Is there a capacitance between the Earth and the Moon, and if there was enough potential difference, could a discharge strike occur?
I remember that - in one of his columnes in "Electronic Design" - the late Bob Pease has shown how to calculate this capacitance. Just now I have found an addendum to the original contribution: Here it comes
Originally published in Electronic Design, September 3, 1996.
I believe the answers are
1) Edit: see other answer about Bob Pease
2) There's no theoretical reason why not, but there are a number of practical reasons:
It is straightforward to calculate the capacitance of any two conductors. Place equal and opposite amounts of charge on each conductor then calculate the voltage between them. By definition, C=Q/V.
In the case of the Earth and Moon the calculation is difficult because the charges are not distributed over perfect spheres but oblate spheroids. To a reasonable approximation though we can assume that they are spheres.
With this approximation, the electric potential difference is roughly (to about 0.3%) equal to the difference of potential of each body at its own surface. This is a bit strange, but because the Moon is so far away the electric potential of say the Earth at the Moon is very small when compared to the electric potential of the Moon itself.
The mutual capacitance is quite small compared with the self capacitance of the Earth and Moon separately. The self capacitance of the Earth is about 709 microFarads and that of the Moon is about 193 microfarads. The effective capacitance of the pair is 1/709+1/193=1/Ceq, so Ceq=152 microfarads. Again, it is odd that the capacitance between the Earth and Moon is not dependent upon the Moon's orbital radius, but that is the answer.
To do this problem exactly requires you to integrate the electric field between the Earth and Moon over any path between them then divide this voltage into the charge that you used to create the field. This will show a small dependence upon separation. As a last comment, this is a nice problem in that it shows that the conductors themselves hold charge and store energy in their respective electric fields. Capacitance must account for all of this energy.
Normally, the mutual capacitance dominates as in a parallel plate capacitor with a small gap between the plates. But the capacitance of a parallel plate capacitor, where the plates-size-to-gap ratio is very small, is just the sum of the capacitance of each plate in isolation!