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To conclude our discussion of multipole fields, let us relate the multipole
moments defined and used above (which are exact) to the ``usual''
static, long wavelength moments we deduced in our earlier studies. Well,

(13.83) 
and
(using the vector identity

(13.85) 
to simplify). Then
Now, (from the continuity equation)

(13.87) 
so when we (sigh) integrate the second term by parts, (by using

(13.88) 
so that

(13.89) 
and the divergence theorem on the first term,
for sources with compact support to do the integration) we get
The electric multipole moment thus consists of two terms. The first
term appears to arise from oscillations of the charge density itself,
and might be expected to correspond to our usual definition. The second
term is the contribution to the radiation from the radial
oscillation of the current density. (Note that it is the axial or
transverse current density oscillations that give rise to the
magnetic multipoles.)
Only if the wavelength is much larger than the source is the second term
of lesser order (by a factor of ). In that case we can write

(13.92) 
Finally, using the long wavelength approximation on the bessel functions,
and the connection with the static electric multipole moments
is complete. In a similar manner one can establish the
long wavelength connection between the and the magnetic moments of
earlier chapters. Also note well that the relationship is not
equality. The ``approximate'' multipoles need to be renormalized in
order to fit together properly with the Hansen functions to reconstruct
the EM field.
Next: Angular Momentum Flux
Up: The Hansen Multipoles
Previous: A Linear CenterFed HalfWave
Contents
Robert G. Brown
20140819