Multipole expansion of electric field

In physics, a static three-dimensional distribution of electric charges offers a potential field to its environment. (An exception being a neutral, spherically symmetric, charge distribution, like a noble gas atom. Such a charge distribution does not create an outside electric field.) Take a fixed point inside the charge distribution as the origin of a Cartesian frame (orthonormal system of axes). The potential Φ(R) in a point R outside the charge distribution can be expanded in powers of 1/R. Here R is the position vector of the point expressed with respect to the Cartesian frame and R = |R| is its length (distance from the outside point to the origin). In this expansion the powers of 1/R are not only multiplied by angular functions depending on the spherical polar angles of R, but also by fixed coefficients. The latter are completely determined by the shape and total charge of the charge distribution and are known as the multipoles of the charge distribution. Therefore the expansion is known as the multipole expansion of the electric potential field—a scalar field.

Two different ways of deriving the multipole expansion can be found in the literature. The first is a Taylor series in Cartesian coordinates, while the second is in terms of spherical harmonics which depend on spherical polar coordinates. The Cartesian approach has the advantage that no prerequisite knowledge of Legendre functions, spherical harmonics, etc., is assumed. Its disadvantage is that the derivations are fairly cumbersome, in fact a large part of it is the implicit rederivation of the Legendre expansion of 1/|r-R|, which was done once and for all by Legendre in the 1780s. Also it is difficult to give closed expressions for general terms of the multipole expansion—usually only the first few terms are given followed by some dots.

Expansion in Cartesian coordinates

For the sake of argument we consider a continuous charge distribution ρ(r), where r indicates the coordinate vector of a point inside the charge distribution. The case of a discrete distribution consisting of N charges q _i follows easily by substituting

$\rho (\mathbf {r} )=\sum _{i=1}^{N}q_{i}\delta (\mathbf {r} _{i}-\mathbf {r} )$

where r _i is the position vector of particle i and δ is the 3-dimensional Dirac delta function. Since an electric potential is additive, the potential at the point R outside ρ(r) is given by the integral

$\Phi (\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{V}{\frac {\rho (\mathbf {r} )}{|\mathbf {r} -\mathbf {R} |}}d\mathbf {r} ,\quad {\hbox{with}}\quad d\mathbf {r} \equiv dxdydz,$

where V is a volume that encompasses all of ρ(r) and ε₀ is the permittivity of the vacuum.

The Taylor expansion of a function v(r-R) around the origin r = 0 is,

$v(\mathbf {r} -\mathbf {R} )=v(\mathbf {R} )+\sum _{\alpha =x,y,z}r_{\alpha }v_{\alpha }(\mathbf {R} )+{\frac {1}{2}}\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {R} )+\cdots$

with

$v_{\alpha }(\mathbf {R} )\equiv \left({\frac {\partial v(\mathbf {r} -\mathbf {R} )}{\partial r_{\alpha }}}\right)_{\mathbf {r} =\mathbf {0} }\quad {\hbox{and}}\quad v_{\alpha \beta }(\mathbf {R} )\equiv \left({\frac {\partial ^{2}v(\mathbf {r} -\mathbf {R} )}{\partial r_{\alpha }\partial r_{\beta }}}\right)_{\mathbf {r} =\mathbf {0} }.$

Note that the function v must be sufficiently often differentiable, otherwise it is arbitrary. In the special case that v(r-R) satisfies the Laplace equation

$\left(\nabla ^{2}v(\mathbf {r} -\mathbf {R} )\right)_{\mathbf {r} =\mathbf {0} }=\sum _{\alpha =x,y,z}v_{\alpha \alpha }(\mathbf {R} )=0$

the expansion can be rewritten in terms of the components of a traceless Cartesian second rank tensor,

$\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}r_{\alpha }r_{\beta }v_{\alpha \beta }(\mathbf {R} )={\frac {1}{3}}\sum _{\alpha =x,y,z}\sum _{\beta =x,y,z}(3r_{\alpha }r_{\beta }-\delta _{\alpha \beta }r^{2})v_{\alpha \beta }(\mathbf {R} ),$

where δ_αβ is the Kronecker delta and r² ≡ |r|². Removing the trace is common and useful, because it takes the rotational invariant r² out of the second rank tensor.

So far we considered an arbitrary function, let us take now the following,

$v(\mathbf {r} -\mathbf {R} )\equiv {\frac {1}{|\mathbf {r} -\mathbf {R} |}},$

then by direct differentiation it follows that

$v(\mathbf {R} )={\frac {1}{R}},\quad v_{\alpha }(\mathbf {R} )={\frac {R_{\alpha }}{R^{3}}},\quad {\hbox{and}}\quad v_{\alpha \beta }(\mathbf {R} )={\frac {3R_{\alpha }R_{\beta }-\delta _{\alpha \beta }R^{2}}{R^{5}}}.$

Define a monopole, dipole and (traceless) quadrupole by, respectively,

$q_{\mathrm {tot} }\equiv \int _{V}\rho (\mathbf {r} )\,d\mathbf {r} ,\quad P_{\alpha }\equiv \int _{V}r_{\alpha }\rho (\mathbf {r} )\,d\mathbf {r} ,\quad {\hbox{and}}\quad Q_{\alpha \beta }\equiv \int _{V}(3r_{\alpha }r_{\beta }-\delta _{\alpha \beta }r^{2})\rho (\mathbf {r} )\,d\mathbf {r}$

and we obtain finally the first few terms of the multipole expansion of the total potential,

$4\pi \varepsilon _{0}\Phi (\mathbf {R} )\equiv \int _{V}v(\mathbf {r} -\mathbf {R} )\rho (\mathbf {r} )\,d\mathbf {r}$

