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Re: Conserving Q ? (long)



On Fri, 20 Nov 98 Ludwik wrote:
... . We often say that in any
short time interval the current in wire #1 is equal to the
current in wire #2. And some add "because charges must be
conserved". Here are some questions.

It's true that electric charge is conserved, but it is false that charge
conservation is the reason for the usual situation where the currents in
both wires tend to be equal to each other. Actually, the reason for the
usual equality of the currents is found in the 2nd law of Thermodynamics
combined with the observation that unbalanced currents would tend to
result in macroscopic charge separations which would result in a
typically non-spontaneous increase in the electrostatic potential energy
of the system (presumably funded by a nonspontaneous negative
dissipation rate).

... . For example, I1=5*I2 and Q1=5*Q2. Would such
data be in conflict with the law of conservation of electric
charges? I do not think so.

Ludwik, you are correct that the answer to this question is no.

Suppose a circuit is very simple, only one branch, a vacuum
capacitor C (parallel plates very close to each other) in
series with R. Plus two ammeters, one inserted into the wire
#1 and another inserted into the wire #2; the R*C is
conveniently long. An experiment shows that I1=I2, at any
moment. How can it be if there is no motion of charges in
the vacuum? The displacement current through C was invented
by Maxwell; is it real or imaginary?

The displacement "current" term (in Ampere's law) really exists and it
(along with the actual current) is implicated the production of the
transverse magnetic field associated with the circuit, but it is *not* a
real *current* in any legitimate sense of the term. It is, at most, a
pretend current describing an actual effect. It actually is the time
rate of change of the electric flux. Nothing flows or is transported
in a pure (vacuum) displacement current.

I recall this topic was extensively discussed as a thread on this list
from Feb. 18 - 21 of this year under the subject headings:
<"Charged" capacitor mis-terminology> and <displacement current>.

Some say the displacement current is a mathematical
abstraction (extremely useful); it is not a real current.
What makes it less real than other abstractions, such as
"ohmic current" in a wire, or "water current" in a tube?

It's a real effect, but that effect is not a current. Water current in
a pipe and the "ohmic" current of charged carriers in a conducting
substance involves the transport of some sort of "stuff" (i.e. water
molecules & electrons). This is not the case with a displacement
current. To be a current, some sort of "stuff" (as tangible or as
abstract as you wish) needs to be transported from place to place.

On Fri, 20 Nov 98 Barlow Newbolt wrote:
... . I don't think that the displacement
current is any less real than an "ohmic current". Among
other things it serves the function of preserving charge
conservation in the following sense. It preserves the
nature of current that there are no true sources or sinks
for electric current.

This is not a property of a current qua current, nor is it a property
of charge conservation. It is merely a statement of the transverse,
i.e. divergence-free, nature of the sum of the displacement "current"
and the actual current, or, equivalently, it is a restatement of
Kirchoff's nodal law for this sum of "currents". Kirchoff's nodal law
does not *in general* characterize charge conservation--it does so only
in the *special case* of steady-state currents involving a time
independent distribution of charge density (a special case for which the
displacement "current" vanishes anyway). When the currents are not in a
steady-state and charges are locally accumulating, then charge
conservation requires that Kirchoff's nodal law be violated, but in this
case, the sum of the actual current and the displacement "current" obeys
this nodal "law" anyway.

Between the plates of a charging
capacitor the current exists and flows along so that if you
integrate the source/sink density over some volume,
including the space between the plates, you always find
that the same amount of current is flowing into the volume
as is flowing out.

This is always true for the sum of the actual current and the
displacement "current", but this property does not necessarily
characterize the concept of a *current* in anything other than a
time independent steady-state.

