The scalar potential due to a static electric monopole is given by
\begin{equation*}
\phi = \frac{1}{4 \pi \epsilon_{0}} \frac{e}{r}
\end{equation*}
Here \(e\) is the electric monopole moment. The electric field due to a static electric monopole is given by
\begin{equation*}
\mathbf{E} \equiv -\nabla \phi = \frac{1}{4 \pi \epsilon_{0}} \frac{e \mathbf{r}}{r^{3}}
\end{equation*}
The scalar potential due to a static electric dipole is given by
\begin{equation*}
\phi = \frac{1}{4 \pi \epsilon_{0}} \frac{(\mathbf{f} \cdot \mathbf{r})}{r^{3}}
\end{equation*}
Here \(\mathbf{f}\) is the electric dipole moment. The electric field due to a static electric dipole is given by
\begin{equation*}
\mathbf{E} \equiv - \nabla \phi = \frac{1}{4 \pi \epsilon_{0}} \frac{3 (\mathbf{f} \cdot \mathbf{r}) \mathbf{r}}{r^{5}} - \frac{1}{4 \pi \epsilon_{0}} \frac{\mathbf{f}}{r^{3}} - \frac{1}{3 \epsilon_{0}} \mathbf{f} \delta(\mathbf{r})
\end{equation*}
The energy associated to the interaction of an electric monopole with an external scalar potential is given by
\begin{equation*}
U = - e \phi
\end{equation*}
Similarly, the energy associated to the interaction of an electric dipole with an external electric field is given by
\begin{equation*}
U = - (\mathbf{f} \cdot \mathbf{E})
\end{equation*}
Consider three cases.
Monopole-Monopole
The interaction energy between two electric monopoles is given by
\begin{equation*}
U_{12} = -e_{1} \phi_{2} = -e_{2} \phi_{1} = - \frac{1}{4 \pi \epsilon_{0}} \frac{e_{1} e_{2}}{r}
\end{equation*}
This is the familiar Coulomb potential energy. Note that this energy is invariant under \(e_{1} \rightarrow -e_{1}\) and \(e_{2} \rightarrow -e_{2}\).
Monopole-Dipole
The interaction energy between an electric monopole and an electric dipole is given by
\begin{equation*}
U_{12} = -e_{1} \phi_{2} = - (\mathbf{f}_{2} \cdot \mathbf{E}_{1}) = - \frac{1}{4 \pi \epsilon_{0}} \frac{e_{1} (\mathbf{f}_{2} \cdot \mathbf{r})}{r^{3}}
\end{equation*}
Note that this energy is invariant under \(e_{1} \rightarrow -e_{1}\) and \(\mathbf{f}_{2} \rightarrow -\mathbf{f}_{2}\).
Dipole-Dipole
The interaction energy between two electric dipoles is given by
\begin{equation*}
U_{12} = - (\mathbf{f}_{1} \cdot \mathbf{E}_{2}) = - (\mathbf{f}_{2} \cdot \mathbf{E}_{1}) = \frac{1}{4 \pi \epsilon_{0}} \frac{3 (\mathbf{f}_{1} \cdot \mathbf{r}) (\mathbf{f}_{2} \cdot \mathbf{r})}{r^{5}} - \frac{1}{4 \pi \epsilon_{0}} \frac{(\mathbf{f}_{1} \cdot \mathbf{f}_{2})}{r^{3}} - \frac{1}{3 \epsilon_{0}} (\mathbf{f}_{1} \cdot \mathbf{f}_{2}) \delta(\mathbf{r})
\end{equation*}
This energy is symmetric in both electric dipole moments. Note that this energy is invariant under \(\mathbf{f}_{1} \rightarrow -\mathbf{f}_{1}\) and \(\mathbf{f}_{2} \rightarrow -\mathbf{f}_{2}\).
