electric-circuitselectricity
In a closed circuit, electrons flow from negative electrode to positive electrode.
In an open circuit, electrons only flow up to the end of the circuit.
Why don't the electrons keep flowing into the vacuum until the negative electrode on the battery drains off electrons ?
A battery is basically just a chemical reaction. At the negative (cathode) end of the battery the reaction releases electrons while at the positive (anode) end of the battery the reaction consumes electrons. As long as the external circuit allows electrons to flow from the cathode to the anode the reaction goes and the battery generates power.
If you break the external circuit then electrons can't flow and the battery stops producing power. But if you can use some kind of instrument (to use your words) to supply electrons to the anode and remove them from the cathode the reaction in the battery will go and the battery will produce power. The battery doesn't care where the electrons are coming from or where they're going.
But you won't be able to do this indefinitely because as you remove electrons from the cathode you end up with a large collection of electrons i.e. a negative charge. In the same way, as you supply electrons to the anode you'll end up with a positive charge. This charge separation generates a potential difference (i.e. a voltage) and as soon as this voltage gets bigger than the battery voltage the electrons will stop flowing. At this point you'll need to let the two collections of charges neutralise by closing the external circuit or the battery will stop producing power.
An extreme example of the open circuit is the battery itself. If you pick up a battery it will have an excess of electrons at the -ve end and a deficit of electrons at the +ve end because the battery has pushed electrons to its ends until it couldn't push any more.
Response to comment:
A very common analogy for an electrical circuit is to image it as a series of pipes with water flowing through them. The electrons are analagous to the water, and the battery is the water pump. So the usual closed circuit is shown on the left, with the battery pumping water (electrons!) out of one end of the battery, round the circuit and back in the other end of the battery.
The open circuit is represented by the diagram on the right, where the battery pumps water up from a closed container at the bottom to another closed container at the top. As the battery pumps water the pressure in the bottom decreases and the pressure at the top increases, and at some point the pressure difference will get bigger than the pump can manage. You can use a more powerful pump (i.e. a higher voltage) but even this will reach a point where it can't pump any more water.
This is why a battery in an open circuit can only pump a certain amount of electrons. As it pumps electrons they generate a reverse voltage that opposes the battery.
Both!
When you first connect the battery the electrons flowing are mainly the electrons that were in the conduction band of the copper wire before you connected the battery. However the electrons obviously have to flow through the battery to complete the circuit. If we take the example of an alkaline battery, electrons flowing into the battery anode react with MnO$_2$ to produce Mn$_2$O$_3$ and hydroxide anions. At the cathode Zinc reacts with the hydroxide ions to produce zinc oxide and electrons, and these electrons flow into the wire.
So while initially most of the electrons flowing are from the copper conduction band, the electrons flowing into the wire from the battery cathode have come from the chemical reaction in the battery. If you wait long enough all the electrons flowing will have come from the battery and all the electrons originally in the copper conduction band will have flowed into the battery anode and reacted there.
Best Answer
If you think about the flow of electricity at the atomic level. The electrons flow by changing the positions from one atom to the other from their valance shells.
The battery just provides the electromotive force or the electric field for the movement of the electrons in the conductor. it is due to the interaction of the electron with this field.
The free electron theory in metals will give you a great idea about the topic. The negative electrode will not contain the electrons but the electro chemical reaction causes the electron to flow and complete the circuit.
The electron may come out of the conductor tip even if the battery is not there by the effect of photoelectric effect. But the work function for that material is what determines whether the electron will flow in the vacuum.
And to give a precise answer to your question the dielectric constant of vacuum is a lot.