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To summarize from the last chapter, two useful Green's functions for the
inhomogeneous wave equation:
|
(18.1) |
are
|
(18.2) |
(the retarded Green's function) and
|
(18.3) |
(the advanced Green's function). The integral equations associated with these
Green's functions were:
|
(18.4) |
and
|
(18.5) |
For the moment, let us ignore Dirac's observations and the radiation
field and focus instead on only the ``normal'' causally connected
retarded potential produced by a single charged particle as it moves in
the absence of external potentials. This potential is ``causal'' in
that the effect (the potential field) follows the cause (the motion of
the charge) in time, where the advanced potential has the effect
preceding the cause, so to speak. Let me emphasize that this is not a
particularly consistent assumption (again, we the theory is manifestly
time symmetric so ``past'' and ``future'' are pretty much arbitrary
namings of two opposed directions), but it yields some very nice
results, as well as some problems. In that case:
|
(18.6) |
where the four-current of a point charge is found from
|
(18.7) |
in the lab/rest frame or (in covariant form):
|
(18.8) |
where
|
(18.9) |
Note that the function in these expressions simply forces the
particle to be found at the correct location at each (proper) time. The
function is the trajectory of the particle. Its derivative
is the four-velocity. This yields (when the 's have all been
accounted for) the rest frame expression.
To do the integral, we need the ``manifestly covariant'' form of the retarded
Green's function. Note that:
(where
). In terms of this, is given by
|
(18.11) |
Again, the second delta-function makes no contribution because of the
opposing -function. Thus
The vector potential at a point gets a contribution only where-when that
point lies on the light cone in the future (picked out by the
function) of the world line of the charge (picked out be the
function). The contribution is proportional to
at that
(retarded) time. It dies off like , although that is obscured by the
form of the function.
To evaluate this (and discover the embedded ), we use the rule (from
way back at the beginning of the book, p. 30 in J1.2)
|
(18.14) |
where the are the non-degenerate zeros of . is
assumed to be ``smooth''. Then if we let
|
(18.15) |
(which is zero when in the past) then
|
(18.16) |
and therefore
|
(18.17) |
From this we see that
|
(18.18) |
where is the proper time in the past of when the light cone of
the charge contains the event . This potential (and its other forms above)
are called the Liénard-Wiechert potentials. In non-covariant form,
they are obtained from the identity
where n is a unit vector in the direction of
and where
as usual.
Recall that
. Thus:
|
(18.20) |
and
|
(18.21) |
where all quantities (e.g.
) must be evaluated at the
retarded time where the event x is on the light cone of a point on the
particle trajectory.
Similarly
where again things must be evaluated at retarded times on the particle
trajectory. Note well that both of these manifestly have the correct
non-relativistic form in the limit
.
We can get the fields from the 4-potential in any of these forms. However,
the last few forms we have written are compact, beautiful, intuitive, and have
virtually no handles with which to take vector derivatives. It is simpler to
return to the integral form, where we can let
act on the
and functions.
|
(18.23) |
where
|
(18.24) |
Again, we let
. Then
|
(18.25) |
This is inserted into the expression above and integrated by parts:
There is no contribution from the function because the derivative of
a theta function is a delta function with the same arguments
|
(18.27) |
which constrains the other delta function to be . This
only gets a contribution at (on the world line of the charge),
but we already feel uncomfortable about the field there, which we
suspect is infinite and meaningless, so we exclude this point from
consideration. Anywhere else the result above is exact.
We can now do the integrals (which have the same form as the potential
integrals above) and construct the field strength tensor:
|
(18.28) |
This whole expression must be evaluated after the differentiation at the
retarded proper time .
This result is beautifully covariant, but not particularly transparent
for all of that. Yet we will need to find explicit and useful forms for
the fields for later use, even if they are not as pretty. Jackson gives
a ``little'' list of ingredients (J14.12) to plug into this expression
when taking the derivative to get the result, which is obviously quite a
piece of algebra (which we will skip):
|
(18.29) |
and
|
(18.30) |
``Arrrgh, mateys! Shiver me timbers and avast!'', you cry out in dismay.
``This is easier? Nonsense!'' Actually, though, when you think about it (so
think about it) the first term is clearly (in the low velocity, low
acceleration limits) the usual static field:
|
(18.31) |
Interestingly, it has a ``short'' range and is isotropic.
The second term is proportional to the acceleration of the charge;
both E and B are transverse and the fields drop off like
and hence are ``long range'' but highly directional.
If you like, the first terms are the ``near'' and ``intermediate''
fields and the second is the complete ``far'' field; only the far field
is produced by the acceleration of a charge. Only this field
contributes to a net radiation of energy and momentum away from the
charge.
With that (whew!) behind us we can proceed to discuss some important
expressions. First of all, we need to obtain the power radiated by a
moving charge.
Subsections
Next: Larmor's Formula
Up: Relativistic Electrodynamics
Previous: Covariant Green's Functions
Contents
Robert G. Brown
2007-12-28