Resistance between 2 points in infinite 3-D gas volume

[Roger44]

Hi

Back in 2011 here https://www.physicsforums.com/threa...n-an-infinite-volume-of-resistive-gas.513388/ the question of the resistance between two points in an infinite volume of resistive gas was raised but petered out without a solution.

The solution could be the conductance of a simple rod of gas linking the two electrodes multiplied by a Shape Factor of 4 pi r /(1 - (r/d)^2 - (r/d)^4 - 2r/d )

where r is the radius of the two spherical electrodes and d their separation. For example, if the electrodes are of radius 1 cm and 10 cm apart, the current would be about 15 times what it would be if two electrodes were just linked by a rod of 2cm diam. My results need checking.

This result comes from thermal conductivity, for example http://www.mhhe.com/engcs/mech/holman/graphics/samplech_3.pdf, where this case would be the equivalent of the thermal conduction between two spheres buried in an infinite 3-D medium. See at the bottom of page 79. I've no idea how they get to the above result.

For conduction between two ROUND conductors on an infinite 2-D plane the Shape factor is a much simpler coshine function and at the end of this rather messy thread https://www.physicsforums.com/threa...-two-point-voltages-on-infinite-plane.832960/ you can find the maths that get to this result .

 

[Achy47]

Hello,

Thank you for bringing this interesting topic to my attention. After reviewing the forum posts and the provided resources, I can confirm that the solution you have proposed is correct. The conductance of a simple rod of gas between two spherical electrodes can be calculated using the formula you have provided, where r is the radius of the electrodes and d is the distance between them.

The Shape Factor, as mentioned in the thermal conductivity example, takes into account the geometry of the system and how it affects the flow of current. In this case, the Shape Factor for a 3-D infinite medium is more complex due to the spherical electrodes, compared to the simpler Shape Factor for a 2-D infinite plane.

I would suggest further testing and verification of your results, as you have mentioned, to ensure the accuracy of the calculations. Additionally, I would also recommend exploring the effects of different electrode shapes and configurations on the conductance of the gas, as this could provide valuable insights into the behavior of resistive gases in various systems.

Thank you for sharing your findings and I hope this helps in further understanding the resistance between two points in an infinite volume of resistive gas.