First I'll quote the relevant passage from John D's post
"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".
Moses F. wrote in part:
|
| | 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.
|
Joel R. wrote:
| >Why do you say these are not currents, but rather current densities?
|
To which Moses F: responded:
| Because the expressions in the brackets are (and can only be)
| obtained using ADDITIONAL information about local current
| density distribution.
| You cannot derive them from CURRENT ONLY, this will give you just
| (1/8)(dQ/dt) for each of them. Unless you introduce by hand
| (as John has done) the corresponding unit vectors sticking
| out of the respective octants.
Regarding a quantity's vector nature, it is immaterial how the quantity
is derived, and it does not matter if additional information is used in
the derivation. If it has a magnitude and direction and satisfies some
vector space properties than it is a vector. To repeat, it does not
matter if you can not derive them from CURRENT ONLY.
|
| >So I do not see any way you can interpret the above as a
| >current density.
(I note in passing that "the above" referred to are the 8 quantities in
brackets.)
|
| I did not interpret anything. I did not say current density.
| Current density averaged over an octant is not identical to
| local current density, as the averaged lifetime is not
| identical to individual lifetime.
I quote from your post of 2/24
"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."
It appears to me that you said they were "current densities".
More to your point, I would agree with you and John that Current density
averaged over an octant is not identical to a local current density.
That's partly our point. However, the current density averaged over an
Octant is a vector! And if one multiplies by an appropriate scalar
quantity it will be a quantity that one would probably want to interpret
as a 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.
|
| You put into my mouth what I have never said, and then you
| dispute with
| it.
We agree you never literally said the above. I was making an analogy
between what you said and another situation to illuminate a facet of
your argument. There is a reason why I wrote "This is a bit like . .
.". And upon reflection the analogy I was making is less than
illuminating to the issue, as it was quite open to varying
interpretations as you pointed out.
More apropos is an analogy between momentum density and momentum
vectors.
|
| >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.
|
| I am not against introducing new definitions - there would be
| no progress
| without it. But each time you do it you should say it explicitly and
| unequivocally: here I introduce a new definition different from the
| conventional one in this or that. In this case it has not
| been done, and
| what is introduced is not new and is not current, but an
| analog of current
| element (see one of my previous posters.)
|
I understood from what John said that he was doing something that wasn't
entirely usual.
However, IIRC, I could understand how one might say that it wasn't
baldly stated.
BTW, and here I speak for myself: I'm not denying that current density
may be a more fundamental and often a more useful concept to use. The
point is that it simply is not contradictory to the idea that one can
construct a vector representation of current.