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Re: [Phys-l] current vector



Hi all-
Let me see if I can understand the problem posed by John M. Roughly, and not very accurately stated: Is current a vector? Somewhat more precisely, is flowing charge always describable as a vector?
The answer is clearly, no, but as one of my favorite comics once said, "True, if interesting." The answer is uninteresting because the question misuses the role of mathematics. The correct question, I insist, is: Are there situations where flow of charge can be described by macroscopic vectors?
John M. focuses on the infiniteseimal description of electromagnetism. Currents, as vectors, occur in the integral version of E&M, but the transition from infinitesimal to integral imposes a debt. The debt is that the description applies only to the kinds of regions to which the integration applies.
Three dimensional current flow in a thin wire is quite amenable to the integral description because there is a well-developed branch of mathematics that deals with one dimensional curves embedded in three dimensional surfaces. Similarly, whenever the physical situation is amenable to a mathematical description, the mathematical description may be used as a proxy for the physical situation. Sometimes we encounter physical situations that do not fit the conditions required for simple mathematical descriptions, in which cases we must invoke other descriptions.
Regards,
Jack


On Tue, 21 Feb 2006, John Mallinckrodt wrote:

John Denker wrote:

On 2/21/06, John Mallinckrodt <ajm@csupomona.edu> wrote:

>While it is true that the notion of current often--as when it flows
>in a thin wire--has a strong directional sense to it very much LIKE a
>one dimensional vector, I don't see how one can rigorously treat it
>as a vector.

I am completely baffled by that statement. How can it be "LIKE"
a vector and not *be* a vector?

I'm surprised--and a little skeptical--that someone as mentally agile
as yourself could be "completely baffled" by such a simple statement.
Cubic zirconia is very much "like" a diamond, but it most decidedly
is *not* a diamond. This kind of thing happens *all* the time in
real life. As you are fond of saying, get used to it.

But perhaps I really wasn't sufficiently clear. Let me try again:

In the real, 3-dimensional world, vector quantities like position,
velocity, acceleration, force, momentum, electric field, current
density, etc., often can be represented as signed scalar quantities
in simple, one-dimensional situations. The current *in a wire* is
similar in that regard. However, very much UNlike the aforementioned
vector quantities, there simply is no 3-dimensional "current vector"
that simplifies in the same way they do and for the same reasons they
do to a signed scalar quantity in simple cases.

Look, I don't mean to make a big deal out of this and I certainly
don't want to pretend that I can't imagine what you are talking
about, but I was responding to your unqualified, emphatic, and, I
believe, incorrect, assertion that "current is a vector." In my
opinion it is usually more useful (and *certainly* far more common in
the literature) to consider current to be the flux of current density
across a specified surface. *That* is a scalar quantity. A *signed*
scalar quantity to be sure, but a scalar quantity nonetheless.

Trust me when I say, as I already have, that I fully understand the
motivation, and even the utility, of treating the current in a wire
as if it were a vector quantity. It even works--*approximately*--in
the Biot-Savart law in cases where the wire is sufficiently thin
compared to the distance to the point at which one is calculating the
magnetic field. In general (meaning what "in general" is supposed to
mean), it seems to me that the concept of a "vector current" is not
merely completely useless, but unavoidably ambiguous. In general,
one must, I think, use "j_vector dV" as the infinitesimal source
element in the Biot-Savart law.

[The ensuing treatise on the uncontroversial fact that current can
flow two different ways through an ammeter and on what it means to
have a negative current flowing from terminal A to terminal B has
been deleted.]

>Suppose I have a REAL wire with some current flowing in it. I
>challenge anyone to give me an operational procedure to rigorously
>associate a useful "vector current" with a specific position (or
>cross section or whatever) of the wire.

Challenge accepted. Here you go:
http://www.av8n.com/physics/straight-wire.htm

I honestly don't see a single thing in this document that looks in
any way shape or form like a response to the challenge even taking
into account my lack of expertise in Clifford Algebra.

BTW, that is definitely NOT to say that I don't find the document
interesting. Indeed, as one who is struggling to better understand
Clifford Algebra, I genuinely appreciate your contributions in this
regard and have printed out several of them for extended study.

But here I was looking for something quite a bit more prosaic. I
asked for a way 1) to rigorously determine a "vector current" in the
general situation where, just for instance it is flowing in a wire
that has a LARGE cross section that is NOT circular and that VARIES
in shape and size as one moves along whatever one may consider to be
its "length," 2) to associate it with anything like a specific
position or surface or anything else, and 3) to have it be of any use
whatsoever.



--
"Trust me. I have a lot of experience at this."
General Custer's unremembered message to his men,
just before leading them into the Little Big Horn Valley