physical bridge but the passage or motion.  In this case, you could say "across" whenever the emphasis was on the crossing (getting from one side to the other) and "over" whenever the emphasis was on the state of being above/distant from the river.  "He needed a bridge to get across the river" means that he is focussed on being on the other side of the river, and it doesn't matter what kind of bridge he ends up using.  "Like a bridge over troubled waters" emphasizes that the bridge is removing you from the dangerous situation.   

A:

Google Ngram shows that "a bridge across/over the river" are variants of almost the same frequency. As already said, the two prepositions don't say anything about the kindforce is not useful because it regards the interaction of a curved surface with a flat one, while we have two flat surfaces. Intuitively, we expect to have increasing capillary force with the radius of the flat tip, but not in proportion respect to the curvature radius of a commercial tip.

Conclusions
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We describe the production and test of truncated cone shaped tips. A noble metal film, deposited on commercial tips, is shaped by pushing them with a constant force on an atomically flat Si(111) wafer in air. The film thickness and the applied force are the main parameters in the front radius control. We make a first attempt to describe the van der Waals interaction forces between the tip and the sample. A comparative test on the capillary forces