Multipole Expansion of the Vector Potential

The electrostatics track built a ladder and climbed it: expand $1/\mathfrak{r}$ in Legendre polynomials, and any far-away charge blob resolves into monopole, dipole, quadrupole — silhouettes of decreasing coarseness, falling off with increasing speed. This article builds the same ladder for magnetism, and the construction pays off in a way the electric version never could: the first rung is empty. Where a charge distribution's far field is usually dominated by its total charge, a steady current's "total current vector" $\int\vec J\,\mathrm{d}\tau'$ vanishes identically — current that flows in must flow out, loops close, and the $1/r$ term dies before it is born. The leading silhouette of every localized current in the universe is therefore the dipole. That empty rung promotes one number to headline status. The magnetic dipole moment $\vec m=\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'$ — which for a flat loop collapses to the unreasonably simple $\vec m=I\vec a$ , current times vector area — is what a distant observer can know about your circuit, and, because the monopole is absent, it comes with a bonus theorem the electric dipole cannot match: $\vec m$ is origin-independent, an intrinsic property of the source rather than an artifact of coordinates. One vector summarizes a MRI coil, an electron's orbit, and the molten dynamo of the Earth, and this article extracts it honestly from the expansion. The electric ladder is the constant companion here — same generating function, same $1/r^{n+1}$ bookkeeping, cross-referenced rung by rung — and the differences are the physics: vector source instead of scalar, conservation constraint instead of free total charge. The article stops, deliberately, at the potential $\vec A_{\rm dip}$ : taking its curl to unwrap the dipole field, with its bar-magnet anatomy and its famous delta-function heart, is the next article's whole subject.

Key takeaways

Far from a localized steady current, expand the kernel of $\vec A=\frac{\mu_0}{4\pi}\int\frac{\vec J}{\mathfrak r}\mathrm{d}\tau'$ with the Legendre generating function: $\vec A=\sum_n\frac{\mu_0}{4\pi r^{n+1}}\int(r')^nP_n(\cos\alpha)\,\vec J\,\mathrm{d}\tau'$ — the magnetic multipole ladder.
The monopole rung ( $n=0$ ) vanishes identically: $\oint\mathrm{d}\vec\ell'=0$ for a loop, $\int\vec J\,\mathrm{d}\tau'=0$ in general (proved via $\nabla'\cdot\vec J=0$ ) — the ladder echo of "no magnetic charge."
The leading term is the dipole: $\vec A_{\rm dip}=\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}$ , with $\vec m=\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'\xrightarrow{\rm loop}I\vec a$ .
$\vec m$ is origin-independent — shifting coordinates changes it by $-\tfrac12\vec a\times\int\vec J\,\mathrm{d}\tau'=0$ — unlike the electric dipole moment of a charged object.
Successive rungs cost $r'/r$ apiece; the quadrupole matters only at moderate distances or when symmetry kills the dipole (opposed coil pairs, anti-Helmholtz).
The dipole field (curl of $\vec A_{\rm dip}$ , contact term included) and dipole mechanics (torque, force, energy) are owned by the next two articles.
KEY IDEA: expand $1/\mathfrak{r}$ in Legendre polynomials; current conservation deletes the monopole term, leaving the origin-free magnetic dipole $\vec m$ as the universal far-field signature of every localized current.

The core insight

Expand $|\vec r-\vec r'|^{-1}$ in Legendre polynomials; the monopole term vanishes by current conservation, leaving the magnetic dipole $\vec A_{\rm dip}=\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}$ with $\vec m=\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'$ as the dominant far-field contribution.

Where this lives today

The most heavily used magnetic multipole expansion on (and around) the planet is the planet's. The International Geomagnetic Reference Field — the model behind every compass app, aircraft navigation system, and directional-drilling rig — is a spherical-harmonic expansion of Earth's field updated every five years: rung one is the tilted dipole ( $\sim30\ \mu$ T scale, currently tilted $\sim11°$ and weakening $\sim5\%$ per century), and the model carries the ladder to degree 13, where core structure gives way to crustal noise. Reading the rungs separately is the science: the dipole tracks the geodynamo's main circulation, the quadrupole and octupole encode its asymmetries — the South Atlantic Anomaly, where satellites take their radiation hits, is a feature of the higher rungs eating into the dipole. At the other end of the length scale, the expansion runs inward. Magnetoencephalography reconstructs neural activity by fitting measured $\sim100$ fT scalp fields to current dipoles — literal $\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'$ of ionic currents in cortex — and particle physics tests the Standard Model at the ladder's quantum rung: the electron's $g-2$ (its dipole moment's deviation from Dirac's prediction) matches theory to twelve digits, while nonzero electric dipole moments of fundamental particles, forbidden rungs by time-reversal symmetry, are hunted as signatures of new physics. One expansion, from planetary cores to the electron's anomaly.

