# Physical Model of Conduction

The conduction of electric current in wires may seem a trivial process of electrons moving through the lattice of the conductor. Yet, this motion is subject to reflections, resonances and other effects that are not easy to visualize nor can they be easily understood from their mathematical description. If you have followed the simulated experiments in the Mathematical Model of Conduction you have encountered some of the mentioned behaviours. We will now take a look behind the scenes of these phenomena and follow the Nature's way of producing them. We will have to consider the mutual interaction of electric charges through force fields which surround each one of them and which take a finite time to propagate between them. The change in the force field of one charge is not sensed by the neighbouring charges instantly but after a delay during which time the offending charge is free to approach or distance itself from them. This is the key to understanding the electric current flow and we will exploit it to the fullest. The three natural principles that we will employ in this pursuit are conservation of momentum, Coulomb forces and special relativity. You will be provided with all the information necessary to follow the discussion.

## Force Field of a Single Electron

J.Priestley in 1767, J.Robison in 1769, H.Cavendish in 1771 and C.A.Coulomb in 1785 have performed experiments which prove stationary electric charges to be surrounded by force fields, spherically symmetric with respect to the charge. One most common representation of such field are radial lines emanating from the charge. Their direction coincides with the direction of the force field and their closeness is a measure of the force field's magnitude. We will use it in the animation which you are encouraged to initiate now by clicking one of the red buttons. You should be seeing in the animation window one point source of force field located at S . We will think of it as an electron.

There is a slider control on the bottom of the window which enables you to control the speed of animation. The default value set now may be satisfactory. The animation is triggered by clicking the left mouse button anywhere within the window. The announcement on top of the window alerts the reader to whatever action will be triggered by the next click. Trigger the animation from the prompt Click for Initial Field which appears initially but also cyclically after every animation. If you wish to arrest the animation at any time during the active phase click the mouse button over the animation window. A second click will cause the action to resume.

Start the animation and observe lines of force coming out from the electron symmetrically in all directions. As they reach out to a radius at which the observation point O is located the time is frozen and the magnitude and direction of the Coulomb force field there are highlighted by heavy red lines. One may legitimately ask if the force field is actually coming out of the electron or was this just an animation artifact. The answer is not easy if the electron is standing still and the field has already formed. But as soon as the electron starts moving we should be able to tell for sure. And all indications are that the field is actually streaming out the electron and is then propagating away from it at the velocity of light c . Let us perform a few experiments relevant to these questions.

Assume first that the field is stationary and tightly coupled to the source. A moving electron would then carry the field along. Click inside the window and observe the electron being jerked into a new position slightly to the right of the initial one. If we believed this picture we would have to accept the notion that the force field everywhere, including at infinity knows instantly what the source is doing. This, in turn, would require infinite speed of information transfer across space. The observations, on the other hand, tell us that nothing can move faster than the light, the latter being itself a force field. Consequently, we must discard the idea that the force field is coupled to its source. The one thing we know for sure is that the force field is symmetric around a stationary source. So our initial picture and the one we are seeing at the moment are correct and in agreement with Coulomb's observations. It is just the way we arrived at it that conflicts with other observations, notably those involving the velocity limitations.

Reset the electron to the starting position by another click inside the window and allow the field to build up again. What we are seeing is in agreement with observations. But so was the final result of our first experiment. Consequently we have two states of the force field that agree with what we know about them. If we then allow the initial field to be gradually replaced by the new one we may have at least a process that allows for finite field propagation velocity and is bracketed by two acceptable states. A click of the mouse over the animation window will jerk the electron into the new position and you can watch the initial force field decay while the new field is building up. Any time during the process you may freeze the action by a click inside the window. As initially, this animation also freezes when the field reaches the observation point.

