Maxwell's equations can be written

(1262) | |||

(1263) | |||

(1264) | |||

(1265) |

with the continuity equation

(1266) |

(1267) |

(1268) | |||

(1269) | |||

(1270) | |||

(1271) |

The curl equations can be combined to give two inhomogeneous Helmholtz wave equations:

(1272) |

(1273) |

Since the multipole coefficients in Eqs. (7.54) are determined according to
Eqs. (7.57) from the scalars
and
, it is sufficient to consider wave equations
for these quantities, rather than the vector fields and
. From Eqs. (7.42), (7.81), (7.82), and the identity

(1274) |

(1275) | |||

(1276) |

Now the Green's function for the inhomogeneous Helmholtz equation (defined
by Eq. (7.17)), subject to the boundary condition of outgoing waves
at infinity, is given by Eq. (7.18). It follows that
Eqs. (7.84) can be inverted to give

(1277) | |||

(1278) |

In order to evaluate the multipole coefficients by means of Eqs. (7.57), we first observe that the requirement of outgoing waves at infinity makes in Eq. (7.45). Thus, we choose in Eqs. (7.54) as the radial eigenfunctions of and in the source free region. Next, let us consider the expansion (7.26) of the Green's function for the Helmholtz equation in terms of spherical harmonics. We assume that the point lies outside some spherical shell which completely encloses the sources. It follows that and in all of the integrations. Making use of the orthogonality property of the spherical harmonics, it follows from Eq. (7.26) that

(1279) |

(1280) | |||

(1281) |

The expressions (7.87) give the strengths of the various multipole
fields outside the source in terms of integrals over the source
densities and . They can be transformed into more
useful forms by means of the following arguments. The results

(1282) | |||

(1283) |

follow from the definition (7.29) of , and simple vector identities. Substituting into Eq. (7.87a), we obtain

(1284) |

where use has been made of Eq. (7.78). Use of Green's theorem on the second term replaces by (since we can neglect the surface terms, and is a solution of the Helmholtz equation). A radial integration by part on the third term (again neglecting surface terms) casts the radial derivative over onto the spherical Bessel function. The result for the

(1285) |

The analogous set of manipulations using Eq. (7.87b) leads to an expression for the

(1286) |

Both the above results are exact, and are valid for arbitrary wavelength and source size.

In the limit in which the source dimensions are very small compared
to a wavelength (*i.e.*, ) the expressions for the
multipole coefficients can be considerably simplified. Using the
asymptotic form (7.14a), and keeping only lowest powers in for
terms involving , , and , we obtain the
approximate electric multipole coefficient

(1287) |

(1288) | |||

(1289) |

The moment has the same form as a conventional electrostatic multipole moment. The moment is an induced electric multipole moment due to the magnetization. It is generally a factor smaller than the normal moment . For the magnetic multipole coefficient the corresponding long wavelength approximation is

(1290) |

(1291) | |||

(1292) |

Note that for a system with intrinsic magnetization the magnetic moments and are generally of the same order of magnitude.

Thus, in the long wavelength limit the electric multipole fields are determined by the charge density , whereas the magnetic multipole fields are determined by the magnetic moment densities and .