$={\frac {q_{\mathrm {tot} }}{R}}+{\frac {1}{R^{3}}}\sum _{\alpha =x,y,z}P_{\alpha }R_{\alpha }+{\frac {1}{6R^{5}}}\sum _{\alpha ,\beta =x,y,z}Q_{\alpha \beta }(3R_{\alpha }R_{\beta }-\delta _{\alpha \beta }R^{2})+\cdots$

This expansion of the potential of a charge distribution is very similar to the one in real solid harmonics given below. The main difference is that the present one is in terms of linear dependent quantities, for

$\sum _{\alpha }v_{\alpha \alpha }=0\quad {\hbox{and}}\quad \sum _{\alpha }Q_{\alpha \alpha }=0.$

Note

If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance d apart, so that d/R >> (d/R)², it is easily shown that the only non-vanishing term in the multipole expansion is

$\Phi (\mathbf {R} )={\frac {\mathbf {P} \cdot \mathbf {R} }{4\pi \varepsilon _{0}R^{3}}}$,

the electric dipolar potential field. Since the non-unit vector R appears in the numerator, the dependence of the field on distance is 1/R², not 1/R³ as it may seem on first sight. A charge distribution is called a point dipole if it consists of two charges of opposite sign and same absolute value at an infinitesimal distance apart. It can be shown that an electrically neutral distribution of four charges at an infinitesimal distance apart (a point quadrupole) gives only a term in the external field proportional to 1/R³. An point octupole requires 8 charges, and so on. This explains the name multipole.

Spherical form

Since the charge distribution is enclosed in volume V that is finite, there is a maximum distance r_max on the boundary of V. The potential Φ(R) at a point R outside V, i.e., |R| > r_max, can be expanded by the Laplace expansion,

$\Phi (\mathbf {R} )\equiv \int _{V}{\frac {\rho (\mathbf {r} )}{4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {R} |}}d\mathbf {r} ={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R} )\int _{V}\rho (\mathbf {r} )\,R_{\ell }^{m}(\mathbf {r} )d\mathbf {r} ,$

where $I_{\ell }^{-m}(\mathbf {R} )$ is an irregular solid harmonics (which is a spherical harmonic function depending on the polar angles of R and divided by R^l+1) and $R_{\ell }^{m}(\mathbf {r} )$ is a regular solid harmonics (a spherical harmonics times r ^l). Both functions are not normalized to unity, but according to Racah (also known as Schmidt's semi-normalization). We define the spherical multipole moment of the charge distribution as follows

$Q_{\ell }^{m}\equiv \int _{V}\rho (\mathbf {r} )\,R_{\ell }^{m}(\mathbf {r} )d\mathbf {r} ,\qquad -\ell \leq m\leq \ell .$

Note that a multipole moment is solely determined by the charge distribution ρ(r).

A unit normalized spherical harmonic function Y^m_l depends on the unit vector ${\hat {R}}$. (A unit vector is determined by two spherical polar angles and conversely.) Thus, by definition, the irregular solid harmonics can be written as

$I_{\ell }^{m}(\mathbf {R} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}{\frac {Y_{\ell }^{m}({\hat {R}})}{R^{\ell +1}}}$

so that the multipole expansion of the field V(R) at the point R outside the charge distribution can be written in two equivalent ways,

$\Phi (\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }(-1)^{m}I_{\ell }^{-m}(\mathbf {R} )Q_{\ell }^{m}$

$={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\ell =0}^{\infty }\left[{\frac {4\pi }{2\ell +1}}\right]^{1/2}\;{\frac {1}{R^{\ell +1}}}\;\sum _{m=-\ell }^{\ell }(-1)^{m}Y_{\ell }^{-m}({\hat {R}})Q_{\ell }^{m},\qquad R>r_{\mathrm {max} }.$

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 expression containing the summation over m 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

$\Phi _{\ell =0}(\mathbf {R} )={\frac {q_{\mathrm {tot} }}{4\pi \varepsilon _{0}R}}\qquad {\hbox{with}}\quad q_{\mathrm {tot} }\equiv \int _{V}\rho (\mathbf {r} )d\mathbf {r}$

This is in fact Coulomb's law again. For the l = 1 term we introduce

$\mathbf {R} =(R_{x},R_{y},R_{z}),\quad \mathbf {P} =(P_{x},P_{y},P_{z})\quad {\hbox{with}}\quad P_{\alpha }\equiv \int _{V}\rho (\mathbf {r} )r_{\alpha }\;d\mathbf {r} ,\quad \alpha =x,y,z.$

Then

$\Phi _{\ell =1}(\mathbf {R} )={\frac {1}{4\pi \varepsilon _{0}R^{3}}}(R_{x}P_{x}+R_{y}P_{y}+R_{z}P_{z})={\frac {\mathbf {R} \cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{3}}}={\frac {{\hat {R}}\cdot \mathbf {P} }{4\pi \varepsilon _{0}R^{2}}}.$

This term is identical to the one found above in Cartesian form.

In order to write the l=2 term, we have to introduce short-hand notations for the five real components of the quadrupole moment and the real spherical harmonics. Notations of the type

$Q_{z^{2}}\equiv {\frac {1}{2}}\sum _{i=1}^{N}q_{i}\;(3z_{i}-r_{i}^{2})\quad {\hbox{or}}\quad Q_{z^{2}}\equiv {\frac {1}{2}}\int _{V}\;\rho (\mathbf {r} )(3z^{2}-r^{2})d\mathbf {r}$

can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation.