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Let us examine some of the properties of the multipole fields (7.48), (7.49),
and (7.51). Consider, first of all,
the socalled near zone, for which .
In this region is proportional to , given by the asymptotic
expansion (7.14b), unless its coefficient vanishes identically. Excluding this
possibility, the limiting behaviour of the magnetic field for an
electric multipole is

(1239) 
where the proportionality coefficient is chosen for later convenience. To
find the electric field we must take the curl of the righthand side.
The following operator identity is useful

(1240) 
The electric field (7.51b) is

(1241) 
Since
is a solution of Laplace's equation, the first term in
(7.59) vanishes. Consequently, the electric field at close distances for
an electric multipole is

(1242) 
This, of course, is an electrostatic multipole field. Such a field is obtained
in a more straightforward manner by observing that
, where
, in the near zone. Solving Laplace's
equation by separation of variables in spherical polar coordinates, and
demanding that be well behaved as
,
yields

(1243) 
Note that the magnetic field (7.58) (normalized
with respect to ) is smaller than the electric field
(7.61) by a factor of order . Thus, in the near zone the magnetic
field associated with an electric multipole is always much smaller
than the corresponding electric field. For magnetic multipole fields it
is evident from Eqs. (7.48), (7.49), and (7.51) that the roles of
and are interchanged according to the transformation
In the socalled far zone or radiation zone, for which
, the multipole fields depend on the boundary conditions
imposed at infinity. For definiteness, let us consider the case of
outgoing waves at infinity, which is appropriate to radiation
by a localized source. For this case, the radial function
is proportional to the spherical Hankel function .
From the asymptotic form (7.16), it is clear that in the radiation zone
the magnetic field of an electric multipole goes as

(1246) 
Using Eq. (7.51b), the electric field can be written

(1247) 
Neglecting terms which fall off faster than , the above expression
reduces to

(1248) 
where use has been made of the identity (7.59), and
is a unit vector pointing
in the radial direction. The second term in square brackets
is smaller than the first term by a factor of order , and can therefore be neglected in the limit . Thus, we find that the electric
field in the radiation zone is given by

(1249) 
where
is given by Eq. (7.64). These fields are typical
radiation fields; i.e., they are transverse to the radius vector,
mutually orthogonal, and fall off like . For magnetic multipoles
we merely make the transformation (7.63).
Consider a linear superposition of electric multipoles with
different values, but all possessing a common value. It follows
from Eqs. (7.54) that
For harmonically varying fields the time averaged energy density is
given by

(1252) 
In the radiation zone the two terms are equal.
It follows that the energy density contained in a spherical shell
between radii and is

(1253) 
where the asymptotic form (7.16) of the spherical Hankel function
has been used. Making use of the orthogonality relation (7.53a), we
obtain

(1254) 
which is clearly independent of the radius. For a general superposition
of electric and magnetic multipoles the sum over becomes a sum
over and , and becomes
.
Thus, the total
energy in a spherical shell in the radiation zone is an
incoherent sum over all multipoles.
The time averaged angular momentum density of harmonically varying
electromagnetic fields is given by

(1255) 
For a superposition of electric multipoles the triple product can
be expanded and the electric field (7.68b) substituted, to
give

(1256) 
Thus, the angular momentum in a spherical shell lying between radii
and in the radiation zone is

(1257) 
It follows from Eqs. (7.27) and (7.52) that

(1258) 
According to Eqs. (7.33), the Cartesian components of
can be written:
Thus, for a general th order electric multipole that consists of
a superposition of different values, only the component of
the
angular momentum takes a relatively simple form.
Next: Sources of multipole radiation
Up: The multipole expansion
Previous: Multipole expansion of the
Richard Fitzpatrick
20020518