Editing Multipole expansion (section)

==Multipole expansion of a potential outside an electrostatic charge distribution==
Consider a discrete charge distribution consisting of {{mvar|N}} point charges {{math|''q''<sub>''i''</sub>}} with position vectors {{math|'''r'''<sub>''i''</sub>}}. We assume the charges to be clustered around the origin, so that for all ''i'': {{math|''r''<sub>''i''</sub> &lt; ''r''<sub>max</sub>}}, where {{math|''r''<sub>max</sub>}} has some finite value. The potential {{math|''V''('''R''')}}, due to the charge distribution, at a point {{math|'''R'''}} outside the charge distribution, i.e., {{math|{{abs|'''R'''}} &gt; ''r''<sub>max</sub>}}, can be expanded in powers of {{math|1/''R''}}. Two ways of making this expansion can be found in the literature: The first is a [[Taylor series]] in the [[Cartesian coordinates]] {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}, while the second is in terms of [[spherical harmonics]] which depend on [[spherical polar coordinates]]. The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required. 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 {{math|1 / {{abs|'''r''' − '''R'''}}}}, which was done once and for all by [[Adrien-Marie Legendre|Legendre]] in the 1780s). Also it is difficult to give a closed expression for a general term of the multipole expansion&mdash;usually only the first few terms are given followed by an ellipsis.

===Expansion in Cartesian coordinates===
Assume {{math|1=''v''('''r''') = ''v''(−'''r''')}} for convenience. The [[Taylor series|Taylor expansion]] of {{math|1=''v''('''r''' − '''R''')}} around the origin {{math|1='''r''' = '''0'''}} can be written as
<math display="block">\begin{align}
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 + \cdots\\
&=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 + \cdots
\end{align}</math>
with Taylor coefficients
<math display="block">v_\alpha(\mathbf{R}) \equiv\left( \frac{\partial v(\mathbf{r}-\mathbf{R}) }{\partial r_\alpha}\right)_{\mathbf{r} = \mathbf 0} \quad\text{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}= \mathbf0} .</math>
If {{math|''v''('''r''' − '''R''')}} satisfies the [[Laplace equation]], then by the above expansion we have
<math display="block">\left(\nabla^2 v(\mathbf{r}- \mathbf{R})\right)_{\mathbf{r}=\mathbf0} = \sum_{\alpha=x,y,z} v_{\alpha\alpha}(\mathbf{R}) = 0,</math>
and the expansion can be rewritten in terms of the components of a traceless Cartesian second rank [[tensor]]:
<math display="block">\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} \left(3r_\alpha r_\beta - \delta_{\alpha\beta} r^2\right) v_{\alpha\beta}(\mathbf{R}) ,</math>
where {{math|''δ''<sub>''αβ''</sub>}} is the [[Kronecker delta]] and {{math|''r''<sup>2</sup> ≡ {{abs|'''r'''}}<sup>2</sup>}}. Removing the trace is common, because it takes the rotationally invariant {{math|''r''<sup>2</sup>}} out of the second rank tensor.

====Example====

Consider now the following form of {{math|''v''('''r''' − '''R''')}}:
<math display="block">v(\mathbf{r}- \mathbf{R}) \equiv \frac{1}{|\mathbf{r}- \mathbf{R}|} .</math>
Then by direct [[differentiation (mathematics)|differentiation]] it follows that
<math display="block">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} .</math>
Define a monopole, dipole, and (traceless) quadrupole by, respectively,
<math display="block">q_\mathrm{tot} \equiv \sum_{i=1}^N q_i , \quad P_\alpha \equiv\sum_{i=1}^N q_i r_{i\alpha} , \quad \text{and}\quad Q_{\alpha\beta} \equiv \sum_{i=1}^N q_i (3r_{i\alpha} r_{i\beta} - \delta_{\alpha\beta} r_i^2) ,</math>
and we obtain finally the first few terms of the '''multipole expansion''' of the total potential, which is the sum of the Coulomb potentials of the separate charges:<ref name="Jackson75">{{cite book| last1=Jackson|first1=John David| title=Classical electrodynamics|date=1975| publisher=Wiley|location=New York| isbn=047143132X|edition=2d|url-access=registration| url=https://archive.org/details/classicalelectro00jack_0}}</ref>{{rp|pages=137–138}}
<math display="block">\begin{align}
4\pi\varepsilon_0 V(\mathbf{R}) &\equiv \sum_{i=1}^N q_i v(\mathbf{r}_i-\mathbf{R}) \\
&= \frac{q_\mathrm{tot}}{R} + \frac{1}{R^3}\sum_{\alpha=x,y,z} P_\alpha R_\alpha +
\frac{1}{2 R^5}\sum_{\alpha,\beta=x,y,z} Q_{\alpha\beta} R_\alpha R_\beta + \cdots
\end{align}</math>

