Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-l] Current as Vector

John Denker wrote:
James McLean wrote:

This whole discussion is a microcosm of the more general fact that all fluxes are signed quantities that depend on a choice of positive direction, but are not vectors.


Help me out here. We have something that is not a scalar. It has
magnitude *and direction*. But is not a vector. So, what is it?
Do mathematicians have a name for this type of non-scalar non-vector
quantity?

The term "flux" seems to be well established, so I would be surprised if mathematicians did not use it. Of course, they may have generalized the concept in some way - I wouldn't know.

I have yet to see any evidence that flux is not a scalar. In general, fluxes do not have directions (read my description carefully). For an example I'm sure we're all familiar with, consider the electric field flux through a Gaussian sphere centered on a point charge. That flux has a sign which depends on whether we choose to call "outward" positive or negative. But I don't see how you can ascribe a direction to that flux (in the sense of an arrow direction).
[We of course have a well established convention to call outward positive. Nevertheless, that convention is arbitrary.]

As you have pointed out, there are cases where it is intuitively appealing to associate a direction with a flux. However, after playing with it a bit, I'm not seeing any way to give such a concept a well defined pairwise addition rule. Here is a challenge:
Consider an isotropic radial vector field, and its flux through the sides of a cube with the same center. Label the "directed flux" through three of the sides with A=(0,1), B=(1,0), C=(0,-1). Devise a mathematical addition rule such that A+B=(sqrt(2),sqrt(2)) and (A+B)+C=(3,0).

Anyway, they aren't vectors.


So current is a signed quantity, and properly dealing with it requires defining a positive direction on each circuit segment. Why not just say that, instead of trying to make it a vector? In fact, even here the vector model defeats the point: one of the salient properties of vectors is that only a single coordinate system choice is required to solve a problem; but for a circuit, you often want to pick a *different* positive direction for each segment.


I don't see much merit in that argument. I can imagine lots of
mechanics problems where I might decide that this force is a multiple
of this unit vector, and that force is a multiple of that unit vector,
with lots of different unit vectors. A spaceship with N thrusters
for maneuvering springs to mind.

Yes, my third sentence wasn't well put. It isn't intended to be an argument. Rather, it is an example of a useful property of vectors that is not helpful in analyzing circuits. The question of the second sentence stands: Even if you can define a vector current, what benefit does that provide that "choosing a positive direction" doesn't already provide?


--James
--
Dr. James McLean phone: (585) 245-5897
Dept. of Physics and Astronomy FAX: (585) 245-5288
SUNY Geneseo email: mclean@geneseo.edu
1 College Circle web: http://www.geneseo.edu/~mclean
Geneseo, NY 14454-1401