The Multipole Expansion

Why the multipole expansion matters

Nature is full of charge distributions too complicated to write down exactly — a thundercloud, a protein, an atomic nucleus, a radio antenna. Solving for the potential at every point of such a source is hopeless in closed form. But almost always we do not need the field inside the source; we need it far away, where the tangled interior detail smooths out. The multipole expansion is the mathematical statement that, viewed from a distance, every charge cloud looks like a hierarchy of ever-simpler shapes: first a point charge, then a pair of opposite charges, then a symmetric arrangement of four, and so on.

The power of this is that the terms decay at different rates. The monopole fades as 1/r, the dipole as 1/r², the quadrupole as 1/r³. So even though the series is infinite, at large distance only the first nonzero term matters — the rest are suppressed by additional factors of (d/r), where d is the physical size of the source. This is why a distant galaxy's gravity is well approximated as a point mass, why the electric field a metre from a molecule is set almost entirely by its dipole moment, and why a nucleus with net charge Ze looks Coulombic to an electron far outside it.

How it works — the 1/r expansion, step by step

Start with the exact potential of a static charge density ρ(r'):

V(r) = (1 / 4πε₀) ∫ ρ(r') / |r − r'| d³r'

The trick is to expand the kernel 1/|r − r'| for r ≫ r'. Writing r = |r|, r' = |r'|, and γ for the angle between r and r', the generating function of the Legendre polynomials gives:

1 / |r − r'| = (1/r) Σ_{ℓ=0}^∞ (r'/r)^ℓ Pℓ(cos γ)

Insert this back and pull the r-dependence outside the integral. The potential becomes a sum of terms, each a fixed power of 1/r times an integral over the source:

V(r) = (1 / 4πε₀) Σ_{ℓ=0}^∞ (1 / r^{ℓ+1}) ∫ (r')^ℓ Pℓ(cos γ) ρ(r') d³r'

Term by term this is the multipole expansion:

• ℓ = 0, monopole: V₀ = (1/4πε₀) Q/r, where Q = ∫ ρ d³r' is the total charge. Falls as 1/r — this is just Coulomb's law for the net charge.
• ℓ = 1, dipole: V₁ = (1/4πε₀) (p·r̂)/r², where p = ∫ r' ρ d³r' is the dipole moment. Falls as 1/r².
• ℓ = 2, quadrupole: V₂ = (1/4πε₀) (1/2r³) Σᵢⱼ Qᵢⱼ r̂ᵢ r̂ⱼ, where Qᵢⱼ = ∫ (3x'ᵢx'ⱼ − r'²δᵢⱼ) ρ d³r' is the quadrupole tensor. Falls as 1/r³.
• ℓ = 3 octupole, ℓ = 4 hexadecapole, … each falling one more power faster.

Symbol definitions. V — electric potential (volts, V = J/C). ε₀ — vacuum permittivity, 8.854 × 10⁻¹² F/m. ρ — charge density (C/m³). r — field point position, r' — source point position (m). Q — total charge (C). p — electric dipole moment (C·m). Qᵢⱼ — components of the traceless quadrupole tensor (C·m²). Pℓ — Legendre polynomial of degree ℓ. r̂ — unit vector toward the field point.

Spherical harmonics and the general moments

Legendre polynomials only depend on the single angle γ between source and field point. To fully separate the geometry it is cleaner to use the spherical-harmonic addition theorem, which rewrites Pℓ(cos γ) as a sum over m of Yℓm(θ,φ) Yℓm*(θ',φ'). The exterior potential then reads:

V(r,θ,φ) = (1 / 4πε₀) Σ_{ℓ=0}^∞ Σ_{m=−ℓ}^{ℓ} (4π / (2ℓ+1)) (qℓm / r^{ℓ+1}) Yℓm(θ,φ)

qℓm = ∫ (r')^ℓ Yℓm*(θ',φ') ρ(r') d³r'

The complex numbers qℓm are the spherical multipole moments. Because m runs from −ℓ to +ℓ there are exactly 2ℓ + 1 of them at each order: 1 monopole, 3 dipole, 5 quadrupole, 7 octupole. These are precisely the same functions Yℓm that describe atomic orbitals — the s, p, d, f shells — which is no coincidence: both come from solving Laplace's equation on the sphere. The angular shape Yℓm you attach to a source is the same angular shape its far field wears.

The multipole hierarchy at a glance

Order ℓ	Name	Potential falloff	Field falloff	# components (2ℓ+1)	Leading term when…
0	Monopole	1/r	1/r²	1	net charge Q ≠ 0
1	Dipole	1/r²	1/r³	3	Q = 0, p ≠ 0 (e.g. H₂O)
2	Quadrupole	1/r³	1/r⁴	5	Q = p = 0 (e.g. CO₂, N₂)
3	Octupole	1/r⁴	1/r⁵	7	lower moments vanish (e.g. CH₄)
4	Hexadecapole	1/r⁵	1/r⁶	9	high-symmetry sources

Notice methane CH₄ is nonpolar (p = 0) and also has vanishing quadrupole by tetrahedral symmetry — its lowest nonzero moment is the octupole. Symmetry, not size, decides which term leads.

