Field of a Magnetized Object

Hand the chapter a magnet — a body with magnetization $\mathbf{M}(\mathbf{r})$ frozen in — and ask its central computational question: what field does this thing actually make? Bound Currents answered at the level of sources, counting loop mismatches into $\mathbf{J}_b = \nabla\times\mathbf{M}$ and $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ . This article answers at the level of fields — and, en route, certifies those same formulas by a completely independent route: summing the dipole vector potentials of the matter and reorganizing the integral by parts splits it, with no further physics, into a volume piece fed by $\nabla\times\mathbf{M}$ and a surface piece fed by $\mathbf{M}\times\hat{\mathbf{n}}$ . Two derivations that share not a single step, one theorem: that is how you know the bound currents are not an artifact of clever counting. Then the construction earns its keep on the chapter's set-piece: the uniformly magnetized sphere. Its only source is a sine-theta ribbon of surface current, and a one-rung ansatz for the vector potential — matched using boundary rules bought in the magnetostatics chapter — delivers two exact miracles: a perfectly uniform interior field $\mathbf{B}_{\rm in} = \tfrac{2}{3}\mu_0\mathbf{M}$ , along the magnetization, and an exterior field that is a pure dipole at every point down to the surface. That $+\tfrac23$ , prosecuted against the polarized sphere's $-\tfrac13$ , is the article's deepest lesson — and it settles a debt: the strange looping $\mathbf{D}$ -field of the electret, which The Electric Displacement D promised "the magnetism chapter will earn," is exactly this sphere's $\mathbf{B}$ . The loop closes here.

Key takeaways

Summing dipole potentials gives $\mathbf{A} = \frac{\mu_0}{4\pi}\int \mathbf{M}\times\hat{\boldsymbol{\mathscr{r}}}/\mathscr{r}^2\,\mathrm{d}\tau'$ ; by parts it splits into $\frac{\mu_0}{4\pi}\left[\int \mathbf{J}_b/\mathscr{r}\,\mathrm{d}\tau' + \oint \mathbf{K}_b/\mathscr{r}\,\mathrm{d}a'\right]$ — a second, independent derivation of $\mathbf{J}_b = \nabla\times\mathbf{M}$ , $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ , agreeing exactly with the counting route of Bound Currents.
The magnetized sphere's one source is the belt $\mathbf{K}_b = M\sin\theta\,\hat{\boldsymbol{\phi}}$ ; a one-rung ansatz $A_\phi \propto \sin\theta$ ( $\propto r$ inside, $\propto 1/r^2$ outside), matched with the $\mathbf{A}$ -rules of Boundary Conditions for the Magnetostatic Field, gives $\mathbf{B}_{\rm in} = \tfrac{2}{3}\mu_0\mathbf{M}$ — exactly uniform, along $\mathbf{M}$ — and an exterior pure dipole with $\mathbf{m} = \tfrac{4}{3}\pi R^3\mathbf{M}$ .
The sign twist: Uniformly Polarized Sphere has $\mathbf{E}_{\rm in} = -\mathbf{P}/3\varepsilon_0$ , against $\mathbf{P}$ . The $+2\mu_0/3$ vs $-1/3\varepsilon_0$ pair is the dipole contact terms of Magnetic Dipole and Magnetic Moment made macroscopic.
Average-field theorem, magnetic edition: $\mathbf{B}_{\rm ave} = 2\mu_0\mathbf{m}/4\pi R^3$ over any sphere enclosing the currents — mirror of Field of a Polarized Object's $\mathbf{E}_{\rm ave} = -\mathbf{p}/4\pi\varepsilon_0R^3$ , opposite sign.
Promise delivered: the electret's looping $\mathbf{D}$ -portrait from The Electric Displacement D *is* this sphere's $\mathbf{B}$ , exactly, under $\mathbf{P}\leftrightarrow\mu_0\mathbf{M}$ .
KEY IDEA: sum the dipole potentials and integrate by parts — the matter disappears into bound currents, the sphere solves in one Legendre rung, and $\mathbf{B}$ comes out along $\mathbf{M}$ with the $+\tfrac23$ that makes magnetism the sign-flipped twin of polarization.

The core insight

The vector potential of magnetized matter, assembled from dipole potentials, integrates by parts into the potentials of $\mathbf{J}_b = \nabla\times\mathbf{M}$ and $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ — and for the uniformly magnetized sphere this machinery yields an exactly uniform $\mathbf{B}_{\rm in} = \tfrac{2}{3}\mu_0\mathbf{M}$ along $\mathbf{M}$ and an exact exterior dipole $\mathbf{m} = \tfrac{4}{3}\pi R^3\mathbf{M}$ .

Where this lives today

Every micromagnetic simulation on Earth runs this article's integral in its inner loop. Codes like mumax3 and OOMMF evolve $\mathbf{M}(\mathbf{r},t)$ over billions of cells and at each time step must compute the "stray field" — the $\mathbf{B}$ the current magnetization produces on itself. That is exactly equation (1), evaluated as an FFT-accelerated convolution of $\mathbf{M}$ with the dipole kernel; the demagnetizing tensor at the heart of those kernels is the sphere's $\tfrac13$ generalized to cuboid cells. Permanent-magnet engineering speaks the same language: a Halbach array — the arrangement in high-end motor rotors, undulators, and portable MRI systems — is a deliberately sculpted $\mathbf{K}_b$ , rotating $\mathbf{M}$ segment by segment so the bound-current sheets add on one face and cancel on the other. The flagship sphere itself has a modern career. Uniformly magnetized spheres are the calibration standard of magnetometry — their exterior is exactly dipolar, so a measured deviation calibrates the instrument, not the source — and the sphere-in-a-uniform-field version of this boundary-value problem (in Linear and Nonlinear Magnetic Media) is the design calculation behind magnetic shielding for quantum-sensor labs. Even planetary science leans on the far-field limit: to first approximation Earth's field is the $\mathbf{m} = \int\mathbf{M}\,\mathrm{d}\tau$ dipole of its core currents, and the multipole corrections geophysicists chart are the terms the sphere's one-rung miracle deletes.

