From one of my favorite books, Code: The Hidden Language of Computer Hardware and Software by Charles Petzold:

Here, he explains how to setup switches and lightbulbs in two locations with the purpose of communicating by turning the lights on and off according to some previously established protocol/code.
My question: is he oversimplifying - because that isn’t the point of this chapter - which route/path the current would take when both switches are closed? As in:

Would the wire, which I awkwardly marked blue, also become charged and/or carry current? Or would there be no current because the cathodes of the two adjacent batteries cancel out any difference/voltage in the “blue wire” that would otherwise cause a current?


Try visualizing the current by using arrows of a proportional length directing from
+to-(or in opposite direction).If the book is trying to explain an Wien bridge, the OP will need to actually put numbers there. Just looking at the sizes won’t be enough.
It doesn’t matter if you use numbers or scaled! vectors.