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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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Potential drop across connecting wires is never really zero. However it is so small compared to other impedances in the circuit, that for practical purposes it can be taken to be zero. –  user10001 Nov 25 '12 at 4:14
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@dushya: maybe that should be an answer. Then I could upvote it. –  Peter Shor Nov 26 '12 at 13:16
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2 Answers

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 is like 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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But isn't V is like what causes the current? –  Revo Nov 25 '12 at 2:00
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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. –  Alfred Centauri Nov 25 '12 at 2:19
    
@Revo, I've updated my answer with a mechanical analogy that may help. –  Alfred Centauri Nov 26 '12 at 12:57
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It is possible because potential gradient defines 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 mobility of career is on 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 this fact,only one thing drive the formulation and that is when a body in inertial frame is moving then it will keep on it`s 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 condition which is called super conductivity.In such superconductor materials current keeps on flowing for sufficiently long duration even after withdrawal of potential difference across the conductor.

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