It should be noted that one can always construct a type of "displacement
current" for any flux of any kind of a locally conserved additive
(i.e. extensive) quantity, whether that quantity is electric
charge, strangeness (for phenomena that do not involve the weak
interaction), lepton number, particle number (in systems not undergoing
chemical or nuclear reactions), energy, (nonrelativistic) mass, the
x-component of momentum, even probability etc. The only requirement
here is that the conserved 'stuff' be localizable enough so that the
concept of the local density and a local flux current density of that
stuff makes sense. For instance, let [rho] (scalar) be the density and
j (vector) be the flux current density for any kind of locally conserved
'stuff' you wish. Since the 'stuff' is locally conserved it obeys the
continuity equation: d[rho]/dt = - div(j) (here d(...)/dt is a partial
derivative). This is just a differentially local statement (via the
Divergence Theorem) that the time rate of accumulation of the amount of
'stuff' inside any volume is the net current of 'stuff' entering that
volume through its boundary surfaces (simply because the conservation of
the 'stuff' prevents any 'stuff' from being created or destroyed in
situ). IOW, the only way for the amount of 'stuff' inside any volume
can change is for it to be transported across the boundaries of that
volume. Because [rho] obeys the continuity equation we know that both
sides of this equation must be divergences (of something). This means
that there must exist some vector field j_d constructed out of [rho]
whose divergence is d[rho]/dt, and the continuity equation can be
rearranged as:
0 = div(j) + d[rho]/dt = div(j) + div(j_d) = div(j + j_d) = div(j_eff)
where j_eff is defined as j_eff == j + j_d. We thus see that j_eff must
always be transverse (divergence-free) and it must, therefore, always
obey Kirchoff's nodal law (even when j alone does not).

It is relatively straightforward to construct a functional expression
out of [rho] that will function as an adequate j_d. Suppose that the
'stuff' is sufficiently localized so that the total amount of 'stuff' in
all of space is finite (i.e. [rho] has a finite volume integral), then
we can define a potential-type field [phi](r) from [rho](r) according to
the equation:
[phi](r) == 1/(4*[pi]))*Integral_(all space){dV'*[rho](r')/|r - r'|}
where r and r' are the vector locations of the point of observation of
the [phi](r) field and the dummy integration coordinates for the
[rho](r') field respectively. Here dV' is the 3-dimensional differential
volume element involving the r' coordinates. If we then define the
vector field e(r) as e(r) == - grad([phi(r)) we can then express j_d
as: j_d = de/dt. Note that our definitions of e and [phi] require
that e obey a version of Gauss' law: div(e(r)) = [rho](r) and whose
time derivative is: div(j_d) = div(de/dt) = d(div(e)/dt = d[rho]/dt.

Since j_eff is divergence-free we know that it must be a curl and there
exists some vector field b(r) whose curl is j_eff, i.e.
curl(b) = j_eff = j + de/dt.

Note that the "displacement current" j_d = de/dt is not a current of any
kind. It is just a particular functional constructed from the density
[rho] whose divergence is d[rho]/dt, and this term in the continuity
equation merely represents the time rate of accumulation of 'stuff' in
some fixed region of space; it does not represent any kind of flow of
any stuff. The flowing current is described by the other term--the j
term.

It should be emphasized though that the fields e & b defined here are
not actual physically significant fields (unless the 'stuff' in question
is electric charge) and are just mathematical constructions. When the
'stuff' *is* electric charge then these fields are significant because
they are implicated in the physical electromagnetic force exerted on the
charges according to the Lorentz force law for the E and B field.

Suppose we return to the case of electric charges (and ignore for now
other kinds of locally conserved 'stuff'). We can write Ampere's law as
curl(B) - (1/c^2)*dE/dt = ([mu]_0)*j. The LHS of this equation (when
written in 4-vector/4-tensor notation for Minkowski space) is actually 3
(spatial) of the 4 components of the exterior derivative of the dual of
the antisymmetric 2nd rank (2-form) electromagnetic field tensor whose
space-space components are the components of B and whose space-time
components are (up to irrelevant factors of c) the components of E. The
4th (time) component of this 4-tensor equation is expressed by the other
inhomogeneous Maxwell equation i.e. Gauss' law
div(E) = [rho]/[epsilon_0]. Note the current is the source for the
composite expression: curl(B) - (1/c^2)*dE/dt which is a part of a
single 4-tensor expression in Minkowski space. We thus see that it is
not so much that the displacement current is a source for the magnetic
field as that the actual current is a source for a composite
combination of *both* the transverse magnetic field *and* part of the
time-dependent electric field, i.e. the displacement "current" term
when it is not time-steady as well.

David Bowman
dbowman@georgetowncollege.edu