Re: Question 07/02 CURRENT IN A WIRE

["John S. Denker" <jsd@MONMOUTH.COM>]

Eric T. Lane wrote:

> In freshman physics we are told that a current in an "infinite"
> cylindrical wire flows with constant density J independent of this
> distance from the center of the wire.

We, Kemosabe?  I don't recall being told any such
thing.  For starters, it's not true.  It's bad luck
to prove things that aren't true.

It's spectacularly untrue at high frequencies.
High-frequency conductors are often tubular, to
save the weight and cost of materials.

> But we also learn how to use
> Ampere's Law to calculate the magnetic field in the wire. Doesn't the
> magnetic field act to make the current density non-uniform?

For magnetics problems, it's probably best to duck
the issue entirely.  The magnetic field outside the
wire is independent of the distribution of current
inside the wire, for any radially-symmetric distribution.

For problems other than magnetics you can't duck the
question, for instance if you want to calculate the
"extensive" resistance from the "intensive" resistivity.

> Answer: 11/02 The Hall Effect explains this question.

I very much doubt it.  In particular, what are you
going to do if you have a conductor with zero Hall
coefficient?

The relativistic argument is also bogus.

============

Here's how I look at it:

0) Nitpickers note we are assuming we start with a
symmetric homogenous structure;  otherwise the
question doesn't make sense.

1) Argue that for a long straight wire, in the
low-frequency limit, the current must flow
everywhere in the direction of the wire.  If
it flowed in any other direction, charge would
pile up somewhere.  There _may be_ some pile-up,
but if the amount is bounded, in the low frequency
limit the current involved (the sideways component
of the current) goes to zero.

2) Therefore you can decompose your main wire into
a bundle of fine parallel sub-wires.  If (!) you
think that all such sub-wires have the same
resistivity, then you can conclude from Ohm's law
that the all carry the same current-density.

But they might not have the same resistivity.

a) For one thing, their resistivity might depend on
their charge density, and we are allowing for possible
charge build-up.  For typical metals, this effect
will not be noticeable, because they have such a
huge density of carriers that a few more or less
won't make a difference.  But the statement of the
question wasn't limited to metals.  In a semiconductor
or a vacuum tube, the number of carriers might be
quite limited, and charging effects could be a big
deal.

b) For another thing, the material might have a
nonzero magnetoresistivity.

This posting is the position of the writer, not that of David Axton,
Leonard Chris, Brian Coffey, Deanna Dwyer, K. R. Dwyer, John Hill, Leigh
Nichols, Anthony North, Richard Paige, Owen West, or Aaron Wolfe.

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.