Field of a Polarized Object

The previous article located the bound charges by counting displaced charge. This one proves the claim that made the counting worthwhile: a polarized object's potential — everywhere, inside and out — is exactly the potential of those bound charges, treated as ordinary charge sitting in vacuum. The proof is two lines of vector calculus doing something almost sly: start from the honest superposition of molecular dipole potentials, integrate by parts once, and watch the integral reorganize itself into a surface term (the potential of $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$ ) plus a volume term (the potential of $\rho_b = -\nabla\cdot\mathbf{P}$ ). The same densities the charge-counting found — arrived at from the opposite direction, with no mention of chains or shipping. Two derivations, one answer: that agreement is the chapter's load-bearing consistency check. The theorem's power is strategic. "Field of polarized matter" sounds like a new subject; the identity says it is a solved subject — every tool of the electrostatics chapters applies verbatim once the bound charges are on the table. Uniform slab? Two charged sheets; done by inspection. Radially graded sphere? Gauss's law on $\rho_b$ . The classic cosine-charged sphere gets its own article next. This article works the slab and a bar electret end to end, and the electret example pays extra: comparing the interior field of a long thin rod ( $\approx 2\%$ of $P/\varepsilon_0$ ), a slab ( $100\%$ ), and — next article — a sphere (exactly $1/3$ ) exposes the geometry-dependent depolarizing field, one of dielectric physics's organizing ideas. One piece of fine print is owed and paid here. The dipole-potential starting point is unimpeachable outside matter, but inside, real molecular fields are ångström-scale chaos — in what sense does the smooth formula hold there? The honest answer (the macroscopic field is a spatial average, and a lovely average-field theorem makes the dipole superposition reproduce it exactly) is this article's second derivation, so that nothing in the chapter rests on a wave of the hand.

Key takeaways

The potential of polarized matter is the superposition of its molecular dipole potentials: $V = \frac{1}{4\pi\varepsilon_0}\int \frac{\mathbf{P}\cdot\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}^2}\,\mathrm{d}\tau'$ — correct but unwieldy.
One integration by parts converts it exactly into $V = \frac{1}{4\pi\varepsilon_0}\left[\oint\frac{\sigma_b}{\mathscr{r}}\,\mathrm{d}a' + \int\frac{\rho_b}{\mathscr{r}}\,\mathrm{d}\tau'\right]$ : the potential of the bound charges, full stop.
This is the second, independent route to $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$ and $\rho_b = -\nabla\cdot\mathbf{P}$ — the charge-counting article got the same densities physically; the agreement is the chapter's consistency check.
Strategy upgrade: "compute the field of polarized matter" = "replace by bound charges, then use E1 tools" — sheets, Gauss, images, separation of variables, whatever fits.
Inside matter the formula computes the macroscopic (averaged) field; the license is the average-field theorem $\mathbf{E}_{\rm ave} = -\mathbf{p}/4\pi\varepsilon_0R^3$ , derived here, not waved at.
Geometry rules the interior field: for uniform $P$ , a thin slab fights itself with the full $P/\varepsilon_0$ , a long needle with almost nothing ( $\approx2\%$ in our bar-electret numbers), a sphere with exactly one third — the depolarizing-field hierarchy.
KEY IDEA: a polarized object is electrostatically indistinguishable from its bound charges $\sigma_b$ and $\rho_b$ — compute their field by ordinary Coulomb superposition and you have the object's field, inside and out.

The core insight

A polarized object is electrostatically indistinguishable from its bound charges $\rho_b=-\nabla\cdot\mathbf{P}$ (volume) and $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ (surface); compute the field of those charges by ordinary Coulomb superposition.