Origins · 1 minAndré-Marie Ampère and how this was discovered

André-Marie Ampère

1775–1836 · Lyon and Paris — the mathematician of the molecular current

Ampère's deepest bet is the premise this article runs on: that all magnetism is circulating current. Within weeks of Ørsted's 1820 news he had not only measured wire–wire forces but proposed that magnets themselves are nothing but arrays of microscopic current whorls — "molecular currents," a century before electrons. His equivalence theorem made it quantitative: a small current loop is magnetically indistinguishable, at a distance, from a thin magnetized lamina — dipole for dipole, exactly the identification $\vec m=I\vec a$ that drops out of this article's expansion. Fresnel, privately, raised the killer objection (why don't the whorls dissipate?) and Ampère, remarkably, answered by fiat: they simply don't — resistance-free molecular motion, vindicated by quantum mechanics a hundred years later. Self-taught from Bernoulli and Euler in his father's library in Lyon — a father guillotined in the Revolution — Ampère carried a life of grief into an almost monastic devotion to unification; Maxwell's verdict stands as the epitaph: "the Newton of electricity."

History

1820

— Ørsted's discovery reaches Paris in September; within weeks Ampère demonstrates current–current forces and conjectures that all magnetism is circulating charge.

1821

— Ampère's molecular-current hypothesis and loop–lamina equivalence: the small current loop is a magnetic dipole, m = Ia in modern dress.

1826

— The Mémoire codifies the mathematics of current elements; the far field of any circuit organizes around its moment.

1839

— Gauss expands the Earth's field in spherical harmonics — the first real-world magnetic multipole fit, ancestor of today's IGRF.

1928

— Dirac's equation makes the electron's dipole moment a theorem (g = 2); the measured anomaly g − 2 becomes the ladder's precision frontier.

The physical picture

Walk away from any current distribution — a crumpled superconducting coil, a neuron's ionic loop, a planet's core — and information about it dies in a strict order. First to go is the fine structure (where exactly each wire bends), then the coarser shape, until at great distance only the crudest summaries survive. The multipole expansion is that dying-of-detail made quantitative: each term is a summary statistic of the source, weighted by one more power of the small ratio $r'/r$ , so the far field is a short list of numbers rather than an integral over geometry. For charges, the crudest summary was the total charge $Q$ — one number, no directional information, field $\propto1/r^2$ . For currents, the corresponding summary would be the net current vector, and here magnetism files its defining constraint: steady currents have no net flow direction. Every ampere heading east somewhere is an ampere heading west somewhere else, because current lines close. The would-be leading term integrates to zero — always, exactly, for any localized steady source — and this is the third time the chapter has met the same fact in a new costume: no magnetic monopoles ( $\nabla\cdot\vec B=0$ ), field lines that close, and now an empty first rung on the ladder.

Figure

The magnetic ladder next to the electric one — the monopole rung is structurally empty, so the dipole leads

So the dipole leads, and its moment deserves a picture before its formula. For a flat loop, $\vec m=I\vec a$ : multiply the current by the enclosed vector area, point it by the right-hand rule. Why area? Because a loop's far field is a near-cancellation — from far away, the two sides of the loop are opposite currents almost on top of each other, and what survives the cancellation is proportional to how much they are offset, integrated around the circuit: an area. The same cancellation logic explains the extra power of falloff ( $1/r^2$ in $\vec A$ , against the monopole's forbidden $1/r$ ) and previews the general moment $\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'$ , in which the "swept area" idea survives as a cross product even when no flat loop exists. The chapter has met this quantity twice already, from other directions: the torque on a rectangular loop packaged itself as $\vec m=I\vec a$ (the force-on-wires article), and the far tail of the circular loop's axial field carried $m=I\pi R^2$ (the catalog article). Here it stops being a pattern and becomes a theorem: $\vec m$ is the $n=1$ coefficient of the universal expansion, the reason those two appearances had to agree.

