The force field emanates from an electron at the speed of light and propagates outward at that same speed. It adopts a life of its own and cannot be affected in any way by what the electron does after that. If you imagine that the emitting electron is jerked into a new position along the direction of the field propagation the field must somehow double up on itself because it cannot move any faster. It then produces some kind of a fold which is preceded and also followed by the field moving at the velocity of light. The fold then rides the field at that same velocity and wherever it arrives it dispatches the information about the acceleration that produced it.
If the electron is jerked away from the observer, i.e., in the direction opposite to field propagation a gap in the field must occur and be preceded and followed by the field traveling at the velocity of light. The gap rides along and signals a deceleration wherever it arrives. From it the magnitude of deceleration of the electron can be inferred just as in the acceleration case.
We have encountered the formation of a "doubling up" of the field and of a gap earlier but did not pay attention to it. It took place in the animation of the realistic Coulomb field following a sudden move of an electron. We are reproducing here only the last phase of that animation as . Initiate the motion but stop it when the old and the new field are about equal in length. Then pay attention to the two horizontal rays and note the gap on the left and the overlap of the field on the right. These are consequences of maintaining the field velocity at constant value regardless of the motion of its source. And this is the extent to which the relativity needs to be considered in this study. Let the animation complete by another click after which you can either repeat it or simply quit. The transition from the old red field vector into the new one has been simplified in this animation. Recall from our previous discussion that at the time the disturbances, i.e., gaps and folds, respectively, reach the observation points they temporarily change the fields there. We show only the steady state values.
One may rightfully ask what happens to the remaining rays of the Coulomb field.
While this course does not penetrate into this question we show on the right the result of a simple calculation of the field trajectory imposed by the relativity principle. A distinctive presence of transversal field components can be observed in the picture, the strength of which increases with the angle away from the direction of motion. To a student of antennas this is a revealing physical insight that is otherwise hidden behind a fairly complex theory of radiation.
Next we ask what happens if the electron has been accelerated and decelerated a few times during the field travel time. Its final position will obviously bear no relationship to the initial acceleration. The idea of "anticipation" breaks down all together because a different anticipated position of the electron would be associated with each acceleration and deceleration involved. The answer is quite straightforward, on the other hand, if we believe that the sequence of accelerations and decelerations of the electron is carried along by the field line in its folds and gaps. Then they themselves determine the time sequence of the magnitude and direction of the field reaching the observer.
An attempt to graphically illustrate the speculation about folds and gaps in the field is given in the animation which you are invited to initiate now. It shows three electrons spaced equally along the horizontal axis. Click over the animation window and observe the field lines streaming out of the three electrons . We assume this process to be continuous so that any disturbance produced by the motion of a particular electron is carried away by the streaming field lines of that electron. We will create such a disturbance by accelerating the electron S to the right. Since the field line cannot stream faster nor slower than the speed of light, a fold is produced to the right and a gap to the left of the accelerated electron. Click over the window and stop the motion when the gap comes near the electron A . The gap and the fold were moving at the speed of light away from the primary electron. When the gap reaches the electron A it creates a pull on it to the right, thus closing the gap. At the same time A produces a gap to its left which continues propagating to the left carried by the streaming field line of A. The electron itself moves to the right. Focus on these processes first and ignore for the moment the events around the electron B . A click will allow the animation to run its course. The gap appears to have moved continuously to the left and the electrons have moved the distance indicated by the yellow trace.
Let us now repeat the whole sequence. When you see the prompt Click for Acceleration of S , do it and wait for the green fold to move about halfway between S and B . Then stop the animation and we can have a moment to speculate about the upcoming events. When the fold will reach the electron B it will push it forward, thus stretching out the fold and leaving nothing to ride backwards toward S on the field line streaming out of B. But a new fold will be created to the right of the pushed electron and this will ride the field line streaming out of B to the right. Observe now this process following a click and pay attention to the recreation of a fold ahead of B as the electron itself starts to move. Note, that all disturbances between moving electrons have been eliminated while they are being progressively reproduced by each newly accelerated electron. Consequently the disturbance appears to be travelling from the primary source S outwards at the speed of light, accelerating the electrons along the way. The electrons themselves move at some slower velocities.
