Multipole Expansion of the Potential

Walk away from any tangle of charge — a molecule, a charged instrument, a thundercloud — and watch its potential. Up close the mess is irreducible: every lump contributes, every asymmetry matters. As you retreat, the blob organizes itself. First you see only its total charge, a $1/r$ glow. If that vanishes, a fainter $1/r^2$ pattern emerges — the dipole whisper of separated plus and minus. Behind that hides a $1/r^3$ quadrupole structure, then an octopole, each rung falling one power of $r$ faster than the last. Distance is a low-pass filter for charge structure, and the multipole expansion is the mathematics of that filter. Potential of a Localized Charge Distribution made a promise it could not keep at the time: it climbed exactly two rungs — monopole and dipole, by brute binomial expansion — and deferred the full ladder to this article. Here we deliver. The engine is the generating function proved in Legendre Polynomials: one substitution converts the inverse separation into an exact power series in $r'/r$ , every coefficient a Legendre polynomial, and the potential of any localized source into a systematic hierarchy of moments. The word exact deserves its italics. Outside the smallest sphere enclosing the source, this is not an asymptotic estimate but a convergent series — as much an identity as the geometric series it descends from. Only truncations are approximate, and their error is controlled. Master the ladder and the next three articles — monopole, dipole, and quadrupole moments — become close inspections of its first three rungs.

Key takeaways

Far from a localized charge distribution, $V$ organizes into a ladder: monopole $\propto 1/r$ , dipole $\propto 1/r^2$ , quadrupole $\propto 1/r^3$ , octopole $\propto 1/r^4$ — each rung weighted by a moment of the charge.
The engine: the law of cosines plus the Legendre generating function (proved in Legendre Polynomials) give the exact ladder $\frac{1}{\mathfrak{s}}=\frac{1}{r}\sum_l\left(\frac{r'}{r}\right)^l P_l(\cos\gamma)$ .
Integrating against $\rho$ rung by rung gives the multipole series $V=\frac{1}{4\pi\varepsilon_0}\sum_l\frac{1}{r^{l+1}}\int r'^{\,l}P_l(\cos\gamma)\,\rho\,\mathrm{d}\tau'$ .
Validity: the series converges to the exact potential everywhere outside the smallest origin-centred sphere enclosing the source ( $r>r'_{\max}$ ) — not merely a far-field formula.
Dominance: the first non-vanishing moment rules the far field; truncation error is one factor $\sim r'_{\max}/r$ per rung skipped — for a physical dipole, the dipole term alone is within $1\%$ at $r=5d$ .
Key idea: expand $1/\mathfrak{s}$ in Legendre polynomials of $r'/r$ ; integrating against $\rho$ term by term turns the potential into a sum over moments, each falling off one power faster in $1/r$ than the last.

The core insight

Expand $1/|\mathbf{r}-\mathbf{r}'|$ in Legendre polynomials of $r'/r$ ; integrating against $\rho$ term by term turns the potential into a sum over moments, each falling off one power faster in $1/r$ than the last.

Where this lives today

The multipole expansion's biggest modern job is not analysis but speed. The fast multipole method (FMM) of Greengard and Rokhlin — routinely listed among the top-ten algorithms of the twentieth century — attacks the $N$ -body problem head on: pairwise potentials cost $O(N^{2})$ , but compressing distant clusters into a handful of moments (Derivation C's error bound saying how few rungs suffice at a given separation) collapses the cost to $O(N)$ . Molecular-dynamics engines breathe this idea, and polarizable force fields such as AMOEBA assign every atom its own permanent monopole, dipole, and quadrupole — a per-atom copy of this ladder, marched through millions of time steps. The ladder also climbs out of statics. When sources oscillate, each moment radiates its own signature — electric dipole first and loudest, quadrupole fainter — and "which multipole leads" becomes the selection-rule logic of antennas and atomic transitions alike; the E9 radiation chapter makes that story quantitative. Gravity runs the identical mathematics one floor down: satellite geodesy expands Earth's field in harmonics ( $J_2$ , the oblateness quadrupole, precesses satellite orbits), and gravitational-wave astronomy exists because a mass distribution's monopole and dipole cannot radiate — the quadrupole is the first rung allowed to sing.