The ladder, assembled

Everything expands from the previous articles' integral (Coulomb-gauge representative, localized steady source):

\vec A(\vec r)=\frac{\mu_0}{4\pi}\int\frac{\vec J(\vec r')}{|\vec r-\vec r'|}\,\mathrm{d}\tau' \qquad (1)

where: all symbols as established; the line dialect replaces $\vec J\,\mathrm{d}\tau'\to I\,\mathrm{d}\vec\ell'$ . The expansion engine is imported, not rebuilt: the Legendre-polynomials article proved the generating function, and the electric multipole article already spent it on this exact kernel —

\frac{1}{|\vec r-\vec r'|} =\frac{1}{r}\sum_{n=0}^{\infty}\left(\frac{r'}{r}\right)^{n}P_n(\cos\alpha), \qquad \cos\alpha=\hat r\cdot\hat r', \qquad (2)

where: the series converges absolutely for $r>r'$ and — a nuance owned by the electric twin and inherited here — is exact, not merely asymptotic, whenever the field point lies outside the smallest sphere enclosing the source. Everything magnetic about this article is what happens when (2) meets a vector, conserved source.

Derivation AThe magnetic multipole series, and the death of the monopole

The goal is the ladder itself — insert (2) into (1), organize by powers of $1/r$ — and the first structural theorem: the $n=0$ term vanishes for every localized steady current, in both the loop and volume dialects. The strategy leans on imported machinery (generating function from the Legendre article; the divergence trick already used once by this chapter's moving-charges assembly) so that the new content, the conservation constraint, stands out alone.

Step 1. Insert and sort. Substituting (2) into (1) and exchanging sum and integral (legitimate inside the convergence sphere):

\vec A(\vec r)=\frac{\mu_0}{4\pi}\sum_{n=0}^{\infty}\frac{1}{r^{n+1}}\int (r')^{\,n}P_n(\cos\alpha)\,\vec J(\vec r')\,\mathrm{d}\tau' \qquad (3)

— monopole ( $n=0$ , would-be $1/r$ ), dipole ( $n=1$ , $1/r^2$ ), quadrupole ( $n=2$ , $1/r^3$ ), each rung a source moment weighted one power of $r'/r$ smaller than the last. For a filamentary circuit the same line reads $\vec A=\frac{\mu_0I}{4\pi}\sum_n\frac{1}{r^{n+1}}\oint (r')^nP_n(\cos\alpha)\,\mathrm{d}\vec\ell'$ .

Step 2. The monopole, loop dialect — one line. With $P_0=1$ , the $n=0$ coefficient is $\oint\mathrm{d}\vec\ell'$ : the sum of the displacement vectors once around a closed circuit, which is the total displacement of a round trip — zero. For circuits, the monopole dies by the geometry of closure alone.

Step 3. The monopole, volume dialect — conservation does it. The general claim is $\int\vec J\,\mathrm{d}\tau'=0$ . For each Cartesian component, use the identity $\nabla'\cdot(x_i'\vec J)=x_i'(\nabla'\cdot\vec J)+J_i$ with the steady-current condition $\nabla'\cdot\vec J=0$ :

\int J_i\,\mathrm{d}\tau' =\int\nabla'\cdot\!\big(x_i'\vec J\big)\,\mathrm{d}\tau' =\oint_S x_i'\,\vec J\cdot\mathrm{d}\vec a' =0 ,

the divergence theorem converting the volume integral to a surface term on any surface enclosing the whole source — where no current crosses. Hence

