Re: mapping out equipotentials

[Roger Haar <haar@PHYSICS.ARIZONA.EDU>]

Carl,
        I think a simple way to look at this is as what
it is: an electric circuit.  In the 2D case, where
does the current flow inside the inner ring.  The
ring is one terminal, but where is the other
terminal.  In terms of field lines and e-fields,
there is a source but there is no sink.

        When you go to the 3D system with the length of
the cylinders less than the depth of the water,
you have created a path for the current.

        Concerning answer #2: When working in a 2D
system, the 3D analog to be used for calculation
with the standard 3D equations is not a ring but
an infinite cylinder. This is the only way that
flux lines would be confined to a plane.

Thanks
Roger Haar

PS.   My apologies to those who hate the concepts:
field, flux, sources, sinks, anything less than
eleven dimensions, apple pie, and heat.

*******************************************************
"Carl E. Mungan" wrote:
> Friends: We do an experiment in our second semester intro course
> which many of you probably do too. Basically we have a shallow tank
> of water in which we place a pair of metal strips of various
> geometries. We then connect the two strips across a function
> generator putting out about 5 V rms at 300 Hz. (We use ac to prevent
> ion migration to the electrodes.) The slight salt content of the tap
> water gives it enough conductivity so that we can measure the voltage
> at any point in the tray (relative to one of the strips which is
> grounded) using a multimeter.
>
> One of the geometries is two concentric cylinders. The strips in this
> case are made of copper, approx. 1 mm thick, about 2 cm high (sitting
> in about 1 cm water depth), and the inner one is about 6 cm in
> diameter and the outer one perhaps double that.
>
> The main part of the experiment is of course to find the
> equipotentials between the two cylinders and of course they turn out
> to be circles concentric with them but unevenly spaced for equal
> voltage steps.
>
> However, a secondary part is to measure the potential at various
> points *inside* the inner cylinder. Suppose for the sake of argument
> it is the 5 V electrode. What one finds is 5 V everywhere inside to
> within a few mV I believe.
>
> The question is: WHY DO YOU FIND THIS? A couple of us have been
> arguing about this.
>
> Answer #1: Because the water inside is a conductor and a conductor is
> an equipotential.
>
> Counter-argument: But you don't find the same potential everywhere
> between the cylinders which is also filled with water! Obviously the
> conductivity of the water is sufficiently low compared to that of the
> copper that you don't reach electrostatic equilibrium.
>
> Answer #2: Because the inner cylinder shields the cavity inside it.
>
> Counter-argument: A hard but soluble problem is to find the electric
> field everywhere on the plane inside a uniform ring of charge. The
> answer is only zero right at the center. In fact, the field diverges
> as you approach the ring if you assume it's a wire of zero diameter
> because it then looks (at least roughly - I'm not worried about exact
> details here) like a finite piece of wire evaluated along its
> perpendicular bisector which is a standard textbook problem.
>
> Answer #3: The water somehow changes the symmetry of the problem so
> that the ring is actually like an infinite cylinder, for which the
> electric field is zero inside.
>
> Response: Well, that's what the experiment seems to say. Some people
> suggest the water somehow "confines" the field lines (like magnetic
> field lines through a transformer core?) to make this so. If this is
> true, please explain why water does this.
>
> If you agree with this explanation, then I venture to say if I
> suspended the ring in the center of an aquarium (to change the water
> from 2D to 3D) and repeated the experiment I would no longer find an
> equipotential inside the inner cylinder, right? Do you think such an
> experiment is feasible or will the function generator get loaded down
> too much by the resulting drop in resistance between the rings? Carl
> --
> Dr. Carl E. Mungan, Asst. Prof. of Physics   410-293-6680 (O) -3729 (F)
> U.S. Naval Academy, Annapolis, MD 21402-5026     mailto:mungan@usna.edu
>              http://physics.usna.edu/physics/faculty/mungan/