Editing Multipole expansion (section)

====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]].