Origins · 1 minSiméon Denis Poisson and how this was discovered

Siméon Denis Poisson

1781–1840 · Pithiviers, Paris

Poisson built the first honest mathematical theory of magnetized bodies. His 1824 memoirs on magnetism did for Coulomb's two-fluid picture what his 1812 electricity memoirs had done for charge: converted intuition into potential theory, expressing the field of a magnetized body as a volume integral plus a surface integral — precisely the two-part anatomy that survives in this article's split of $\mathbf{A}$ into $\int\mathbf{J}_b/\mathscr{r}\,\mathrm{d}\tau'$ and $\oint\mathbf{K}_b/\mathscr{r}\,\mathrm{d}a'$ , and the direct ancestor of the equivalent-sources idea. He introduced the magnetic moment per unit volume — this chapter's $\mathbf{M}$ — and solved the induced magnetization of a sphere, finding the $\tfrac13$ geometry factor a century before anyone called it a demagnetizing coefficient. The electric chapter's bound-charge article carries his band as a deliberate twin: Poisson stands at the head of both columns of the dictionary, charge-caps and current-belts alike. That his magnetic "fluids" were fictitious while Ampère's currents were real does not diminish the mathematics — Kelvin and Maxwell rebuilt the theory on Ampère's physics but kept Poisson's integrals nearly untouched.

History

1820

— Ampère proposes that all magnetism is microscopic circulating currents, the physical basis of the equivalent-current picture

1824

— Poisson's magnetism memoirs cast the field of a magnetized body as volume-plus-surface potential integrals, with the sphere solved

1845

— William Thomson (Kelvin) rederives Poisson's theory field-theoretically, distinguishing the two magnetic vectors inside matter

1873

— Maxwell's Treatise consolidates magnetization theory; the magnetized sphere becomes the canonical solved example

1948

— Néel's fine-particle theory puts the uniformly magnetized sphere at the center of permanent-magnet physics

1963

— Brown's Micromagnetics formalizes stray-field computation — this article's integral as the workhorse of magnetic simulation

The physical picture

Start with what the magnetized sphere's sources look like, because they explain both miracles before any algebra. Uniform $\mathbf{M} = M\hat{\mathbf{z}}$ makes no volume current ( $\nabla\times\mathbf{M} = 0$ — the silent bulk of Bound Currents), so the object collapses to its surface sheet, $\mathbf{K}_b = M\sin\theta\,\hat{\boldsymbol{\phi}}$ : a current ribbon circling the sphere like lines of latitude, strongest at the equator, dying to zero at both poles. That $\sin\theta$ profile is special. A solenoid makes a uniform field because its windings are evenly spaced along a cylinder; the $\sin\theta$ belt is the winding density that plays the same trick on a sphere — the pattern a uniformly charged shell would carry if you spun it — and it manufactures a perfectly uniform interior field. Not approximately, not just near the center: uniform to the last point, a result the derivation below gets exactly and worked example 1 puts at $1.42\,\mathrm{T}$ for an iron-register sphere.

Figure

The flagship - a sine-theta current belt makes a uniform interior field and an exact exterior dipole

Now put this sphere next to its electric twin, because the comparison is where the physics lives. The uniformly polarized sphere of Uniformly Polarized Sphere has sources of the opposite character: charge caps, $\sigma_b = P\cos\theta$ . Its interior field runs from positive cap to negative cap — against $\mathbf{P}$ , the depolarizing $\mathbf{E}_{\rm in} = -\mathbf{P}/3\varepsilon_0$ . The magnetized sphere's source is not caps but a belt, and a belt is a solenoid: its field threads through the loop, along $\mathbf{M}$ , and $\mathbf{B}_{\rm in} = +\tfrac{2}{3}\mu_0\mathbf{M}$ comes out reinforcing the magnetization. Same geometry, same mathematics rung for rung, opposite sign and doubled magnitude — the $\pm$ twist prosecuted in full below, down to its root in the dipole contact terms. One more portrait before the mathematics. The electret's $\mathbf{E}$ -lines start and end on its bound-charge caps. A magnet's $\mathbf{B}$ -lines can do no such thing: $\nabla\cdot\mathbf{B} = 0$ has no exceptions inside matter, so every field line of the sphere — and of any bar magnet — is a closed loop, running up through the body, out the north face, around, and back in the south. No $\mathbf{B}$ -line is ever born. The iron-filing picture shows only the exterior half of these loops; the interior half, where $\mathbf{B}$ marches along $\mathbf{M}$ , is the part the electric analogy would get exactly backwards. The same portrait scales to the magnet in your hand: a uniformly magnetized bar is a solenoid wound in bound current — $\mathbf{K}_b$ wraps the four long faces, the end faces are silent — and the backwards-running interior field the pole picture draws is not $\mathbf{B}$ but $\mathbf{H}$ , which gets its own article next.

