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Resistance between two points in an infinite volume of resistive gas

  1. Jul 11, 2011 #1
    Imagine for a moment that there is a box filled with some resistive gas. You are holding two probes a distance of X units away from each other in the center of the box. There should be some finite resistance between the two probes. Now imagine that the box is infinitely large, so that the gas extends infinitely in every direction.

    1) In what units would one define the resistivity of the gas?
    2) How would one calculate the resistance between two points in this infinite expanse?

    This has been bugging me for months!
  2. jcsd
  3. Jul 11, 2011 #2


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    I don't see this as any different than an infinitely large gridwork of finite value resistors with ohmeter probes poked onto several different nodes. So, I would say model it that way.
  4. Jul 11, 2011 #3
    Right. Except it would be a 3-dimensional network.

    So, how do I model that mathematically? I really don't know.
  5. Jul 11, 2011 #4


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    Last edited by a moderator: Apr 26, 2017
  6. Jul 11, 2011 #5


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    IIRC, in a fluid the resistance is largely determined by the geometry of the probes.

    This is very old memories from E&M class over 30yrs ago.
  7. Jul 11, 2011 #6


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    I know I saw a paper where someone solved this once. I did a quick search for "resistor network 3d" and got this. I just glanced through it but it looks close anyway.

  8. Jul 11, 2011 #7
    How do you explain the physics of electrical conduction in a gas, like helium for example? Conduction requires electron transport.

    Are you thinking about conduction in gases above the critical point? It is a supercritical fluid, not a gas.

    Bob S
  9. Jul 12, 2011 #8


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    :uhh: Rather distant from my competence so this may be wrong. However I have the impression that your question is a small branch of mathematics! That the question was first investigated or at least asked by Rayleigh*, who was a long time ago.

    That they are happy if they can put bounds on the values of resistance. The problem is dealt with in the languages of graph theory and random walks - you have to know the translation to electricalese though that is not very difficult. There is a book about it which you can read online www.math.dartmouth.edu/~doyle/docs/walks/scan/walks.rtf As far as I can make out the answer to your question is an exercise for the reader at the end! :devil:

    I am a bit surprised it is not easier than this but I wouldn't want to spend like you months trying. Perhaps I am just wrong, there should be someone along who knows.

    * (I am unable to view these papers myself.) [Broken]

    Last edited by a moderator: May 5, 2017
  10. Jul 12, 2011 #9
    Electrical conductivity in a gas first requires something to knock an electron off a neutral atom. Then there is a free electron and a positive ion that can drift in an electric field. For the energies (volts) needed to knock off an electron, see ionization potential table of elements (including gases) at


    If there are no free electrons, then the resistance of the gas is essentially infinite.

    If you put a gas in a hollow tube (Geiger tube) about 2 cm diameter and 10 cm long, with a thin wire down the middle, and put +1000 volts on the center wire, the resistance is still infinite, until a cosmic ray (or similar) hits it. The cosmic ray causes ionization in the gas, and conduction (a spark) occurs between the wire and the tube wall. So gases are not a good example of a wire network. Measuring a current in gas is a way of detecting ionizing radiation.

    Verrry roughly, the density of a gas is about 0.001 times the density of a liquid. This means that the intermolecular spacing in a gas is about 10 times the molecular spacing in a liquid. The space in between is vacuum. How can a vacuum conduct electricity?

    Bob S
  11. Jul 12, 2011 #10


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    I think he is just trying to get a handle on the 3-D nature of problems like this. He probably should have specified it as a conducting liquid, to keep the problem statement physical.
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