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[Phys-L] the "current" in a capacitor --> pseudo-current



I received (off-list) a highly technical question about the
"current" flowing through a capacitor. That's a tricky concept.

As I understand it, the so-called "current" flowing through a
capacitor is sometimes called a /displacement current/ ...
but it is not really a current within the technical meaning
of the term, just as a "Jerusalem artichoke" is not really an
artichoke.

There are actually two independent conservation laws involved
here:
-- There is the conservative flow of charge, as measured
by the four-vector J = [ρ, j]; this appears as a source
term in the Maxwell equations. We know charge is conserved;
this is a corollary of the Maxwell equations. It is an
amusing exercise to prove this.
-- There is Gauss's law, which says the amount of charge
in a region minus the number of field lines crossing the
boundary of the region is a conserved quantity, namely
zero.
++ If we combine those, we get a third conservation law.
This is linearly dependent on the other two, so it
doesn't really tell us anything new. We can define a
/pseudo-current/ that involves the charge plus the
number of field lines. Strictly speaking, a current
cannot flow through the capacitor gap, but a pseudo-
current can. In the gap, there is a changing number
of field lines, and this is what carries the pseudo-
current.

========

If you re-arrange the Maxwell equations in a certain way,
you see that the pseudo-current can be considered the
source term for ∇ × B, the curl of the magnetic field.
This is a reasonable and useful thing to do ... in the
laboratory frame.

In contrast, from a spacetime / four-vector / bivector
point of view, this is the wrong way to arrange the
equations. We shouldn't even be talking about the
magnetic field separately from the electric field.

Suppose we have a temporarily-steady DC current flowing
through a capacitor. At some level, if we don't look
inside the capacitor, we think of this as a steady
situation: steady magnetic field, negligible electric
field. However, if we think about things more closely,
we find it cannot be quite that simple. Every time a
unit of charge flows into the capacitor, a field line
somehow flows into the capacitor-gap. The charge flows
in along the wire, but the field line does not; it
comes from somewhere far away, far off the axis of the
wire. This tells us that in some sense there is no
such thing as an ideal capacitor; there must always
be some fringing field.

As DC current flows through the capacitor, the field
lines in the fringing field move in a self-similar way,
so that the field strength remains the same even as the
field lines move. There is probably some clever way to
understand this, but I'm not seeing it at the moment.
(There's a general rule that if a complicated process
produces a simple result, you don't fully understand
what's going on.)