Figure

Bar magnet - every B line is a closed loop threading the body, none starts or ends

Assembling the potential from dipoles

The starting point is bought machinery. Multipole Expansion of the Vector Potential established that a point dipole $\mathbf{m}$ has vector potential $\mathbf{A}_{\rm dip}(\mathbf{r}) = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\hat{\mathbf{r}}}{r^2}$ — the leading and, for a neutral current loop, dominant term of the expansion. A magnetized body is a continuum of such dipoles: the element at $\mathbf{r}'$ carries $\mathrm{d}\mathbf{m} = \mathbf{M}(\mathbf{r}')\,\mathrm{d}\tau'$ , and superposition sums the contributions, with the separation vector $\boldsymbol{\mathscr{r}} = \mathbf{r} - \mathbf{r}'$ doing the bookkeeping:

\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_V \frac{\mathbf{M}(\mathbf{r}')\times\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}^2}\,\mathrm{d}\tau' \qquad (1)

where: $\mathbf{M}$ — magnetization ( $\mathrm{A/m}$ ); $\boldsymbol{\mathscr{r}} = \mathbf{r} - \mathbf{r}'$ — vector from source element to field point, magnitude $\mathscr{r}$ , direction $\hat{\boldsymbol{\mathscr{r}}}$ ; $\mathbf{A}$ — the vector potential, from which $\mathbf{B} = \nabla\times\mathbf{A}$ . Equation (1) is exact for the macroscopic field and complete — but as unwieldy as every direct superposition integral, the same complaint that drove the electric chapter from the dipole-potential sum of Field of a Polarized Object to bound charges. The cure is also the same: integrate by parts and watch the matter reorganize itself into currents.

Derivation AThe by-parts identity that certifies the bound currents

The goal is to transform equation (1) into potentials sourced by $\nabla\times\mathbf{M}$ in the volume and $\mathbf{M}\times\hat{\mathbf{n}}$ on the surface — deriving $\mathbf{J}_b$ and $\mathbf{K}_b$ a second time, by a route sharing no step with the loop-counting argument of Bound Currents. The strategy is the primed-gradient trick: rewrite $\hat{\boldsymbol{\mathscr{r}}}/\mathscr{r}^2$ as a gradient with respect to the source coordinate, unpack the cross product with a product rule, and convert the total-curl term to a surface integral. The watch-out is signs, twice: the primed gradient of $1/\mathscr{r}$ comes with a plus, and the curl analogue of the divergence theorem carries a minus that students reliably drop. Both are owned below.

Step 1. Trade the kernel for a gradient. Since $\mathscr{r} = |\mathbf{r} - \mathbf{r}'|$ ,

\nabla'\left(\frac{1}{\mathscr{r}}\right) = +\frac{\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}^2},

with the plus sign because moving the source toward the field point shrinks $\mathscr{r}$ and raises $1/\mathscr{r}$ fastest along $+\hat{\boldsymbol{\mathscr{r}}}$ (the unprimed gradient gives the minus). Equation (1) becomes $\mathbf{A} = \frac{\mu_0}{4\pi}\int \mathbf{M}\times\nabla'(1/\mathscr{r})\,\mathrm{d}\tau'$ .

Step 2. Unpack with the product rule $\nabla'\times(f\mathbf{M}) = f\,(\nabla'\times\mathbf{M}) + (\nabla' f)\times\mathbf{M}$ , taking $f = 1/\mathscr{r}$ :

\mathbf{M}\times\nabla'\left(\frac{1}{\mathscr{r}}\right) = \frac{1}{\mathscr{r}}\left(\nabla'\times\mathbf{M}\right) - \nabla'\times\left(\frac{\mathbf{M}}{\mathscr{r}}\right).

Why this split: the first term already has the shape of a current-over- $\mathscr{r}$ potential; the second is a total curl, and total derivatives under integrals become boundary terms — the exact vector cousin of the divergence split in Field of a Polarized Object.

Step 3. Convert the total curl using $\int_V \nabla'\times\mathbf{F}\,\mathrm{d}\tau' = -\oint_S \mathbf{F}\times\mathrm{d}\mathbf{a}'$ (apply the divergence theorem to $\mathbf{F}\times\mathbf{c}$ , $\mathbf{c}$ constant). With $\mathbf{F} = \mathbf{M}/\mathscr{r}$ ,

-\int_V \nabla'\times\left(\frac{\mathbf{M}}{\mathscr{r}}\right)\mathrm{d}\tau' = +\oint_S \frac{\mathbf{M}\times\hat{\mathbf{n}}}{\mathscr{r}}\,\mathrm{d}a'.

Why the minus matters: it is what turns $\hat{\mathbf{n}}\times\mathbf{M}$ into $\mathbf{M}\times\hat{\mathbf{n}}$ — drop it and the belt circulates backwards, and the sphere's field would come out against $\mathbf{M}$ .

Step 4. Read off the theorem:

\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_V \frac{\nabla'\times\mathbf{M}}{\mathscr{r}}\,\mathrm{d}\tau' + \frac{\mu_0}{4\pi}\oint_S \frac{\mathbf{M}\times\hat{\mathbf{n}}}{\mathscr{r}}\,\mathrm{d}a'.

These are exactly the potentials a volume current and a surface current would produce in vacuum. The identification is forced: $\mathbf{J}_b = \nabla\times\mathbf{M}$ , $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ .

Step 5. Cross-check against the counting route. Bound Currents got the same pair by tallying adjacent-loop mismatches — a mesoscopic argument with no potentials in it. Here the pair emerged from an integral identity with no loops and no counting. The two routes share no assumption beyond "magnetized matter is a density of dipoles," and they agree term for term. That concordance is the theorem's real credential: the bound currents belong to the physics, not to either derivation's scaffolding. Two quick checks: in the uniform- $\mathbf{M}$ limit the volume term vanishes (zero curl) and only the surface sheet survives — the solenoid result of Bound Currents — and the counting route of that article delivers the identical $\mathbf{J}_b$ , $\mathbf{K}_b$ pair. Dimensions confirm: $M/\mathscr{r}^2$ times volume gives $\mathrm{(A/m)/m^2\cdot m^3 = A}$ , and $\mu_0$ times that is $\mathrm{T\cdot m}$ — the unit of $\mathbf{A}$ .

