The key idea is that air does not obey ohm's law. The current is not simply proportional to the electric field. At low electric
fields, there's almost nothing in the air that can carry current. At
high electric fields, the air starts to break down, forming
current-carrying ions.
The points are important because the electric field near the tip builds up to large values. The air is therefore more likely to ionize near the tip than somewhere else with a smaller field. That means that current can start flowing there, rather than somewhere else, so it flows directly to the conducting rod.
Mike W.
(published on 11/18/2011)
