How does electricity flow in conductor when potential difference is applied? Electrons move from higher potential to lower potential. When a conductor is connected to battery, electron move from negative terminal to positive terminal.
But the battery itself forms a Electric field like below

If a free electron were there on negative terminal it would follow electric field to the positive terminal.
My Question is how is Electric Field set up in the wire so that electrons pass from negative terminal to positive terminal in a conductor?
Does it follow external electric field like below or set's up it's own electric field?

how can i visualize Electric Field inside conductor when Potential difference is applied across it. How can current be explained in terms of Electrostatics ( I hope it does not sound funny)
I think my problems is this http://www.physicsforums.com/showthread.php?t=159205
EDIT:: suppose two highly charges plates is connected by the conductor (that allows low current), Can i presume that the internal current inside conductor C is independent of external current? If the electric field is inside conductor, then electric field is in same shape as conductor.

 A: Electrons will flow against the electric field lines because their charge is negative, and the electric field thus exerts a force $\mathbf{F}=q\mathbf{E}$ on them which is in the opposite direction. Thus electric field lines inside the wire go from the positive to the negative terminal and the electron flow goes from the negative to the positive terminal. Electric current goes, consistently with both of the above (because the electron charge is negative), from the positive to the negative terminal.
The electric field lines will twist with the conductor if you bend it into some weird shape. (This is due to slight charge buildups on the wire bends and is beautifully explained by Purcell.) For the situation you describe, the electric field lines and the wire pretty much match already so just draw some more lines. You've already explained current flow in terms of electrostatics in a circuit like this! the only snag is what the state of affairs is inside the battery, but that's another story.
A: The wire sets up charges on the surface to channel the electric field exactly along the path of the wire. This isn't surprising, if the electric field doesn't exactly follow the path of the wire, charges are shunted to the surface, and these charges will then move the electric field so that it is parallel to the wire. The amount of charge required to do this shunting is tiny, it's a negligible capacitance of the wire that depends in a crazy nonlocal way on the shape of the wire, the type of battery, and the other conductors around.
But the answer is just yes: the metal conducts charges to the surface just until the electric field is going parallel to the wire along the entire length of the wire, no matter how many times it doubles back.
