Have attached a diagram of a sample circuit with emphasis placed on the valence and conduction band electrons of the copper wires. I know that there exists a voltage difference between the wires and can calculate current, etc. but now I'm trying to understand this circuit in terms of vector quantities (Electric Fields, Forces, Charge Separation, Drift Velocity). Is my portrayal of the charge differences (charge density?) between the two conducting wires accurate? If so, since the current is equal in them, the drift velocity of the electrons in the High Potential Wire must be higher to compensate for the different in charge densities. Is this right? Please bear with me, because I'm trying to get ILLUSTRATIVE INTUITION about circuits like this (without resorting to scalar quantities and vague/simplistic analogies of "voltage pressure"). Can anyone explain how the VOLTAGE difference is represented in circuits like this? I know the mathematics of voltage, electric field, electric force (Coloumb's law), etc. and I know how to do scalar calculations with these circuits, but the connection between these two entities gets blurred once I stare at a circuit like this. Any help will be appreciated. Please no answers that talk about water hoses, cliffs, "pressure", or "energy drops." I am well aware of these analogies and ideas(mathematically)- what I am looking for now is a particle level illustrative answer/explanation. Thanks to all.