\vec A_{\rm mono}=\frac{\mu_0}{4\pi r}\int\vec J\,\mathrm{d}\tau'=\vec 0 \qquad (4)

identically: no configuration of steady currents, however baroque, radiates a $1/r$ potential. Why this is the ladder version of a familiar law: "no magnetic charge" has now appeared as a field equation ( $\nabla\cdot\vec B=0$ ), as geometry (closed field lines), and as an expansion statement (empty first rung). All three are one fact: sources of the Coulomb type would populate the monopole rung, and magnetism has none. The electric ladder's leading generic term is exactly the one magnetism forbids. *Verified: dimension check — rung $n$ carries $[\mu_0 J\,r'^{\,n}\,\tau/r^{n+1}]=\mathrm{T\,m}$ for every $n$ ✓; Step 3's identity is the same maneuver the current-density article used to convert carrier sums, recalled not reinvented; loop and volume dialects agree — squeeze $\vec J\,\mathrm{d}\tau'\to I\,\mathrm{d}\vec\ell'$ and Step 3 collapses to Step 2 ✓.*

Before extracting the surviving leader, one geometric object needs its portrait: the vector area. The moment about to emerge is $I$ times $\tfrac12\oint\vec r'\times\mathrm{d}\vec\ell'$ , and that integral has a picture — the circuit fanned into thin triangles from the origin, each $\tfrac12\vec r'\times\mathrm{d}\vec\ell'$ one triangle's vector area, the sum tiling the loop:

Figure

The vector area as a fan of swept triangles — and the shadow rule for warped loops

Go deeper · Derivation B · 2 minExtracting the dipole — the vector-area identity

Derivation BExtracting the dipole — the vector-area identity

The goal is the leading surviving term in closed form: from the $n=1$ rung to

\vec A_{\rm dip}(\vec r)=\frac{\mu_0}{4\pi}\,\frac{\vec m\times\hat r}{r^2}, \qquad \vec m=I\vec a\;\;(\text{loop}) \;=\;\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'\;\;(\text{general}),

with the two dialects of $\vec m$ reconciled. The strategy runs the filament version in full — its engine is a beautiful, provable identity about vector areas — then obtains the volume version by the symmetric/antisymmetric split, flagging what each move spends.

Step 1. Isolate the rung. With $P_1(\cos\alpha)=\cos\alpha$ and $r'\cos\alpha=\hat r\cdot\vec r'$ , the $n=1$ term of the loop series is

\vec A_{\rm dip}=\frac{\mu_0 I}{4\pi r^2}\oint(\hat r\cdot\vec r')\,\mathrm{d}\vec\ell' .

Everything now rides on the loop integral $\oint(\hat c\cdot\vec r')\,\mathrm{d}\vec\ell'$ for a constant unit vector $\hat c$ .

Step 2. The vector-area lemma. Claim: $\oint(\hat c\cdot\vec r')\,\mathrm{d}\vec\ell'=\vec a\times\hat c$ , where $\vec a=\tfrac12\oint\vec r'\times\mathrm{d}\vec\ell'$ is the loop's vector area. Proof by components via Stokes' theorem: for the $i$ -th component, $\oint(\hat c\cdot\vec r')\,\mathrm{d}\ell_i'$ is the circulation of the vector field $\vec v=(\hat c\cdot\vec r')\,\hat e_i$ , whose curl is $\nabla'\times\vec v=\nabla'(\hat c\cdot\vec r')\times\hat e_i=\hat c\times\hat e_i$ — a constant. Stokes then gives $\oint\vec v\cdot\mathrm{d}\vec\ell'=\int(\hat c\times\hat e_i)\cdot\mathrm{d}\vec a'=(\hat c\times\hat e_i)\cdot\vec a=(\vec a\times\hat c)_i$ , the last step by cyclic permutation of the scalar triple product. Assembling components proves the lemma. Why $\vec a$ deserves its name: $\tfrac12\vec r'\times\mathrm{d}\vec\ell'$ is the area of the thin triangle swept from the origin to the element $\mathrm{d}\vec\ell'$ — the loop integral fans the circuit into triangles and adds them vectorially. For a flat loop, $|\vec a|$ is the enclosed area and $\hat a$ the right-hand normal; for a warped loop, $\vec a$ is the shadow rule — its component along any direction is the loop's projected area onto the perpendicular plane.

Step 3. Assemble the loop dipole.

