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Imagine a circuit with only a 12 Volts battery and a wire connecting the ends of the battery. Point A and point B lies on the wire. What is then the potential difference between point A and B if the wire is:

  1. Ideal (Resistivity = 0)
  2. Realistic (Resistivity > 0)

Circuit

For the ideal case, we actually had some problems solving it. The first method uses the fact that the potential difference across a wire is zero. Consequently, the potential difference between A and B is zero. However, if we consider points infinitesimally near the battery terminals, we should found the the potential difference is 12 Volts, as one could similarly use a voltmeter to measure the voltage. This clearly contradicts the "zero potential difference" theory. How does one explain this inconsistency?

For the realistic case, assuming that the wire doesn't melt, I guess that all we have to do is consider the whole wire a long resistor.

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Situation 1., the ideal wire with an ideal voltage source, is an idealization: it is not a situation that can occur in nature. One should not be surprised that physics, which aims to describe nature, cannot describe a situation that cannot occur in nature.

I will note that you specified a battery, not an ideal voltage source. In this case one can develop an answer to the question. But since this sounds like a homework problem, I'll leave it at that.

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If the wire has a non-zero resistivity, then there will be a finite resistance between the points $A$ and $B$, say $R_{AB}$. However, to determine the p.d. between these points you would have to know the resistance of the whole wire (or do you just mean that the wire between the points $A$ and $B$ has finite resistivity?), otherwise you can use Ohm's Law to determine the current since you know the size of the battery (the emf) and the resistance of the circuit (assuming a perfect battery with zero internal resistance).

As an aside, note that if you have a circuit with a resistor connected between a perfect battery and connected with zero resistivity wires, then even though the p.d. across the perfect wires are zero, you can still have a current flow. This follows from $V_{wire} = I \times R_{wire}$ and $I$ can't be determined since $V_{wire} = 0$ and $R_{wire} = 0$.

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