Our speculation led us to an answer that is correct in principle provided the electron does not move into the new position faster than the field is moving away from it. Nothing can move faster than the force field in vacuum. Nevertheless there are a few more open questions. What happens if the electron moves slowly rather than it being jerked into the new position? Would we get the correct shape of the field at the transition from the old one to the new one by simply connecting them together? And finally, the jerking of the electron into the new position involved an acceleration followed by a deceleration. It is reasonable to expect that both of these would leave some signature in the shape of the field transition. These and other questions will be answered in the next section by means of the principles of special relativity. The mention of relativity may sound discouraging to the uninitiated but you will find it quite natural in the present context. At this time you are offered a repeat of the animation and you may either accept it or quit by clicking over the "quit" button in the upper left corner of the animation window.

## Relativistic Principles Applied to a Moving Electron.

The famous experiment performed by Albert A. Michelson and Edward W. Morley in 1881 with follow-up experiments in 1887 and 1904 has triggered an intense scientific activity which culminated in Albert Einstein's formal theory of special relativity published in 1905. One of the most consequential conclusions of the special relativity is the fact that force fields propagate at velocities independent of the motion of the source which produced them and independent of the motion of the interceptor. Our latest experiment in the previous section certainly does not conflict with this requirement because we allowed the field to move independently of the source. But in order to satisfy this condition we had to allow the field to stream continuously out of the electron and away from it into the surrounding space. There is no other way to progress from the starting state of the field into its final state while at the same time enforcing the finite field propagation velocity. We are therefore answering the question posed at the outset affirmatively, i.e., that the force field must be streaming continuously out of the electron. In addition we must conclude that the kink, which somehow connects the old and the new field, is carried away at the field propagation velocity since it is attached at both ends to it. We will now focus on the nature of the kinks themselves.

Electric force field propagates away from an electron at constant absolute velocity regardless of whether the electron is at rest or is in motion. The direction and magnitude of the field, on the other hand, are highly dependent on what the electron is doing. We know that for a stationary electron the field is radial and aligned with the direction of its propagation. But for an accelerated electron the emitted kink may contain a stronger or a weaker field which may have no alignment at all with the direction of its propagation. The kinks caused by the change of velocity carry the signature of the electron acceleration vector. Both the magnitude and direction of acceleration determine what direction and magnitude the emitted field will have. Once emitted, the established field configuration propagates away from the electron at constant velocity as though it had a life of its own. You may quit the animation at this point if you have not done so earlier.

We defer the study of the general case of arbitrary electron motion to an upcoming course and concentrate here on special conditions which suffice for our study of electric current flow in wires. This is the case of electron motion being aligned with the direction of field propagation while both coincide with the direction of the electric current flow in the wire. Our purpose will be served by the two horizontal rays of the Coulomb field which we access via the animation. Click a red button if you have not done so already and note a window which contains a source of force field at the point S. You may again think of it as an electron if you wish. We will observe the effects of the source status at points O1 and O2. Any force imposed on the electron from the outside will result in emission of disturbances which will always propagate away from the electron in all directions at the same velocity regardless of what actually happened to the electron. In our special case we will show these field disturbances as two green dots propagating towards O1 and O2 , respectively.

First we will establish the stationary Coulomb field lines in the horizontal direction assuming that the electron has just been miraculously placed into position. Click the left mouse button over the animation window and watch the field fronts move towards the observers. As they reach O1 and O2 the animation is frozen. The direction of the force field and its magnitude are represented, respectively, by the direction and the length of the heavy red lines as we remember from the previous animation. No surprises here, just a plain radial Coulomb force field. The potentials at points O1 and O2 are inversely proportional to their respective distances from the source S. If the concept of potential is not firmly embedded in your knowledge base do not concern yourself. We intend to use it here only for comparison purposes. Let it be said only that there is a direct link between the potential and the force field at a given point, the latter being the slope of the former as long as the object is at rest.