One sentence of appreciation before the identity is spent: nothing about matter entered Steps 1–4 — the split is pure vector calculus — yet the output is physics. The field of magnetized matter is the field of its bound currents, exactly, everywhere (for interior points in the macroscopic-average sense; the average-field theorem below is the license). Delete the magnet, keep $\mathbf{J}_b$ and $\mathbf{K}_b$ in vacuum, and no measurement of $\mathbf{B}$ can tell the difference.

The flagship: the uniformly magnetized sphere

Now the payoff problem — Griffiths's Example 6.1, the magnetic set-piece the rest of the chapter leans on. Take a sphere of radius $R$ , freeze in $\mathbf{M} = M\hat{\mathbf{z}}$ , and find $\mathbf{B}$ everywhere. The sources collapse immediately: uniform $\mathbf{M}$ has zero curl, so $\mathbf{J}_b = 0$ , and the surface keeps everything:

\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}} = M\hat{\mathbf{z}}\times\hat{\mathbf{r}} = M\sin\theta\;\hat{\boldsymbol{\phi}} \qquad (2)

where: $\theta$ — polar angle from the magnetization axis; $\hat{\boldsymbol{\phi}}$ — the azimuthal direction, so the current circulates like lines of latitude. Read the geometry: the belt vanishes at both poles ( $\mathbf{M}\parallel\hat{\mathbf{n}}$ there — the silent-face rule of Bound Currents), peaks at the equator at full $M$ , and its $\sin\theta$ profile is the single $\ell = 1$ rung of the angular ladder — the vector cousin of the polarized sphere's $\sigma_b = P\cos\theta$ on the scalar ladder. One-rung source, one-rung response: that is the entire solution strategy.

Go deeper · Derivation B · 2 minThe magnetized sphere solved end to end

Derivation BThe magnetized sphere solved end to end

The goal is $\mathbf{B}$ everywhere from the single source (2). The strategy is the boundary-value method: propose the minimal ansatz $A_\phi(r,\theta) = f(r)\sin\theta$ in each region — justified because the source is pure $\sin\theta$ and the Legendre machinery of Separation of Variables in Spherical Coordinates (recalled, not re-derived; the honest kill of all higher rungs is quoted from the twin construction in Uniformly Polarized Sphere) guarantees rungs do not mix — then spend the matching rules bought in Boundary Conditions for the Magnetostatic Field: $\mathbf{A}$ continuous across a surface current, normal derivative jumping by $-\mu_0 K$ . Watch regularity (which radial term survives where) and the direction of the normal in the jump rule (outside minus inside).

Step 1. The ansatz. Inside, $f(r) = Cr$ (the $1/r^2$ companion is singular at the origin); outside, $f(r) = D/r^2$ (the growing solution violates decay). So

A_\phi^{\rm in} = C\,r\sin\theta, \qquad A_\phi^{\rm out} = \frac{D}{r^2}\sin\theta.

Why these shapes mean something: $A_\phi \propto r\sin\theta$ is precisely the potential of a uniform field ( $\mathbf{A} = \tfrac12\mathbf{B}\times\mathbf{r}$ ), and $A_\phi \propto \sin\theta/r^2$ is precisely the point-dipole potential recalled from Multipole Expansion of the Vector Potential. The ansatz already announces both miracles; the constants only calibrate them.

Step 2. Spend rule one — continuity of $\mathbf{A}$ at $r = R$ :

C\,R = \frac{D}{R^2} \qquad\Longrightarrow\qquad D = C\,R^3.

Step 3. Spend rule two — the jump $\left(\partial A_\phi^{\rm out}/\partial r - \partial A_\phi^{\rm in}/\partial r\right)_{r=R} = -\mu_0 K_b$ with $K_b = M\sin\theta$ . Both derivatives share the factor $\sin\theta$ , which cancels:

-\frac{2D}{R^3} - C = -\mu_0 M \qquad\Longrightarrow\qquad C + \frac{2D}{R^3} = \mu_0 M,

and with $D = CR^3$ : $3C = \mu_0M$ , so $C = \mu_0M/3$ and $D = \mu_0MR^3/3$ .

Step 4. Curl inside. With $A_\phi^{\rm in} = \tfrac{\mu_0 M}{3}r\sin\theta$ : $B_r = \frac{1}{r\sin\theta}\partial_\theta(A_\phi\sin\theta) = \tfrac{2\mu_0 M}{3}\cos\theta$ and $B_\theta = -\frac{1}{r}\partial_r(rA_\phi) = -\tfrac{2\mu_0 M}{3}\sin\theta$ , which assemble into a constant vector:

\mathbf{B}_{\rm in} = \frac{2}{3}\,\mu_0\,\mathbf{M},

exactly uniform everywhere inside, pointing along the magnetization — a perfectly homogeneous field manufactured by a surface current alone.

Step 5. Read the exterior. Compare $A_\phi^{\rm out} = \frac{\mu_0MR^3}{3}\frac{\sin\theta}{r^2}$ against the recalled dipole potential $A_{\rm dip} = \frac{\mu_0}{4\pi}\frac{m\sin\theta}{r^2}$ : identical, with

\mathbf{m} = \frac{4}{3}\pi R^3\,\mathbf{M} = V\mathbf{M}.

