What keeps an electric circuit from stopping?

[Heatherfield]

I follow a high school Physics course which is the very opposite of rigorous: special relativity was put somewhere halfway the first year of the curriculum and we've been studying electricity twice now without ever really explaining definitions: things like potential differences were magical quantities.

Now we're studying electromagnetism and Lorenz forces we're finally getting into the more rigorous part of the curriculum and we're starting to define potential difference. We've found that the potential difference means that electrons are more pulled towards a pole due to a difference in electric p.d. For example, when we pull a electric circuit (with no battery attached, so no current flowing) through a perpendicular magnetic field, the Lorenz force pushes electrons towards a side of the circuit, causing a potential difference. This causes electromagnetic induction which causes a current until the electrons are divided up 'fairly' throughout the circuit again.

Knowing this definition and example of the potential difference I'm having a hard time relating this to the electric circuits we studied before: electrons always seemed to be divided equally throughout the circuit. Yet we know a circuit still flows. What causes the potential difference in an electric circuit?

 

[axmls]

Where there is an electric field, there is a potential difference. The potential difference in a circuit is set up by the (ideal) battery, which always maintains a constant (in DC) potential difference across itself. Then once we connect circuit components up to this battery, there is a potential difference across them that "pushes" the electrons towards the drop (really it is the accompanying electric field that pushes them). A 9 V battery sets up a potential difference of 9 V across its terminals. If we connect a resistor to its top (which is at 9 V relative to its bottom) and its bottom (0 V), then there is a 9 V potential difference across the resistor, and this causes a current according to Ohm's law.