\vec A_{\rm dip} =\frac{\mu_0 I}{4\pi r^2}\,(\vec a\times\hat r) =\frac{\mu_0}{4\pi}\,\frac{\vec m\times\hat r}{r^2}, \qquad \vec m=I\vec a . \qquad (5)

The dipole potential circulates azimuthally around the moment's axis ( $\vec m\times\hat r$ ), falls as $1/r^2$ , and vanishes on the axis itself — $\vec A$ inheriting, one last time, the geometry of its circulating source.

Step 4. The volume dialect. For a general $\vec J$ , the $n=1$ integrand $(\hat r\cdot\vec r')\vec J$ splits into symmetric and antisymmetric parts under $\vec r'\leftrightarrow\vec J$ . The symmetric part is a total divergence — $\int[(\hat r\cdot\vec r')J_i+(\hat r\cdot\vec J)x_i']\,\mathrm{d}\tau'=\int\nabla'\cdot[x_i'(\hat r\cdot\vec r')\vec J]\,\mathrm{d}\tau'=0$ by the same surface-term kill as the monopole proof — and the antisymmetric part is a disguised cross product via BAC–CAB, $\tfrac12[(\hat r\cdot\vec r')\vec J-(\hat r\cdot\vec J)\vec r']=-\tfrac12\hat r\times(\vec r'\times\vec J)$ . Hence

\vec A_{\rm dip} =\frac{\mu_0}{4\pi r^2}(-\hat r\times\vec m) =\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}, \qquad \vec m=\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau' ,

and squeezing $\vec J\,\mathrm{d}\tau'\to I\,\mathrm{d}\vec\ell'$ shows $\tfrac12\oint\vec r'\times I\,\mathrm{d}\vec\ell'=I\vec a$ : the dialects agree exactly where they overlap. Verified: matches Griffiths Eqs. 5.84–5.86 and Jackson §5.6; on-axis cross-check — for $\vec m=m\hat z$ , (5) gives $A_\phi=\mu_0m\sin\theta/4\pi r^2$ , whose curl's far-field axial value $\mu_0m/2\pi z^3$ is precisely the loop tail the catalog article computed from raw Biot–Savart ✓ (the full curl is next article's opening act); units — A·m² × T·m/A ÷ m² = T·m ✓.

Go deeper · Derivation C · 1 minThe moment is origin-independent — because the monopole died

Derivation CThe moment is origin-independent — because the monopole died

The goal is the bonus theorem: $\vec m$ does not depend on where you put the origin, making it an intrinsic label of the source — the magnetic mirror of the electric shift theorem, which found $\vec p$ origin-dependent whenever $Q\neq0$ . The strategy is one substitution; the physics is watching the monopole's corpse do the work.

Step 1. Shift. Move the origin by $\vec c$ , so source positions are re-labeled $\vec r'\to\vec r'-\vec c$ :

\vec m' =\tfrac12\int(\vec r'-\vec c)\times\vec J\,\mathrm{d}\tau' =\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau' -\tfrac12\,\vec c\times\int\vec J\,\mathrm{d}\tau' .

Step 2. Spend the empty rung. The second integral is the monopole moment — zero by Derivation A —

\vec m'=\vec m .

Step 3. Read it against the electric twin. The E2 origin-dependence article proved $\vec p'=\vec p-Q\vec c$ : the electric dipole is a matter of convention unless the monopole $Q$ vanishes. Magnetism has no choice in the matter — its monopole always vanishes — so the general principle ("the lowest nonvanishing moment is origin-independent") lands here with full force on the first occupied rung. The Earth's $8\times10^{22}\ \mathrm{A\,m^2}$ , the proton's $1.41\times10^{-26}\ \mathrm{J/T}$ : these are properties of the objects, not of anyone's coordinate choice, which is what licenses tabulating them. Verified: consistent with Zangwill §11.2; the loop dialect gives the same one-liner, since $\tfrac12\oint(\vec r'-\vec c)\times\mathrm{d}\vec\ell'=\vec a-\tfrac12\vec c\times\oint\mathrm{d}\vec\ell'=\vec a$ by round-trip closure ✓.

