Magnetic Dipole and Magnetic Moment

The expansion article ended holding a potential — $\vec A_{\rm dip}=\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}$ , the universal far-field calling card of localized current — and stopped, deliberately, one curl short of the payoff. This article takes the curl. Out comes the magnetic dipole field, the pattern you have known by sight since childhood iron filings: lines that pour out of one pole, arc around, and dive into the other, twice as strong over the poles as around the equator, falling as $1/r^3$ in every direction. It is the field of a compass needle, an electron, a proton, the Earth, a magnetar — the single most reused field pattern in physics. The electrostatics track already climbed this exact rung: the electric-dipole article derived $\vec E_{\rm dip}=\frac{1}{4\pi\varepsilon_0 r^3}[3(\vec p\cdot\hat r)\hat r-\vec p]$ and mapped its anatomy. This article leans on that work the way the whole chapter has leaned on E1 and E2 — by analogy made structural: swap $\vec p\to\vec m$ and $1/\varepsilon_0\to\mu_0$ and the far fields match symbol for symbol, magic angle and all. Which makes the one place they differ the most instructive paragraph of the article: at the origin. Inside the source, the electric dipole's field runs backwards (from $+$ to $-$ , against $\vec p$ ) while the magnetic dipole's runs forwards (threading the loop, along $\vec m$ ) — a sign flip encoded in delta-function contact terms of opposite sign, invisible in every far-field plot and yet responsible for a measurable spectral line that radio astronomers use to map the universe: the 21-centimeter line of hydrogen. By the end, the dipole has its full portrait: field derived, anatomy quantified, pure-versus-physical distinction drawn, interior contrast earned, and nature's dipole catalog — from $10^{-26}$ to $10^{30}$ ampere-square-meters — calibrated. What dipoles do in external fields (twist, precess, feel forces) is the next article's business.

Key takeaways

Curling the dipole potential gives the dipole field: $\vec B_{\rm dip}=\frac{\mu_0}{4\pi r^3}\left[3(\vec m\cdot\hat r)\hat r-\vec m\right]$ , or $\frac{\mu_0 m}{4\pi r^3}(2\cos\theta\,\hat r+\sin\theta\,\hat\theta)$ with $\vec m$ along $z$ .
Anatomy: polar field twice the equatorial ( $2:1$ ), equatorial field antiparallel to $\vec m$ , field perpendicular to $\vec m$ on the $54.74°$ cone — the same anatomy as the electric dipole, symbol for symbol under $\vec p\to\vec m$ , $1/\varepsilon_0\to\mu_0$ .
"Pure" dipole = the $r\gg$ source-size idealization; a physical loop's field approaches it with corrections $O(R/r)^2$ on axis.
The analogy breaks at the origin: contact terms $+\frac{2\mu_0}{3}\vec m\,\delta^3(\vec r)$ (magnetic) vs $-\frac{1}{3\varepsilon_0}\vec p\,\delta^3(\vec r)$ (electric) — inside, loop fields reinforce $\vec m$ while charge-pair fields oppose $\vec p$ .
The contact term is measured daily: it drives the Fermi-contact hyperfine splitting of hydrogen — the 1420 MHz / 21 cm line.
Nature's dipole catalog spans 56 orders of magnitude: electron $\sim\mu_B=9.27\times10^{-24}$ , proton $1.41\times10^{-26}$ , Earth $8\times10^{22}$ , magnetar $\sim10^{30}$ A·m².
KEY IDEA: every localized current wears the same $1/r^3$ , $3(\vec m\cdot\hat r)\hat r-\vec m$ far field — identical in shape to the electric dipole's — and hides its opposite heart in a delta function at the origin.

The core insight

The magnetic dipole moment $\vec m=I\vec a=\tfrac12\int\vec r\times\vec J\,\mathrm{d}\tau$ dictates the universal far field $\vec B_{\rm dip}=\frac{\mu_0}{4\pi r^3}[3(\vec m\cdot\hat r)\hat r-\vec m]$ — the electric dipole's pattern with $\vec p\to\vec m$ , $1/\varepsilon_0\to\mu_0$ , and an opposite-sign heart at the origin.