The exterior is a pure dipole — no higher multipoles — at every point down to the surface, with moment equal to magnetization times volume, exactly as summing the constituent dipoles demands: the magnetic cousin of the shell theorem's point-charge impersonation.

Step 6. Audit the seams, then name what is deferred. At the poles the interior value $\tfrac23\mu_0M$ equals the exterior polar value $\tfrac{\mu_0}{4\pi}\tfrac{2m}{R^3} = \tfrac23\mu_0M$ — normal $\mathbf{B}$ continuous, as it must be with no monopoles. At the equator the tangential component jumps by exactly $\mu_0K_b = \mu_0M$ — the current sheet's signature (worked example 1 puts numbers on it). And one line for the road: $\mathbf{B}_{\rm in}/\mu_0 - \mathbf{M} = -\mathbf{M}/3$ — this same interior viewed through magnetism's bookkeeping field, pointing against $\mathbf{M}$ . That field and the whole $\mathbf{H}$ apparatus belong to The Auxiliary Field H, next; here it is only named. Three independent checks: the rotating-charged-shell correspondence ( $\sigma\omega R \mapsto M$ ) reproduces the same $\tfrac{2}{3}$ interior field; $B^\perp$ continuity at the poles and the $\mu_0 K_b$ tangential jump at the equator are exactly what the boundary rules demand; and $\mathbf{m} = V\mathbf{M}$ agrees with direct integration of $\mathbf{M}$ over the volume.

The sign twist, prosecuted — and a promise kept

Set the two flagship spheres side by side — every constant is now on the table:

\mathbf{E}_{\rm in} = -\frac{1}{3\varepsilon_0}\,\mathbf{P} \qquad\text{versus}\qquad \mathbf{B}_{\rm in} = +\frac{2\mu_0}{3}\,\mathbf{M} \qquad (3)

where: the left result is Uniformly Polarized Sphere's, recalled, not re-derived; the right is Step 4 above. Two differences demand prosecution — the sign and the factor of two — and neither is an accident of the sphere. Magnetic Dipole and Magnetic Moment established that the point-dipole fields carry delta-function contact terms, $\mathbf{E}_{\rm dip} \supset -\frac{1}{3\varepsilon_0}\mathbf{p}\,\delta^3(\boldsymbol{\mathscr{r}})$ but $\mathbf{B}_{\rm dip} \supset +\frac{2\mu_0}{3}\mathbf{m}\,\delta^3(\boldsymbol{\mathscr{r}})$ — opposite signs, factor of two, encoding the fact that inside an electric dipole the field runs from $+$ to $-$ against $\mathbf{p}$ , while inside a current loop it threads the loop along $\mathbf{m}$ . Pack dipoles shoulder to shoulder into a sphere and each dipole's contact term is what its neighbors sit in; the interior fields are those coefficients with $\delta^3 \to$ moment density: $-\mathbf{p}/3\varepsilon_0 \to -\mathbf{P}/3\varepsilon_0$ , $+2\mu_0\mathbf{m}/3 \to +2\mu_0\mathbf{M}/3$ . The set-piece spheres are the contact terms made walk-in size.

Go deeper · Derivation C · 2 minThe average field theorem and the contact pair made macroscopic

Derivation CThe average field theorem and the contact pair made macroscopic

The goal is the magnetic average-field theorem — the average of $\mathbf{B}$ over any sphere enclosing all the currents depends only on their total dipole moment — plus its corollary, the license to use equation (1) inside matter. The strategy is economical: the heavy lifting (the swap trick) is owned by Field of a Polarized Object, which proved $\mathbf{E}_{\rm ave} = -\mathbf{p}/4\pi\varepsilon_0R^3$ ; we state the magnetic mirror, fix its coefficient with a one-line calibration against the flagship sphere, and read off the twin structure. Watch that the two theorems have the same $1/R^3$ anatomy but opposite signs — the contact pair again, in averaged form.

Step 1. State the theorem. For steady currents entirely inside a sphere of radius $R$ ,

\mathbf{B}_{\rm ave} \equiv \frac{1}{\tfrac43\pi R^3}\int_{\rm ball} \mathbf{B}\,\mathrm{d}\tau = \frac{\mu_0}{4\pi}\,\frac{2\mathbf{m}}{R^3},

with $\mathbf{m}$ the total moment of the enclosed currents. Everything beyond $\mathbf{m}$ averages away — the same forgiving-average miracle as the electric version, provable by the same swap trick with the Biot–Savart kernel in place of Coulomb's.

Step 2. Calibrate on the flagship, one line. The sphere's currents (the belt) live on radius $R$ and its interior field is uniform, so the ball average is $\mathbf{B}_{\rm in} = \tfrac23\mu_0\mathbf{M}$ ; the theorem's right side with $\mathbf{m} = \tfrac43\pi R^3\mathbf{M}$ gives $\frac{\mu_0}{4\pi}\frac{2}{R^3}\cdot\frac{4\pi R^3}{3}\mathbf{M} = \tfrac23\mu_0\mathbf{M}$ — equal ✓.

Step 3. Put the mirror pair on the record:

\mathbf{E}_{\rm ave} = -\frac{1}{4\pi\varepsilon_0}\,\frac{\mathbf{p}}{R^3} \qquad\text{versus}\qquad \mathbf{B}_{\rm ave} = +\frac{\mu_0}{4\pi}\,\frac{2\mathbf{m}}{R^3}.

