Comments by "Jeff Huffman" (@tejing2001) on "How Electricity Actually Works" video.
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After digesting this for a bit, I've settled on the idea that the electrons are a thin interface between the fields and the energy source/sink. The chemically-forced movement of electrons against the E field in the battery moves energy into the fields, and the tendency of electrons to vibrate nuclei in the resistive load provides a means for them to transmit the energy from the fields into heat. In both cases, though, the electrons themselves carry almost none of the energy at any time. They are dragged around by the fields and whatever the source or sink of energy is, like a piece of paper in between a heavy pushing object and another being pushed by it.
One illuminating thought experiment along these lines is to consider the value of the poynting vector inside the conductor. It's near zero because the electric fields are small compared to outside, but what power flow there is actually points into the wire, not along it. Energy from the fields is entering the wire to be converted into heat due to the wire's resistance, but the power flow along the wire is all outside. Really the only power flow that's actually inside the wire is the (non-thermal) kinetic energy of the electrons, which is minuscule, both because the drift velocity is quite low, and because electrons weigh almost nothing. That minuscule amount of energy is the transfer medium all the power goes through when interfacing with some other physical system, but carries basically no energy itself.
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