The physical picture

Stand far from a clump of charge and you cannot resolve its shape. What you can see is a hierarchy of increasingly delicate features, revealed one at a time as the coarser ones vanish. Net charge is the loudest signal — a spherically symmetric $1/r$ potential, indistinguishable from a point charge. Neutralize it and the next signal through is the lopsidedness of plus against minus: a two-lobed $1/r^2$ dipole pattern. Kill that too and a four-lobed $1/r^3$ quadrupole shape emerges. Every distribution is a stack of ideal silhouettes, and walking away peels them off the top down.

Figure

The far-field ladder — a messy blob resolves into a hierarchy of ideal shapes

Two facts make this quantitative. First, the small parameter is geometric: $r'/r$ , source size over distance, and each rung carries one more power of it — why successive silhouettes fade one power of $r$ faster. Second — the nuance most treatments blur — outside the smallest sphere enclosing the charge, the infinite ladder is the potential: an exact, convergent series. "Approximate" describes only the act of keeping finitely many rungs, and even that carries a computable error tag. The naming is a doubling game: the $l$ -th rung is a $2^l$ -pole — one charge (monopole), two opposite charges (dipole), four in alternation (quadrupole), eight (octopole) — each the cheapest arrangement whose first non-vanishing moment sits at order $l$ . The anatomy of the first three moments is the business of the next three articles, Monopole Moment, Electric Dipole Moment, and Electric Quadrupole Moment. This article builds the ladder they hang from.

From separation to ladder

Everything starts from the master formula of Potential of a Localized Charge Distribution: carve the source into lumps, credit each a scalar $\mathrm{d}q/4\pi\varepsilon_0$ over its separation from the field point, and add. Writing $\mathfrak{s}=|\mathbf{r}-\mathbf{r}'|$ for that separation,

V(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho(\mathbf{r}')}{\mathfrak{s}}\,\mathrm{d}\tau' \qquad (1)

where: $\rho(\mathbf{r}')$ — the charge density at the source point $\mathbf{r}'$ , ranging over the bounded distribution; $\mathbf{r}$ — the fixed field point; $\mathrm{d}\tau'$ — the source volume element. The formula is exact and, for any interesting $\rho$ , miserable: $\mathfrak{s}$ tangles source and field coordinates inside a square root. The whole multipole expansion is one idea — untangle them. The geometry is a triangle: origin (planted inside the source), source point at distance $r'$ , field point at distance $r$ , with $\gamma$ the angle between $\mathbf{r}$ and $\mathbf{r}'$ :

Figure

The triangle behind the expansion — field point, source point, and the enclosing sphere

Derivation AThe exact ladder for the inverse separation

The goal is the identity this whole family of articles runs on: the inverse separation $1/\mathfrak{s}$ , written as a power series in $r'/r$ whose coefficients are Legendre polynomials of $\cos\gamma$ . Two moves — use the law of cosines to squeeze the tangled square root into a function of the single ratio $r'/r$ , then recognize that function as the Legendre generating function, which Legendre Polynomials derived and which we recall by name rather than re-prove. Watch how the convergence condition $r'<r$ is not imposed by hand: it walks in as the radius of convergence of the generating function itself.

Step 1. Apply the law of cosines to the triangle with sides $r$ , $r'$ , and $\mathfrak{s}$ :

\mathfrak{s}^2 = r^2 + r'^2 - 2rr'\cos\gamma = r^2\left[1 - 2\left(\frac{r'}{r}\right)\cos\gamma + \left(\frac{r'}{r}\right)^{2}\right]

Why factor out $r^2$ : the bracket now depends on the source only through the ratio $\varepsilon\equiv r'/r$ and the angle $\gamma$ — the tangle of $\mathbf{r}$ with $\mathbf{r}'$ has been compressed into two clean variables.

Step 2. Invert and take the square root:

\frac{1}{\mathfrak{s}} = \frac{1}{r}\,\frac{1}{\sqrt{1-2\varepsilon\cos\gamma+\varepsilon^{2}}}

Look hard at that square root. With $t=\varepsilon$ and $x=\cos\gamma$ it is precisely $\left(1-2xt+t^{2}\right)^{-1/2}$ — the generating function of the Legendre polynomials, whose expansion $\sum_l P_l(x)\,t^l$ for $|t|<1$ was derived in Legendre Polynomials. We spend that owned result without re-deriving it.