Where this lives today

The equivalence theorem is the daily working method wherever polarized matter meets field solvers. Every finite-element electrostatics package handles electrets, ferroelectrics, and piezo elements exactly this way: the mesh never sees "polarization," only the $\sigma_b$ and $\rho_b$ computed from $\mathbf{P}$ , fed to a vacuum Poisson solver — this article's identity, industrialized. MEMS designers sizing electret energy harvesters, and the acoustics engineers behind electret microphones, live on the slab and rod formulas of the worked examples, depolarizing factors included; the general-ellipsoid depolarization tensor (of which slab, sphere, and needle are the milestones) is tabulated in every reference on magnetics and dielectrics alike, since the identical mathematics runs magnetized bodies. The inside-the-matter half — the average-field theorem — quietly underwrites two modern industries of calculation. Molecular simulation validates its force fields by comparing coarse-grained averages of explicit molecular electrostatics against continuum predictions; that the two match is precisely Derivation B operating in silico. And the optics of nanoparticles leans on the sphere's uniform interior field: a subwavelength particle in light behaves as a single dipole whose response is set by the $\tfrac13$ depolarizing geometry — the starting point of plasmonics, biosensing colorimetry (why gold sols are ruby red), and the effective-medium theories that design metamaterials. A two-line integration by parts from 1828-vintage potential theory, holding up sensors, simulations, and stained glass.

Origins · 1 minGeorge Green and how this was discovered

George Green

1793–1841 · Sneinton windmill, Nottingham; Caius College, Cambridge

Green is the patron saint of exactly the manipulation this article turns on: moving derivatives across an integral and harvesting the boundary term. His 1828 Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism — self-published by subscription, written by a miller with one year of formal schooling, studied by almost nobody on release — introduced the potential function by that name, the theorems relating volume and surface integrals now filed under his name, and the kernel-function method (Green's functions) that turns boundary-value electrostatics into integrals. The parts-and-boundary move that converts dipole soup into $\sigma_b$ and $\rho_b$ is Green's essay operating at textbook scale; Poisson's equivalent distributions for magnetism (the previous article's band) and Kelvin's and Maxwell's field theory all drew on it once Kelvin — who tracked down the forgotten essay in 1845, four years after Green's death — had it reprinted and evangelized. Green worked most of his life in the family windmill, entered Cambridge at forty, and died before his tools conquered physics; the mill in Sneinton is now a science museum with his theorems on the wall.

History

1812

— Poisson computes fields of electrified bodies via the potential, setting the integral-representation stage

1824

— Poisson's equivalent surface-plus-volume distributions for magnetized matter — the theorem in magnetic dress

1828

— Green's Essay supplies the potential, the boundary theorems, and the parts machinery of the equivalence identity

1845

— Kelvin rediscovers and republishes Green, wiring the essay into the emerging field theory

1873

— Maxwell's Treatise makes bound-charge equivalents standard apparatus for media

1912

— Lorentz's averaging program (with the average-field theorem) makes the interior claim rigorous

The physical picture

Think of the theorem as a licensed demolition. On the left stands the real object: $10^{28}$ molecular dipoles per cubic meter, each contributing its little $\mathbf{p}\cdot\hat{\boldsymbol{\mathscr{r}}}/4\pi\varepsilon_0\mathscr{r}^2$ to the potential — a sum no one will ever perform. On the right stands the replacement: the same shape, empty of matter, wearing a film of surface charge $\sigma_b$ and (if $\mathbf{P}$ is non-uniform) filled with a fog of volume charge $\rho_b$ . The theorem says the two produce identical potentials at every field point. Not similar — identical, as an exact mathematical identity. So demolish the hard problem and keep the easy one: fields of surface and volume charge are the bread and butter of the electrostatics chapters, with Gauss's law, superposition, and boundary-value machinery all standing by.

Figure

Two descriptions, one field — the equivalence theorem

Why should such a reorganization exist at all? Because a dipole is two displaced charges: writing the object as dipoles or as (slightly separated) charge distributions is the same physics in different groupings, and integration by parts is precisely the mathematics of regrouping — it moves the derivative from the $1/\mathscr{r}$ kernel onto $\mathbf{P}$ , trading "density of displaced pairs" for "where the displacement leaves unbalanced charge." The boundary term that calculus produces automatically is the surface charge; the derivative term is the volume charge. It is deeply satisfying — and no accident — that the mathematics dumps out exactly the two densities the physical chain-counting predicted: the previous article and this one are the two sides of one bookkeeping identity, charge conservation on one side, the divergence theorem on the other.