Because the electron S had been moving while the disturbance was traveling towards A and B it has traversed a longer path as indicated by its longer yellow trace. Therefore it has reduced its distance to B and increased that to A. You may also notice that the trace of electron A is slightly longer than that of B . This is to indicate that A moves faster than B . We will say more about this in the next section.
Assume that S is accelerated to velocity v in the direction of the resting electron B and that it launches a fold that propagates at velocity c towards it. As the fold reaches B this is accelerated and let us first assume that the acceleration is identical to that that propelled S . In this case the electron B will start moving at v and the accelerating fold will experience a "calibrated tug" which will stretch it out into a straight field line. If the tug were lesser it would leave behind a fold remnant and if the tug was stronger it would generate a gap in the field line. For an exactly calibrated tug which arises when the acceleration of B is equal to that, that launched S initially, there is no fold and no gap between the two electrons to propagate back to S . Electrons S and B would move from now on undisturbed as a pair at a reduced distance. Of course, B will have produced a fold ahead of itself at the time of acceleration. Should this fold encounter another electron in its path it would reproduce the process just described. We would end up with a brigade of electrons moving together at velocity v while their front is progressing at the velocity c, i.e. at the velocity at which the folds are propagating.
Assume now that the electron B was accelerated to something other than v . The stretching out of the fold would not take place and there would be either a residual gap or fold riding the field line streaming out of B towards S . This would tend to modify the velocity of S . If you think about this in quantitative terms you can conclude quickly that S would receive a deceleration fold if B were slower than S and an accelerating gap if B were faster. This means that soon a correction would be made tending to equalize the velocities of both electrons. Reflections on wires arise from this mechanism. They are generated whenever an electron moving in a "gapless and foldless" brigade experiences a change of velocity.
The processes just discussed can only happen in neutral environments. This means that the density of positive neutralizing charges must be equal to the density of electrons. But this is exactly the situation in a neutral wire which we are studying. In such a medium the average spacing of electrons is equal to the average spacing of neutralizing atoms. As the brigade of electrons is forming according to the above speculation it is clear that during the time, the fold needs to propagate between two electrons, the emitting electron has a chance to get nearer its upstream neighbour. Consequently the brigade consists of electrons that are all moving at the same velocity but at a reduced spacing. The front of the brigade is progressing at the speed of light at which the fold is moving. The net charge density in the region where the brigade has already formed is negative and this is the process by which the wire looses its neutrality.
While the moving electrons do not sense any pull nor push in the direction of their inertial motion, as we described above, they do sense each other's presence in the lateral direction via the net negative charge. Their repelling static charges force them to distance themselves from each other in the lateral direction which they best achieve on the surface of the wire. Their parallel motion, on the other hand, results in an attracting force which aids the moving electrons to stay at the surface of the wire. In absence of any other effects the current carrying electrons would always tend to move on the surface of the wire. It is the collisions with the atomic lattice that counteract this tendency and are responsible for current carrying electrons to occupy the whole diameter of a resistive wire. In our ideal case, all the conduction processes are confined to the surface.
This is certainly a legitimate question. The initial acceleration of the electron S launches a gap towards the resting electron A . While this gap travels towards A , S distances itself from it. Upon sensing the gap, A is accelerated forward and starts its motion with a distance from S that is larger than the distance between positive atoms. We are not ready to pursue a detailed explanation of what exactly takes place between the electron S and A so we urge the reader to believe for the time being that the electron A will be accelerated to a velocity higher than that of S. But the rest of the processes we can follow right away. As A accelerates to the right it creates a gap behind itself which is carried by its streaming field line further to the left. Should this gap encounter another electron in its path, it would accelerate it to exactly the same velocity at which A is moving. But because A was on its way during the gap travel time, the pair now moves at the same velocity but at an increased spacing. By this process a brigade of rarefied electrons is forming and consequently the net charge of the wire is positive. The locus of positive net charge which represents the tail of the brigade travels to the left at the velocity of the gaps, i.e., at the velocity of light, while the brigade itself moves slower to the right.