Step 3. Substitute the expansion and restore the physical variables:

\frac{1}{\mathfrak{s}} = \frac{1}{r}\sum_{l=0}^{\infty}\left(\frac{r'}{r}\right)^{l}P_l(\cos\gamma) \qquad (2)

Why $|t|<1$ becomes $r>r'$ : the generating function converges only inside its unit disk, so the expansion holds whenever the field point is farther from the origin than the source point. An extended source needs this for every source point at once — where $r'_{\max}$ enters, in Derivation C. *Verified: binomial expansion to second order gives $\frac{1}{r}\left[1+\varepsilon\cos\gamma+\varepsilon^{2}\tfrac{1}{2}\left(3\cos^{2}\gamma-1\right)+\cdots\right]$ — exactly $P_0$ , $P_1$ , $P_2$ . Numerically, at $x=0.6$ , $t=0.3$ : closed form $1/\sqrt{0.73}=1.170411$ ; partial sums $\sum_{l\le6}P_l(0.6)(0.3)^l$ reach $1.170406$ — five decimals by six terms.*

Equation (2) is the article. The left side couples the two position vectors inseparably; the right side is a sum of products — a power of $r'$ (source only), a power of $r$ (field only), a Legendre polynomial of the angle between. That untangling lets the integral in (1) act on the source factors alone, leaving pure numbers: the moments.

The multipole series

Go deeper · Derivation B · 2 minAssembling the series and naming its rungs

Derivation BAssembling the series and naming its rungs

The goal is the multipole expansion proper: substitute the ladder (2) into the master formula (1) and read off, order by order, what number each rung attaches to its power of $1/r$ . The strategy is honest bookkeeping — swap the sum and the integral (legality settled by Derivation C), then interrogate the first terms until their identities snap into focus. Watch how the field point survives only through $1/r^{l+1}$ and the direction hiding inside $\gamma$ .

Step 1. Substitute and exchange sum with integral:

V(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\sum_{l=0}^{\infty}\frac{1}{r^{\,l+1}}\int r'^{\,l}\,P_l(\cos\gamma)\,\rho(\mathbf{r}')\,\mathrm{d}\tau' \qquad (3)

This is the multipole expansion of the potential — the payoff equation of the chapter. Why each integral is (almost) a property of the source alone: $r'$ and $\rho$ are pure source data, but $\gamma$ involves the field direction $\hat{\mathbf{r}}$ ; repackaging it into genuinely field-independent numbers is what the moment definitions of the next three articles accomplish.

Step 2. Read the $l=0$ rung. $P_0=1$ , so the integral is $\int\rho\,\mathrm{d}\tau' = Q$ , the total charge, and the term is $\frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$ — the monopole: from far enough away, everything is a point charge. Its anatomy (and the origin-independence of $Q$ ) is owned by Monopole Moment.

Step 3. Read the $l=1$ rung. $P_1(\cos\gamma)=\cos\gamma$ and $r'\cos\gamma=\hat{\mathbf{r}}\cdot\mathbf{r}'$ , so the integral is $\hat{\mathbf{r}}\cdot\int\mathbf{r}'\rho\,\mathrm{d}\tau' = \hat{\mathbf{r}}\cdot\mathbf{p}$ — the dipole term $\frac{1}{4\pi\varepsilon_0}\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^{2}}$ , exactly the second rung Potential of a Localized Charge Distribution climbed by binomial force. The vector $\mathbf{p}$ , and the celebrated field it produces, belong to Electric Dipole Moment.

Step 4. Read the $l=2$ rung and stop unpacking. With $P_2(u)=\tfrac{1}{2}(3u^{2}-1)$ , the quadrupole term is

V_2(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\,\frac{1}{r^{3}}\int r'^{\,2}\,P_2(\cos\gamma)\,\rho(\mathbf{r}')\,\mathrm{d}\tau'

Why we leave it in scalar form: recasting the direction-dependence as a traceless second-rank tensor is a construction with its own payoffs and pitfalls, owned wholesale by Electric Quadrupole Moment. For axially symmetric sources, this scalar integral is all you need.

Step 5. Name the hierarchy. Assembling the first rungs,

V(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\left[\frac{Q}{r}+\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^{2}}+\frac{1}{r^{3}}\int r'^{\,2}P_2(\cos\gamma)\,\rho\,\mathrm{d}\tau'+\cdots\right] \qquad (4)

Each successive rung carries one more factor of $r'$ inside the moment and one more $1/r$ outside — the moment gains a length, the falloff pays for it: each term is smaller than its predecessor by roughly $r'_{\max}/r$ . Verified: the $l=0$ and $l=1$ terms reproduce, coefficient for coefficient, the binomial climb in Potential of a Localized Charge Distribution (Griffiths §3.4.1–3.4.2); the $l=2$ scalar integral matches Griffiths Eq. 3.95's third term with $P_2$ written out.