Figure

Geometry's tax — the depolarizing field at identical P, three shapes

The remaining subtlety is inside the material, and it deserves respect rather than fine print. Between the molecules the true microscopic field thrashes at $10^{11}\,\mathrm{V/m}$ ; at a nucleus it diverges. The smooth $V$ our formula produces obviously isn't that. The claim — made precise by the second derivation — is that it is the macroscopic average of that: average the microscopic chaos over the mesoscopic cells of the polarization article, and the result agrees exactly with the bound-charge formula. The key input is a small theorem with a big reach: the average field inside any sphere, due to the charges within it, depends only on their total dipole moment — all the near-field violence cancels on averaging. Matter's chaos is self-erasing at the scale instruments see; that erasure is why macroscopic electrostatics exists.

The identity

Start from what superposition licenses directly. Each volume element $\mathrm{d}\tau'$ carries dipole moment $\mathbf{P}\,\mathrm{d}\tau'$ , whose potential is the point-dipole formula owned by Electric Dipole Moment:

V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\int_{\mathcal{V}} \frac{\mathbf{P}(\mathbf{r}')\cdot\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}^2}\,\mathrm{d}\tau' \qquad (1)

where: $\boldsymbol{\mathscr{r}} = \mathbf{r} - \mathbf{r}'$ — the separation vector from source point to field point, $\mathscr{r}$ its magnitude, $\hat{\boldsymbol{\mathscr{r}}}$ its direction (Griffiths's script-r). Correct, and nearly useless as it stands: the dipole kernel tangles the geometry of $\mathbf{P}$ with the angular factor at every point. The identity untangles it:

V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\oint_{\mathcal{S}} \frac{\sigma_b(\mathbf{r}')}{\mathscr{r}}\,\mathrm{d}a' + \frac{1}{4\pi\varepsilon_0}\int_{\mathcal{V}} \frac{\rho_b(\mathbf{r}')}{\mathscr{r}}\,\mathrm{d}\tau' \qquad (2)

where: $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$ and $\rho_b = -\nabla'\cdot\mathbf{P}$ — the bound charges, now appearing as outputs of calculus rather than inputs from physics. Equation (2) is ordinary Coulomb potential of ordinary charge densities: everything after this line is the electrostatics you already own. The derivation is short enough to know by heart.

Derivation AThe integration by parts — dipole soup becomes bound charge

The goal is the exact passage from (1) to (2). The strategy is a single integration by parts in the source variable, enabled by one kernel identity; the reward is the chapter's central theorem plus, as a corollary, an independent derivation of both bound-charge densities. Watch which variable the derivatives act on — everything here differentiates with respect to the source point $\mathbf{r}'$ (primed), with the field point $\mathbf{r}$ frozen; mixing them up is the classic way to lose a sign.

Step 1. The kernel identity. Differentiate $1/\mathscr{r}$ with respect to the source coordinate:

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

with a plus sign — moving the source toward the field point shrinks $\mathscr{r}$ and raises $1/\mathscr{r}$ (the unprimed gradient gives the familiar minus). Substituting turns (1) into

V = \frac{1}{4\pi\varepsilon_0}\int_{\mathcal{V}} \mathbf{P}\cdot\nabla'\left(\frac{1}{\mathscr{r}}\right)\mathrm{d}\tau'.

Step 2. The product rule (E0's toolkit, rule for $\nabla\cdot(f\mathbf{A})$ ), rearranged to isolate our integrand:

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

This is the parts step: the derivative migrates off the kernel and onto the polarization, at the cost of a total divergence.

Step 3. The divergence theorem converts the total-divergence piece into a surface integral over the object's boundary:

V = \frac{1}{4\pi\varepsilon_0}\oint_{\mathcal{S}} \frac{\mathbf{P}\cdot\hat{\mathbf{n}}}{\mathscr{r}}\,\mathrm{d}a' + \frac{1}{4\pi\varepsilon_0}\int_{\mathcal{V}} \frac{(-\nabla'\cdot\mathbf{P})}{\mathscr{r}}\,\mathrm{d}\tau'.

