Related: Tubelights+power lines pictures?
I would've edited this into the above question, but I realized that there' enough to it to qualify as a new one. Plus this seems to be a confusion of many people (including me).
I've already been able to explain part of this, so I'll put that explanation first just to make the question clearer.
The original question is as follows: In a circuit, you can have potential, but what is the origin of the field associated with it? Conducting wires have no net field, just some tiny opposing fields popping in and out of existence which facilitate current.
Explaining this for a simple capacitor is easy: With a charged (parallel plate) capacitor, one can see that we have a field between the plates but not in the rest of the circuit (for simplicity we can assume that the capacitor is just being shorted. So, traversing a different path in the line integral for potential gives a zero p.d, clearly a contradiction. After thinking about it, I realized that there is a significant fringe field near the terminals of the capacitor, which contributes heavily to the potential. SO the rest of the wire is equipotential without a field, and we have a p.d. only at the terminals due to the fringe fields. IMO there will be similar fringe fields that explain where the battery's potential comes from.
Now, I can't seem to get a similar explanation for current through a resistor. I fail to see any field being formed.
And my main issue is this: Let's say we set up a power line parallel to the ground. It may carry AC or DC current. Either way, it has a p.d with the ground at every point in time, which may vary. P.d. $\implies$ field, but I don't see any. In the AC situation, it could be from EMI, though I doubt that EMI is strong enough to produce the required field strength. In DC, @akhmateli mentioned charge being distributed on the wire surface, but I doubt that as well.
So where does the field come from? Is my explanation for a capacitor correct?
Oh, and in your explanation, I'd prefer no "this comes from the potential" or "these charges move due to the p.d.". I want one that talks about fields and charges only. I've had too many explanations of this which talk about "the field comes from the potential", which is IMO cheating.