Here's the conundrum I have been facing. Yesterday I asked the question: how can batteries constantly motivate electrons to complete a circuit? (What maintains constant voltage in a battery?) After some thought I realized that the concept is that of an energy source. The batteries are a source of energy, and regardless of the mechanism, they will constantly 'push' for a goal. My solution finally came when I realized that speaking about someone's 'motivation' was identical to the constant push of the battery. Here's the issue:
My sister wanted to get me coffee yesterday for my birthday. She was attempting to give it to me before my class started. Unfortunately, she had difficulty with traffic, and so was able to deliver only after my class started. I realized when she told me that she tried as hard as she could, and that what really mattered was her motivation. The push she gave was constant regardless of the resistance she encountered. The parallel here to the constant push of a battery was strong; voltage didn't matter in respect to resistance for a given battery.
Now I have a different issue. The voltage drop is described as the per-capita potential energy difference of electric charges at points a and b right? The total energy would then be the amount of 'capitas' multiplied by this per capita potential energy difference. More capitas mean more total energy.
Here's the problem. If the motivation of a battery is described by voltage, then if we look at a circuit with a resistor and compare it to another circuit with an equal-length resistor but with a wider surface area, the potential difference is supposed to stay the same in both cases. However, in taking the charge from one surface to the other, more energy is required for the resistor with larger surface area! The amount of energy supplied is different in either case, yet from the same 'constant' energy source. This seems so wrong to me right now.
Yet I know that Power=Voltage*(Current)^2, so this actually makes sense in terms of the equation. I wonder what is lacking in my fundamental understanding of the potential difference in comparison to the idea of motivation/constant energy source. I'm really fundamentally misunderstanding something complex here that I don't quite have my finger put on yet. Please help clear up my view of a battery as a constant pusher to line up with this changing energy ouput a battery can give.