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According to Ohm's law, if there is a potential difference, $V$, across a resistor then there is a current, $I$, flowing through it.

Since we assume that points along the connecting wire are at the same potential, how can current, $I$ flow between points at the same potential, $V$?

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When there is no resistance, as is the case with an ideal wire, any value of current satisfies Ohm's Law:

$V = I R$,

since both $V=0$ and $R=0$.

UPDATE:

But isn't V what causes the current?

Perhaps a mechanical analogy of the resistor will help. Consider the dashpot where the velocity of the arm is analogous to current while the force acting on the arm is analogous to voltage.

The relationship between the force and velocity for a dashpot with impedance $\mu$ is:

$F = \mu v $

This has the form of Ohm's law and is in fact its mechanical analog.

If the dashpot impedance is zero, the arm can have any velocity even though the force is zero. Physically, this seems reasonable since, when there is no external or damping force acting on the arm, we expect that the motion will be unchanging.

Similarly, if there is a steady current through a zero resistance (an ideal wire), we shouldn't expect that a voltage is required to maintain that current, we should expect that the current will be unchanging.

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    $\begingroup$ But isn't V what causes the current? $\endgroup$
    – Revo
    Commented Nov 25, 2012 at 2:00
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    $\begingroup$ The electric field accelerates charge. A flow of charge does not require an electric field in the absence of resistance. Think of it this way: in a resistor, the flowing charge carriers interact with the structure of the resistor, losing kinetic energy in the process. To maintain the current, the flow of electric charge, an electric field is required to accelerate the charge. In the absence of these interactions, when there is no resistance to the flow, the charge carriers loose no energy to the wire and thus, require no electric field to maintain the flow. $\endgroup$
    – Alfred Centauri
    Commented Nov 25, 2012 at 2:19
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    $\begingroup$ @Revo, I've updated my answer with a mechanical analogy that may help. $\endgroup$
    – Alfred Centauri
    Commented Nov 26, 2012 at 12:57
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    $\begingroup$ Charges have mass. Once you start a mass moving its motion will not change if there are no forces acting on it. So you start the charges moving and as the wires are ideal and offer no impediment to the motion of the charges, the charges will carry on moving. $\endgroup$
    – Farcher
    Commented Feb 20, 2017 at 12:04
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It is possible because potential gradient defines the electric field and if the gradient is absent then the electric field is also absent so there is no force that the charge career can feel and if the mobility of the career is at its maximum (which can be achieved when electrical resistance is vanishing) AND there was already a current before the started being equipotential. Through classical physics arriving at this fact, only one thing drives the formulation and that is when a body in an inertial frame is moving then it will keep on its same pace though there is no force required. Actually, the electrical version of it is achieved by imposing statistical laws on this law of inertia of classical mechanics. Even it is achieved yet in a condition which is called superconductivity. In such superconductor materials current keeps on flowing for a sufficiently long duration even after withdrawal of potential difference across the conductor.

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