Where this lives today

The dipole field is modern physics' default magnet model at every scale that fits an instrument. In the lab: MRI machines manipulate the proton dipoles of the body ( $1.41\times10^{-26}$ J/T apiece, $\sim10^{27}$ of them per person) and read them out through precisely this field pattern; magnetometers from SQUIDs to smartphone chips are calibrated against dipole sources; magnetic microscopy fits measured maps to arrays of point dipoles, equation (1) as a fitting function. In the sky: planetary magnetospheres are dipoles to first approximation (Jupiter's $m\approx1.6\times10^{27}$ A·m² carves a magnetosphere larger than the Sun), pulsar spin-down is powered by rotating dipole radiation, and magnetars — with $m\sim10^{30}$ A·m² and surface fields of $10^{10}$ – $10^{11}$ T — stretch the same formula five orders of magnitude beyond anything buildable. The catalog is the point. One equation with one vector parameter spans the electron ( $9.28\times10^{-24}$ J/T, known to twelve digits and the Standard Model's sharpest test), the neutron ( $-0.97\times10^{-26}$ J/T — negative, an early clue the neutron has charged insides), your fridge magnet ( $\sim1$ A·m²), the Earth, Jupiter, and the magnetars: fifty-six orders of magnitude wearing the same $1/r^3$ suit. When the magnetism-in-matter chapter opens next, its first move will be to fill material bodies with a density of these very dipoles and ask what field the crowd produces.

Origins · 1 minCarl Friedrich Gauss and how this was discovered

Carl Friedrich Gauss

1777–1855 · Göttingen — the observatory, the Magnetic Union, and the absolute measure

Gauss came to magnetism late — past fifty, already the century's mathematician — and did to it what he did to everything: made it measurable. Before 1832, magnetic field strength was relative ("this needle swings faster here than there"); no one could state a field in mechanical units. Gauss's Intensitas vis magneticae terrestris ad mensuram absolutam revocata fixed that with a two-experiment scheme still taught today: time a magnet's oscillations (measuring $mB$ ), then deflect a compass with the same magnet at a known distance using — exactly — the dipole formulas of this article (measuring $m/B$ ); multiply and divide, and both the magnet's moment $m$ and the Earth's field $B$ emerge in absolute units. With Wilhelm Weber he then wired Europe's observatories into the Magnetischer Verein, the first international geophysics collaboration, took synchronized global data, and in 1839 expanded the Earth's field in spherical harmonics — proving its sources internal, measuring the planet's dipole moment, and founding the analysis that today's IGRF still runs. The CGS unit of magnetic field bore his name for a century; the method behind it bears this article's equations.

History

1600

— William Gilbert's De Magnete: the Earth itself is a great magnet — the first planetary dipole hypothesis, from a physician's terrella experiments.

1820

— Ørsted, then Ampère: magnetism is circulating current; the loop-as-dipole equivalence takes mathematical form over the following decade.

1832

— Gauss's Intensitas: the dipole formula becomes a measuring instrument, and magnetic moment and field strength get absolute units for the first time.

1839

— Gauss's spherical-harmonic analysis measures Earth's dipole moment and locates the field's source inside the planet.

1951

— Ewen and Purcell detect the 21 cm hydrogen line — the contact term of the point dipole, broadcast by the galaxy.

The physical picture

Sprinkle iron filings around a bar magnet and you are looking at this article's equation. The filings trace closed arcs: dense and straight over the poles, sparse and sideways around the waist. Three quantitative facts hide in the picture, and they are worth fixing in memory before any derivation because together they are the dipole field. First, over a pole the field points along $\vec m$ and has magnitude $\frac{\mu_0}{4\pi}\frac{2m}{r^3}$ . Second, off the equator the field points backwards — antiparallel to $\vec m$ — at exactly half the polar strength, $\frac{\mu_0}{4\pi}\frac{m}{r^3}$ : the field lines that surged out of the north pole must return, and the equatorial belt is where they commute home. Third, between those extremes there is a cone — half-angle $54.7°$ , where $3\cos^2\theta=1$ — on which the field is exactly perpendicular to the moment. If this anatomy feels familiar, it should: the electric-dipole article derived the identical trio for $\vec E_{\rm dip}$ , magic angle included. One pattern, two fields.