Why this is the license the article needed: equation (1) was built from far-field dipole potentials, yet molecules are not in each other's far fields. Split the sources at a mesoscopic sphere around the field point: distant matter is legitimately far-field, and the theorem says the near matter's wild true field averages to exactly what its dipole moment dictates — which is what (1) assigns it. So (1) computes the macroscopic $\mathbf{B}$ correctly everywhere, interior included — Field of a Polarized Object's argument, transplanted whole.

Step 4. Close the electric displacement's loop. The Electric Displacement D, derivation 2, mapped the electret's $\mathbf{D}$ -field — $\nabla\cdot\mathbf{D} = 0$ everywhere, curl carried on the surface, $\mathbf{D}_{\rm in} = +\tfrac23\mathbf{P}$ , pure dipole outside, lines closing into loops "like the B-field of a bar magnet" — and promised the correspondence would be earned by the magnetism chapter. Earned here: this sphere's $\mathbf{B}$ is divergence-free everywhere, carries its curl on the belt, has $\mathbf{B}_{\rm in} = +\tfrac23\mu_0\mathbf{M}$ , is a pure dipole outside, and loops through the body. Under the dictionary $\mathbf{P}\leftrightarrow\mu_0\mathbf{M}$ , the electret's $\mathbf{D}$ is this sphere's $\mathbf{B}$ , field line for field line. And the correspondence runs both ways: the electret's curl-free $\mathbf{E}$ maps onto this sphere's other field, the one named forward to The Auxiliary Field H. Two checks: the Step 2 calibration closes exactly — $\tfrac{2}{3}\mu_0\mathbf{M}$ on both sides — confirming the theorem is self-consistent on the flagship sphere; and the $\mathbf{D}$ -to- $\mathbf{B}$ dictionary holds component by component, with $\mathbf{D}_{\rm in} = \tfrac{2}{3}\mathbf{P}$ mapping onto $\mathbf{B}_{\rm in} = \tfrac{2}{3}\mu_0\mathbf{M}$ under $\mathbf{P}\leftrightarrow\mu_0\mathbf{M}$ .

Limiting cases

Case	Result	Meaning
uniform $\mathbf{M}$ , any shape	$\mathbf{J}_b = 0$ ; only $\mathbf{K}_b$ survives	the volume term dies with the curl; surface keeps all
sphere, inside	$\mathbf{B}_{\rm in} = \tfrac23\mu_0\mathbf{M}$ , exactly uniform	one-rung source, one-rung response
sphere, outside	pure dipole, $\mathbf{m} = \tfrac43\pi R^3\mathbf{M}$	exact at every exterior point, not asymptotically
far field, any magnetized body	dipole with $\mathbf{m} = \int\mathbf{M}\,\mathrm{d}\tau$	every magnet is a dipole from far away
poles of the sphere	$B^\perp$ continuous ( $\tfrac23\mu_0M$ both sides)	no monopoles, no flux jump
equator of the sphere	$B^\parallel$ jumps by $\mu_0K_b = \mu_0M$	the current sheet's signature (worked example 1)
electric twin	$\mathbf{E}_{\rm in} = -\mathbf{P}/3\varepsilon_0$ , against $\mathbf{P}$	the sign twist: caps vs belt, contact $-\tfrac{1}{3\varepsilon_0}$ vs $+\tfrac{2\mu_0}{3}$
average over enclosing sphere	$\mathbf{B}_{\rm ave} = 2\mu_0\mathbf{m}/4\pi R^3$	mirror of $\mathbf{E}_{\rm ave} = -\mathbf{p}/4\pi\varepsilon_0R^3$

Common trapInside a magnet B does not run from north pole to south pole

The electret picture — field lines leaving the positive cap, landing on the negative, running against the moment inside — is correct for $\mathbf{E}$ in a polarized body and flatly wrong for $\mathbf{B}$ in a magnet. $\nabla\cdot\mathbf{B} = 0$ forbids endpoints: every $\mathbf{B}$ -line closes through the magnet's interior, where it runs along $\mathbf{M}$ with the $+\tfrac23\mu_0M$ of the flagship sphere. The backwards-running interior field you may have seen drawn between poles is $\mathbf{H}$ ( $= -\mathbf{M}/3$ inside the sphere), a useful bookkeeping field belonging to The Auxiliary Field H — but it is not $\mathbf{B}$ , and confusing the two flips the sign of every interior conclusion. When in doubt, draw the belt: a magnetized body is a solenoid, and a solenoid threads its own windings.

Common trapThe exterior dipole is exact only for the sphere — and never valid inside

Two abuses of the beautiful result. First: for a cube, cylinder, or actual bar magnet, the exterior field is a dipole only asymptotically — near the body, higher multipoles contribute at the tens-of-percent level. Only the sphere's one-rung source guarantees a one-rung exterior everywhere. Second: even for the sphere, the dipole formula holds strictly outside $R$ . Inside, the field is the uniform $\tfrac23\mu_0\mathbf{M}$ — extrapolating the dipole formula inward gives a field that diverges at the center and points the wrong way over most of the volume. The two forms meet at $r = R$ with continuous $B^\perp$ and a $\mu_0K_b$ tangential jump; neither is valid in the other's territory.

Worked examples

1 · The iron-register sphere (confidence builder)

A sphere of radius $R = 1.0\,\mathrm{cm}$ carries uniform magnetization at iron's saturation register, $M = 1.7\times10^{6}\,\mathrm{A/m}$ (the number recalled from Magnetization). Find the surface current, the interior field, the dipole moment, and audit both boundary seams.

Step 1. The belt. $K_b = M\sin\theta$ peaks at the equator at $1.7\times10^{6}\,\mathrm{A/m}$ — 1.7 kA per millimeter of arc. Total current wrapping the sphere: $I_{\rm tot} = \int_0^\pi M\sin\theta\,R\,\mathrm{d}\theta = 2MR = 3.4\times10^{4}\,\mathrm{A}$ — thirty-four thousand ampere-turns around a marble, no battery. Bound currents are not small.