The dominance logic is worth a slogan: the first non-vanishing moment rules the far field. If $Q\neq0$ , the source is asymptotically a point charge. If $Q=0$ (a neutral molecule), the dipole leads and $V\sim1/r^{2}$ . If $Q=0$ and $\mathbf{p}=\mathbf{0}$ (carbon dioxide, or the array of worked example 2), the quadrupole leads and $V\sim1/r^{3}$ . Symmetry often tells you where the ladder starts before you compute anything — finding that first rung is frequently the entire problem.

$l$	name	falloff	moment integral	anatomy owned by
0	monopole	$1/r$	$Q=\int\rho\,\mathrm{d}\tau'$	Monopole Moment
1	dipole	$1/r^{2}$	$\mathbf{p}=\int\mathbf{r}'\rho\,\mathrm{d}\tau'$	Electric Dipole Moment
2	quadrupole	$1/r^{3}$	$\int r'^{\,2}P_2(\cos\gamma)\,\rho\,\mathrm{d}\tau'$	Electric Quadrupole Moment
3	octopole	$1/r^{4}$	$\int r'^{\,3}P_3(\cos\gamma)\,\rho\,\mathrm{d}\tau'$	rarely needed in statics

One disclosure about generality. Equation (3) is most convenient for azimuthally symmetric sources ( $\gamma$ then reduces to the polar angle $\theta$ for sources on the axis). With no symmetry, the clean bookkeeping uses the addition theorem stated in Spherical Harmonics, splitting $P_l(\cos\gamma)$ into products of $Y_{lm}$ 's of source and field directions separately. Each rung then carries $2l+1$ complex moments $q_{lm}=\int r'^{\,l}\,Y_{lm}^{*}(\theta',\phi')\,\rho\,\mathrm{d}\tau'$ , and the series becomes

V(\mathbf{r})=\frac{1}{\varepsilon_0}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\frac{1}{2l+1}\,\frac{q_{lm}}{r^{\,l+1}}\,Y_{lm}(\theta,\phi) \qquad (5)

where: $q_{lm}$ — the spherical multipole moments, $2l+1$ numbers exhausting the order- $l$ information; $\theta,\phi$ — the field point's angles. Nothing conceptually new: same ladder, direction-dependence stored in an orthonormal basis instead of a single $P_l(\cos\gamma)$ . We state it, use it never (in this article), and move on.

Exact, not asymptotic

The claim that separates careful practitioners from formula users: outside the smallest origin-centred sphere containing all the charge, the multipole series does not merely approximate $V$ — it converges to $V$ . Exactly. The proof costs one bound and one geometric series.

Go deeper · Derivation C · 2 minWhy the series is exact outside the enclosing sphere

Derivation CWhy the series is exact outside the enclosing sphere

The goal is the validity statement with its precise boundary: the series (3) converges absolutely to the true potential at every field point with $r>r'_{\max}$ , the radius of the smallest origin-centred sphere enclosing the source. The strategy is domination: cap the Legendre polynomials by $1$ , whereupon the ladder is bounded rung by rung by a geometric series in $r'_{\max}/r$ ; uniform convergence then licenses the sum-integral swap Derivation B performed on credit. Watch what the argument does not deliver — speed. Convergent is a promise about the infinite sum, not your two-term truncation.

Step 1. Recall the bound $|P_l(x)|\le1$ for $-1\le x\le1$ (catalogued in Legendre Polynomials). Then each rung of the ladder (2) obeys

\left|\left(\frac{r'}{r}\right)^{l}P_l(\cos\gamma)\right| \le \left(\frac{r'_{\max}}{r}\right)^{l} \quad\text{for every source point } r'\le r'_{\max}

Step 2. Dominate by the geometric series. For $r>r'_{\max}$ , set $\epsilon = r'_{\max}/r<1$ :

\sum_{l=0}^{\infty}\epsilon^{\,l} = \frac{1}{1-\epsilon} < \infty

The ladder converges absolutely and uniformly over all source points at once. Why uniformity is the working part: one $\epsilon$ handles the whole distribution, so the interchange of $\sum_l$ and $\int\mathrm{d}\tau'$ in Derivation B is legitimate — the summed series equals the exact integral (1), not an approximation to it.