Limiting cases

case	result	meaning
$r\gg r'$	$\vec A\to\vec A_{\rm dip}$ , relative error $O(r'/r)$	the universal far field; already $\sim1\%$ by $r\approx10\,r'$ on-axis (worked below)
$r>r'_{\max}$ , any ratio	series exact, converges	inherited nuance from the electric twin — expansion, not approximation, outside the source sphere
planar loop	$\vec m=I\vec a$ , $a=$ enclosed area	the workhorse; $N$ turns: $\vec m=NI\vec a$
$\vec m=0$ by symmetry	quadrupole ( $1/r^3$ in $\vec A$ ) leads	opposed coil pairs, anti-Helmholtz; worked below
on the moment's axis	$\vec A_{\rm dip}=0$ , $\vec B\neq0$	$\vec A$ circulates around the axis; the field there comes from the curl, not the value
near zone $r\lesssim r'$	ladder invalid	use Biot–Savart / exact integrals; "pure dipole" is a far-field idealization

Common trapThe electric analogy is load-bearing — and it breaks in exactly two places

The magnetic ladder is deliberately built on electric scaffolding — same generating function, same $1/r^{n+1}$ rungs — and the analogy will carry you far. It breaks, by design, twice. First: the monopole. An electric source generically has one; a magnetic source never does, so magnetic far fields start one rung higher, fall one power faster, and their leading moment is origin-independent where the electric $\vec p$ (of a charged body) is not. Treating a bar magnet's far field as if some "magnetic charge" contributed a $1/r^2$ Coulomb piece is the classic error this kills. Second, deferred but flagged: at the origin, the point-dipole fields carry delta-function contact terms of opposite character ( $+\tfrac{2\mu_0}{3}\vec m\,\delta^3$ magnetic vs $-\tfrac{1}{3\varepsilon_0}\vec p\,\delta^3$ electric) — the loop-versus-charge-pair fingerprint, derived with the dipole field next article. Where the analogy holds, use it hard; where it breaks, the break is the physics.

Worked examples

1 · How fast the far field becomes "dipole" (confidence builder)

A circular loop, $R=5.0$ cm, $I=2.0$ A. Find $\vec m$ , the dipole prediction for the axial field at $z=50$ cm, and its error against the exact result.

Step 1. The moment. $m=I\pi R^2=2.0\times\pi\times(0.05)^2=1.57\times10^{-2}\ \mathrm{A\,m^2}$ , along the axis by the right-hand rule.

Step 2. Dipole prediction. On the axis, the curl of (5) gives the tail the catalog article already derived: $B_{\rm dip}=\mu_0m/2\pi z^3=(2\times10^{-7}\times1.57\times10^{-2})/(0.5)^3$ = 25.1 nT.

Step 3. Exact comparison. $B_{\rm exact}=\mu_0IR^2/2(z^2+R^2)^{3/2}$ = 24.8 nT. The dipole overshoots by +1.5% — and the expansion predicts exactly that: the next correction on-axis is the factor $(1+R^2/z^2)^{-3/2}\approx1-\tfrac32(R/z)^2$ , and $\tfrac32(R/z)^2=\tfrac32(0.1)^2=1.5\%$ ✓. At $z=10$ source-radii the silhouette is already accurate to a percent; by $z=5R$ the error is $6\%$ ; inside $z\sim2R$ the ladder is the wrong tool.

Step 4. A planetary calibration. The same two formulas run the Earth: $m_\oplus\approx8.0\times10^{22}\ \mathrm{A\,m^2}$ gives an equatorial surface field $\mu_0m/4\pi R_\oplus^3\approx31\ \mu\mathrm{T}$ — the compass-needle number — with $r'/r$ corrections supplied by the core's finite size and higher moments (about which more in the today-band).

2 · When the dipole dies: an opposed pair (exam level)

Two coaxial loops as in example 1 ( $m=1.57\times10^{-2}\ \mathrm{A\,m^2}$ each) are separated by $d=10$ cm and carry opposite currents. Find the far axial field at $z=1.0$ m from the midpoint.

Step 1. The dipole rung empties. Total moment: $\vec m+(-\vec m)=0$ . The $1/r^3$ far field is gone — by symmetry, not accident — and the next rung leads. (This is the anti-Helmholtz/MOT geometry from the catalog article, now read through the ladder.)

Step 2. Superpose two displaced dipoles. Far on the axis, each loop is a valid dipole ( $z\gg R,d$ ): $B(z)=\frac{\mu_0m}{2\pi}\left[\frac{1}{(z-d/2)^3}-\frac{1}{(z+d/2)^3}\right]$ .