Step 2. Interior field. $B_{\rm in} = \tfrac23\mu_0M = \tfrac23(1.257\times10^{-6})(1.7\times10^{6}) = 1.42\,\mathrm{T}$ — MRI-class strength, uniform to the last interior point, along $\mathbf{M}$ .

Step 3. Dipole moment. $m = \tfrac43\pi R^3M = \tfrac43\pi(0.010)^3(1.7\times10^{6}) = 7.1\,\mathrm{A\cdot m^2}$ — on the axis at $r = 10\,\mathrm{cm}$ this delivers $\frac{\mu_0}{4\pi}\frac{2m}{r^3} = 1.4\,\mathrm{mT}$ , thirty times Earth's field a hand-width away.

Step 4. Audit the pole seam. Just outside the north pole: $B = \frac{\mu_0}{4\pi}\frac{2m}{R^3} = \tfrac23\mu_0M = 1.42\,\mathrm{T}$ , radial — equal to the interior value. $B^\perp$ continuous ✓.

Step 5. Audit the equator seam. Just inside: $1.42\,\mathrm{T}$ along $+\hat{\mathbf{z}}$ , tangent to the surface there. Just outside: the dipole formula gives $\frac{\mu_0}{4\pi}\frac{m}{R^3} = \tfrac13\mu_0M = 0.71\,\mathrm{T}$ along $-\hat{\mathbf{z}}$ . Tangential jump: $1.42 - (-0.71) = 2.14\,\mathrm{T} = \mu_0K_b$ ✓ — exactly the current sheet's signature. Note the exterior equatorial field opposes the interior: beside the sphere's waist you stand in the return path of the loops.

2 · The neodymium sphere magnet (exam level)

Sphere magnets are sold as toys: NdFeB, remanent field $B_r = 1.30\,\mathrm{T}$ , radius $R = 5.0\,\mathrm{mm}$ . Model the magnet as uniformly magnetized with $\mu_0M = B_r$ . Find $M$ , the interior field, the moment, the field on the axis at $2.0\,\mathrm{cm}$ and $5.0\,\mathrm{cm}$ from the center, and check the average-field theorem.

Step 1. Magnetization from the datasheet. $M = B_r/\mu_0 = 1.30/(1.257\times10^{-6}) = 1.03\times10^{6}\,\mathrm{A/m}$ — within a factor of 1.6 of iron's saturation, from a sintered lattice that (unlike soft iron) keeps its $\mathbf{M}$ with no applied field; that is why the uniform- $\mathbf{M}$ model is honest for NdFeB (the why belongs to Ferromagnetism).

Step 2. Interior field. $B_{\rm in} = \tfrac23\mu_0M = \tfrac23B_r = 0.87\,\mathrm{T}$ — the sphere geometry's neat shortcut: exactly two-thirds of the quoted remanence, no other numbers needed.

Step 3. Moment. $m = \tfrac43\pi R^3M = \tfrac43\pi(5.0\times10^{-3})^3(1.03\times10^{6}) = 0.54\,\mathrm{A\cdot m^2}$ .

Step 4. Axial exterior field — exact, since the sphere's exterior is a pure dipole at every point. At $z = 2.0\,\mathrm{cm}$ (four radii): $B = \frac{\mu_0}{4\pi}\frac{2m}{z^3} = 10^{-7}\times\frac{1.08}{8.0\times10^{-6}} = 14\,\mathrm{mT}$ . At $z = 5.0\,\mathrm{cm}$ : $8.7\times10^{-4}\,\mathrm{T} = 0.87\,\mathrm{mT}$ — still seventeen times Earth's field, which is why the toy wrecks compasses across a desk. The collapse between the two points ( $\times\tfrac{1}{15.6}$ for $\times2.5$ the distance) is the $1/z^3$ scaling in the flesh.

Step 5. Average-field check. Over the ball $r \leq R$ the theorem predicts $B_{\rm ave} = \frac{\mu_0}{4\pi}\frac{2m}{R^3} = 10^{-7}\times\frac{1.08}{1.25\times10^{-7}} = 0.87\,\mathrm{T}$ — equal to $B_{\rm in}$ , as it must be, since the interior is uniform and the sourcing belt sits on the ball's rim ✓. The toy magnet is, point for point, the flagship problem: $0.87\,\mathrm{T}$ inside, exact dipole outside, $\pm$ twist and all.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.2–6.3 · Jackson, Classical Electrodynamics, 3rd ed., §5.9–5.11 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, §13.1–13.4. The uniformly magnetized sphere has $\mathbf{B}_{\text{in}}=\tfrac23\mu_0\mathbf{M}$ , $\mathbf{H}_{\text{in}}=-\tfrac13\mathbf{M}$ , demagnetizing factor $N=1/3$ , and $\mathbf{m}=\tfrac43\pi R^3\mathbf{M}$ . Demagnetizing factors for the three principal geometries are $0$ , $1/3$ , and $1$ . Boundary conditions follow from $\nabla\cdot\mathbf{B}=0$ and $\nabla\times\mathbf{H}=\mathbf{J}_{\text{free}}$ . Worked numbers: iron-register sphere ( $M = 1.7\times10^{6}$ A/m, $R = 1$ cm) gives $B_{\rm in} = 1.42$ T, $m = 7.1$ A·m², belt total $2MR = 3.4\times10^{4}$ A, equatorial tangential jump $\mu_0K_b = 2.14$ T; NdFeB sphere ( $B_r = 1.30$ T, $R = 5$ mm) gives $B_{\rm in} = 0.87$ T, $m = 0.54$ A·m², axial field 14 mT at 2 cm and 0.87 mT at 5 cm, with $B_{\rm ave} = 2\mu_0m/4\pi R^3 = 0.87$ T confirming the average-field theorem.

