Pre-Quantum Electrodynamics
Monopole and Dipole Termsems.ca.me.md
- PM 2.7, 10.2
- Gr 3.4.2
The series p_Leg is organized in increasing powers of inverse distance. The leading term is called the monopole term, and is
\[ \phi_m ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm} Q = \int_{\cal V} d\tau_s \rho({\bf r}_s). \tag{p_mono}\label{p_mono} \]
For a point charge, the monopole term gives the exact potential.
The next term is the dipole term: by using \(P_1 (x) = x\), we have \[ \phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^2} \int_{\cal V} d\tau_s \hat{\bf r} \cdot {\bf r}_s \rho({\bf r}_s) \] This can be written
\[ \phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2}, \hspace{10mm}{\bf p} \equiv \int_{\cal V} d\tau_s ~{\bf r}_s ~\rho({\bf r}_s). \tag{p_di}\label{p_di} \]
in terms of the dipole moment \({\bf p}\). Note that the dipole moment is a property related to the internal distribution 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.
In the literature, one comes across the notion of a mathematical dipole or synonymously a pure dipole: this is simply an abstraction in which the charges are moved infinitesimally close together, while their charges are proportionally increased such that the dipole moment remains the same. In such cases, the multipole expansion terminates at the dipole term.
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Created: 2024-02-27 Tue 10:31