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Let us evaluate this term. It is (c. f. J9.13):

(13.45) 
(note:
). If
we integrate this term by parts (a surprisingly difficult chore that
will be an exercise) and use the continuity equation and the fact that
the source is harmonic we get:

(13.46) 
where

(13.47) 
is the electric dipole moment (see J4.8). Note that if we
define
to be a ``probability density'' for the
electrons during a transition this expression is still valid.
This is wonderfully simple. If only we could quit with the vector
potential. Alas, no. We must reconstruct the electromagnetic
field being radiated away from the source from the expressions
previously given
and
After a tremendous amount of straightforward but nonetheless difficult
algebra that you will do and hand in next week (see problems) you will
obtain:

(13.48) 
and
E = 14&pi#pi;&epsi#epsilon;_0 {k^2(n ×
) ×n e^ikrr
+ [3n(n ·)  ]
( 1r^3  ikr^2 ) e^ikr}
The magnetic field is always transverse to the radial vector. Electric dipole radiation is therefore also called transverse
magnetic radiation. The electric field is transverse in the far
zone, but in the near zone it will have a component (in the
direction) that is not generally perpendicular to n.
Subsections
Next: Asymptotic properties in the
Up: Electric Dipole Radiation
Previous: Radiation outside the source
Contents
Robert G. Brown
20170711