Step 3. Extract the free gift: an explicit truncation error. Keeping rungs through $l=L$ , the discarded tail is bounded by

|\Delta V| \le \frac{1}{4\pi\varepsilon_0}\,\frac{\int|\rho|\,\mathrm{d}\tau'}{r}\;\frac{\epsilon^{\,L+1}}{1-\epsilon}

Why this is the practical face of exactness: each extra rung buys a factor of $\epsilon$ . At $r=10\,r'_{\max}$ a rung is worth a factor of ten; at $r=1.05\,r'_{\max}$ it buys almost nothing and you may need dozens. Convergent everywhere outside the sphere, fast only well outside it.

Step 4. Mark the boundary honestly. At $r<r'_{\max}$ the geometric bound fails and so does the series: for charge at radius $r'>r$ the correct move is to split the integral and expand the near part in the interior ladder $r^{\,l}/r'^{\,l+1}$ — the other radial family from Separation of Variables in Spherical Coordinates. The exterior series diverges there because $1/\mathfrak{s}$ genuinely has no power series in $1/r$ inside the source region. Verified: for the physical dipole of worked example 1 evaluated brutally close in, at $r=2d$ , $\theta=60^\circ$ — exact potential $531.45\ \mathrm{V}$ ; the ladder through $l=9$ agrees to $1.5\times10^{-7}$ relative, through $l=20$ to machine precision. The dipole term alone is off by $5.7\%$ : exactness of the series, sloppiness of the truncation, in one number.

Limiting cases

case	result	meaning
$r\gg r'_{\max}$ , $Q\neq0$	$V\to\frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$	keep $l=0$ : every distribution impersonates a point charge
neutral system ( $Q=0$ )	$V\sim\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{4\pi\varepsilon_0 r^{2}}$	dipole term leads — the far field of every polar molecule
$Q=0$ and $\mathbf{p}=\mathbf{0}$	$V\sim1/r^{3}$	quadrupole rules: worked example 2 in numbers
point charge at the origin	series truncates to $\frac{q}{4\pi\varepsilon_0 r}$ exactly	$r'=0$ kills every rung above $l=0$ : the only one-rung ladder
on-axis field point, axial source ( $\gamma=0$ )	every $P_l(1)=1$	plain powers of $1/r$ — the on-axis trick behind the sphere problems of Separation of Variables in Spherical Coordinates
$r<r'_{\max}$	exterior series diverges	split the source; expand the near part in the interior ladder $r^{\,l}/r'^{\,l+1}$

Common trapThe multipole expansion is only valid far away — a half-truth that misleads in both directions

Wrong twice. Direction one: the full series is not a far-field caricature — it is exact everywhere outside the smallest enclosing sphere, even a hair's breadth outside, where "far away" is laughable. What degrades near the sphere is convergence speed, not validity. Direction two: the word "valid" seduces people into trusting truncations too close in — quoting the dipole term at $r=2d$ ( $5.7\%$ off in Derivation C's check) as if it were the dipole series. The same confusion makes a physical dipole "be" a pure dipole: two charges at finite separation carry an infinite tail of higher rungs that a point dipole lacks, agreeing only in the limit $d\to0$ at fixed $qd$ — a story Electric Dipole Moment finishes. Say it precisely: the series is exact for $r>r'_{\max}$ ; every truncation carries error $\sim(r'_{\max}/r)^{L+1}$ ; and inside the sphere the exterior series is not slow — it is wrong.

Worked examples

1 · The physical dipole, exactly versus its dipole term (Griffiths Ex. 3.10 with numbers)

Charges $\pm q$ sit at $z=\pm d/2$ , with $q=1.0\ \mathrm{nC}$ , $d=2.0\ \mathrm{mm}$ . Compare the exact potential with the dipole term $V_1=\frac{1}{4\pi\varepsilon_0}\frac{qd\cos\theta}{r^{2}}$ at $r=5d,\ 10d,\ 50d$ , on-axis and at $\theta=60^\circ$ .

Step 1. Set up the exact potential — two point charges, no expansion:

V_{\text{exact}} = \frac{q}{4\pi\varepsilon_0}\left[\frac{1}{\sqrt{r^{2}-rd\cos\theta+d^{2}/4}}-\frac{1}{\sqrt{r^{2}+rd\cos\theta+d^{2}/4}}\right]

On the axis this collapses to $V_{\text{exact}}=\frac{1}{4\pi\varepsilon_0}\frac{qd}{r^{2}-d^{2}/4}$ , which telegraphs the answer: the dipole term differs from it by the factor $1-(d/2r)^{2}$ .

