resistivity - Can the resistance between two points in air be calculated? How?

If I have just a battery, the terminals are separated by air, a very good insulator. We can say, practically, that no current flows.

However, air has a very large, but finite resistivity. Wikipedia gives a range of \$1.3\cdot 10^{16}\$ to \$3.3\cdot 10^{16} \Omega \mathrm m\$, which I'm sure varies by pressure, temperature, humidity, pollutants, and so on.

From that, can we calculate the resistance between the battery terminals, knowing the dimensions of the battery, and assuming that there is an infinite space of air around the battery? What's the math?

Could this be extended for calculating the resistance between two points in an arbitrary material, of infinite volume, with a known resistivity? The practicality of obtaining an infinite volume of a thing aside, I'm wondering how resistivity relates mathematically to idealized volumes of things that aren't extruded blocks of things. (That is, not \$R = \rho L / A\$.)

Answer

Just a note: the resistance between two points in space is highly dependent on the geometry of the space. Even things far away affect the electromagnetic properties of things close by, which means the approximation isn't very good except for wide open spaces.

Ok, so to get the resistance in the setting you mentioned you also need to factor the current emitter itself. It makes no sense to ask the resistance between two points if the full geometry of the problem isn't specified (and this includes the battery).

In practice (or is it theory?), this means you have to e.g. model your two points as charged spheres with a constant +I/-I into them and solve Maxwell's equations. If you shrank you spheres too much, the current would inevitably be concentrated in a too small area, and you get rapidly increasing resistance. In other words, the resistance between any two points assuming pointwise sources is infinite -- which shows mathematically that asking "what is the resistance of the air" is incomplete -- you need to specify everything.

Edit: I forgot to add that because of the linearity of Maxwells equations in simple media with uniform conductivity like the models I exemplified, the final resistance is going to be simply proportional to the conductivity (a simple subdivision argument will convince you of this), so you can compare different (linear/homogeneous/etc) conductors directly.