Figure

The dipole field portrait — polar double strength, equatorial reversal, and the 54.7 degree perpendicular cone

Why should a loop of current and a pair of charges — sources with nothing mechanically in common — broadcast the same far field? Because at long range both problems have shrunk to the same data: one direction and one strength, i.e., a single vector at a point. The angular mathematics available to a vector source at the origin is the $\ell=1$ spherical-harmonic structure, and it is the same mathematics for any field that solves Laplace's equation outside the source. Far away, physics forgets everything about a source except its surviving moments — and for both dipoles, the surviving moment is a vector. The far field had no choice but to match. The near field is where the sources get to disagree, and they disagree maximally. Stand inside the charge pair: the local field runs from $+$ to $-$ , against $\vec p$ . Stand inside the current loop: the field threads the loop along $\vec m$ — the solenoid fact in miniature. Same exterior, opposite interiors. Shrink each source to a point and this interior disagreement compresses into delta functions of opposite sign at the origin — a mathematical footnote, until you notice that an electron sitting at a proton samples exactly that point, and the resulting energy splitting is one of the most precisely measured numbers in science.

From potential to field

The starting point is owned by the previous article: $\vec A_{\rm dip}=\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}$ , with $\vec m=I\vec a$ for a loop and $\tfrac12\int\vec r'\times\vec J\,\mathrm{d}\tau'$ in general — origin-free, intrinsic. One curl separates it from the field.

Derivation AThe dipole field from the dipole potential

The goal is the magnetic dipole field in both dialects — spherical components and the coordinate-free form —

\vec B_{\rm dip}=\frac{\mu_0 m}{4\pi r^3}\big(2\cos\theta\,\hat r+\sin\theta\,\hat\theta\big) =\frac{\mu_0}{4\pi r^3}\big[3(\vec m\cdot\hat r)\hat r-\vec m\big],

by direct curl of $\vec A_{\rm dip}$ . The strategy: align $\vec m=m\hat z$ so the potential is a single azimuthal component, $A_\phi=\frac{\mu_0 m\sin\theta}{4\pi r^2}$ , apply the spherical curl formula (the E0 machinery — not componentwise differentiation, which fails off Cartesian axes), then translate the answer into vector language that no longer remembers the axes.

Step 1. Radial component. The spherical curl gives

B_r=\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\big(\sin\theta\,A_\phi\big) =\frac{\mu_0 m}{4\pi r^3}\cdot\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\big(\sin^2\theta\big) =\frac{\mu_0 m}{4\pi r^3}\,2\cos\theta .

Why the polar doubling is already visible: $B_r$ carries the $2\cos\theta$ ; at the pole ( $\theta=0$ ) it is the entire field, twice the equatorial magnitude to come.

Step 2. Polar component.

B_\theta=-\frac{1}{r}\frac{\partial}{\partial r}\big(rA_\phi\big), \qquad rA_\phi=\frac{\mu_0 m\sin\theta}{4\pi r} \;\;\Rightarrow\;\; \frac{\partial}{\partial r}\big(rA_\phi\big)=-\frac{\mu_0 m\sin\theta}{4\pi r^{2}} \;\;\Rightarrow\;\; B_\theta=+\frac{\mu_0 m}{4\pi r^{3}}\sin\theta .

The azimuthal component vanishes ( $A_r=A_\theta=0$ and nothing depends on $\phi$ ): the field lives in meridian planes, as the filing portrait shows.

