Re: [Phys-l] current vector

[John Mallinckrodt <ajm@csupomona.edu>]

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.

--
John "Slo" Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://www.csupomona.edu/~hsleff/OoPs.html>