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Maxwell's equations in free space reduce to



(1205) 



(1206) 



(1207) 



(1208) 
assuming an
time dependence of all field
quantities. Here, .
Eliminating between Eqs. (7.36c) and (7.36d),
we obtain the following equations for :
with given by

(1211) 
Alternatively, can be eliminated to give
with given by

(1214) 
It is clear that each Cartesian component of and
satisfies the Helmholtz wave equation (7.3). Hence, these components
can be written in a general expansion of the form

(1215) 
where stands for any Cartesian component of or .
Note, however, that the three Cartesian components of or
are not entirely independent, since they must also satisfy
the constraints
and
. Let us examine how these constraints can be satisfied
with the minimum labour.
Consider the scalar
, where
is a well behaved vector field. It is easily verified that

(1216) 
It follows from Eqs. (7.37) and (7.39) that the scalars
and
both satisfy the Helmholtz wave equation:
Thus, the general solutions for
and
can be written in the form (7.41).
Let us define a magnetic multipole field of order by
the conditions
where

(1221) 
The presence of the factor is for later convenience.
Equation (7.40) yields

(1222) 
where is given by Eq. (7.29). With
given by Eq. (7.44a), the electric field associated with a magnetic
multipole must satisfy

(1223) 
and
. Note that the operator
acts only on the angular variables
.
This means that the radial dependence of
must be given by . Note also, from Eqs. (7.33), that the
operator acting on transforms the value
but does not change the value. It is easily seen from Eqs. (7.27)
and (7.31)
that the solution to Eqs. (7.44b) and (7.47) can be written in the form

(1224) 
Thus, the angular dependence of
consists
of some linear combination of , , and
. Equation (7.48),
together with

(1225) 
specifies the electromagnetic fields of a magnetic multipole of
order . Note from Eq. (7.31) that the electric field given
by Eq. (7.48) is transverse to the radius vector. Thus, magnetic multipole
fields are sometimes termed transverse electric (TE)
multipole fields.
The fields of an electric or transverse magnetic (TM)
multipole of order are specified by the conditions
It follows that the fields of an electric multipole are given by
The radial function is given by an expression like
(7.45).
The two sets of multipole fields (7.48), (7.49), and (7.51), form a
complete set of vector solutions to Maxwell's equations in free space.
Since the vector spherical harmonic
plays an important
role in multipole fields, it is convenient to introduce the normalized
form

(1230) 
It can be demonstrated that the vector spherical harmonics possess
the orthogonality properties
for all , , , and .
By combining the two types of fields we can write the general solution
to Maxwell's equations in free space in the form
where the coefficients and specify the
amounts of electric and magnetic multipole fields.
The radial functions and are of the form (7.45). The
coefficients and , as well as the relative
proportions in (7.45), are determined by the sources and
the boundary conditions.
Equations (7.54) yield
and
where use has been made of Eqs. (7.27), (7.29), and (7.31).
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 and in and
.
Next: Properties of multipole fields
Up: The multipole expansion
Previous: Multipole expansion of the
Richard Fitzpatrick
20020518