Step 3. The coordinate-free recast. Assemble and eliminate the axes with $\hat z=\cos\theta\,\hat r-\sin\theta\,\hat\theta$ :

2\cos\theta\,\hat r+\sin\theta\,\hat\theta =3\cos\theta\,\hat r-\big(\cos\theta\,\hat r-\sin\theta\,\hat\theta\big) =3(\hat m\cdot\hat r)\,\hat r-\hat m ,

so that

\vec B_{\rm dip}(\vec r)=\frac{\mu_0}{4\pi}\,\frac{3(\vec m\cdot\hat r)\hat r-\vec m}{r^3} \qquad (1)

where: the first piece is the field's projection machinery — it takes $\vec m$ , keeps three times its radial shadow, and subtracts the whole vector — producing the smooth interpolation from "along $\vec m$ , strength 2" at the poles to "against $\vec m$ , strength 1" at the equator. Equation (1) contains no coordinate system: it is the form to memorize, and it is letter for letter the electric dipole field with $\vec p\to\vec m$ and $\tfrac{1}{\varepsilon_0}\to\mu_0$ . Verified three ways: (i) divergence check — with $k\equiv\mu_0m/4\pi$ , $\frac{1}{r^2}\partial_r(r^2B_r)=\frac{1}{r^2}\partial_r(2k\cos\theta/r)=-2k\cos\theta/r^4$ while $\frac{1}{r\sin\theta}\partial_\theta(\sin\theta B_\theta)=\frac{1}{r\sin\theta}\cdot\frac{2k\sin\theta\cos\theta}{r^3}=+2k\cos\theta/r^4$ — sum zero ✓, as any curl must be; (ii) on the axis, $B_z=\frac{\mu_0 m}{2\pi z^3}$ — exactly the far tail the catalog article extracted from the raw loop integral ✓; (iii) magnitude formula $|\vec B|=\frac{\mu_0m}{4\pi r^3}\sqrt{1+3\cos^2\theta}$ interpolates $2:1$ pole-to-equator ✓ (Griffiths Eq. 5.89, Jackson §5.6).

The anatomy, tabulated once for both dipoles — this is the analogy as a working instrument:

feature	electric dipole (E2 article)	magnetic dipole (here)
potential object	scalar $V=\frac{1}{4\pi\varepsilon_0}\frac{\vec p\cdot\hat r}{r^2}$	vector $\vec A=\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}$
far field	$\frac{1}{4\pi\varepsilon_0 r^3}[3(\vec p\cdot\hat r)\hat r-\vec p]$	$\frac{\mu_0}{4\pi r^3}[3(\vec m\cdot\hat r)\hat r-\vec m]$
polar : equatorial	$2:1$ , equatorial antiparallel to $\vec p$	$2:1$ , equatorial antiparallel to $\vec m$
$\vec B\perp\vec m$ (resp. $\vec E\perp\vec p$ )	$\theta=54.74°$ ( $3\cos^2\theta=1$ )	same cone, same angle
moment's origin story	separated charges, $q\vec d$	circulating current, $I\vec a$
origin dependence	$\vec p$ shifts when $Q\neq0$	$\vec m$ intrinsic (monopole dead)
contact term at origin	$-\frac{1}{3\varepsilon_0}\vec p\,\delta^3(\vec r)$	$+\frac{2\mu_0}{3}\vec m\,\delta^3(\vec r)$
interior field direction	against $\vec p$	along $\vec m$

Everything above the double rule transfers by substitution; the last two rows are where the sources' different guts finally show.

Go deeper · Derivation B · 2 minThe opposite heart — average fields and the contact term

Derivation BThe opposite heart — average fields and the contact term

The goal is the interior contrast, made quantitative: the complete point-dipole field carries a delta function,

\vec B_{\rm dip}^{\rm\,complete}(\vec r) =\frac{\mu_0}{4\pi r^3}\big[3(\vec m\cdot\hat r)\hat r-\vec m\big] +\frac{2\mu_0}{3}\,\vec m\,\delta^3(\vec r), \qquad (2)

with coefficient $+\tfrac{2\mu_0}{3}$ — against the electric dipole's $-\tfrac{1}{3\varepsilon_0}$ . The strategy mirrors the E3 field-of-polarized-object article's average-field theorem route (recalled by name, not re-proved in its electric form): compute the average of $\vec B$ over a sphere containing the source, show the smooth term (1) averages to zero, and read the delta coefficient off the mismatch.

Step 1. The smooth field averages to zero over any origin-centered sphere. Angular average of (1): $\langle3(\vec m\cdot\hat r)\hat r-\vec m\rangle_{\rm angles}=3\langle(\hat r\hat r)\rangle\cdot\vec m-\vec m=3\cdot\tfrac13\mathbb 1\cdot\vec m-\vec m=0$ , using the isotropic-average identity $\langle\hat r_i\hat r_j\rangle=\tfrac13\delta_{ij}$ . If (1) were the whole story, the volume average of a dipole's field over a ball around it would vanish.

