Current flow in infinite sheet, and shape of isocurrents

[AR01075]

I am trying to determine the size of a conductive 2-D sheet that has a specified degree of increased resistance (or reduced conductivity) compared to an infinite sheet.

Imagine that electrons enter the infinite sheet and exit the sheet at 2 points which are 1 unit of distance apart and aligned on the X axis. The sheet has a certain conductivity, for simplicity let's say it is 1.

There is a certain finite resistance that could be measured between the points (I don't think it would be zero!). What is the size and shape of an area that encloses 90% or 99% of the total current (again, for simplicity, let's say the voltage is 1)?

I have thought long and hard about the SHAPE of the "isocurrent" lines: Circular arcs? Canenarys? Parabolas? Elliptical arcs? And I have no idea what the answer is. Should the curve have a minimum length for a given enclosed area? Should any derivatives have a zero value or no discontinuities?

I had thought that if I could get a function for the shape of the isocurrent lines, then I could have a function for the length of the isocurrent line between the endpoints. Current is inversely proportional to the length of the isocurrent line, and I would hopefully be able to integrate the current as the center of the isocurrent line sweeps from a Y value of zero to infinity.

But I have come to a complete block. I thought that the isocurrent curves may be catenary lines, but the integral does not appear to converge on a finite value.

I am looking for guidance or a source for the answer.

-Tony

 

[tech99]

It is important to define your return path.

 

[AR01075]

The return path is not the plane of the sheet. Imagine a 1 volt battery with the positive end connected to one of the points, as described in the first post, through a zero resistance wire and the negative end connected to the other point.

 

[tech99]

I think that the return wire and the sheet form a transmission line. The current in the sheet will tend to follow the position of the return wire, by attraction. You might be able to gain an insight by drawing a cross section, say half way along, showing the return wire, then drawing in the E field lines. As an extension of this, if you imagine the return wire is imaged in the sheet, the wire and its image form a two wire line. The field lines in a cross section of a two wire line are now easily drawn. The sheet is in the neutral plane of this diagram.

 

[AR01075]

Well, this is a totally hypothetical, ideal world question. So let's say the point where electrons enter and exit the infinite sheet are infinitely small... points. And that the return wire is infinitely far away. The electrons, in effect, just teleport in and out, subject to the constraint that there is no net change in charge of the sheet. And let's say the voltage between the two points is 1 volt.

So, in summary there is a 2-D plane of infinite extent, with a conductivity of 1, with 2 point source electrodes one unit of distance apart, and 1 volt potential between the points.

What is the conductivity of the infinite sheet, and what is the size and shape of an area with 90% or 99% of the conductivity of the infinite sheet.

All the formulae I can try suggest that the conductivity of the infinite conductor is infinite. So,it makes no sense to ask for an area with 99 % of the conductivity of infinity. But I really think the conductivity of the infinite sheet is a finite value. So, I think I have no idea how to approach this problem properly.

-Tony