Read on before you do the next click. If you have inadvertendly triggered the animation, let it run through its cycles until you induce the prompt Click for Horizontal Field Lines with subsequent clicks. You will observe what happens when the source S is accelerated in the direction of O2. After the initial acceleration the source will be allowed to move freely by inertia at the acquired velocity in conformance with the principle of conservation of momentum. The disturbances emitted by the source at the moment of acceleration will be shown as two green dots moving towards the observation points. Prepare to stop the animation when the right hand green dot is about halfway to its destination. You do this by a second click over the window following the one you are encouraged to do now. The source S has traveled a shorter distance because it has been given a smaller velocity than the disturbance. This is in agreement with the speed limitations of all material objects. The length of the trailing yellow line is a direct measure of the velocity v of the electron and of the time passed since acceleration. But because the electron travels at constant velocity, the yellow line length is also a measure of the acceleration which propelled the electron on its way. The field disturbance, which is represented by the green dot and that was produced by the acceleration travels at velocity c . It carries with it the information about the acceleration and as it reaches the observer modifies the field value at that point in accordance with this information. Watch this process take place as you do another click and allow the completion of this phase of animation. Note that the field magnitude at O2 has increased as indicated by the longer red line. At the same time the field intensity at O1 has diminished. Their magnitudes are directly related to the potential which is again determined by the inverse distances S-O1 and S-O2 , respectively.

Reset the animation with another click and prepare this time to observe the case of higher acceleration. Start it and let it continue uninterrupted. While the travel time of the disturbance is the same as before, the faster electron has traveled a longer distance during that time interval. The field magnitude at O2 is even larger because the distance S-O2 has diminished. At the same time the distance O1-S has increased which caused the field magnitude at O1 to be much smaller. At first glance it may seem obvious that the effects at O1 and O2 are functions of their distances from the source S . Until you consider that the field disturbances arriving at O1 and O2 were emitted at the time when the electron was at the origin. Yet, the fields have now values conforming to the present position of the electron. One is then faced with the conclusion that the present position of the electron was somehow anticipated by the destinations O1 and O2 . The scientific literature on this subject relies heavily on the acceptance of the "anticipatory" behaviour of effects surrounding moving charges. Yet, the concept of anticipation cannot withstand much scientific scrutiny and we will therefore reject it in favor of the more appealing concept of the field disturbances carrying along the information about their causes. The direction and magnitude of acceleration of the source at emission time are sufficient information from which its distant effects can be inferred. Furthermore, this concept allows for non uniform motion, where changes of velocity are captured as a temporal sequence of disturbances. On the other hand, the concept of anticipation breaks down completely for non uniform motion of a charge.

Imagine now that another electron is positioned at the location O2 and that neutralizing positive charges are arranged so that the mutual net force is zero. Such neutral states are commonly found in currentless wires. As the acceleration disturbance reaches the electron at O2 it tilts the balance of forces in such a way that the electron is accelerated to exactly the velocity of the primary electron. For the time being the reader is urged to accept this as a fact.

What about an electron at O1 ? Assume again that neutralizing charges are placed so that no net force acts on that electron initially which means that the force pushing to the right is equal to the force pushing to the left. As the disturbance from S reaches the electron at O1 the pushing field is reduced and as a result the electron accelerates to the right. Now the acceleration on this electron is not exactly identical to that experienced by the primary electron but is somewhat larger. There is no mystery involved in such behaviour but the proof is not elementary. Therefore we will postpone it for later when we have absorbed more insight into the processes within a wire. You may at this point repeat the sequence of animations or just stop it by clicking over the "quit" button.

Before we continue, we owe it to the reader to justify at least some of the claims presented. For one we must defend the rejection of "anticipation" in favor of information transfer via the field disturbances. We offer one possible speculation inviting a more satisfactory answer in the Thinking aid . The reader will find there some qualitative proof to the claim that the "pushed" and "pulled" electrons do not move at the same velocity and that the acceleration tends to be reproduced from electron to electron.

This much background in first principles which govern the electric force fields of accelerated electrons is sufficient for us to address the interactions of multiple electrons. If you have not yet done so remove the animation window by clicking over the "quit" button. Then proceed with the section Interaction of Multiple Electrons .