Another word for potential difference, is apparently, "voltage drop", although I've heard potential difference be interchangeable with just voltage, as well, a bit confusingly. However, if I understand it properly, the battery has an electric field. Things closest to it are being pushed with the greatest force.
However, my understanding of potential difference needs a bit of refining.
Then the difference in voltage between any two points, connections or junctions (called nodes) in a circuit is known as the Potential Difference, ( p.d. ) commonly called the Voltage Drop. The Potential difference between two points is measured in Volts with the circuit symbol V, or lowercase “v“, although Energy, E lowercase “e” is sometimes used to indicate a generated emf (electromotive force). Then the greater the voltage, the greater is the pressure (or pushing force) and the greater is the capacity to do work.
The greater the voltage drop, the greater the capacity to do work? Does this have some similarity to closely spaced contour lines? If there's a big voltage drop in a small distance, this implies we are in close proximity to the battery? Because the electrostatic force obeys the inverse square law, so the electrostatic force will be exponential with distance, and thus a high voltage drop implies high force difference in short distance, implying a close spatial proximity to the battery? Is my understanding correct? If not, could someone help refine it?