Pre-Quantum Electrodynamics
Monopole and Dipole Termsems.ca.me.md
The series (\ref{Gr(3.95)}) is organized in increasing powers of inverse distance. The leading term is called the {\bf monopole} term, and is
\[ V_{\mbox{\tiny mono}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm} Q = \int_{\cal V} d\tau_s \rho({\bf r}_s). \label{Gr(3.97)} \]
For a point charge, the monopole term gives the exact potential.
The next term is the {\bf dipole} term: by using \(P_1 (x) = x\), we have \[ V_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^2} \int_{\cal V} d\tau_s \hat{\bf r} \cdot {\bf r}_s \rho({\bf r}_s) \] This can be written
\[ V_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2}, \hspace{10mm}{\bf p} \equiv \int_{\cal V} d\tau_s ~{\bf r}_s ~\rho({\bf r}_s). \label{eq:electric_dipole} \]
in terms of the {\bf dipole moment} \({\bf p}\). Note that the dipole moment is an internal property of the source charges, and that it in general depends on the chosen point of origin (more on this later).
Since dipole moments are vectors, they are summed following vector addition rules.
{\bf Pure dipole:} two charges closer and closer together, but charges higher and higher such that \({\bf p}\) remains finite.
Created: 2022-02-08 Tue 06:55