Read the two integrals: they are the Coulomb potentials of a surface density $\mathbf{P}\cdot\hat{\mathbf{n}}$ and a volume density $-\nabla'\cdot\mathbf{P}$ — equation (2), with $\sigma_b$ and $\rho_b$ derived, independently of last article's charge-counting. Why the two routes had to agree: charge conservation (the counting route) and the divergence theorem (this route) are the integral and differential faces of the same statement. Still, running both is not redundancy — it is how a physicist knows the formalism is watertight, and each route carries intuition the other lacks: the counting says where the charges physically are; the calculus says their fields need no correction terms.

Step 4. Scope and small print. The manipulation needs $\mathbf{P}$ differentiable inside $\mathcal{V}$ — where $\mathbf{P}$ jumps (layered media, interfaces), run the identity piecewise and each internal interface contributes its own $\sigma_b = (\mathbf{P}_1 - \mathbf{P}_2)\cdot\hat{\mathbf{n}}$ . Nothing restricts the field point: (2) holds for $\mathbf{r}$ outside or inside the object — but "inside" means the macroscopic field, and that claim gets its own derivation next.

Go deeper · Derivation B · 2 minThe inside story — why the smooth formula gives the true macroscopic field

Derivation BThe inside story — why the smooth formula gives the true macroscopic field

The goal is the license to use equation (2) inside matter, where the starting point (1) — built from far-field dipole potentials — looks least trustworthy: a molecule's neighbors are not in each other's far field. The strategy is Griffiths's §4.2.3 argument, hinging on the average-field theorem, which we derive rather than cite: the average electric field inside a sphere of radius $R$ , produced by an arbitrary charge arrangement within it, is $\mathbf{E}_{\rm ave} = -\mathbf{p}/4\pi\varepsilon_0R^3$ with $\mathbf{p}$ the total dipole moment of the interior charge. The chaos averages out; only the dipole moment survives — which is exactly what our smooth formula assumed matter to be.

Step 1. The swap trick. Average, over a sphere $\mathcal{B}$ of volume $\tfrac43\pi R^3$ , the field of a single interior point charge $q$ at position $\mathbf{s}$ :

\mathbf{E}_{\rm ave} = \frac{1}{\tfrac43\pi R^3}\int_{\mathcal{B}} \mathbf{E}_q(\mathbf{r})\,\mathrm{d}\tau.

By Coulomb's law this integral has the same form as the field at the charge's location produced by a uniform ball of charge density $\rho = -q/(\tfrac43\pi R^3)$ filling $\mathcal{B}$ — integrating the kernel over field points versus over source points is the same integral with roles swapped (and one sign from $\hat{\boldsymbol{\mathscr{r}}}$ reversing).

Step 2. Evaluate with the Gauss catalog. The field inside a uniform ball is the interior-linear workhorse: at radius $\mathbf{s}$ from the center, $\mathbf{E}_{\rm ball} = \rho\,\mathbf{s}/3\varepsilon_0$ . So

\mathbf{E}_{\rm ave} = \frac{-q/(\tfrac43\pi R^3)\;\mathbf{s}}{3\varepsilon_0} = -\frac{q\,\mathbf{s}}{4\pi\varepsilon_0R^3}.

Step 3. Superpose. For many interior charges, $\sum q_i\mathbf{s}_i = \mathbf{p}$ , the total dipole moment (about the sphere's center — and by Origin Dependence of Multipole Moments, for the neutral molecular clusters at issue, no origin ambiguity arises):

\mathbf{E}_{\rm ave} = -\frac{\mathbf{p}}{4\pi\varepsilon_0R^3}.

Every detail of the interior arrangement beyond $\mathbf{p}$ — the ångström-scale spikes, the near-molecule divergences — has averaged to zero exactly.

Step 4. Assemble the license. To find the macroscopic field at an interior point, average over a mesoscopic sphere there and split the sources: distant molecules (outside the sphere) are legitimately in the far-field regime, so formula (1) treats them correctly, and averaging their smooth field over the small sphere changes nothing. Near molecules (inside the sphere) contribute, by the theorem, an average field $-\mathbf{p}_{\rm in}/4\pi\varepsilon_0R^3 = -\mathbf{P}/3\varepsilon_0$ (using $\mathbf{p}_{\rm in} = \mathbf{P}\cdot\tfrac43\pi R^3$ ) — and a short calculation shows the smooth formula (1) assigns the same value to the sphere's interior contribution (it is precisely the interior field of a uniformly polarized sphere, the next article's headline result, recalled here in preview). Piece by piece, the smooth formula reproduces the honest average: the macroscopic field of equation (2) is the mesoscopic average of the microscopic truth.

