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Re: [Phys-l] current vector



Fayngold, Moses wrote:

It looks like the disagreement between John Denker and John Mallinckroft
has been caused by interchangeable use of the two different concepts: current
and current density.

I think both JM and I are smart enough to know the difference
between current and current density, and to not use them
interchangeably.

Current (as a rate of charge transfer through a surface) is not a vector. This can be made crystal clear by considering, say, electrical discharge of a spherical capacitor. We have non-zero charge transfer with no direction to single out.

Now suppose I get out a felt-tip marker and delineate eight
octants on the sphere in the obvious way. Now there are
eight identifiable currents, each carrying 1/8th of the
total current. The directions of these currents are as
follows (in some basis of my choosing):
[ +1, +1, +1 ]
[ +1, +1, -1 ]
[ +1, -1, +1 ]
[ +1, -1, -1 ]
[ -1, +1, +1 ]
[ -1, +1, -1 ]
[ -1, -1, +1 ]
[ -1, -1, -1 ]

These eight currents have direction and magnitude. That's
why they're vectors.

For each of these currents, its direction is anticorrelated
with the location it's coming from. That is, they are all
directed radially inbound.

Sure, the _sum_ of these vectors adds up to zero, but that
doesn't mean they are zero individually.

At this point you might be wondering, how does the sphere
get discharged, if the sum of the current vectors is zero?
Well, the sum of the currents does not appear in the
charge conservation law. What actually appears is the
divergence of the current. That's nonzero in this case.
Indeed it has eight equal contributions, all of the same
sign, because of the aforementioned anticorrelation between
location and direction of current flow.

For more about conservation in general, see
http://www.av8n.com/physics/conservative-flow.htm

===========================

Here are the real reasons I think of current as a vector:

1) Do you believe in such a thing as discrete charge, as opposed
to charge density? Most people do. For purposes of classical
electromagnetism, an electron is considered a delta function
of charge density. Charge is quantized. So discrete charge
is evidently more "real", more "physical" than charge density.

Now suppose a parade of electrons flows past my observation
post, single file. Each electron has a velocity. I can
define a current vector (representing the average current
or macroscopic current) as the product of three factors:

electron velocity
* number of electrons per unit length
* elementary charge (i.e. charge per electron)

I'm pretty sure velocity is a vector. I'm pretty sure the
other two factors are scalars.
(For instance, in any given length of space, I can count
the number of electrons per unit length starting from the
left or starting from the right; direction doesn't matter.)
Therefore dimensional considerations alone suffice to show that
my current vector really is a vector.

(The foregoing argument can be made much fancier-sounding by
re-expressing it in four-dimensional spacetime terms, but the
3D version gets the main idea across.)

It is often possible to construct a scalar current *also*,
especially when flow is restricted to one dimension. There's
nothing surprising about this; much of first-semester high-
school kinematics is restricted to one dimension, making it
hard to distinguish vectors from scalars.

2) IN ANY CASE, we keep coming back to where this all started: it
is important to draw circuit diagrams, and to label the diagram
so as to show the physical significance of whatever current
variables (and other variables) you intend to use. An arrow
is required to define a _basis vector_ for each current. The
actual current may or may not flow in the direction of this
basis vector.

Even if you treat current as a scalar for every other purpose,
treating it as a vector just long enough to explain the meaning
of that arrow on the diagram solves a significant conceptual
and pedagogical problem that is hard to solve any other way.

It's a catch-22 situation: You are free to treat the current
as a scalar _only after_ you have tagged the diagram with a
basis vector arrow:

(scalar current) = (vector current) · (basis vector) [1]

I've seen plenty of smart people who failed to figure this out
on their own. Equation [1] works for me. Recent discussions
on this list suggest that typical textbook approaches don't
work.

If somebody has other approaches that seem to work, please
speak up.