Step 3. Expand to leading order. With $(z\mp d/2)^{-3}\approx z^{-3}(1\pm\tfrac{3d}{2z})$ :

B(z)\approx\frac{\mu_0 m}{2\pi z^3}\cdot\frac{3d}{z}=\frac{3\mu_0\,md}{2\pi z^4}

— a $1/z^4$ law: the axial signature of a magnetic quadrupole, whose strength is the dipole pair's moment times its lever arm, $md$ .

Step 4. Numbers and audit. $B\approx3\times(4\pi\times10^{-7})\times1.57\times10^{-2}\times0.10/(2\pi\times1)$ = 0.94 nT, versus the unexpanded two-dipole sum $0.95$ nT (−0.8%, consistent with the next correction $O(d^2/z^2)$ ) ✓. Contrast a single loop at the same distance: 3.1 nT falling as $1/z^3$ — an order of magnitude of suppression already, growing with distance. The ladder's practical moral: to hide a magnet, cancel its lowest moments; to detect a hidden one, look for the residual higher-rung falloff. (Magnetically "quiet" instrument platforms and compensated research magnets are engineered exactly this way.)

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.4.3 (Magnetic Dipole) and §6.1 · Jackson, Classical Electrodynamics, 3rd ed., §5.6 (Magnetic Multipole Expansion) · Zangwill, Modern Electrodynamics, §11.1–11.2 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §5.4.3 (Eqs. 5.83–5.89) and Jackson Classical Electrodynamics 3rd ed. §5.6 (Eqs. 5.50–5.57). A verifier should reproduce: (1) the Legendre generating-function expansion of $|\mathbf{r}-\mathbf{r}'|^{-1}$ ; (2) the symmetric/antisymmetric split yielding $\mathbf{m}=\tfrac12\int\mathbf{r}'\times\mathbf{J}\,\mathrm{d}^3r'$ ; (3) the curl computation giving $\mathbf{B}_{\mathrm{dip}}$ ; (4) the standard constants $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ , $\mu_B=9.274\times10^{-24}\,\mathrm{J/T}$ , $\mu_N=5.051\times10^{-27}\,\mathrm{J/T}$ (CODATA). The $\frac{2}{3}$ contact-term coefficient is from Jackson Eq. (5.64). (Scope note: items (3)–(4) — the dipole field, its curl derivation and contact term — are owned by the next article per the chapter's division of labor; this article carries the expansion, the moment, and origin-independence. Worked numbers: $m=1.57\times10^{-2}\ \mathrm{A\,m^2}$ , 25.1 vs 24.8 nT (+1.5% $=\tfrac32(R/z)^2$ ), opposed pair 0.94 vs 0.95 nT, Earth 31 μT — recomputed.)

Builds on: Magnetic Vector Potential · Biot-Savart Law · Continuity Equation · Electric Current and Current Density · Divergence of B and Absence of Magnetic Monopoles

Leads to: Magnetic Dipole and Magnetic Moment · Magnetic Field of Steady Current Configurations · Lorentz Force Law

Is the vanishing of the monopole term the same thing as saying magnetic monopoles don't exist?

Same physics, sharper logical status. "No magnetic monopoles" as a law of nature is the empirical statement $\nabla\cdot\vec B=0$ , tested by searches that have all come up empty. The vanishing of the ladder's $n=0$ term is a theorem about steady currents: given that magnetic fields are made by currents (Biot–Savart), the $1/r$ term integrates to zero — no experiment needed. The two connect because Biot–Savart itself encodes $\nabla\cdot\vec B=0$ (the divergence article proved that), so the theorem is the law's far-field shadow. If a genuine monopole were ever found, both would fail together: $\nabla\cdot\vec B$ would acquire a source term and far fields would gain a Coulombic $q_m/r^2$ rung below the dipole.

Why is the magnetic moment defined with that factor of one half?