Should there be parallel brigades of positive net charges moving along the wire each of them would sense a positive net charge all around. Therefore they would have the tendency to seek the largest possible distance from each other. And this is again at the surface of the wire where the brigades of rarefied electrons do form. The attraction between moving electrons also aids in keeping them there. We have concluded earlier that in absence of folds and gaps there are no forces acting between moving electrons in the direction of their motion. Consequently, as long as the electrons maintain their acquired spacing they can maintain their flow forever. Because there is no friction, only collisions with the lattice prevent superconduction to be a normal occurrence.
We must say at least something about the leading electron in the rarefied brigade. This would be the electron A in our case. The initial acceleration of the electron S can only be accomplished by a voltage or a current source located in the center of our animation window. It is the property of any source to maintain the balance between the inflow and the outflow and to modify either the spacings, velocities or both in accordance with the source type. A current source, for example, will maintain the ratio of velocity and spacing on its terminals at the nominal value without regard to their makeup. The spacings may be identical for that matter. A voltage source, on the other hand, will always enforce a spacing difference between its terminals and this regardless of the respective velocities provided they balance the flow. But this is only possible when the wider spaced electrons move faster then the dense ones. The ratio of velocities to spacing must be identical at both terminals or else a surplus or deficiency of electrons arises which soon restores the balance. In our case the electron A is distanced from S more than B and must therefore move faster. As such it reduces its distance to S during the joint travel of S and B. A quantitative analysis shows that A reaches exactly the center of our picture when its distance from S is same as that of B. Our envisioned driving source is also at this same position and it forces the velocity of electron to v . The electron A now gracefully joins the brigade of compressed electrons with the same velocity v and the same spacing. In reality we are dealing with large numbers of electrons and their pool on one source terminal is being replenished by inflowing fast electrons as in our example. The source than draws on this pool and launches electrons more frequently but at a reduced velocity out the other terminal. While speaking of reality let us look at some real numeric values of interest.
In copper wire the density of electrons is about 10 23 per cubic centimeter. The atom spacing and consequently the mean electron spacing is about 0.23 nanometers. The travel time of a gap or a fold between two electrons inside the wire is then in the order of a micro-picosecond.
The acceleration of an electron to its travel velocity happens typically in the order of a pico-pico second. The acceleration therefore happens in one millionth of the propagation time and can be considered instantaneous.
The travel velocity of electrons in a wire of 1mm diameter at 1 Ampere of current is in the order of fractions of millimeters per second. As we have seen and will see again, the velocity of electrons is no direct measure of the current they are carrying. If the wire geometry is not uniform we may have a constant current flowing in the wire but the velocities of electrons are different in each portion of the wire that has a different geometry. As a matter of fact the electron velocities in the positive wire of a closed circuit are larger than those in the negative wire because of their different spacings. These differences, while real, are very small and reside in the thirteenth decimal place for a current of one Ampere in a copper wire of one millimeter thickness. Such small differences exist also between the spacings of electrons in the positive and the negative wire yet they are sufficient to produce 300 Volt of potential difference between two one millimeter copper wires spaced six millimeters apart and carrying one Ampere of current.
Lastly let us ponder the question of what would happen if the electron were accelerated or decelerated a few times during the disturbance propagation time. What would the neighbouring electrons sense? We have already concluded that the sequence of accelerations and decelerations would be reproduced in the folds and gaps of the propagating field and when reaching another electron would impart the same sequence of accelerations and decelerations on it. But what is the likelihood of being able to change the acceleration during the micro-pico second it takes for the field to propagate from one electron to the other? Only X-rays are fast enough to do it and because our discussion is concerned with much slower events we will justly assume that the electron velocity remains constant during the field travel time.
You may at this point terminate the animation by clicking the "quit" button and either return to the previous text or directly continue with the Interaction of Multiple Electrons in which we will apply our conclusions to a numerical simulation of electron interactions.