Re: [Phys-l] current vector

[John Denker <jsd@av8n.com>]

Julie Quah wrote:

> just a thought. Since we define current as the rate of flow of charges,
> would it be "right" to say that current flows? As in " the rate of flow of
> charges flow"? I am not nickpicking. It is the precision and accuracy of
> term used in classroom i am thinking about.

This sort of innocent redundancy ("flowing current") is not a problem.

  By way of contrast, a NON-flowing current would be a contradiction,
  and therefore something to worry about.

By way of analogy, we say a river flows.  A river is flowing water.
So that means the flowing water flows.  That's not a problem.


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?  In my book, vector _is_ as vector
_does_.  Something either satisfies the vector-space axioms, or it
doesn't.

If I want to measure a current, then on the circuit diagram -- and
on the physical wire -- I need to specify a _direction_ for attaching
the ammeter.  Here's a snippet from a circuit diagram:

             I
       ----->>>------        direction is required

This has simple but profound conceptual and pedagogical consequences.
The most crucial point is that we need not (and must not!) redraw
the diagram to reverse the direction of the >>> arrow if the ammeter
reading comes out negative.  That's because the >>> does not depict
the direction of the current;  it depicts the direction of the
_basis vector_.

An ammeter measures _scalar current_.  The _vector current_ is
this scalar multiplied by the relevant basis vector.

This same concept shows up in many, many disciplines, not just in
electronics.  This certainly includes physics.  Sometimes you want
to diagram the basis vectors, and sometimes you want to diagram the
actual physical quantities.

  Once upon a time during my first week of graduate school, there
  was a sadistic TA who marked wrong the homework of about half the
  people in the class, because we had diagrammed a vector in one
  direction when the actual force was in the other direction ...
  even though we had all gotten the correct bottom-line answer.
  He could not (or would not) accept the idea that it was OK to
  establish a basis vector on page 1 without prejudicing the
  direction of the bottom-line solution on page 3.

I explicitly duck the question of whether hooking up a voltmeter
the "wrong" way around is the same as hooking up an ammeter the
"wrong" way around.  It's partly the same and partly different.
There are actually two symmetries involved: one of them treats
E and B the same, and the other treats E and B differently.

> > 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

We can start by looking at the first equation,
  http://www.av8n.com/physics/straight-wire.htm#eq-J1
wherein the current density (which people seem ready to accept as
a vector quantity) is specified in terms of the current vector:
  J = I / π R^2        inside the wire 	
  J = 0                outside

IMHO this vector current (I) definitely counts as a "useful" quantity.
We have just seen one use;  its usefulness can be seen even more
clearly in the final equation in the aforementioned document:
  http://www.av8n.com/physics/straight-wire.htm#eq-straight-wire
which gives the electromagnetic field of a long straight wire in
terms of the radius vector and the current vector:
  F = (2/r) /\ I

This can be considered a simple introductory version of the Biot-Savart
law;  if you want more rigor and/or if your wire is not sufficiently
long and straight, you need the full-blown Biot-Savart setup.  The
formula is practically the same, but the proof is more involved.