# Charges and Electric Energy Density Fields

### Keywords

electron, Feynman, permittivity, polarization of space, superimposition, polarization field, polarisation of space, polarisation field, potential difference, electric field strength, mathematical artefact

## From Coulomb to Harvey

Electric fields were first investigated by Coulomb in the days when cats fur, pith balls and ebony rods were the essential tools in his study of static electricity. By hanging his pith balls on threads and charging them, he was able to work to a remarkable degree of accuracy and was able to define a unit of charge in terms of the force it exerted on another charge. We no longer use his "electrostatic units" but with the insertion of a constant Coulomb's law lies at the heart of our understanding of nature. Coulomb observed that if a number of charged balls were involved, the force on one charge would be equal to the sum of the forces exerted by each of the other charges. We still regard this principle as being of the greatest fundamental importance.

To understand just how the charges exerted forces on one another, Faraday argued that a similar process to that which he had described for magnetism must exist. Thus there must be some medium carrying the lines of force which formed an electric field. Ever since people have wondered about the nature of the electric field and the nature of its transmission from place to place. In more recent times, Richard Feynman drew a nice little picture in which the interaction of two electrons was accomplished by the exchange of a photon. This let the quantum physicists loose and we now have an explanation in which the individual charged particles each have a quantum field which explores space for the presence of other charges enabling them to exchange photons thus transmitting the force.

After much thought, my own understanding is that each elementary charged particle has its own unique electric field extending to infinity-ishness and that in the absence of motion, these electric fields are able to coexist in space without distorting or interfering with each other in any way. (When there is relative motion between these electric fields a magnetic field is generated as a direct result.)

## Creating an electric charge

We know that an electron can be created when two gammer rays meet a neutrino. It is not impossible to imagine a time when God, or chance random fluctuation created the very first electron. It seems reasonable to suppose that the electrons we are familiar with evolved through a number of prototype stages. I imagine that the first attempts to create finite charged particles were dogged by the problem that they immediately got lost in the infinite emptiness of space. Finite particles are next to useless, some way of knowing where they are is needed. The answer is to give space a nature from which we can create the particle.

We know that empty space has physical properties. We normally say that space has a permittivity and a permeability . The fact that empty space has a permittivity implies that it has an electric nature consisting of both electric positiveness and electric negativeness which coexist. In the presence of an electric field, the two are pulled apart. Radio waves are able to travel through space from a distant probe because real electric currents flow as the two layers of positiveness and negativeness slip through each other. The degree of their movement is limited and a strong force is created attempting to restore their former relative positions. Energy is stored in their pulled-apart-ness. An electric field has been created.

Let us think how an elementary charged particle might be made using this electrical property of space. The secret is to realise that we do not have to enquire too deeply into the workings of the particle. We define a point in space at the centre of the particle and imagine a spherical shell centred on that point and larger than the charge. As the charge is created, we notice that space is polarised and a quantity of charge flows through our surface. As an electron is created, -1.6 x 10-19 C flows into this spherical region. I think it quite meaningless to ask if it might be that +1.6 x 10-19 flowed out or that +8 x 10-20 flowed out as -8 x 10-20 flowed in. The strength of the electric field at the surface of our sphere is related to the area of the sphere, the amount of charge and the ability of space to resist being polarised. What is happening as the charge is being created is that space is being pulled apart, its positiveness and its negativeness being displaced relative to each other. This is fiercely resisted and work has to be done thus storing energy in the fabric of space. The result is an energy density field described by the equation. Which we can expand as: The strength of the electric field associated with the polarisation of space is: It is this energy density field which is the presence of the charge throughout all space. The distribution of this energy density outside our spherical surface is symmetric and well behaved, so we can perform a volume integral and determine the total amount of energy outside of the surface.   We notice that the energy outside of our spherical region is inversely proportional to its radius. This poses an interesting question. How small can we make our spherical region? If Einstein is right and then we are surely limited by the electrons mass. My Casio calculator gives me a value for the radius of the electron which equates nicely to an energy equivalent to half of the mass of the electron. Wherever it does actually terminate, the inner limit of the polarisation field forms a raw edge of charge. This is the electrons charge.

