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Let's imagine that we have a tridimensional metal object of infinite size, and decide to calculate the resistance between two arbitrary points. How would we go about doing this?

I have thought of two possibilities for calculating this, the first one being an adaptation of the conductivity (S/m) using the Pythagorean theorem to find a distance and isolating the resistance from there, treating it like we would treat an electrolytic liquid.

The second possibility I have been thinking about would be a 3-dimentional adaptation of the infinite-resistors problem (which I have only seen in 2d).

Would one of these methods result in giving a decent approximation, or would another method give better results?

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Hi Acebulf - this is a site for conceptual questions about physics, not for getting people to solve homework(-like) problems for you. If you can edit your question to ask about the specific physics concept that is giving you trouble, I'll be happy to reopen it. See our FAQ and homework policy for more information. – David Zaslavsky♦ Jan 5 at 23:40
I disagree with your assessment of this being a "homework question". Maybe a better way to put it would have been "how would you calculate the electrical resistance between two points in a metal object that is large enough for the boundaries not to matter?" – Acebulf Jan 5 at 23:47
Homework-like, really, but anyway, the point is that you've just asked us to solve a problem for you. That's not what this site is for, regardless of whether the question has anything to do with a homework assignment. As I said, if you edit it to show your progress and ask about the conceptual issue that is giving you trouble, I'll be happy to reopen this. – David Zaslavsky♦ Jan 5 at 23:49
Is this an acceptable format for my question? – Acebulf Jan 6 at 0:01
Yep, that's good enough to be reopened. – David Zaslavsky♦ Jan 6 at 0:19
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1 Answer

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For a homogeneous material characterized by a conductivity $\sigma$ (in S/m) the resistance between any two points is unbounded. Such "infinite" resistance even applies if one point is replaced by a spherical surface centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $ 1/(4 \pi r^2 \sigma)$ from zero to any finite radial distance.

Physically what is happening is that the electrical field strength diverges towards a current injection point. You have to assume finite electrodes to obtain a meaningful answer.

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This doesn't seem correct. Ohm meters work: the result doesn't diverge or go to 0 if you put the two probes of an ohm meter on two positions of a sheet conductor - you get a nonzero reading for the resistance that does not depend sensitively on the shape of the contact area. – FrankH Jan 6 at 12:23
In 2D the divergence is milder (logarithmic), so you won't see a strong dependence on the size of the contact area. – Johannes Jan 6 at 12:33

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