Limiting cases

Case	Result	Meaning
uniform $\mathbf{P}$ , any shape	$\rho_b = 0$ : surface problem only	slab → sheets; sphere → cosine shell (next article)
thin slab ⊥ $\mathbf{P}$	$E_{\rm in} = -P/\varepsilon_0$ , zero outside	full-strength depolarizing field (factor 1)
long needle ∥ $\mathbf{P}$	$E_{\rm in} \approx 0$ (ends far away)	depolarizing factor $\to 0$ ; our bar: 2%
sphere	$E_{\rm in} = -P/3\varepsilon_0$	the 1/3 — next article's construction
far from any polarized object	$V \to \mathbf{p}_{\rm tot}\cdot\hat{\mathbf{r}}/4\pi\varepsilon_0r^2$	net dipole rules (zero total charge, always)
$\mathbf{P}$ jumps at internal interface	extra $\sigma_b = (\mathbf{P}_1 - \mathbf{P}_2)\cdot\hat{\mathbf{n}}$	run the identity piecewise
field point inside matter	(2) gives the macroscopic average	licensed by the average-field theorem

Common trapThe interior field of a polarized object opposes P — and depends on the object's shape, not just its material

Two habits to install. First, the depolarizing sign: for uniform polarization the bound charges arrange themselves so their interior field points against $\mathbf{P}$ (heads pile on the face $\mathbf{P}$ aims at, and fields run from heads back to tails inside). An electret's interior field fights its own polarization; in a linear dielectric this self-opposition is exactly why screening is partial. Second, the geometry dependence: "the field inside polarized material" has no material-only answer — identical $P$ gives $-P/\varepsilon_0$ in a slab, $-P/3\varepsilon_0$ in a sphere, nearly zero in a needle polarized lengthwise. Quoting an interior field without naming the shape is as meaningless as quoting a higher multipole moment without naming the origin — and the exam version of this trap is applying the slab formula to a sphere.

Worked examples

1 · The uniformly polarized slab, both ways (confidence builder)

A large slab of thickness $t = 2.0\,\mathrm{mm}$ carries uniform polarization $P = 1.0\times10^{-4}\,\mathrm{C/m^2}$ along its normal. Find the field everywhere and the potential step across the slab — first by bound charges, then by stacking dipole layers.

Step 1. Bound-charge route. $\rho_b = 0$ (uniform); $\sigma_b = +P$ on the exit face, $-P$ on the entry face: a parallel-plate pair. By the sheet-superposition of E1: outside, the sheets cancel — $\mathbf{E} = 0$ ; inside, they reinforce:

E_{\rm in} = \frac{\sigma_b}{\varepsilon_0} = \frac{P}{\varepsilon_0} = \frac{10^{-4}}{8.854\times10^{-12}} = 1.13\times10^{7}\,\mathrm{V/m},

directed from the $+$ face to the $-$ face — i.e. against $\mathbf{P}$ : $\mathbf{E}_{\rm in} = -\mathbf{P}/\varepsilon_0$ , the full-strength depolarizing field.

Step 2. Potential step: $\Delta V = E_{\rm in}t = (1.13\times10^{7})(2.0\times10^{-3}) = 2.3\times10^{4}\,\mathrm{V}$ — twenty-three kilovolts across two millimeters of un-electrified wax, from frozen-in molecular alignment alone.

Step 3. Dipole-layer route. Slice the slab into laminae of thickness $\mathrm{d}z$ : each is a dipole sheet with moment per area $P\,\mathrm{d}z$ , whose textbook signature is zero external field and a potential jump $\mathrm{d}V = P\,\mathrm{d}z/\varepsilon_0$ across itself. Stack the laminae: outside, still nothing; the total step is $\Delta V = \int_0^t (P/\varepsilon_0)\,\mathrm{d}z = Pt/\varepsilon_0 = 2.3\times10^{4}\,\mathrm{V}$ ✓.