This expansion of the potential of a discrete 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 linearly dependent quantities, for
<math display="block">\sum_{\alpha} v_{\alpha\alpha} = 0 \quad \hbox{and} \quad \sum_{\alpha} Q_{\alpha\alpha} = 0 .</math>

'''Note:'''
If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance {{mvar|d}} apart, so that {{math|''d''/''R'' ≫ (''d''/''R'')<sup>2</sup>}}, it is easily shown that the dominant term in the expansion is
<math display="block">V(\mathbf{R}) = \frac{1}{4\pi \varepsilon_0 R^3} (\mathbf{P}\cdot\mathbf{R}) ,</math>
the electric [[Dipole#Field from an electric dipole|dipolar potential field]].

===Spherical form===
The potential {{math|''V''('''R''')}} at a point {{math|'''R'''}} outside the charge distribution, i.e. {{math|{{abs|'''R'''}} > ''r''<sub>max</sub>}}, can be expanded by the [[Laplace expansion (potential)|Laplace expansion]]:
<math display="block">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^{-m}_\ell(\mathbf{R}) \sum_{i=1}^N q_i R^m_\ell(\mathbf{r}_i),</math>
where <math>I^{-m}_{\ell}(\mathbf{R})</math> is an irregular [[solid harmonic]] (defined below as a [[spherical harmonic]] function divided by <math>R^{\ell+1}</math>) and <math>R^m_{\ell}(\mathbf{r})</math> is a regular solid harmonic (a spherical harmonic times {{math|r<sup>''ℓ''</sup>}}). We define the ''spherical multipole moment'' of the charge distribution as follows
<math display="block">Q^m_\ell \equiv \sum_{i=1}^N q_i R^m_\ell(\mathbf{r}_i),\quad\ -\ell \le m \le \ell.</math>
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 <math>\hat{R}</math>. (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as
<math display="block">I^m_{\ell}(\mathbf{R}) \equiv \sqrt{\frac{4\pi}{2\ell+1}} \frac{Y^m_{\ell}(\hat{R})}{R^{\ell+1}}</math>
so that the ''multipole expansion'' of the field {{math|''V''('''R''')}} at the point {{math|'''R'''}} outside the charge distribution is given by

<math display="block">\begin{align}
V(\mathbf{R})
& = \frac{1}{4\pi\varepsilon_{0}}\sum_{\ell=0}^{\infty}
\sum_{m=-\ell}^{\ell}(-1)^{m} I^{-m}_{\ell}(\mathbf{R}) Q^{m}_{\ell}\\
& = \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^{-m}_{\ell}(\hat{R}) Q^{m}_{\ell}, \qquad R > r_{\mathrm{max}}
\end{align}</math>

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 {{math|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 [[Solid harmonics#Real form|unitary transformation]], we can simply substitute real irregular solid harmonics and real multipole moments. The {{math|1=''ℓ'' =&thinsp;0}} term becomes
<math display="block">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.</math>
This is in fact [[Coulomb's law]] again. For the {{math|1=''ℓ'' =&thinsp;1}} term we introduce
<math display="block">\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.</math>
Then
<math display="block">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}.</math>
This term is identical to the one found in Cartesian form.

In order to write the {{math|1=''ℓ'' =&thinsp;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
<math display="block">Q_{z^2} \equiv \sum_{i=1}^N q_i\; \frac{1}{2}(3z_i^2 - r_i^2),</math>
can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation.