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Mathematics for Physics

75 Multipole Expansion

A series expansion of the potential of a localized charge distribution that is used to organize contributions by monopole, dipole, quadrupole, and higher moments, where a localized charge distribution is a charge collection confined to a finite region.

The multipole expansion of the potential. Far from a localized charge distribution the potential can be expanded in inverse powers of distance. This principle is used to replace a complicated source by a sequence of simpler terms that fall off faster and faster.

The multipole expansion of the potential is

\[ V(\mathbf{r}) = \dfrac{1}{4\pi\epsilon_{0}}\sum_{n=0}^{\infty}\dfrac{1}{r^{n+1}}\displaystyle\int (r')^{n}P_{n}(\cos\alpha)\,\rho(\mathbf{r}')\,d\tau' \]

where

  • \(V\) is the electric potential.
  • \(r\) is the distance to the field point.
  • \(r'\) is the distance of a source element from the origin.
  • \(P_{n}\) is the \(n\)th Legendre polynomial.
  • \(\alpha\) is the angle between \(\mathbf{r}\) and \(\mathbf{r}'\).
  • \(\rho\) is the charge density.
  • \(d\tau'\) is the volume element.
  • \(\epsilon_{0}\) is the permittivity of free space.

The monopole term. The first term is the monopole potential of the total charge. This principle is used to treat a distant charge collection as a point charge when the net charge is not zero.

The monopole term is

\[ V_{\mathrm{mon}}(\mathbf{r}) = \dfrac{1}{4\pi\epsilon_{0}}\dfrac{Q}{r} \]

where

  • \(V_{\mathrm{mon}}\) is the monopole potential.
  • \(Q\) is the total charge.
  • \(r\) is the distance to the field point.
  • \(\epsilon_{0}\) is the permittivity of free space.

The dipole term. The next term is the dipole potential of the dipole moment. This principle is used to describe a neutral charge collection whose opposite charges are slightly separated.

The dipole term is

\[ V_{\mathrm{dip}}(\mathbf{r}) = \dfrac{1}{4\pi\epsilon_{0}}\dfrac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^{2}} \]

where

  • \(V_{\mathrm{dip}}\) is the dipole potential.
  • \(\mathbf{p}\) is the electric dipole moment.
  • \(\hat{\mathbf{r}}\) is the unit vector toward the field point.
  • \(r\) is the distance to the field point.
  • \(\epsilon_{0}\) is the permittivity of free space.