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Re: Flat conductors

Perhaps I should have been more explicit about the suggested
experiment with an electrolyte. Put the copper wire into the
flat bath and allow a small current, say 1 mA, to flow through
it. I suspect the equipotential lines emerging from the wire will
be more or less like those I described in Appendix 2 (see below).

The advantage of doing this in an electrolyte is that the resistivity
of the surrounding medium can be controlled by changing the
concentration. Start with distilled water and keep adding salt
or acid. Does the line pattern change or not? How does it
change? Does the wire thickness (resistance) have an effect
on the line pattern? And, as suggested by Bernard, a deep
bath can be used to explore the 3D patterns. A project for a
student looking for something to explore? Why not?

Ludwik Kowalski wrote:

P.S.
Asking about the "hanging wire" I had in mind a wire
whose resistance can be changed at will to control the
current. How large must the resistance be to replace
the lines due to a flowing current by lines due to static
charges?

Ludwik Kowalski wrote:

In trying to compose a summary of this thread I realized the
message became too long. Therefore I decided to post it on
my website. It now Appendix 2 of the document at:

http://blake.montclair.edu/~kowalskil/elec/strip1.html

Most of those who are interested have already seen the
rest of the document; go directly to appendices. What
was added is essentially a list of simple Pasco sheet
activities for an introductory physics course plus some
questions about the observed equipotential lines. The
item 6 of Appendix 2 is shown below. Please answer
my questions, if you can.

6) Here are some questions I still have. Would the shapes
of the equipotential lines OUTSIDE THE SILVER STRIP,
painted on glass or plastic, be the same as outside the
strip pained on Pasco paper? I would expect the shapes to
change according to the resistivity of the medium on which
the silver strip is painted. But my relaxation code makes
correct predictions without knowing anything about the
materials involved. The only thing it knows is the boundary
condition on the surface of the strip. We are solving the
current electricity problems by pretending that we are
solving a static electricity problem. I would be more
comfortable if the lines were predicted by using the laws
of steady currents and not the low of static potentials.

The relaxation method is a numerical approach to Laplace
equation of electrostatics. The equation shows how the
potential V, due to static charges, is distributed in space.
Where is the corresponding equation governing the distribution
of steady current densities j in a conducting medium? I agree
with JohnD that "identical equations must lead to the same
solutions" but he sees two equations while I see only one.

And here is another way to ask the same question.
Consider a parallel plates capacitor represented by two
silver lines painted on glass. Let the distance between the
plates be one inch while each plate is one inch wide.
Connect the plates to a d.c. power supply. We all know
what kind of equipotential lines will exist on the glass
surface. They are expected to have the left-right symmetry
and the up-down symmetry. Now suppose a circular silver
path is painted representing a "hanging wire" connecting the
plates. A current flows through that wire but the power supply
is able to maintain a constant DOP.

We know that the equipotential lines are no longer the same;
the left-right symmetry of lines still exists but the up-down
symmetry goes away. Equipotential lines crossing the wire
are still squeezed into the space between the plates but they
quickly turn above the plates and go back to cross the wire
again. How large should the current be to destroy the up-down
symmetry of the equipotential lines? It is clear to me that
the equipotential lines do not move along the wire loop when
the current is changed from 1 mA to 10mA or 100 mA. In other
words, the potential gradient along the wire is a constant
which does not depend on the current.

But will this still be true when the current is reduced to 1nA or
1pA or 1 fA (for example, by changing the resistance)? In other
words, is the transition from the electrostatic to the electrokinetic
pattern sudden or gradual? I do not think that a current of 1.6e-19 A,
for example, will impose a new distribution of field lines with respect to
that which exists in electrostatics. What about 1 fA; would this be enough?
Ludwik Kowalski