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Re: Conserving Q ? (long)



David Bowman is exactly correct. If one goes to the 4-D covariant form
in empty space, the right hand sides of both Maxwell equations equations
are zero. But in order to interpret in terms of what we see in the lab
(3=space fields evolving in time), the time derivatives are moved to the
right side and play exactly the role of sources (i.e. the same as the
current in the curl B case). The 4-d formulation is much neater and
ultimately more profound, but it is no substitute for understanding what
we see in the lab in terms of things moving in space.
Jerry Epstein

David Bowman wrote:

Regarding Bob Sciamanda's observation:
Note that since there is no real "magnetic charge" current, the
corresponding equation for the curl sources of E has ONLY a "magnetic
displacement" current term:

curl(E) = -dB/dt

There is no real current to be interpreted as both the curl source of E
and the time derivative source of B !

Yes, this is true. In the off chance that anyone may be interested, let
me mention that I commented about this in the previous incarnation of the
'displacement current' discussion last February. These 2 posts both had
the subject: <Re: "Charged" capacitor mis-terminology> and had the
respective dates & message-ids:
<19 Feb 1998 15:43:54 -0500>
<199802192043.PAA29932@tiger.gtc.georgetown.ky.us> and also
<20 Feb 1998 12:10:41 -0500>
<199802201710.MAA04101@tiger.gtc.georgetown.ky.us>.
(The archives of old phys-l posts are still available at wfu.edu even
though the list has recently moved to nau.edu.)


How to consistently give such an interpretation to this equation?

Well, one could, as you and Jerry have done, interpret it just as you
say having the time derivative terms (the so-called 'displacement
current' terms) of E & B act as the source terms for the curl equations
for the B & E fields respectively. This view (essentially non-
relativistic, or at least, not manifestly relativistically covariant)
effectively treats time as an external global secular parameter t with
the geometric domain of the PDEs as ordinary Euclidean 3-space with E & B
being 3-vector fields on that space. In this view these source terms
are essential for EM wave propagation as mentioned by Jerry.

One could also (as I often prefer to do for conceptual purposes) put the
time derivative terms in with the curl terms leaving the true source terms
on the RHSs of these equations completely absent. The composite
expressions involving the mixed curls and the partial time derivatives
along with the other Maxwell equations (involving the divergences of E &
B) are themselves just the various components of the exterior derivative
(generalized curl) and 4-divergence of the EM field strength tensor when
they are written in the relativistically covariant notation of 4-tensors
and forms on Minkowski spacetime (where time has been demoted from a
global parameter to another coordinate direction in spacetime). As such
these, now homogeneous, equations do *not need* (or have in regions free
of real charges and real currents) any source terms. The propagating
wave-like motions do not require such source terms since the propagating
nature of the solutions of the equations is supplied by the (hyperbolic)
effects of the indefinite signature of the metric for spacetime.

David Bowman
dbowman@georgetowncollege.edu