Editing Dipole (section)

== Field from an electric dipole ==
<!-- This section is linked from [[Intermolecular force]] -->
The [[electrostatic potential]] at position '''r''' due to an electric dipole at the origin is given by:
: <math> \Phi(\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\,\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^2}</math>
where '''p''' is the (vector) [[Electric dipole moment|dipole moment]], and ''є''<sub>0</sub> is the [[permittivity of free space]].

This term appears as the second term in the [[Multipole expansion#Expansion in Cartesian coordinates|multipole expansion]] of an arbitrary electrostatic potential Φ('''r'''). If the source of Φ('''r''') is a dipole, as it is assumed here, this term is  the only non-vanishing term in the multipole expansion of Φ('''r'''). The [[electric field]] from a dipole can be found from the [[gradient]] of this potential:
: <math> \mathbf{E} = - \nabla \Phi =\frac {1} {4\pi\epsilon_0} \ \frac{3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{r^3} - \delta^3(\mathbf{r})\frac{\mathbf{p}}{3\epsilon_0}.</math>

This is of the same form of the expression for the magnetic field of a point magnetic dipole, ignoring the delta function. 
In a real electric dipole, however, the charges are physically separate and the electric field diverges or converges at the point charges. 
This is different to the magnetic field of a real magnetic dipole which is continuous everywhere. The delta function represents the strong field pointing in the opposite direction between the point charges, which is often omitted since one is rarely interested in the field at the dipole's position. 
For further discussions about the internal field of dipoles, see<ref name=":0" /><ref>{{Cite book|title=Classical Electrodynamics, 3rd Ed.|last=Jackson|first=John D.|publisher=Wiley|year=1999|isbn=978-0-471-30932-1|pages=148–150}}</ref> or ''{{slink|Magnetic moment#Internal magnetic field of a dipole}}''.