Now let's to untangle the first (symmetric) piece. This will turn out
to be a remarkably unpleasant job. In fact it is my nefarious and
sadistic plan that it be *so* unpleasant that it properly motivates
a change in approach to one that handles this nasty tensor stuff
``naturally''.

We have to evaluate the integral of the symmetric piece. We get:

(13.74) |

The steps involved are:

- integrate by parts (working to obtain divergences of ).
- changing into a times whatever from the continuity equation (for a harmonic source).
- rearranging and recombining.

The vector potential is thus:

(13.75) |

Note that appears

To get the fields from this expression by taking its curl, and then the
curl of its curl, is - ahem - most unpleasant. *Jackson* wimps
out! Actually, taking the curls is no more difficult than it was for
the magnetic term, but untangling the integrals with the result is,
because of the tensor forms that appear. Consequently we too will wimp
out (in the comforting knowledge that we will shortly do this *right* and *not* wimp out to arbitrary order in a precise
decomposition) and will restrict our attention to the far zone.

There we need only consider the lowest order surviving term, which
always comes from the curl of the exponential times the rest:

(13.76) | |||

(13.77) |

If we keep only the lowest order terms of

(13.78) |

If we recall (from the beginning of Chapter 4) the discussion and definition of multipole moments, in particular the

(13.79) |

whose various components can be related to the five spherical harmonics with (!) we can simplify matters. We can write the one messy integral in terms of another:

(13.80) |

where

(13.81) |

Note that the ``vector''
(and hence the fields) depends in
both the magnitude and direction on the direction to the point of
observation **n** as well as the properties of the source. With these
definitions,

(13.82) |

which looks (except for the peculiar form of

(13.83) |

and computing the flux of the Poynting vector through a sphere of radius as a function of angle that the angular power distribution is:

(13.84) |

The angular distribution is too complicated to play with further unless you
need to calculate it, in which case you will have to work it out. The total
power can be calculated in a ``straightforward'' way (to quote Jackson).
First one changes the cross product to dot products using the second relation
on the front cover and squares it. One then writes out the result in tensor
components. One can then perform the angular integrals of the products of the
components of the **n** (which is straightforward). Finally one term in the
resulting expression goes away because
is traceless. The
result is

(13.85) |

(note frequency dependence). For the numerologists among you, note that there is almost certainly some sort of cosmic significance in the 1440 in the denominator as this is the number of seconds in a day.

Just kidding.

For certain symmetric distributions of charge the general quadrupole moment tensor simplifies still further. A typical case of this occurs when there is an additional, e. g. azimuthal symmetry such as an oscillating spheroidal distribution of charge. In this case, the off-diagonal components of vanish and only two of the remaining three are independent. We can write

(13.86) |

and the angular distribution of radiated power is

(13.87) |

which is a four-lobed radiation pattern characteristic of azimuthally symmetric sources. In this case it really is straightforward to integrate over the entire solid angle (or do the sum in the expression above) and show that:

(13.88) |

At this point it should be clear that we are off on the wrong track. To quote Jackson:

The labor involved in manipulating higher terms in (the multipolar expansion ofSome would say that we should have quit after the electric dipole or magnetic dipole.A()) becomes increasingly prohibitive as the expansion is extended beyond the electric quadrupole terms.

The problem has several roots. First, in the second and all succeeding terms in the expansion as written, the magnetic and electric terms are all mixed up and of different tensorial character. This means that we have to project out the particular parts we want, which is not all that easy even in the simplest cases. Second, this approach is useful only when the wavelength is long relative to the source ( ) which is not (always) physical for radio antennae. Third, what we have done is algebraically inefficient; we keep having to do the same algebra over and over again and it gets no easier.

Understanding the problem points out the way to solve it. We must start again
at the level of the Green's function expansion, but this time we must
construct a generalized *tensorial* multipolar expansion to use in the
integral equation. After that, we must do ``once and for all'' the necessary
curl and divergence algebra, and classify the resulting parts according to
their formal transformation properties. Finally, we will reassemble the
solution in the new **vector** multipoles and glory in its formal
simplicity. Of course, the catch is that it is a lot of work at first. The
payoff is that it is *general* and *systematically extendable* to all
orders.

As we do this, I'm leaving you to work out the various example problems
in Jackson (e.g. section J9.4, 9.5) on your own. We've already covered
most of J9.6 but we have to do a bit more review of the angular part of
the Laplace operator, which we largely skipped before. This will turn
out to be key as we develop Multipolar Radiation Fields *properly*.