Step 2. Identify the ladder structure before computing. $Q=q-q=0$ : no monopole. The dipole moment is $qd\,\hat{\mathbf{z}}$ . And the elegant bit — by the up-down antisymmetry of the pair, every even rung vanishes. The first correction to the dipole term is the octopole ( $l=3$ ), so the truncation error scales as $(d/2r)^{2}$ relative — two powers, not one.

Step 3. Compute, with $\frac{q}{4\pi\varepsilon_0}=8.988\ \mathrm{V\,m}\times10^{-9}$ :

$r$	direction	$V_{\text{exact}}$	dipole term	error of truncation
$5d=10$ mm	on-axis	$181.57$ V	$179.75$ V	$-1.00\%$
$5d=10$ mm	$\theta=60^\circ$	$89.09$ V	$89.88$ V	$+0.88\%$
$10d=20$ mm	on-axis	$45.05$ V	$44.94$ V	$-0.25\%$
$10d=20$ mm	$\theta=60^\circ$	$22.42$ V	$22.47$ V	$+0.22\%$
$50d=100$ mm	on-axis	$1.798$ V	$1.798$ V	$-0.010\%$
$50d=100$ mm	$\theta=60^\circ$	$0.8987$ V	$0.8988$ V	$+0.009\%$

Step 4. Read the table like a physicist. The error falls by $4$ from $5d$ to $10d$ and by $100$ from $5d$ to $50d$ — the $(d/2r)^{2}$ octopole scaling, exactly $1.000\%$ , $0.250\%$ , $0.010\%$ on-axis. Even the sign is diagnostic: on-axis the dipole term underestimates ( $P_3(1)=1$ adds), while at $60^\circ$ it overestimates because $P_3(\cos60^\circ)=-0.4375$ flips the correction — predicted error $(d/2r)^{2}\times0.875=0.875\%$ against the observed $0.88\%$ . At five source-lengths the single dipole term is already a one-percent theory.

2 · Three charges in a line — the ladder starts at the third rung (exam level)

Charges $+q$ , $-2q$ , $+q$ sit at $z=+d$ , $0$ , $-d$ with $q=1.0\ \mathrm{nC}$ , $d=1.0\ \mathrm{mm}$ . Show the monopole and dipole moments vanish, find the leading term, test it at $r=10d$ .

Step 1. Climb the ladder from the bottom. Monopole: $Q=q-2q+q=0$ . Dipole: $p_z = q(d)+(-2q)(0)+q(-d)=0$ . Two rungs gone by construction — the standard model of a linear quadrupole. The far potential must open at $l=2$ with $V\sim1/r^{3}$ .

Step 2. Evaluate the $l=2$ scalar integral of equation (4) — for point charges, a sum. The sources sit on the $z$ -axis, so for the charge at $z=\pm d$ the angle obeys $\cos\gamma=\pm\cos\theta$ , and $P_2$ , being even, gives $P_2(\cos\theta)$ for both:

\int r'^{\,2}P_2(\cos\gamma)\,\rho\,\mathrm{d}\tau' \;\longrightarrow\; q\,d^{2}P_2(\cos\theta)+0+q\,d^{2}P_2(\cos\theta) = 2qd^{2}P_2(\cos\theta)

so the leading term is $V_2 = \frac{1}{4\pi\varepsilon_0}\,\frac{2qd^{2}}{r^{3}}P_2(\cos\theta)$ — the $1/r^{3}$ falloff, with the $3\cos^{2}\theta-1$ four-lobe pattern (positive on the axis, negative on the equator, vanishing at $54.7^\circ$ ).

Step 3. Test on-axis at $r=10d=10$ mm, $P_2(1)=1$ . Truncated: $V_2 = 8.988\times10^{9}\times2\times(10^{-9})(10^{-3})^{2}/(10^{-2})^{3} = 17.98\ \mathrm{V}$ . Exact, from three Coulomb terms $\frac{q}{4\pi\varepsilon_0}\left[\frac{1}{r-d}+\frac{1}{r+d}-\frac{2}{r}\right]$ : $18.16$ V. Error: $-1.00\%$ — and the on-axis closed form $\frac{1}{4\pi\varepsilon_0}\frac{2qd^{2}}{r(r^{2}-d^{2})}$ shows the truncation factor is exactly $1-(d/r)^{2}$ : the next surviving rung is $l=4$ (odd rungs cancel by symmetry), again two powers per step.

