A rigorous treatment of resistive wires and of general resistive phenomena (such as skin effect and dispersion, for example) will appear in a separate course which will be released at a later time. This section addresses the resistors only superficially as needed for understanding of the electron motion in terminated but otherwise lossless wires. The simulation relates to a simple circuit consisting of a source, two wires connected to it and terminated in a resistor which is shown as a red rectangle. Its resistance was chosen to match the characteristic impedance of the particular wire geometry.
Electrons are shown as yellow balls spread uniformly in the wires. The source is either a voltage or current type and its only function is to force the electrons to move upwards. We will say more about the nature of the source as the need arises. Start the simulation by clicking somewhere inside the window and note the almost instantaneous start of electron motion all around. The velocity of electrons has been set to be 15 times slower than that of the electric field disturbances. In typical situations the electrons move 12 orders of magnitude slower. It is therefore not surprising that an instantaneous start of motion of all electrons is assumed to take place, Ohm's Law being an expression of such thinking. But the fact is that the disturbance must propagate from one electron to the other before they are set into motion. Inside copper wire the travel time between two electrons is about one micro-pico second. During this time the moving electron narrows the spacing between itself and the neighbor by a small amount as we have seen in the many preceding simulations. We repeat here the numbers stated elsewhere which reveal that for the case of a 1mm copper wire carrying 1 Ampere of current the change in spacing is one part in 10 13. A small amount considering that the standard spacing is about one quarter of a nanometer. Yet this change of spacing is sufficient to produce 300 Volts of potential difference between two wires spaced 6mm apart.
On the very bottom of the window is a slider for controlling the ratio of field to electron velocity labeled "C over V". It indicates that ratio to be 15 as already stated earlier. Set that control all the way to the left which results in a ratio of 2 to 1. Now rerun the simulation and observe the familiar compression take place in the top wire and a rarefaction in the lower one. (Should the simulation progress too fast for observation reduce the speed with the bottom slider). Stop the simulation with a mouse click when the compression in the upper wire has propagated about halfway to the resistor. If you missed it use the Reset and start over. Because the Reset returns the "C over V" ratio to 15, do not forget to set it to 2 again. Then you will see that in the lower wire the rarefaction has progressed the same distance as the compression in the upper. Let us now ponder this situation.
The right hand side has the same spacing of electrons in both wires and consequently no net charge difference. This implies zero potential difference or voltage, for that matter. The left half, on the other hand, has a net negative charge on the upper wire and a net positive charge on the lower one. Thus a potential difference exists between the two wires. In addition, the electrons are moving in this portion of the wires and do as such produce a current flow. The electrons are moving in the first half of the wire to the right in the upper and to the left in the lower one. But there is no current flow in the rest of the wire.
The rarefied electrons in the lower wire are collected by the source and expelled as compressed electrons into the upper wire. In order to be able to do that, the rarefied electrons must be faster than those leaving on the top. The function of the source is to aid in this process but the actual velocities and spacings of electrons, which happen to be proportional to each other, are dictated by the amount of current flow in the wire. There is not much significance to that fact so we will not pursue it any further. Resume now the simulation by a mouse click within the window and let it run.
While watching the electrons moving around the loop ponder the following question: The electrons at the bottom of the resistor are leaving faster than they are entering on the top. Didn't we hear that the resistor "resists" the flow of current? Doesn't that mean that the electrons should be slowed down inside it rather than accelerated? This legitimate question is answered in the following way. Yes, it is true that a resistor is characterized by collisions of electrons with the lattice resulting in their slowdown. But slowdown also means accumulation. The repelling forces exerted by such localized concentrations propel those electrons that make it through the resistor without collision. Consequently they exit faster but are less frequent. Their accumulation is largest on the top of the resistor and smallest on the bottom. This is how a potential gradient develops across a resistor. Our simulation is vague about this complex process which is statistical in nature and requires more than solitary electrons for a visual presentation. But it becomes clear in a hurry that it takes some very special circumstances inside the resistor to produce at its output exactly the correct electron velocity and spacing. This pair of quantities must satisfy the demands of the source for a steady arrival of electrons necessary to feed the upper wire. It is therefore not surprising that there is only one special value of the terminating resistor which can perform such a delicate balancing act. Any deviation from it produces mismatches of electron velocity or spacing at the resistor terminals. These result in propagating disturbances which tend to rearrange the electrons along the wire and are sensed as reflections. As mentioned earlier, the resistive phenomena and the partial reflections from mismatched terminations will be some of the subjects of a separate course. For now you may exercise the simulation some more or you may quit it by clicking the relevant button.
We move now on to pursue the explanation of quarter wave resonances and start with the case of a voltage source driving open ended wires. The link Open Line brings on the relevant text.