Step 4. Same answers, opposite bookkeeping — route one groups the charge by where it is unbalanced (the two faces), route two by which molecules produced it (every lamina) — a miniature of the whole chapter's two readings of polarization.

2 · The bar electret's interior field — and the shape hierarchy (exam level)

The electret cylinder of the bound-charge article ( $P = 1.0\times10^{-4}\,\mathrm{C/m^2}$ axial, radius $R = 1.0\,\mathrm{cm}$ , length $L = 10\,\mathrm{cm}$ ) has bound charge only on its end faces ( $\pm P$ ). Find the field at the cylinder's center and compare with the slab.

Step 1. Replace and recognize. The equivalent problem: two oppositely charged disks, $\sigma = \pm10^{-4}\,\mathrm{C/m^2}$ , radius $R$ , separated by $L$ . The on-axis disk field is E1 stock: $E_{\rm disk} = \frac{\sigma}{2\varepsilon_0}\left(1 - \frac{z}{\sqrt{z^2 + R^2}}\right)$ at distance $z$ .

Step 2. At the center, each disk sits $z = L/2 = 5.0\,\mathrm{cm}$ away with $R = 1.0\,\mathrm{cm}$ : the bracket is $1 - 5/\sqrt{26} = 0.0194$ . Both disks push the same direction (from $+$ toward $-$ , i.e. against $\mathbf{P}$ ):

E_{\rm center} = 2\times\frac{\sigma}{2\varepsilon_0}(0.0194) = \frac{10^{-4}}{8.854\times10^{-12}}\times0.0194 = 2.2\times10^{5}\,\mathrm{V/m}.

Step 3. Compare shapes at identical $P$ : slab $1.13\times10^{7}$ ; this 10:1 rod $2.2\times10^{5}$ — the rod's interior field is only $1.9\%$ of $P/\varepsilon_0$ , because its bound charges are parked far away on small end-caps. Define the ratio $N \equiv E_{\rm in}/(P/\varepsilon_0)$ : slab $N = 1$ , our rod $N \approx 0.02$ , needle $N \to 0$ — and the sphere, worked in full next article, lands at exactly $N = \tfrac13$ .

Step 4. Read the design lesson. The depolarizing field is geometry's tax on polarization, and the taxonomy ( $N$ between 0 and 1, shapes interpolating) governs real devices: electrets are made long and thin along $\mathbf{P}$ so their own field doesn't destabilize the frozen alignment, while the sphere's exact $\tfrac13$ — the simplest member of the ellipsoid family, all of which have a uniform interior field at uniform $P$ — is the linchpin of the local-field and Clausius–Mossotti constructions later in the chapter.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §4.2 · Jackson, Classical Electrodynamics, 3rd ed., §4.3 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10 · Zangwill, Modern Electrodynamics, §6.2. The slab field follows from the parallel-plate result $E=\sigma/\varepsilon_0$ ; the uniformly polarized sphere gives $\mathbf{E}_{\rm in}=-\mathbf{P}/3\varepsilon_0$ . Using $\varepsilon_0=8.854\times10^{-12}$ F/m, the key worked numbers are: slab — $E_{\rm in} = 1.13\times10^7$ V/m, $\Delta V = 2.3\times10^4$ V across 2 mm, with the dipole-layer route giving the same result; bar electret — disk bracket $1 - 5/\sqrt{26} = 0.0194$ , $E_{\rm center} = 2.2\times10^5$ V/m $= 0.019\,P/\varepsilon_0$ , compared with the slab's factor of 1 and the sphere's factor of 1/3.

Builds on: Bound Charges: Surface and Volume · Polarization Density P · Electric Dipole Moment · Gauss's Law: Applications and Symmetry

Leads to: Uniformly Polarized Sphere · The Electric Displacement D · Gauss's Law in Dielectrics · Local Field and the Lorentz Cavity

If the bound-charge description is exact, is anything lost compared to keeping the dipoles?