Step 4. Off-axis check at $\theta=60^\circ$ , where $P_2(0.5)=-0.125$ : truncated $V_2=-2.247$ V against exact $-2.298$ V — the quadrupole term captures the negative lobe correctly, off by $2.2\%$ . Control: at $r=5d$ on-axis the error grows to $-4.00\%$ , quadrupling as $(d/r)^{2}$ demands. A neutral, dipole-free source is invisible at order $1/r$ and $1/r^{2}$ — the quadrupole rung is its loudest voice; the tensor machinery generalizing $2qd^{2}$ to arbitrary shapes is built in Electric Quadrupole Moment.

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Edinburgh, Aberdeen, King's College London, Cambridge

Maxwell is filed in memory under the four equations, but the Treatise on Electricity and Magnetism (1873) is also where the multipole family acquired its modern spine. Maxwell organized the far field through what he called "poles": start with the point charge, then differentiate its $1/r$ potential along chosen axes — once for a dipole, twice for a quadrupole — so the hierarchy becomes successive directional derivatives of a single function, each falloff one power steeper, each angular pattern one notch more intricate. The construction is exactly equivalent to the Legendre ladder built here, and characteristically it came wrapped in geometry rather than algebra. He wielded the tools in earnest: his 1857 Adams Prize essay dismantled the stability of Saturn's rings with expansion methods, and his molecular theory of gases leaned on spherical-harmonic decompositions decades before quantum mechanics made them compulsory. A man who could see a hierarchy of fading silhouettes inside a derivative — and then measure a molecule with it.

History

1782

— Legendre, computing the gravitational attraction of spheroids, introduces the polynomials

P_l

; within three years Laplace embeds them in his general potential theory.

1799

— Laplace's Mécanique Céleste industrializes the expansion for planetary figures — the ladder's first large-scale customer is gravity, not electricity.

1873

— Maxwell's Treatise constructs the multipole hierarchy as successive derivatives of

1/r

— "poles" — tying spherical harmonics to the geometry of the far field.

1987

— Greengard and Rokhlin publish the fast multipole method, converting the ladder's error bound into an order-N algorithm for N-body fields.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §3.4 (Multipole Expansion) · Jackson, Classical Electrodynamics, 3rd ed., §4.1–4.2 · Zangwill, Modern Electrodynamics, Ch. 4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §10.4

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §3.4 (multipole series, dipole field Eq. 3.104, quadrupole) and Jackson Classical Electrodynamics 3rd ed. §4.1–4.2 (spherical moments $q_{\ell m}$ Eq. 4.3, potential Eq. 4.1, origin dependence). Verifier should recompute: (1) the Legendre generating-function expansion of $1/|\mathbf{r}-\mathbf{r}'|$ ; (2) the dipole field by taking $-\nabla(\mathbf{p}\cdot\hat{\mathbf{r}}/r^2)$ ; (3) the origin-shift transformations $\mathbf{p}'=\mathbf{p}-Q\mathbf{a}$ and the $Q'_{ij}$ relation; (4) the linear-quadrupole on-axis potential. The water dipole moment $1.85\,\mathrm{D}=6.2\times10^{-30}\,\mathrm{C\,m}$ is the standard tabulated value. Worked numbers recomputed for this edition: physical dipole $q=1$ nC, $d=2$ mm — exact vs dipole term $(181.57, 179.75)$ V at $r=5d$ on-axis ( $-1.00\%$ ), $(89.09, 89.88)$ V at $\theta=60^\circ$ ( $+0.88\%$ ); linear quadrupole $q=1$ nC, $d=1$ mm — at $10d$ on-axis $18.16$ V vs $17.98$ V ( $-1.00\%$ ); convergence at $r=2d$ , $\theta=60^\circ$ : exact $531.45$ V, ladder through $l=9$ within $1.5\times10^{-7}$ relative; generating-function spot value $1/\sqrt{0.73}=1.170411$ at $x=0.6$ , $t=0.3$ .

Builds on: Legendre Polynomials · Separation of Variables in Spherical Coordinates · Poisson's Equation · Laplace's Equation Leads to: Boundary Conditions for the Potential · Image Charges for a Conducting Sphere · Harmonic Functions and the Mean-Value Theorem

Is the multipole expansion an approximation or an exact statement?