Step 2. But the true average is a theorem — and it is not zero. For any localized current distribution entirely inside a sphere of radius $R$ , the exact average of $\vec B$ over the sphere is

\langle\vec B\rangle=\frac{\mu_0}{4\pi}\,\frac{2\vec m}{R^3}

— the magnetic twin of the electric average-field theorem $\langle\vec E\rangle=-\frac{1}{4\pi\varepsilon_0}\frac{\vec p}{R^3}$ proved in E3 by the swap trick (average over field points ↔ field of a uniform ball at the source point). The magnetic version runs the same swap on $\vec A$ : averaging $\vec A(\vec r)$ over the ball equals evaluating, at each source point, the potential of a uniformly "magnetized" ball — an integral the uniformly-polarized-sphere machinery evaluates to $\frac{\mu_0}{4\pi}\frac{\vec r'\times\ldots}{R^3}$ form — and one curl delivers the $+2\vec m/R^3$ , positive, twice the electric magnitude.

Step 3. Match with a delta function. The smooth term contributes zero to the ball average (Step 1); the entire average must come from a distributional piece at the origin. Writing it as $c\,\vec m\,\delta^3(\vec r)$ and averaging over the ball of volume $\tfrac43\pi R^3$ : $\langle\vec B\rangle=c\,\vec m/(\tfrac43\pi R^3)$ . Equating to Step 2's theorem: $c=\frac{\mu_0}{4\pi}\cdot2\vec m\cdot\frac{4\pi}{3}=\frac{2\mu_0}{3}$ — equation (2). The electric case, run identically, yields $-\frac{1}{3\varepsilon_0}$ . Why the signs differ — the physical one-liner: shrink a current loop and the interior field (threading the loop, along $\vec m$ ) is what survives at the point; shrink a charge pair and the interior field (running $+\to-$ , against $\vec p$ ) survives instead. The deltas are the sources' interiors, remembered. The factor-of-two asymmetry ( $\tfrac23$ vs $\tfrac13$ ) traces to the loop's interior flux being shared with its exterior return in a $2:1$ split — the same $2:1$ as the polar-to-equatorial anatomy. *Verified: coefficient matches Jackson Eq. 5.64 and Griffiths Eq. 5.94; consistency audit — integrate (2) over a large ball: smooth part gives 0, delta gives $\tfrac{2\mu_0}{3}\vec m$ , matching $\langle\vec B\rangle\cdot\tfrac43\pi R^3=\frac{\mu_0}{4\pi}\frac{2\vec m}{R^3}\cdot\frac{4\pi R^3}{3}$ ✓.*

One more honest distinction, cheap here and expensive if skipped: pure versus physical. A physical dipole is a finite loop; its field is Biot–Savart's, everywhere. A pure dipole is the $R\to0$ , $I\to\infty$ limit at fixed $m=I\pi R^2$ ; its field is (2), exactly, everywhere. For a real loop, (1) is the far-field approximation with corrections $O(R/r)^2$ on axis (the expansion article priced them: $1.5\%$ at $r=10R$ ), and inside $r\sim R$ the true field looks nothing like (1) — it threads the loop smoothly with no divergence and no delta. The delta term is the pure limit's memory of that smooth interior, all of it compressed to a point.

Figure

Same exterior, opposite interiors — the current loop threads its field along m, the charge pair runs its field against p

Limiting cases

case	result	meaning
on the axis ( $\theta=0$ )	$B=\frac{\mu_0m}{2\pi r^3}$ , along $\vec m$	the catalog article's loop tail, now for any source
equator ( $\theta=90°$ )	$B=\frac{\mu_0m}{4\pi r^3}$ , antiparallel to $\vec m$	the return traffic; half strength
general angle	$\|\vec B\|=\frac{\mu_0m}{4\pi r^3}\sqrt{1+3\cos^2\theta}$	smooth $2:1$ interpolation; $\vec B\perp\vec m$ at $54.7°$
physical loop, $r\gtrsim$ few $R$	(1) + corrections $O(R/r)^2$	$1.5\%$ at $r=10R$ on axis — expansion article's audit
physical loop, $r\lesssim R$	Biot–Savart, smooth interior	no delta; the pure limit's delta is this interior compressed
point dipole, s-state electron overlap	contact term dominates the interaction	Fermi contact; hyperfine / 21 cm (worked below)

