Does an electric current flow in an open circuit? When the positive and negative terminals of a battery is connected through a wire, an electric current flows across the circuit. Generally, electrons are the ones that flow (from negative terminal to positive terminal). What would happen when a conductor is connected only to the negative terminal of the battery? Do the electrons flow from the negative terminal to the free end of the conductor?
I think they should flow since the negative end of the battery has a surfeit of electrons and the conductor has less electrons than the positive terminal of the battery. And if they do not flow, does that mean it is because of zero potential difference (since the conductor is neutrally charged)? What if I connected a positively charged conductor only to the negative terminal of the battery? Does any current flow through the conductor until the potential difference becomes zero? 
 A: 
I think they should flow since the negative end of the battery has a surfeit of electrons

A transient current appears until the end of the conductor is at the voltage of the battery

and the conductor has less electrons than the positive terminal of the battery. 

It is not a count of electrons, but of voltages.
You can think of it as the way a capacitor charges, one "plate" is the end of your conductor, the other "plate" the other pole of the battery. A very inefficient capacitor.

When a battery is connected to a series resistor and capacitor, the initial current is high as the battery transports charge from one plate of the capacitor to the other. The charging current asymptotically approaches zero as the capacitor becomes charged up to the battery voltage.

A: In a sense you are now asking about situations in the realm of electrostatics and the transient electric currents which flow to redistribute changes and hence potentials.  
A battery does have an electric field outside it because there has been a redistribution of charge so that there is a potential difference across its terminals, the negative terminal having a surfeit of elections and the positive terminal having a deficit of electrons.  
This means that just placing a conducting wire in the field of the battery, and not even touching a terminal, will lead to a movement of electrons along the wire so that the wire then becomes an equipotential.
In electrostatics you might call these induced charges and the process charging (the wire) by induction.
As the wire gets closer to a terminal more and more electrons move along the wire to keep the wire an equipotential until in the end when the wire touches the negative terminal there is a net negative charge on the end of the wire remote from the terminal and the potential of the wire is the same along its length.  
If you a simple series electric circuit consisting of a battery, a resistor, connecting wires and an open switch there will be a potential difference across the switch contacts and induced charges on those contacts.
The switch is in effect a capacitor.
In the act of closing the switch, even before the contact is made, charges flow along the wires ie there is a current in the circuit,  which is not usually noticed because the currents are so small.
A: Current flows until the wire is at the same electrical potential as the battery terminal, then current flow stops.  This occurs almost instantaneously
