Does 0 resistance lead to constant current? Consider a wire connected with a battery to complete the circuit. Between any two points on wire, the potential difference is 0. But current however is not zero. It does flow through the circuit. Can some one please explain this?
 A: All real wires have resistance (with the exception of supercooled conductors). The potential difference between any two points in the wire due to the current in the wire equals the current times the resistance of the wire between the two points, per Ohm's law.
Since the resistance of the wires in a circuit is generally far less than the resistance of the components in the circuit it connects, the resistance, and any potential difference due to the resistance, is ignored.
A: Real wires are usually not superconducting and the potential difference between any given two points in the wire will therefore usually not be zero.  But assume that you have got your hands on a superconducting wire. Then the issue with your understanding is that an ideal DC voltage source (unlike an ideal wire) is not possible. The battery can be modeled as a ideal source in series with a resistance, such that connecting a perfect wire across it brings the voltage across it to zero.
Once you have found the resistance in the circuit (either in the battery or both the battery and wire), then it is that resistance that sets the current by Ohm's law $V=IR.$ If you do have a superconducting segment somewhere, then yes, current will flow from one point in it to another without any potential difference between those points. This is not mysterious: from mechanics we know that objects in motion remain in motion unless acted on by a force, and a current is just the motion of charges. A (super)conductor is that kind of environment where flowing charges feel (no) little "drag" force from the material. So in your circuit, the resistive parts where there is a voltage difference set the current, and the conductive parts with low resistance just pass that current without a significant voltage across them.
A: 
Consider a wire connected with a battery to complete the circuit.  Between any two points on wire, the potential difference is 0.

This is true only in the static case, not the dynamic case.  A wire may have zero resistance, but it cannot have zero inductance.  As the voltage source is applied, the change in current in the wire will drive a voltage across the wire in proportion to the inductance.
If you had an ideal (magical) voltage source, then the system would never reach steady-state.  The current would rise indefinitely at a rate determined by the voltage source and the circuit inductance.
In the real world one of several things would happen:

*

*The current would exceed the amount able to be supplied by the voltage source and the circuit would reach steady state.

*The current would exceed the superconducting limit of your wire.  The resistance would rise suddenly, dropping the current and probably destroying things.

*Power losses in parts of the circuit other than the zero resistance wire would cause temperature increases and destroy parts of the circuit.

A: A short  is a special case of a part of a circuit and the current through the short  is controlled by other electrical elements of the bigger circuit it belongs.