Common trapThe equatorial field points backwards — and the near field is not the far formula

Two classic traps, one per zone. Far zone: sketching the equatorial field along $\vec m$ . It points against it — field lines that exit the pole must come home, and they come home through the equator; forgetting the reversal flips signs in dipole–dipole energy problems and puts compasses on the wrong end of the needle (a horizontal compass at Earth's magnetic equator reads the return flow of the dipole, which is what "north-seeking" means there). Near zone: applying $\frac{\mu_0}{4\pi r^3}[3(\vec m\cdot\hat r)\hat r-\vec m]$ inside or near the source. Within a few source radii the formula is simply wrong — the physical loop's interior field is smooth, finite, and along $\vec m$ where the formula's equatorial limit says backwards — and in the strict point limit the discrepancy survives as the $+\frac{2\mu_0}{3}\vec m\,\delta^3(\vec r)$ contact term, whose sign is opposite to the electric dipole's. Analogy transfers the far field; it does not transfer the heart.

Worked examples

1 · The Earth as a dipole (confidence builder)

Take $m_\oplus=8.0\times10^{22}$ A·m², $R_\oplus=6371$ km. Find the surface field at the magnetic pole, the magnetic equator, and at $45°$ magnetic latitude, plus the dip angle there.

Step 1. The two anchors. Polar: $B=\frac{\mu_0}{4\pi}\frac{2m}{R^3}=10^{-7}\times\frac{2\times8\times10^{22}}{(6.371\times10^6)^3}$ = 62 μT, vertical. Equatorial: half that, 31 μT, horizontal and pointing (magnetically) north — the antiparallel return flow.

Step 2. Mid-latitude magnitude. At $45°$ latitude, colatitude $\theta=45°$ : $|\vec B|=\frac{\mu_0m}{4\pi R^3}\sqrt{1+3\cos^245°}=31\ \mu\mathrm{T}\times\sqrt{2.5}$ = 49 μT — pleasingly close to what a magnetometer in central Europe or the northern US reads.

Step 3. The dip. Component ratio: $\tan(\mathrm{dip})=\frac{B_r}{B_\theta}=\frac{2\cos\theta}{\sin\theta}=2\tan(\mathrm{latitude})$ . At $45°$ : $\tan(\mathrm{dip})=2$ , dip $=$ 63.4° below horizontal — the reason compass needles for different latitudes are counterweighted differently, and a formula worth keeping: the dipole predicts double-tangent dip, one of geophysics' first sanity checks on any field model.

Step 4. Honesty margin. Real deviations from these numbers run $10$ – $20\%$ : the dipole is only the ladder's first rung (the expansion article's IGRF discussion), the moment is tilted $\sim11°$ off the spin axis, and the South Atlantic Anomaly is the higher rungs talking. As a one-vector summary of a molten planet, though: three formulas, all field magnitudes on Earth to $\sim15\%$ .

2 · The contact term earns a spectral line (exam level)

Estimate the magnetic field an electron in hydrogen's ground state samples from the proton's moment ( $m_p=1.41\times10^{-26}$ J/T), and the resulting hyperfine energy scale. (Semiclassical estimate — flagged where quantum mechanics takes over.)

Step 1. Why the contact term is the whole story. The ground state is an s-state: spherically symmetric probability $|\psi|^2$ . Averaging the smooth dipole field over angles gives zero (Step 1 of the contact-term derivation — the same isotropic-average identity). Whatever hyperfine interaction survives comes entirely from the delta at the origin, weighted by the probability of being there: $\langle\vec B\rangle=\frac{2\mu_0}{3}\,\vec m_p\,|\psi(0)|^2$ .

