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If we have a wire connecting two ends of a battery, a current runs through it. If the movement of electrons is due to an electric field along the wire, then how could the potential difference between two points in the wire be zero? And how could the current be constant, unless the resistance in the wire provides some sort of opposing force to the electric field?

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The potential difference between two points in a real wire is not zero. The resistance of a real wire is not zero. I think you are basing your reasoning on an idealized model of a wire, not a realistic one.

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