Why neutral molecules still interact

An object with Q = 0 has no monopole, so its potential starts at the dipole term. That term is small and short-ranged compared with a bare charge, but it is not zero — and it lets neutral matter hold itself together. The interactions between molecular multipoles are collectively the van der Waals forces:

• Keesom (dipole–dipole): two permanent dipoles orient to attract; thermally averaged energy ∝ −p₁²p₂² / (kT r⁶).
• Debye (dipole–induced dipole): a permanent dipole polarizes a neighbor; energy ∝ −p²α / r⁶, with α the polarizability.
• London dispersion: quantum fluctuations create instantaneous dipoles that correlate; energy ∝ −1/r⁶ even between two noble-gas atoms with zero permanent moments.

All three scale as 1/r⁶ in energy — dramatically shorter-ranged than the 1/r Coulomb energy of net charges, which is exactly why they are weak. Yet they are what condense argon into a liquid at 87 K, let geckos climb glass, and set the fold of every protein. The multipole expansion is the language that makes all of this quantitative.

Worked example — the field of a physical dipole

Take two charges +q and −q separated by a small vector d pointing from − to +. The total charge is zero, so the monopole term vanishes and the dipole leads. The dipole moment is:

p = q d        (magnitude p = q·d, direction − to +)

The potential far away (r ≫ d) is

V(r,θ) = (1 / 4πε₀) (p cos θ) / r²

where θ is measured from the dipole axis. Taking the gradient gives the field, which has the characteristic two-lobe shape:

E_r = (1 / 4πε₀) (2p cos θ) / r³
E_θ = (1 / 4πε₀) (p sin θ) / r³

Numbers for water. Water has p = 1.85 D = 1.85 × 3.336 × 10⁻³⁰ = 6.17 × 10⁻³⁰ C·m. At r = 1 nm along the axis (θ = 0), the field magnitude is E ≈ (1/4πε₀)(2p/r³) = (8.99 × 10⁹)(2 × 6.17 × 10⁻³⁰)/(10⁻⁹)³ ≈ 1.1 × 10⁸ V/m — enormous up close, but because it falls as 1/r³ it drops to a few hundred V/m by 100 nm.

Radiation multipoles

When charges and currents oscillate, they radiate, and the same ℓ index classifies the outgoing waves into electric multipoles (E1 dipole, E2 quadrupole, …) and magnetic multipoles (M1, M2, …). The dominant channel is usually electric dipole (E1). For an oscillating dipole of amplitude p at frequency ω, the time-averaged radiated power is the Larmor-type result:

P = ω⁴ |p|² / (12 π ε₀ c³)

where c = 2.998 × 10⁸ m/s. Each higher multipole is weaker by roughly (a/λ)², with a the source size and λ the wavelength — for atoms a/λ ~ 10⁻³, so E2 and M1 transitions are typically a million times slower than E1. When E1 is forbidden by symmetry (as in many atomic and nuclear transitions), the next allowed multipole takes over, which is exactly what atomic selection rules encode: they state which change in angular momentum ℓ and parity a given multipole can carry away.

Common misconceptions

• "The dipole moment always depends on the origin." Only if there is net charge. For a neutral distribution (Q = 0) the dipole moment is origin-independent; shifting the origin adds Q times a constant vector, which is zero when Q = 0.
• "The expansion works everywhere." It converges only outside the smallest sphere enclosing all the charge (r > d). Inside, the series diverges — you need the interior expansion in powers of r/d.
• "More terms always means more accuracy." Adding terms helps only where the series converges. Near the source, convergence is slow and truncating early can be badly wrong; the leading-term shortcut is a far-field statement.
• "A neutral object has no external field." Zero monopole ≠ zero field. Dipole, quadrupole and higher moments produce real, measurable fields — the whole basis of intermolecular forces.
• "Quadrupole moment is a single number." It is a symmetric traceless 3×3 tensor with 5 independent components, not a scalar. Only for axially symmetric bodies does one number Q suffice.
• "Multipole moments are unique labels." Two very different-looking charge distributions can share the same low-order moments; the moments only capture the far field, discarding interior detail on purpose.

A short history

The mathematics predates electromagnetism. Adrien-Marie Legendre introduced his polynomials in 1782–1785 while studying the gravitational attraction of spheroids, and Pierre-Simon Laplace developed the spherical harmonics (1782) in the same celestial-mechanics context — the gravitational multipole expansion came first. The electromagnetic version matured through the 19th century and was cast in its modern spherical-harmonic form in the 20th, becoming a cornerstone of Jackson's Classical Electrodynamics. Nuclear physicists measured the first electric quadrupole moment of a nucleus (the deuteron) in 1939, proving nuclei are not perfect spheres — a direct, experimental encounter with the ℓ = 2 term of this very series.