The half belongs to the geometry, not the physics: it is the triangle in "swept area." The element $\tfrac12\,\vec r'\times\mathrm{d}\vec\ell'$ is exactly the vector area of the thin triangle with vertices at the origin, $\vec r'$ , and $\vec r'+\mathrm{d}\vec\ell'$ — base $\mathrm{d}\vec\ell'$ , lever arm $\vec r'$ , and the cross product's magnitude is base times height, i.e., twice the triangle. Summing triangles around the circuit tiles the enclosed surface once, so $\tfrac12\oint\vec r'\times\mathrm{d}\vec\ell'=\vec a$ lands precisely on the enclosed area with no stray factor. The volume definition $\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'$ inherits the same half so that the two dialects agree on their common cases.

For a non-planar loop, what does the vector area even mean?

It is the projection rule. Take a warped circuit — a coil bent over a cylinder, say — and compute $\vec a=\tfrac12\oint\vec r'\times\mathrm{d}\vec\ell'$ . Component by component, $a_z$ equals the area of the loop's shadow on the $xy$ -plane, and likewise for the others: the vector area is the triple of shadow areas, independent of which spanning surface you imagine stretched across the loop. It is generally smaller in magnitude than the actual surface area (shadows foreshorten), and two visually different warped loops with equal shadows have identical dipole moments and identical far fields. This is also why a flat estimate often suffices in practice: only the projections enter $\vec m$ .

The electric dipole moment depends on origin choice but the magnetic one doesn't — what's the real reason?

One principle covers both: the lowest nonvanishing multipole moment is origin-independent; everything above it shifts. Shifting the origin mixes each moment with lower ones (the electric shift theorem worked this out rung by rung: $\vec p'=\vec p-Q\vec c$ , quadrupole mixing with dipole and monopole, and so on). The electric dipole sits on the second rung, so it inherits contamination from the first whenever $Q\neq0$ . The magnetic dipole also sits on the second rung — but magnetism's first rung is structurally empty, so there is nothing to mix in, ever. Note the fine print carried over from the electric article: the magnetic quadrupole is origin-dependent whenever $\vec m\neq0$ , by exactly the same mixing.

How good is the dipole approximation for a real, messy source like a motor or the Earth?

Set by $r'/r$ with the source's effective size as $r'$ . For the Earth (core radius $\approx3500$ km, field measured at $r\approx6400$ km) the ratio is $\sim0.55$ — not small! — which is why the IGRF needs a dozen harmonic degrees at the surface: the dipole is only $\sim90\%$ of the story locally, with quadrupole and octupole carrying real structure like the South Atlantic Anomaly. Climb to geostationary orbit and the ratio drops to $\sim0.08$ ; the field is dipole to a percent. For a bench motor ( $r'\sim5$ cm) at 1 m, corrections are percent-level. Rule of thumb from the worked example: on-axis error $\approx\tfrac32(r'/r)^2$ for a simple loop; for generic geometry assume first-power $r'/r$ until symmetry proves better.

If the dipole term of A vanishes on the moment's axis, why is the field strongest there?

Because the field is the curl of the potential, and a curl reads spatial variation, not magnitude. On the axis $\vec A_{\rm dip}\propto\vec m\times\hat r=0$ — the potential circulates around the axis like water around a drain, and at the drain itself the circulation velocity is zero while the swirl (the curl) is maximal. The same lesson appeared inside the solenoid, where $\vec A$ vanished on the centerline while $\vec B$ was full strength, and it is a good working antidote to reading $\vec A$ 's magnitude as if it were field strength. Where $\vec A_{\rm dip}$ is largest — the equatorial plane at fixed $r$ — the field is actually at its weakest ( $\tfrac12$ the polar value, as the next article's anatomy will show).

When do the quadrupole and higher terms actually matter in practice?

Three standard triggers. First, symmetry kills the dipole: opposed coil pairs (the worked example, and every anti-Helmholtz trap), certain nuclei whose spin structure forbids a dipole but allows a quadrupole, and deliberately compensated magnets — then the quadrupole is the far field. Second, precision at moderate distance: geomagnetic navigation, beam-optics multipole magnets (quadrupoles focus, sextupoles correct chromaticity — accelerator lattices are the ladder built in hardware), and gradiometry, which differentiates away the dipole to expose higher structure. Third, radiation: an oscillating source radiates through every multipole channel, and when selection rules close the dipole channel (as in many atomic and nuclear transitions), quadrupole radiation sets the lifetime — the radiation chapter's ladder is this one with time dependence switched on.