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Sorry if this is in the wrong section.

I'm currently doing some work on models for electric circuits, but my own are not particularly complete! If anyone could shed some light on the following, I would be very grateful:

How do the electrons transfer the energy in a circuit? I'm fully up to speed with e.m.f. and p.d. and I can derive drift velocity, but the mechanism for the transfer of energy eludes me. I tried to work out the total kinetic energy of the electrons in a wire based on their drift velocity, but this was massively lower than the energy transferred to the charge in the same wire. I'm particularly intrigued as to why P=I^2R. If we double the current we quadruple the power, which would seem possibly to be related to kinetic energy as K.E.=1/2mv^2. I'm sure this is down to something quantum, so if it is, please be gentle with me!

Secondly, how does the electric field vary within a series circuit? I'm looking for the reason as to why the p.d. is larger across resistors than it is across wires. I understand we can predict this using V=IR or through the potential divider formula, but I'm interested in the mechanics of it. What is it about a resistor compared to a conductor that creates a larger p.d. across it (and I guess a stronger electric field through it).

Any help would be very welcome, thanks.

Graham