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I want to model the resistance between two points on a large mass of uniform resistivity.

There are two cases I am particularly interested in:

a) a cuboid (e.g. a tank of conductive fluid), we could say it is axis-oriented with (0,0,0) and (w,h,d) being diagonally opposite corners, I put two pointlike probes in at known coordinates and measure the voltage between them.

b) a thin sheet whose shape can be decomposed into uniform squares, for example:

enter image description here

For a) it might be acceptable to treat the cuboid as infinitely large. How big would the error be? Or, under what size conditions is the error "small" (within some specified tolerance, i.e. a few percent)?

For b) it would be acceptable to treat the probes as being squares of the same size and alignment as the tiles which comprise the whole shape, if that simplifies things.

How can I calculate the resistance between the probes in these situations?

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    \$\begingroup\$ I think this falls under some form of finite element analysis (which tends to use triangles, not squares). I'm uncertain if you can do it by hand with an acceptable degree of accuracy. For a given gradient in the local current distribution, the error decreases as the size of the elements decreases such that the gradient within each element approaches a uniform value throughout the element (i.e. the current distribution throughout each individual element is uniform and each element approximates a point rather than an area). \$\endgroup\$ – DKNguyen Jul 6 '19 at 18:20
  • \$\begingroup\$ The intent is to subdivide an arbitrary shape into squares/triangles/whatever in software and make them "small enough" to get an suitably accurate answer. \$\endgroup\$ – spraff Jul 6 '19 at 18:21
  • \$\begingroup\$ Instead of modelling it as an area you might model it as a square grid (or triangle if you really wanted to but that might be trickier) of very small resistors instead. Then you could use more directly use circuit analysis methods instead of electromagnetics. Might be simpler. You could probably just use regular circuit simulation software like like LTspice if you took that route as long as it could handle than many hundreds or thousands of resistors without crashing. For a sheet anyways. It might be a nightmare of netnames connecting different 2D layers to do it for a 3D object. \$\endgroup\$ – DKNguyen Jul 6 '19 at 18:22
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    \$\begingroup\$ See How to solve similar sheet questions #1, How to solve similar sheet questions #2, Infinite Grid of Resistors, and An Infinite Series for Resistor Grids. You can use these "red/black"/"checkerboard" techniques for arbitrary shapes, including shapes with holes, wires, and any other crazy shape you want. \$\endgroup\$ – jonk Jul 6 '19 at 18:26
  • \$\begingroup\$ The resistance will strongly depend on the size of the inner-most square. Suppose you had 1 meter square sheet, with 10cm contacts? or 1mm contacts? Big difference, as the current-crowding needed to neck down to 1mm contacts causes lots of voltage drop. \$\endgroup\$ – analogsystemsrf Jul 7 '19 at 16:39
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a is a calculus problem and the result will depend on surface the area of the probes and the conductivity of the tank in addition ot the given paramters.

B is probably best approached as a finite element problem, you could try using femm to solve it. http://www.femm.info/wiki/HomePage

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  • \$\begingroup\$ The tank itself is a perfect insulator and the probes are pointlike. How do I solve this calculus problem? Or, is there a reasonable approximation I can use? \$\endgroup\$ – spraff Jul 7 '19 at 7:49
  • \$\begingroup\$ If point-like, the resistance is infinite. You must specify the diameter of the probes. \$\endgroup\$ – analogsystemsrf Jul 7 '19 at 16:40

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