assuming an time dependence of all field quantities. Here, . Eliminating between Equations (1453) and (1454), we obtain

with given by

(1455) |

Alternatively, can be eliminated to give

with given by

It is clear that each Cartesian component of and satisfies the homogeneous Helmholtz wave equation, (1413). Hence, according to the analysis of Section 11.2, these components can be written as a general expansion of the form

where stands for any Cartesian component of or . Note, however, that the three Cartesian components of or are not entirely independent, because they must also satisfy the constraints and . Let us examine how these constraints can be satisfied with the minimum of effort.

Consider the scalar , where is a well-behaved vector field. It is easily verified that

It follows from Equations (1455)-(1456) and (1458)-(1459) that the scalars and both satisfy the homogeneous Helmholtz wave equation: that is,

(1461) | ||

(1462) |

Thus, the general solutions for and can be written in the form (1461).

Let us define a *magnetic multipole* field of order
as the solution of

where

The presence of the factor in Equation (1465) is for later convenience. Equation (1460) yields

(1466) |

where is given by Equation (1438). Thus, with taking the form (1465), the electric field associated with a magnetic multipole must satisfy

as well as . Recall that the operator acts on the angular variables only. This implies that the radial dependence of is given by . It is easily seen from Equations (1436) and (1442) that the solution to Equations (1466) and (1469) can be written in the form

It follows from the analysis Section 11.3 that the angular dependence of consists of a linear combination of , , and functions. Equation (1470), together with

specifies the electromagnetic fields of a magnetic multipole of order . According to Equation (1442), the electric field (1470) is transverse to the radius vector. Thus, magnetic multipole fields are sometimes termed

The fields of an *electric*, or *transverse magnetic* (TM),
multipole of order
satisfy

(1470) | ||

(1471) |

It follows that the fields of an electric multipole are

Here, the radial function is an expression of the form (1467).

The two sets of multipole fields, (1470)-(1471), and (1474)-(1475), form a complete set of vector solutions to Maxwell's equations in free space. Because the vector spherical harmonic plays an important role in the theory of multipole fields, it is convenient to introduce the normalized form

It can be demonstrated that these forms possess the orthogonality properties

for all , , , and .

By combining the two types of multipole fields, we can write the general solution to Maxwell's equations in free space as

where the coefficients and specify the amounts of electric and magnetic multipole fields. The radial functions and are both of the form (1467). The coefficients and , as well as the relative proportions of the two types of Hankel functions in the radial functions and , are determined by the sources and the boundary conditions.

Equations (1479) and (1480) yield

(1479) |

and

(1480) |

where use has been made of Equations (1436), (1438), (1442), and (1476). It follows from the well-known orthogonality property of the spherical harmonics that

Thus, knowledge of and at two different radii in a source-free region permits a complete specification of the fields, including the relative proportions of the Hankel functions and present in the radial functions and .