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I'm taking an undergraduate electromagnetism course, just to be clear.

Take a closed circuit with a battery of voltage V.

The potential difference between any two points of the wire is zero, if we consider it to be an ideal conductor with no resistivity. Yet there are electrons flowing along that wire with constant drift velocity. A fair amount of teachers insist on the "water pipeline" analogy for circuits, implying that the pipeline has a certain inclination and replacing the battery with a pump. But an inclined pipeline suggests a changing potential across the pipe, which is not the case in electrical circuits.

Where does the potential jump? If it's inside the battery, then it should come down and then up again. This obviously has to do with the intrinsic mechanism of the battery itself, which I don't know anything about in any possible case, so a fairly simple one is welcome.

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    $\begingroup$ If it is an ideal circuit, the battery will burn almost immediately when you short it! $\endgroup$
    – Prasad Mani
    Commented Nov 3, 2016 at 5:59

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Most circuits can be modeled assuming ideal wires because the resistance in some other part of the circuit is so much greater than the relatively low resistance in the wire, that ignoring the wire doesn't make a significant difference.

If you instead create a circuit where there isn't any other resistance (a short circuit), then the minor resistance can no longer be ignored.

A real battery attached to a real wire will drive the current very high until one of the following things happens:

  • Although low, the resistance might be high enough to limit the current to a reasonable level (coin cells might fit here with their high internal resistance)
  • The current outstrips the chemistry so that voltage cannot be maintained (this could happen with a nearly dead battery)
  • Current rises until heat in some part of the circuit causes failure

An ideal voltage source is one that maintains a particular voltage difference between two points. An ideal wire is a device that maintains zero voltage difference between two points. You can't attach such ideal devices together and have them both work.

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