Potential energy.

Imagine trying to move an electron from the positive end of the battery along the wire to the negative end. This would mean moving the electron against a constant repelling force for all of the journey. We would have to do work and the potential energy of the electron would increase. Electrons at the negative end are at a higher potential energy than the positive end. Another way of looking at this is to say that a higher concentration of electrons means the electrons are more squashed together. It takes a force moving a distance to squash the electrons together, so this means an increase in potential energy - like squashing a spring. The higher the concentration of delocalised electrons, the higher their potential energy.

The potential energy of an electron at any point along the wire is determined by the concentration of the delocalised electrons at that point. One could say that concentration and potential energy are two ways of looking at the same thing. As an electron moves along the wire its potential energy decreases in the same way that a ball loses potential energy as is rolls down a gravitational hill (and increases in kinetic energy). In our circuit, the electrons are ‘falling down an electrical hill’. Graphs of concentration of delocalised electrons and potential energy of electrons are equivalent:

A heating effect.

Unlike a ball rolling down a hill, the delocalised electrons do not move faster and faster as they move along the wire. Why? We must remember the ‘atom cores’ - these constitute most of the wire. The moving electrons will collide with the atom cores. These collisions will impede the movement -the electrons will be stopping and starting all the way down the electrical hill. In the collisions, the accelerating electrons will transfer their KE (gained by the loss of PE) to the atom cores. It is rather like a marble rolling down a slope dotted with pins (a pinball machine). Although the slope is always there (the overall force causing the acceleration of the marble), the marble stops and starts (even rebounds) as it makes its way down the slope. This stopping and starting means that the marble cannot build up a speed. The drop in PE of the marble is converted to KE of the pins (they are left wobbling) as the marble descends. (On a clear run the PE would convert to KE of the marble.) Similarly, in our electric circuit, the drop in PE of the electrons as they move along the wire is converted into KE of the atom cores. The evidence for this is that the wire warms up.