Builds on: Magnetization · Bound Currents · Magnetic Dipole Moment of Atoms · Multipole Expansion of the Vector Potential · Boundary Conditions for the Magnetostatic Field

Leads to: The Auxiliary Field H · Ampère's Law for H · Magnetic Boundary Conditions · Linear and Nonlinear Magnetic Media · Ferromagnetism

Why derive the bound currents twice — was the counting argument not enough?

It was enough to establish the formulas; the second route establishes their robustness. The counting argument of Bound Currents models matter as a tidy array of square loops — a skeptic could worry the answer depends on that cartoon. The by-parts route assumes no microstructure at all: only "magnetized matter is a density of dipole moments" plus vector calculus, and it lands on the identical pair. When two derivations share no steps and agree exactly, the result belongs to the physics, not the scaffolding. The practical dividend: the by-parts route hands you the vector potential itself, ready-made for boundary-value problems like the flagship sphere — the counting route never mentions potentials.

How can a surface current produce a perfectly uniform field inside the sphere — not just near the center?

Because the belt's $\sin\theta$ profile is the unique single-rung source on a sphere. A uniform interior field has potential $A_\phi \propto r\sin\theta$ , the $\ell = 1$ regular solution; a pure $\sin\theta$ surface current excites that member and no other, because the Legendre rungs do not mix across the matching conditions (the machinery of Separation of Variables in Spherical Coordinates). Any other profile — or any non-ellipsoidal shape — excites higher rungs and ruins the uniformity. The physical cousin: a spinning uniformly charged shell carries exactly this $\sin\theta$ pattern, which is why the rotating-shell problem and this one are the same problem wearing different clothes.

If B inside points along M, why do people say magnets demagnetize themselves?

Because the field that does the demagnetizing bookkeeping is not $\mathbf{B}$ . Form the combination $\mathbf{B}/\mu_0 - \mathbf{M} = \tfrac23\mathbf{M} - \mathbf{M} = -\tfrac13\mathbf{M}$ : that quantity — the auxiliary field $\mathbf{H}$ , developed in The Auxiliary Field H — points against $\mathbf{M}$ inside, and it is the field against which a material's willingness to stay magnetized is measured. Both statements are true simultaneously: the physical field $\mathbf{B}$ threads the magnet along $\mathbf{M}$ (this article), while the bookkeeping field $\mathbf{H}$ opposes it (next article), with shape controlling how strongly. Keep the two ledgers separate and the paradox evaporates.

Is the correspondence with the electret's D-field exact or just a resemblance?

Exact, structure for structure. The electret's $\mathbf{D}$ obeys $\nabla\cdot\mathbf{D} = 0$ everywhere (no free charge) with its curl concentrated where tangential $\mathbf{P}$ jumps; this sphere's $\mathbf{B}$ obeys $\nabla\cdot\mathbf{B} = 0$ everywhere with its curl concentrated on the belt where tangential $\mathbf{M}$ jumps. Same divergence, same curl structure, same sphere: uniqueness forces the same field under the dictionary $\mathbf{P}\leftrightarrow\mu_0\mathbf{M}$ — hence $\mathbf{D}_{\rm in} = +\tfrac23\mathbf{P}$ maps onto $\mathbf{B}_{\rm in} = +\tfrac23\mu_0\mathbf{M}$ , loops onto loops. The Electric Displacement D promised this correspondence would be earned rather than asserted; the flagship derivation is the earning. The dual statement also holds: the electret's curl-free $\mathbf{E}$ corresponds to the magnet's $\mathbf{H}$ .

Does the exterior-exact-dipole result mean I can always replace a magnet by a point dipole?

For a sphere, yes, exactly, at every exterior point — that is the miracle. For any other shape, only asymptotically: a cube or bar has higher multipole moments, and near the body they matter — which is why refrigerator-magnet force estimates from the dipole formula fail at contact. Two safe general statements: far away, every magnetized body is a dipole with $\mathbf{m} = \int\mathbf{M}\,\mathrm{d}\tau$ ; and everywhere outside, the exact field is that of the bound currents, which you can integrate numerically when the geometry is unkind. Worked example 2 exploits sphericity precisely so that "exact" is available at $z = 4R$ , where a cube would already need corrections.

Where does the factor of two between the electric and magnetic sphere results actually come from?

From what the two dipole types do inside themselves — the contact terms of Magnetic Dipole and Magnetic Moment. Between the charges of an electric dipole the field runs from $+$ to $-$ , against $\mathbf{p}$ : contribution $-\tfrac{1}{3\varepsilon_0}\mathbf{p}\delta^3$ . Through the center of a current loop the field threads along $\mathbf{m}$ : contribution $+\tfrac{2\mu_0}{3}\mathbf{m}\delta^3$ . The magnitudes differ by two because the loop's interior lines continue into its exterior in a divergence-free pattern, while the charge pair's interior field must terminate on the charges — work the averaging integrals and the $2$ falls out of $\nabla\cdot\mathbf{B} = 0$ having no endpoints to spend flux on. The spheres inherit the coefficients wholesale: a filled sphere of dipoles is, for averaging purposes, one big dipole — $\delta^3 \to \mathbf{P}$ or $\mu_0\mathbf{M}$ , the $-\tfrac13$ and $+\tfrac23$ riding along unchanged.