"I think both JM and I are smart enough to know the difference
between current and current density, and to not use them
interchangeably".
I think we are all smart enough, - this is what makes the
discussions here very illuminating - at least, for me.
Only smart people can make errors. I am afraid, the following
excerpts show that you do use the concepts of current and
current density interchangeably.
"Now suppose I get out a felt-tip marker and delineate eight
octants on the sphere in the obvious way. Now there are
eight identifiable currents, each carrying 1/8th of the
total current. The directions of these currents are as
follows (in some basis of my choosing):
[ +1, +1, +1 ]
[ +1, +1, -1 ]
[ +1, -1, +1 ]
[ +1, -1, -1 ]
[ -1, +1, +1 ]
[ -1, +1, -1 ]
[ -1, -1, +1 ]
[ -1, -1, -1 ]
These eight currents have direction and magnitude. That's
why they're vectors".
The 8 currents in this example are NOT vectors. The vectors
indicated in 8 brackets, are NOT currents. They are current
densities averaged each over its respective octant.
"Sure, the _sum_ of these vectors adds up to zero,"
I agree.
"... but that doesn't mean they are zero individually".
I never said the averaged current densities are zero
individually.
"At this point you might be wondering, how does the sphere
get discharged, if the sum of the current vectors is zero?
Well, the sum of the currents does not appear in the
charge conservation law."
Wrong. The charge conservation law in its integral form
says I = dQ/dt, where I is the net current through a closed
surface containing the charge Q. Current explicitly figures
in this law, and it is obviously not a vector. This is why
treating current as a vector has lead you to the statement
that the net current iz zero and at the same time the sphere
is being discharged.
"What actually appears is the divergence of the current".
Here is where you explicitly use current interchangeably with
current density. What you can find in any good textbook or
monograph on electricity, is divergence of CURRENT DENSITY,
figuring in the differential form of conservation law.
Divergence of current is a meaningless concept, because current
is not a local characteristic, and not a vector.
"Here are the real reasons I think of current as a vector:
1) Do you believe in such a thing as discrete charge, as opposed
to charge density? Most people do."
So do I.
"For purposes of classical electromagnetism, an electron is
considered a delta function of charge density. Charge is quantized."
The first part of this statement is wrong. The opposite is true:
charge density of an electron is a delta function of position and time,
and not because the electron charge is quantized, but because it is
considered to be squized into a point.
"Now suppose a parade of electrons flows past my observation
post, single file. Each electron has a velocity. I can
define a current vector (representing the average current
or macroscopic current) as the product of three factors:
electron velocity
* number of electrons per unit length
* elementary charge (i.e. charge per electron)
I'm pretty sure velocity is a vector. I'm pretty sure the
other two factors are scalars.
(For instance, in any given length of space, I can count
the number of electrons per unit length starting from the
left or starting from the right; direction doesn't matter.)
Therefore dimensional considerations alone suffice to show that
my current vector really is a vector."
John, you have introduced your own definition of current.
What you define as a current (product qv or nqv) is a vector
but not a current, even though it is indeed tempting to regard
it as current. But actually it is a product of current density
and volume element containing it. It appears as current, but
it is not. The current is a dot product of current density and
area element - a dot product of two vectors. The analysis will
show that it is equal to dq/dt. I am pretty sure the charge
is a scalar, and the same is true for time in a fixed reference
frame. Therefore dimensional considerations alone in this case
prove my point. Apart from the fact that my point is just
conventional definition.