John can (and I hope will) respond better to this, but I'll make the
following "neutral" comments.
Moses F. wrote in part:
|
| John Denker wrote:
|
| "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 agree, I don't think anyone has claimed that current density and
current are interchangeable.
Therefore, no one has claimed that I_f = dQ/dt is a vector; everyone
agrees that the above is a flux through a surface, and if the surface
becomes an infinitesimal surface element then the I_f becomes j DOT dA .
And I think we all agree that this is what is conventionally called
"current", and is not a vector, but is rather a signed scalar.
However, in response to John M.'s challenge regarding the discharging
spherical capcacitor, John D. wrote:
______________
| "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.
Why do you say these are not currents, but rather current densities? The
way I see it John is saying, e.g., that we have 1/8th Amps flowing out
of octant I, he did not say we have 1/8th Amps per meter^2 flowing out
of octant I. So I do not see any way you can interpret the above as a
current density. He's given a magnitude, he's given a direction for
each of the currents, sure seems to me that they are vectors.
Whether or not this is a useful identification is another matter; but he
was just providing an example of making the identification, and
succeeded as far as I can tell.
| "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.
The integral form of charge conservation is explicitly a flux through a
surface, so quoting the integral form doesn't answer the question at all
as to whether or not it is possible to identify a vector with current.
This is a bit like saying that Gauss law proves the electric field is
not a vector because the r.h.s is proportional to Q_enclosed which is
clearly not a vector. It only proves that electric flux is not a
vector. And your statement only proves that flux of current density is
not a vector.
______________
| "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.
I think you *may* have caught John in a minor slip of the tongue.
The way I interpret what John says is as follows:
We can make a sometimes useful notion of current that is truly a vector.
How? by simply using j in the infinitesimal case, which seems natural
enough, i.e. dI_vec = j_vec x |dA|.
For a finite case superpose dI_vec appropiately over a region of
interest.
The local statement regarding charge conservation could then be written:
(Div dI_vec)/|dA| + d rho/dt = 0 , where |dA| is the infinitesimal area
of a closed surface inside of which you are measuring d rho/dt
Or to paraphrase John, conservation of charge involves something
proportional to the divergence of current, not proportional to its sum.
Hence, the summing of the current vectors to zero is not a contradiction
to charge conservation.
| "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.
I don't think John would deny defining something different from the I_f
that occurs in the expression I_f = dQ/dt. So what. Its not all that
conventional; but it may be a useful definition in certain
circumstances, like getting the signs correct in the discharging
capacitor problem that sparked this thread.
| 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.
Exactly! I think this a quite natural way to look at it. Analogous to
how momentum is actually the product of a momentum density and a volume
element containing it. I see very little difference
|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.
Above, you have simply defined the current I to be the flux through and
area, of course, that definition is not a vector, no one has argued
otherwise; but this does not constitute proof that one can not in a
reasonably natural way associate a vector to the current concept. It
would be odd if we couldn't make such an association, since the current
is moving charges and therefore involves the idea of a directed
magnitude.