How does the current change?

A few days ago a friend of mine proposed this question :

How does the current increase or decrease in a specific circuit ?

and I couldn't help but to post the question here.

Consider, for example, the case of a DC adjustable voltage generator connected to a resistor:

Now when we increase the emf of the generator, the current in the circuit will increase proportionally according to Ohm's law:$$V=RI$$

So the question is, shall the current increase:

1. by an increase of the drift velocity of electrons moving across the circuit, or

2. by an increase of the linear density of electrons forming the current?

• Hmm, I think you need some other assumptions/constraints to go with this. If I assume that # of electrons supplied by my circuit is fixed, then clearly the only thing that charges is the drift velocity, since it is proportional to the electric field and the charge density is assumed fixed. – Ben S Jan 20 '17 at 19:19
• @BenS I think we only need the conductivity of the circuit as a function of the distance traveled from one pole of the battery to the other one. And also the EMF. The other remaining thing is the shape of the conductor. Here, for thin wires, we can assume that the conductor is one dimensional. There are sufficient for solving the boundary value problem, I think. – AHB Jan 20 '17 at 19:23