Is Voltage Drop Across a Resistor Related to the Divergence of Charge Density?

Is a voltage drop across a component a function of the scalar divergence of charge through it?

Okay, this question sounds complicated, and it might be completed nonsense, but here is my reasoning:

A voltage is a difference in electrical potential energy a unit charge would have at two different places. For example if a unit charge had a electrical potential energy of 6J at one side of a battery and a potential energy of 3J at the other side of the battery, then the voltage of the battery is 3J per unit charge, more commonly expressed as 3 volts.

So across a component (for example a resistor) there must be more electrical potential energy per unit charge at one side than the other. Since the charge going in one side of the resistor will be the same as the charge coming out of the other side, the only way there will be more electrical potential energy per unit charge at one side is if the charge is more spread out at one side of the resistor.

This would mean that the charge would be travelling at different speeds at each side of the resistor. (the current would be the same at each side, because the charge will be more spread out at the side of higher speed, so the charge per second at any point will still be constant).

Since this spreading out of charge should be able to be mathematically described by a divergence, can a voltage drop across two places be calculated using the divergence of charge across these two places.

Thanks!

I first asked this question on yahoo answers, and I received this reply:

"You completely miss the point that what travels is energy, not charge. The drift velocity of the charges in a conductor, to which a potential difference is applied, is negligible"

Which is an interesting point, but as yahoo answers does not allow for the ability to further discuss the answers I have moved the question to here.
I would have replied asking if the drift velocity of the electrons is negligible, how does the energy travel? Does it travel as a longitudinal wave just like sound? (the electrons being analogous to the air particles).
I would have also asked again how is it that there is a different electrical potential energy a unit charge would have at each side of a resistor, if the electrons aren't even moving?

Thanks for taking the time to read!

I think you may be confusing the external voltage source (which admittedly comes down to an imbalance of charge) with the charge carriers within a circuit.

Imagine a 3 ohm resistor hooked up to a 3 volt battery. There would be a slight drop, say .01 volt, across the wire leading into the resistor, and a similar drop across the wire leading out, leaving 2.98 volts across the resistor itself.

What you saying is that the 1 amp of current should be moving much faster through the wires than through the resistor. Seductive reasoning, but aren't you basically making the resistor into some kind of a capacitor?

I'm not saying you're wrong. I believe electrons do slow down as they fight their way through an insulator (hence heating), but by how much? I don't know enough about drift velocity to say whether your Yahoo answerer was correct or not.

This is a surprisingly hard question. I asked a somewhat related question about DC current here about a year ago that you might be interested in. The final answer that time boiled down to "possibly ... maybe".

http://www.eeweb.com/electronics-forum/is-dc-current-directionality-ever-important