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Re: [Phys-l] Current as Vector



Rauber, Joel wrote:

"I don't know why I didn' look before, but for those who appreciate an
arguement "from authority"; not particularly me I might add, though I do
put some added weight to the pronouncements of authority.

Griffith's popular upper division textbook on E&M, IMO, plainly says
that current can be considered a vector quantity.

See eqn 5.14 page 208 of third edition, or discussions on pages 211 and
212 of the text."


I think, the whole discusison about current boils down to this:
1) either we all accept the conventional defifnition of current I = dQ/dt
or
2) some of us do not accept this definition

In case 1), there can be no controversy, since dQ/dt is a scalar in a fixed reference frame; in case 2) there may be arguments, but they would not make much sense if each participant uses a different definition.
As to Griffiths, the same example on p. 208 opens with a statement that "current...is the charge per unit time". Since this is a scalar, it is in flat contradiction with the statement at the bottom of this page. So Griffiths (whose textbook is, in my opinion, one of the best, and I use it in my class) in this specific case contradicts himself within half page of the text. Even worse, in the opening statement itself he says that current is the charge "...passing a given point". This statement is, at best, ambiguous, and may be totally misleading, or just meaningless. What does it mean "passing a point?" Passing exactly through the point, or passing in some vicinity of the point? If passing exactly through the point, then there is, in fact, no such thing as current. If in the vicinity, then what determines the vicinity? Of course, current is the charge per unit time passing through a specified surface, not a point on the surface. This becomes clear if we ask what is the current in the same example, when the line charge moves perpendicular to itself rather than along itself? We cannot answer this question unless we specify a surface in the way of motion.
The moment one starts talking about a point, one actually switches the subject - from a global characterisitic to a local one, in this case - from current to current density.
Another side of this story: I think everybody agrees that electric potential is a scalar. Hence a voltage V, being potential difference between two points, is also a scalar. Similarly, everybody agrees that resistance R of a conductor is a scalar. However, if one insists that current is a vector, then one should also insist that voltage is a vector. Indeed, in Ohmic conductors, according to Ohm's law, we have V = R I, and if I is a vector, so must be V. We can make situation even worse by reversing the argument, - starting from voltage as a scalar. Then, writing the Ohm's law as R = V/I, please, explain to me how we must interpret the division by a vector, and what kind of quantity will the result (R) be?
I am afraid that if we stick with the notion that current itself is a vector, we should rewrite many pages in our textbooks.

As Joel himself remarks, a reference to an authority may be a good argument, but never a proof. I can bring a couple examples later when I find the exact references.

Moses Fayngold,
NJIT