdot\mathbf{B}$, whose minimum is the aligned state and whose maximum is the opposed one, with a gap of $2mB$ between them. That gap is the natural scale of the problem, and comparing it with something else is where the description stops being about one loop.

Every atom in a paramagnetic solid carries a moment of about a Bohr magneton, $9.27 \times 10^{-24}$ A m². In a laboratory field of one tesla the alignment energy $mB$ is therefore about $9 \times 10^{-24}$ J. At room temperature $k_BT$ is $4.1 \times 10^{-21}$ J — four hundred times larger. So the field is trying to line the moments up and the thermal jostling is untidying them four hundred times harder, and the fraction of moments that end up aligned is of order $mB/k_BT$, about a quarter of a per cent.

A moment of one Bohr magneton in a one-tesla field has an alignment energy of about $6\times10^{-5}$ eV, and $kT$ at room temperature is 0.025 eV — four hundred times larger. So the population difference across that gap is a fraction of a per cent, and hardly any moments are aligned at any field a laboratory can produce. Magnetism in matter is a small bias on a large disorder.

That comparison is the whole of why ordinary materials are so feebly magnetic and why ferromagnets are startling. A ferromagnet’s moments are aligned by an interaction between neighbours whose energy is not $mB$ but a fraction of an electronvolt — comparable with $k_BT$ at a thousand kelvin, which is why iron has a Curie temperature of 1043 K and why it loses its magnetism above it rather than gradually.

An energy the magnetic force did not supply #

There is something odd about $U = -\mathbf{m}\cdot\mathbf{B}$ that is worth facing, because the rung below this one is the reason it is odd.

A magnetic force never does work: it is always perpendicular to the velocity of whatever it acts on, so it can redirect a charge and never speed it up. Yet a loop released across a field turns, gains kinetic energy, and the energy accounted for is exactly the drop in $-\mathbf{m}\cdot\mathbf{B}$. If the magnetic force did none of that work, who did?

Whatever holds the current constant. As the loop turns, the flux through it changes, so an emf is induced round it that opposes the current, and the source maintaining the current must work against that emf. The energy delivered to the rotation comes from the source, routed through the induction, and the magnetic force acts as a broker exactly as it does for a moving conductor. Every joule is accounted for and none of it passed through a magnetic force doing work.

The consequence is a caution about the sign and the constancy. $-\mathbf{m}\cdot\mathbf{B}$ is the correct energy for computing the torque and the force on a dipole of fixed moment, and it is not the total energy of the field-plus-circuit system, which differs by the work the source did. For a permanent magnet — whose moment is maintained by nothing external — the accounting is simpler and the expression is the whole of it. For an electromagnet it is not, and treating it as such is the usual route to a sign error.

Why a field cannot hold a magnet still #

The gradient force above invites an obvious idea: if a non-uniform field pulls a magnet, arrange one that pulls it toward a point and the magnet is trapped. The idea fails, and it fails for a reason that is a theorem rather than a difficulty.

A dipole of fixed moment aligned with the field has energy $-mB$, so it is drawn toward wherever $|\mathbf{B}|$ is largest. But in a region with no currents in it, $|\mathbf{B}|$ has no local maxima: the field satisfies Laplace’s equation componentwise, and a solution of Laplace’s equation takes its extreme values on the boundary rather than inside. Any apparent maximum in free space is a saddle, and the magnet escapes along the direction that goes downhill. That is Earnshaw’s result applied to magnets, and it is why every stable magnetic suspension either moves, feeds back, or cheats.

The interesting half is that local minima of $|\mathbf{B}|$ are perfectly allowed. So anything whose energy rises with the field — which means anything diamagnetic, whose induced moment opposes the field rather than aligning with it — can be trapped by a static arrangement, stably, with no feedback and no motion. That is the basis of the field’s most quoted demonstration: an ordinary object made mostly of water, which is weakly diamagnetic, held in mid-air at the minimum of a strong field with nothing touching it.