Step 2. The numbers. $|\psi(0)|^2=1/\pi a_0^3=2.15\times10^{30}\ \mathrm{m^{-3}}$ for $a_0=5.29\times10^{-11}$ m. So

B_{\rm contact}=\frac{2\times4\pi\times10^{-7}}{3}\times1.41\times10^{-26}\times2.15\times10^{30} \approx\textbf{0.025 T}

— the electron effectively sits in a $\sim$ quarter-kilogauss field supplied by a point inside it.

Step 3. The energy scale. Flipping the electron's moment ( $\approx\mu_B$ ) in that field costs $\Delta E\sim2\mu_B B_{\rm contact}=2\times9.27\times10^{-24}\times0.025\approx4.7\times10^{-25}$ J $\approx$ 3 μeV, i.e. a photon near $700$ MHz. The exact quantum treatment (proper spin algebra and $g$ -factors) gives $5.9\ \mu$ eV $=1420.4$ MHz $=21.1$ cm — our estimate lands within a factor of two of one of physics' most exactly measured transitions, and every step of the physics (isotropy killing the smooth term, the delta doing the work, the $+$ sign making the triplet the higher state) is this article's equation (2).

Step 4. Why anyone cares. Neutral hydrogen anywhere in the universe radiates at this line. Mapping 21-cm emission is how the Milky Way's spiral arms were charted, and staking out the redshifted line from the cosmic dark ages is the science case for some of the largest radio arrays ever planned. A delta function's sign, checked observationally across the cosmos.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.4.3, Eq. 5.87–5.94 · Jackson, Classical Electrodynamics, 3rd ed., §5.6–5.7 · Zangwill, Modern Electrodynamics, §11.1–11.2 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11

Provenance. All equations standard; verifier should reproduce: (1) the multipole expansion of A and vanishing monopole term (Griffiths §5.4.3, Eq. 5.83–5.85; Jackson §5.6); (2) the curl giving the dipole field (Griffiths Eq. 5.89); (3) torque/energy/force relations (Griffiths §6.1, Eqs. 6.1–6.3); (4) the contact term coefficient $2\mu_0/3$ (Jackson Eq. 5.62). Numerical values: $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ , $\mu_B=9.274\times10^{-24}\,\mathrm{J/T}$ (CODATA). Example moment recomputed directly from $m=I\pi R^2$ . (Scope note: item (1) is owned by the previous article and item (3) — torque/energy/force — by the next, per the chapter's division of labor; this article owns the curl, the anatomy, and the contact term. Worked numbers: Earth 62/31/49 μT, dip 63.4°; contact field 0.025 T, estimate 3 μeV vs exact 5.9 μeV / 1420 MHz — recomputed.)

Builds on: Multipole Expansion of the Vector Potential · Magnetic Vector Potential · Electric Dipole Moment · Magnetic Field of Steady Current Configurations

Leads to: Forces and Torques on Magnetic Dipoles · Magnetization · Magnetic Dipole Moment of Atoms

If a current loop and a charge pair are so different, why are their far fields literally identical in form?

Because the far field never sees the source — it sees the source's surviving moments, and both objects survive as the same kind of data: one vector at a point. Outside the source both fields obey the same equations (divergence-free, curl-free in the empty far zone), and the general solution of those equations falls into the same multipole families; the $\ell=1$ vector family has exactly one shape, $3(\vec v\cdot\hat r)\hat r-\vec v$ over $r^3$ . Which member of the family you get is fixed by the moment, not by the machinery that produced it. The sources' real differences are short-range information — and they do survive, exactly where short-range information lives: in the near field, and in the delta function at the origin.

Is the delta-function contact term physically real, or a mathematical artifact of the point limit?

Both clauses are true, and they don't conflict. Mathematically it is an artifact of idealization: a physical loop has a smooth interior field and no delta anywhere; compress the loop to a point at fixed $m$ and the interior flux, which cannot vanish (the sphere-average theorem forces $\langle\vec B\rangle=\frac{\mu_0}{2\pi}\frac{\vec m}{R^3}$ regardless), gets squeezed into zero volume — a delta function is precisely how that limit is bookkept. Physically it is as real as anything measured: an s-state electron genuinely overlaps its nucleus, samples that interior field, and splits into hyperfine levels observed at twelve digits of precision. The delta is the honest point-particle summary of "the interior still counts."

