Re: [Phys-l] Current as Vector

[James McLean <mclean@geneseo.edu>]

John Denker wrote:

> Herbert Schulz wrote:
>
>  
>
> > Vectors behave in certain ways under addition, etc. 
> >    
>
> That is an important consideration, and a good way to frame
> the discussion.
>
> I recognize the existence of both:
>  -- A vector current J, and
>  -- A scalar current I.
>
> Arguing about which of these should be called "the" current reminds
> me of the holy war between the big-endians and the little-endians.
>
>  
>
> > I don't see how  
> > current can be a vector since it doesn't add like vectors; e.g., two  
> > currents do NOT add like vectors at branch points in circuits.
> >    
>
> Really?  I would say that they DO behave like vectors;  in particular
> the physical significance of "I" cannot be established except by reference
> to a chosen basis vector.
>
> For details, see
>   http://www.av8n.com/physics/resistance.htm#sec-diag
>
>  
That web reference explicitly fails to consider a branch point, as 
suggested by Herb.  Consider a specific example (ASCII art: fixed width 
font required):
           I1=2A                         I2=1A
--------->---------------------->------------
                          |
                         V  I3=1A
                          |
If you try to make vectors out of these, I2=(1A)i-hat, I3=-(1A)j-hat, 
and I2+I3 has a magnitude of 1.414A, which has no useful physical 
meaning that I can detect.  It certainly doesn't make the "junction 
rule" any easier to handle.

No question that when analyzing circuits, one needs to establish a 
"positive direction" for each segment of a circuit.  But that doesn't, 
in an of itself, make the quantity a vector.

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.  The only cases in which you can 
meaningfully associate a flux with a "flux vector" are:
(1) the ultra specific case of uniform flow through all integration 
areas of interest, and
(2) in the limit of a small integration area, in which case all you have 
done is returned to the flow vector from which the flux can be calculated.
Even in these cases, I can't imagine the utility of defining such a 
vector.  It complicates the discussion, and sets up a barrier for 
extension of the concepts to more general cases.

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.

BTW, at the above web link, it says "To label the voltage of a node in 
the diagram, it suffices to put a symbol next to the node."  While this 
is true, I think it might be a tad misleading.  To obtain proper voltage 
*differences*, reference must be made to the same orientation choices 
that are made for description of the currents.  See 
<http://www.geneseo.edu/~mclean/AnalytII/demo/CircuitAnalysis.pdf>.
--

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