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Let us consider the angular momentum radiated away with the
electromagnetic field. The angular momentum flux density is basically
crossed into the momentum density
or:

(13.95) 
Into this expression we must substitute our expressions for
and
:
If we try to use the asymptotic far field results:
we get:
With some effort this can be shown to be a radial result  the
Poynting vector points directly away from the source in the far field to
leading order. Consequently, this leading order behavior contributes
nothing to the angular momentum flux. We must keep at least the
leading correction term to the asymptotic result.
It is convenient to use a radial/tangential decomposition of the Hansen
solutions. The
are completely tangential (recall
). For the
we have:

(13.101) 
Using our full expressions for
and
:
with this form substituted for
and the usual form for
we get:
All the purely radial terms in the outermost
under the
sum do not contribute to the angular momentum flux density. The
surviving terms are:
The lowest order term in the asymptotic form for the spherical bessel
functions makes a contribution in the above expressions. After
untangling the cross products and substituting the asymptotic forms, we
get:
The angular momentum about a given axis emitted per unit time is
obtained by selecting a particular component of this and integrating its
flux through a distant spherical surface. For example, for the
component we find (noting that cancels as it should):

(13.107) 
where the brackets indicate the expression above. We look up the
components of the vector harmonics to let us do the dot product and
find:
Doing the integral is now simple, using the orthonormality of the
spherical harmonics. One obtains (after still more work, of course):

(13.109) 
Compare this to:

(13.110) 
term by term. For example:
(where in the fraction is the spherical harmonic , not the
multipole ). In other words, for a pure multipole the rate of
angular momentum about any given axis transferred is times
the rate of energy transferred, where is the angular momentum
aligned with that axis. (Note that if we chose some other axis we could,
with enough work, find an answer, but the algebra is only simple
along the axis as the multipoles were originally defined with their
index referred to this axis. Alternatively we could rotate frames
to align with the new direction and do the entire computation over.)
This is quite profound. If we insist, for example, that energy be
transferred in units of , then angular momentum is also transferred in units of !
Next: Concluding Remarks About Multipoles
Up: The Hansen Multipoles
Previous: Connection to Old (Approximate)
Contents
Robert G. Brown
20140819