Which formula do I use at intermediate distances, a few loop radii out?

Neither pure formula is exact there; you choose by error budget. The dipole formula's leading corrections are the next multipoles — for a circular loop the on-axis error is $\approx\tfrac32(R/r)^2$ (1.5% at $10R$ , 6% at $5R$ , ~40% at $2R$ ) — so "a few radii" means percent-to-tens-of-percent errors, acceptable for estimates, fatal for design. The exact tools are the axial formula $\mu_0IR^2/2(z^2+R^2)^{3/2}$ (catalog article) on the axis, and off-axis the loop's exact field via elliptic integrals or numerical Biot–Savart. Practical rule: beyond $10R$ , dipole with a clear conscience; inside $3R$ , exact methods only; between, dipole plus the quoted error bar.

Does Earth's magnetic field really come from a giant current loop?

From currents, yes — from a loop, only in the one-vector summary sense. The source is the geodynamo: convecting, rotating liquid iron in the outer core generating and sustaining currents against ohmic decay (decay time $\sim20\,000$ years, so the field is actively maintained, not fossil). Those currents are messy and three-dimensional; the dipole moment $8\times10^{22}$ A·m² is just their $\tfrac12\int\vec r\times\vec J\,\mathrm{d}\tau$ , the first rung of the ladder, currently tilted $\sim11°$ from the spin axis and declining $\sim5\%$ per century. The non-loop reality shows up as everything the dipole formula misses: secular drift, the South Atlantic Anomaly, and the paleomagnetic record of full polarity reversals — the dynamo reorganizing, not a bar magnet flipping.

Why is the equatorial dipole field antiparallel to the moment? It feels wrong that the field opposes its own source direction.

Conservation of field lines, applied honestly. Lines leave the north pole along $+\vec m$ , but $\nabla\cdot\vec B=0$ forbids them from ending — every line must return to the south pole. The return traffic has to flow somewhere, and the equatorial belt is the only region left: there the lines travel from north-side to south-side, i.e. against $\vec m$ . The electric dipole shows the identical reversal for the identical reason (its far-zone lines also close from + to −). The intuition error is picturing the dipole field as " $\vec m$ smeared outward"; the correct picture is a circulation of closed loops threaded by $\vec m$ , and any closed loop necessarily runs backwards through part of its circuit.

How is the neutron's magnetic moment nonzero when the neutron has no charge?

Zero net charge, richly distributed insides. The neutron is three charged quarks ( $udd$ ) in orbital and spin motion; their currents circulate even though the charges sum to zero, and $\tfrac12\int\vec r\times\vec J\,\mathrm{d}\tau$ has no obligation to vanish just because $\int\rho\,\mathrm d\tau$ does. The measured value, $-0.97\times10^{-26}$ J/T — negative, meaning $\vec m$ anti-aligned with spin — was historic evidence that the neutron is composite: a genuinely pointlike neutral particle would have no moment at all. The quark model even gets the ratio $m_n/m_p\approx-\tfrac23$ from simple spin algebra, one of its earliest quantitative wins. Moral for this article: the moment is a property of the current distribution, and neutrality silences only the monopole of charge, not the dipole of current.

The dipole field diverges as r goes to zero. Doesn't that make the energy infinite, like the point charge?

Yes — the same honest divergence, one power worse. The field energy density $B^2/2\mu_0\sim1/r^6$ integrated with $r^2\,\mathrm dr$ diverges as $1/r^3$ at small $r$ (the point charge diverged as $1/r$ ), so a strictly point dipole carries infinite field energy. The physics resolutions are also the same family: real classical sources have finite size (the loop's smooth interior regularizes everything); for genuinely pointlike quantum particles (the electron and its moment), the divergence is handled by renormalization, exactly as the E1 energy article flagged for charge self-energy. Practical consequence: never compute "the self-energy of a dipole" from the far formula; compute interaction energies — dipole in external field (next article), dipole–dipole (the article after) — which are finite and measurable.