The two cases differ only in a sign, and the sign is decided by whether the moment is permanent or induced. Which makes magnetic levitation a good test of whether the expressions above have been understood as statements about energy rather than about attraction.

Making the moment bigger #

Since $m = NIA$, there are three ways to increase it, and their costs differ.

More current heats the wire, and the heat goes as $I^2$ while the moment goes as $I$ — so doubling the moment this way costs four times the power. More area needs more room, and the field a solenoid produces on its axis does not care about its length. More turns is the cheap one, and it is why every electromagnet is a coil rather than a hoop.

Stacking loops multiplies the moment without changing its character. A coil of $N$ turns has $N$ times the moment of one, and inside it the lines run parallel and evenly spaced — uniform, and therefore a place where a magnet feels a torque and no force. Outside, they close round and the whole object is a dipole again, of a size the winding decides. That is the entire engineering of the subject: the shape of the far field is fixed and only its magnitude is a design variable.

Where the model stops #

The moment is the first term of an expansion, not the whole field. Beyond the dipole term come the quadrupole and higher, falling as $1/r^4$, $1/r^5$ and so on, and every one of them is present. Close to a real magnet they dominate, which is why the pattern of iron filings near a bar magnet does not look like the dipole picture and why a magnet placed against another does not obey the fourth power at all.

A magnetised body is not a current loop, quite. Ampère’s picture — that magnetisation is circulating microscopic currents — reproduces every magnetostatic prediction and is what makes the moment of an electron’s orbital motion come out right. It does not account for spin, whose moment is not the moment of any circulating current, and whose gyromagnetic ratio is twice what the orbital picture would give.

The alternative bookkeeping, in which a magnet is a pair of magnetic charges separated by a distance, reproduces the field outside a magnet just as faithfully and gets the field inside it wrong — the two pictures give opposite directions there. Since nothing outside can distinguish them, the disagreement is only decidable by a measurement made within the material, and the current picture is the one that survives it. That is a useful thing to know about a description that is otherwise indistinguishable from a correct one: the place two equivalent models differ is the place at least one of them is wrong. The dipole arithmetic here is indifferent to which of the two supplies the moment, which is a strength of the description and a warning against reading it as a mechanism.

Nothing above holds if the second body is soft iron. A permanent magnet approaching a steel plate induces a moment in the plate proportional to the field it brings, so the force acquires an extra factor and grows faster than $1/r^4$. The fourth power is the law for two fixed moments, and it is measured by holding two permanent magnets apart, not by letting one approach a fridge.

What the pictures cannot show #

The torque curve is drawn against angle and is silent on time. A needle released across the field does not settle at $\theta = 0$; it swings, and if the damping is light it oscillates about the minimum with a period set by $\sqrt{I_{\text{rot}}/mB}$ — which is the pendulum’s period with $mB$ in place of $mgL$, and is how a magnetometer measures $B$.

The inset showing four forces on a rectangle draws a particular shape, and the cancellation is general. Nothing in the figure distinguishes “these two happen to cancel” from “the sum round any closed path is zero”, which is the actual reason.

And the far-field figure plots the departure on the axis, where the ratio has a closed form. Off-axis the discrepancy is larger, and there is no direction in which a real loop matches a point dipole exactly at any finite distance.

Where the ladder goes next #

The path through this anchor has gone from a field with no ends, to a force that does no work, to a single number that summarises a whole current distribution. Each step has removed detail: first the sources, then the energy, and now the shape.

Two rungs follow. One is magnetisation as a bulk quantity: moments per unit volume, the bound currents they are equivalent to, and why the field inside a magnet needs two vectors rather than one. The other is what a dipole does when it is made to rotate — a rotating moment radiates, and the power is the magnetic counterpart of the Larmor formula, one of the two ways an antenna can be built.

The transferable move is the multipole habit. When a source is complicated and the observer is far away, do not solve for the source: expand the field in powers of the size divided by the distance and keep the first term that survives. What survives is usually one number, and the whole of the object’s identity at a distance is that number.