Both, in carefully separated senses. The infinite series is exact: outside the smallest origin-centred sphere enclosing the charge it converges absolutely to the true potential — Derivation C's geometric-series domination is the whole proof. What is approximate is any truncation, and even that carries an error tag: $(r'_{\max}/r)^{L+1}/(1-r'_{\max}/r)$ after rung $L$ . Textbooks introduce the expansion "far away" because that is where one or two rungs suffice, and the shorthand hardens into the false belief that the series itself is a far-field caricature. It is not. It is the Coulomb integral, rewritten.

What happens if the field point is inside the charge distribution?

The exterior series diverges there — the generating function is being used outside its unit disk. The honest procedure splits the source at the field radius $r$ : charge closer in ( $r'<r$ ) expands in the exterior ladder as usual, while charge farther out ( $r'>r$ ) uses the interior ladder $r^{\,l}/r'^{\,l+1}$ — the other radial family from Separation of Variables in Spherical Coordinates, radii swapped. Both pieces converge in their own domains; this split is how one computes potentials inside planets, nuclei, and electron clouds. The exterior-only form here assumes you stand wholly outside the source.

Where do the names monopole, dipole, quadrupole, octopole come from?

From a doubling game: the $l$ -th rung is a $2^l$ -pole, after the minimal point-charge model whose ladder starts at rung $l$ . One charge starts at the monopole; two opposite charges cancel it and start at the dipole; two opposed dipoles (four charges) start at the quadrupole; two opposed quadrupoles (eight) at the octopole. Maxwell's Treatise made the pattern structural: the $2^l$ -pole is the $l$ -th directional derivative of $1/r$ along $l$ axes. The names describe the cheapest specimen, not the general case — a lumpy neutral molecule with no dipole moment is "a quadrupole" in the far field however many charges it contains.

Is the dipole term of the series the same thing as the field of two opposite charges?

No — and the distinction matters at every finite distance. Two charges $\pm q$ separated by $d$ (a physical dipole) have a ladder whose $l=1$ rung is the dipole term, but higher odd rungs are present too; worked example 1 measures them at the percent level at $r=5d$ . The pure (point) dipole is the idealization $d\to0$ with $p=qd$ fixed, which kills every higher rung and leaves the $1/r^{2}$ term alone, exact everywhere except the origin. The two agree only far away; Electric Dipole Moment — which owns the dipole field pattern — makes the limit precise. When an exam trap is set, this is usually the tripwire.

How many terms should I keep in practice?

Let $\epsilon=r'_{\max}/r$ : each rung retained buys one factor of $\epsilon$ . At $r=10\,r'_{\max}$ the first non-vanishing rung alone is a $10\%$ theory, two rungs give $1\%$ , three $0.1\%$ . At $r=1.1\,r'_{\max}$ the series converges so slowly that direct integration may be cheaper — the crossover the fast multipole method engineers around by compressing only well separated clusters (typically $\epsilon\le1/2$ , so ten rungs buy $2^{-10}\approx10^{-3}$ ). Symmetry first, though: identify which rungs vanish identically before counting any.

Do the moments change if I move the origin?

In general, yes — the integrals are taken from the origin you chose, and shifting it reshuffles structure between rungs. The saving theorem: the first non-vanishing moment is origin-independent — $Q$ always, $\mathbf{p}$ when $Q=0$ , the quadrupole moment when both vanish. That is why chemistry can tabulate "the dipole moment of water" (neutral) without specifying an origin, while the dipole moment of an ion is convention-dependent. The physical potential never changes — only its decomposition into rungs does. The checks and transformation rules live in the next articles, beginning with Monopole Moment.

Does the same expansion work for gravity?

Line for line — the same $1/\mathfrak{s}$ integral with $\rho$ the mass density and $-G$ replacing $1/4\pi\varepsilon_0$ . Two differences give it flavor. Mass has one sign, so the monopole never vanishes: every body is asymptotically a point mass, and the physics lives in the correction rungs — Earth's oblateness is the quadrupole coefficient $J_2\approx1.08\times10^{-3}$ , which precesses satellite orbits. And because a mass dipole about the center of mass vanishes identically, gravity's ladder effectively skips $l=1$ — the deep reason gravitational radiation opens at the quadrupole. Legendre and Laplace invented all of this for gravity in the 1780s; electrostatics moved in later.