I can do circuit problems, but conceptually I cannot figure out what is going on from a basic physics perspective, with respect to the following: Say you have a circuit with a battery (emf = V) and a resistor (resistance = R). Then, electrons on one side of the resistor are in a constant potential, and electrons on the other side of the resistor are at a different constant potential. Since E = grad V, there is no force on the electrons in the perfectly conducting wire (on either side of the resistor). We know the battery is supplying an EMF and that the electrons are losing thermal energy through the resistor. Yet they cannot lost any kinetic energy in the direction of motion, since the current has to be the same on both sides of the wire. So my understanding is that just after the battery is turned on, electrons flow unimpeded to the resistor, then they slow down as they encounter resistance. That then slows down the electrons behind them (Coulomb interaction) and speeds up the electrons in front of them. This ends up working out to a constant velocity. But with all these electric interactions in play, I don't see how the electrons in the top wire are actually at a constant potential V from the battery relative to the electrons in the bottom wire, since they are already being slowed due to the resistor. I am phrasing this poorly but I am very confused. Thanks!