For macroscopic fields, nothing — that is the theorem. What the bound-charge picture deliberately discards is sub-mesoscopic information: the true field between molecules, at lattice sites, inside atoms. Those need the microscopic description (and quantum mechanics). The subtle middle ground is the local field — the field a specific molecule actually experiences, which is neither the macroscopic average nor the bare applied field; extracting it from the macroscopic solution is the Lorentz-cavity construction later in this chapter. Rule of thumb: fields that instruments at human scales measure — bound charges suffice, exactly; fields that individual molecules feel — one more layer of theory.

Why does the slab produce no external field while a polarized sphere produces a full dipole field outside?

Geometry of cancellation. The slab's two sheets are (idealized as) infinite and uniformly charged: outside, their fields are uniform, opposite, and equal — exact cancellation everywhere. The sphere's cosine-distributed surface charge has no such planar cancellation; its exterior survives as a pure dipole field (next article's derivation). Real finite slabs leak at the edges — fringing fields — and far away every polarized object, slab included, shows the dipole field of its total moment $\mathbf{p} = \mathbf{P}V$ , as the multipole expansion demands for a neutral source.

Which route should I actually use on a problem — bound charges or direct dipole integration?

Bound charges, almost always: the whole point of the identity is that (2) hands the problem to the strongest tools you own. Uniform $\mathbf{P}$ kills $\rho_b$ and leaves a surface-charge problem (sheets, disks, cosine shells — often Gauss or standard integrals). Symmetric $\rho_b$ invites Gauss directly (the radial sphere of the previous article). Awkward geometry still benefits: numerical solvers want charge densities, not polarizations. Direct dipole integration (1) wins only in special cases — genuinely point-like sources, or when you want the far field, where "total dipole moment times the point-dipole formula" is a one-liner.

How can the average-field theorem hold regardless of how wildly the interior charges are arranged?

Because averaging a $1/r^2$ field over a ball is a very forgiving operation. The swap trick converts "average the field over the ball" into "field at the source's location due to a uniform ball" — and the interior of a uniform ball has the linear field for which only the source's position (weighted by charge) matters. Summing positions weighted by charge is the dipole moment; every higher moment of the arrangement enters the average with coefficient exactly zero. It is the mirror image of the sphere's exterior magic (a shell acts as a point at its center): spherical averaging annihilates all multipoles but one, inside as well as out.

What happens to the equivalence at the sharp edge of the object, where n-hat is undefined?

The identity is fine; the idealization is what's strained. At edges and corners the surface normal jumps, $\sigma_b$ is well defined on each face up to the edge, and the potential remains continuous — but the field of an idealized sharp edge can diverge (mildly, integrably), exactly as it does for charged conductor edges in the E1 chapters. Real materials round every edge at some scale, capping the field; computations either round the geometry or handle the edge singularity with local analytic solutions. None of this is special to bound charge — it is the standard price of sharp-corner idealizations, paid in the same currency as everywhere in potential theory.

Does the equivalence theorem work for time-dependent polarization too?

With well-defined upgrades. Static bound charges generalize: $\sigma_b$ and $\rho_b$ still source fields, but a changing $\mathbf{P}$ additionally carries the polarization current $\mathbf{J}_p = \partial\mathbf{P}/\partial t$ (from the polarization article), which sources magnetic fields; the full time-dependent equivalence is "bound charge plus bound current," and it slots straight into Maxwell's equations — that is in fact how the macroscopic Maxwell equations are derived. At optical frequencies the machinery becomes the theory of refractive index: light in glass is light in vacuum plus the re-radiation of oscillating bound charges. Same theorem, faster clock.

Is there an intuition for why a needle has almost no depolarizing field but a slab has the maximum?

Track where the bound charges end up relative to the interior. Polarize along the long axis of a needle: the charges sit on two tiny, distant end-caps — from the middle of the needle they subtend almost no solid angle, so their field there is negligible. Polarize across a thin slab: the charges form two vast planes hugging every interior point — maximal solid angle, maximal field, $P/\varepsilon_0$ . The sphere sits exactly in between at $\tfrac13$ (a solid-angle average with the cosine weighting, as the next article shows). Mnemonic: the depolarizing field measures how much of the sky, seen from inside, is papered with your own bound charge.