Multipole expansion
Spherical form

The potential V (R) at a point R outside the charge distribution, i.e. | R | > r, can be expanded by the Laplace expansion: V (R) ≡ ∑ i = 1 N q i 4 π ε 0 | r i − R | = 1 4 π ε 0 ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ (− 1) m I ℓ − m (R) ∑ i = 1 N q i R ℓ m (r i), {\displaystyle V(\mathbf {R})\equiv \sum _{i=1}^{N}{\frac {q_{i}}{4\pi \varepsilon _{0}|\mathbf {r} _{i}-\mathbf {R} |}}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R})\sum _{i=1}^{N}q_{i}R_{\ell }^{m}(\mathbf {r} _{i}),} where I ℓ − m (R) {\displaystyle I_{\ell }^{-m}(\mathbf {R})} is an irregular solid harmonic (defined below as a spherical harmonic function divided by R ℓ + 1 {\displaystyle R^{\ell +1}}) and R ℓ m (r) {\displaystyle R_{\ell }^{m}(\mathbf {r})} is a regular solid harmonic (a spherical harmonic times r). We define the spherical multipole moment of the charge distribution as follows Q ℓ m ≡ ∑ i = 1 N q i R ℓ m (r i), − ℓ ≤ m ≤ ℓ. {\displaystyle Q_{\ell }^{m}\equiv \sum _{i=1}^{N}q_{i}R_{\ell }^{m}(\mathbf {r} _{i}),\quad \ -\ell \leq m\leq \ell.} Note that a multipole moment is solely determined by the charge distribution (the positions and magnitudes of the N charges).

A spherical harmonic depends on the unit vector R ^ {\displaystyle {\hat {R}}}. (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as I ℓ m (R) ≡ 4 π 2 ℓ + 1 Y ℓ m (R ^) R ℓ + 1 {\displaystyle I_{\ell }^{m}(\mathbf {R})\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}{\frac {Y_{\ell }^{m}({\hat {R}})}{R^{\ell +1}}}} so that the multipole expansion of the field V (R) at the point R outside the charge distribution is given by

V (R) = 1 4 π ε 0 ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ (− 1) m I ℓ − m (R) Q ℓ m = 1 4 π ε 0 ∑ ℓ = 0 ∞ 1 / 2 1 R ℓ + 1 ∑ m = − ℓ ℓ (− 1) m Y ℓ − m (R ^) Q ℓ m, R > r m a x {\displaystyle {\begin{aligned}V(\mathbf {R})&={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R})Q_{\ell }^{m}\\&={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\left^{1/2}\;{\frac {1}{R^{\ell +1}}}\sum _{m=-\ell }^{\ell }(-1)^{m}Y_{\ell }^{-m}({\hat {R}})Q_{\ell }^{m},\qquad R>r_{\mathrm {max} }\end{aligned}}}

This expansion is completely general in that it gives a closed form for all terms, not just for the first few. It shows that the spherical multipole moments appear as coefficients in the 1/ R expansion of the potential.

It is of interest to consider the first few terms in real form, which are the only terms commonly found in undergraduate textbooks.Since the summand of the m summation is invariant under a unitary transformation of both factors simultaneously and since transformation of complex spherical harmonics to real form is by a unitary transformation, we can simply substitute real irregular solid harmonics and real multipole moments. The ℓ = 0 term becomes V ℓ = 0 (R) = q t o t 4 π ε 0 R with q t o t ≡ ∑ i = 1 N q i. {\displaystyle V_{\ell =0}(\mathbf {R})={\frac {q_{\mathrm {tot} }}{4\pi \varepsilon _{0}R}}\quad {\hbox{with}}\quad q_{\mathrm {tot} }\equiv \sum _{i=1}^{N}q_{i}.} This is in fact Coulomb's law again. For the ℓ = 1 term we introduce R = (R x, R y, R z), P = (P x, P y, P z) with P α ≡ ∑ i = 1 N q i r i α, α = x, y, z. {\displaystyle \mathbf {R} =(R_{x},R_{y},R_{z}),\quad \mathbf {P} =(P_{x},P_{y},P_{z})\quad {\hbox{with}}\quad P_{\alpha }\equiv \sum _{i=1}^{N}q_{i}r_{i\alpha },\quad \alpha =x,y,z.} Then V ℓ = 1 (R) = 1 4 π ε 0 R 3 (R x P x + R y P y + R z P z) = R ⋅ P 4 π ε 0 R 3 = R ^ ⋅ P 4 π ε 0 R 2. {\displaystyle V_{\ell =1}(\mathbf {R})={\frac {1}{4\pi \varepsilon _{0}R^{3}}}(R_{x}P_{x}+R_{y}P_{y}+R_{z}P_{z})={\frac {\mathbf {R} \cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{3}}}={\frac {{\hat {\mathbf {R} }}\cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{2}}}.} This term is identical to the one found in Cartesian form.

In order to write the ℓ = 2 term, we have to introduce shorthand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type Q z 2 ≡ ∑ i = 1 N q i 1 2 (3 z i 2 − r i 2), {\displaystyle Q_{z^{2}}\equiv \sum _{i=1}^{N}q_{i}\;{\frac {1}{2}}(3z_{i}^{2}-r_{i}^{2}),} can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation.
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