I am not in a position to have reached a definite opinion on the exact nature of the electron. The big question is how do we hold its surrounding energy density field in existence. One possibility is that the electron is a kind of superconducting generator in which internal movements of its charge generates a magnetic field which in turn keeps the charge together. Another is that some entity hooks the electric polarisation field to some other structural property of space, but these are matters which we do not need to go into to understand the nature of the electric forces which hold matter together. Whatever the exact nature of the electron, we will find an inner surface to the polarization of space which we can call the charge's surface.

What I am certain of is that we can account for the inertial and gravitational properties of our electron purely in terms of the electric and magnetic properties of space. The creation of the spherically symmetric electric polarisation energy density field is sufficient to explain both its electric field and the presence of an electric charge. I reject the idea that the electron is a thing with mass and negative charge which influences surrounding space creating an electric field. The main problem I have with this picture is that the creation of an electron within my spherical region will result in an equal an opposite amount of charge flowing into the region as the polarization field is formed. If I consider a region into which an existing electron drifts, the movement of Maxwell's displacement current through the boundary of my region plus the charge itself is equal to zero. To me that seems a nonsense.

## Creating more electric charges

It is my contention that once we have created a charge, the energy stored in its electric polarisation energy density field gives it a real existence of its own allowing it to move freely through space. It was created from space, but now it exists in space. It has a separate identity and we can at a philosophical level, if not an experimental level distinguish between the polarisation field of the charge and space. This is important because it means that we are free to create as many charged particles as we like from space. Each will then have its own existence extending throughout all space. The polarisation fields of the separate charges will coexist in space superimposed on one another. This is the Principle of Superimposion.

It is impossible to think directly about space and time and the existence of matter and the laws of nature; we need to put together a whole set of analogies which lead us to the internalisation of the concepts. I like to think that space is like the beaker of two liquids used in the production of polymer fibres. The polymer is produced at the surface where the two liquids meet. There is no limit to the amount of fibre which can be produced and once created the fibres each exist quite separately. This meeting of the two liquids is like the interface between the positiveness and the negativeness of space. We can polarise space to form a charge and because we have to do work to stretch the two apart, the energy density field which is created exists in its own right and the positive and negative layered-ness of space remains waiting for the next electric energy density field to be created.

## The addition of electric fields

Let us now go back to the original problem of explaining how several charges interact. When a second charge is created, each consists of a polarization field extending throughout all space from a finite surface of charge which is the raw edge of the polarization field. The surface of each charge is pervaded by the polarisation field of the other and the infinitesimal elements of charge within the surface feel the internal tension of the other charge's polarisation field. This is similar to the force on a small metal sphere immersed in an insulating liquid which is subject to a strong electric field. But we must not take the similarity too far because the liquid is corpuscular in nature and its molecules present charged ends to the metal sphere. Polarised space on the other hand pervades the electron and the result is that the two phenomena exert forces in the opposite direction. Let us say that the polarisation is such that positive has moved to the right. The metal sphere finds molecules on its left side presenting positive ends. If the metal sphere is negatively charged, the force will be to the left. An element of the negative charge of the electron's surface sits in space and the polarisation moves positiveness to its right and negativeness to its left. The force then is to the right. Of all the new ideas that you will meet in these pages, this is the hardest concept to grasp.

When we create the next billion billion billion charges, the basic picture remains the same. Each element of charge in the surface of an electron is pervaded by the polarization fields of all the other charges. Each exerts a force and the forces combine by simple vector addition. The total, the absolute, the overwhelming beauty of this is its wonderful simplicity.

Of all existing theories, this is the one which un-raps the onion to reveal the simplest possible of mechanisms.

## Of batteries and capacitors

There are two possible ways of explaining what happens when we connect an air filled parallel plate capacitor to a battery. The older theory which predates the discovery of electrons says that the wires connect the emf of the battery to the capacitor plates which causes the space in between the plates to be polarised. This polarisation then transmits the electrical force from one plate to the other causing a positive charge to appear on the positive plate and a negative charge to appear on the negative plate. With the discovery of electrons, the theory was modified by substituting the phrases "surplus of electrons" for "negative charge" and "deficit of electrons" for "positive charge".

The other way of looking at it would be to see the electrons in the conduction band of the metal of the wires as behaving a little like a gas. In the battery, the chemical processes of the battery robs the positive plates of electrons and produce a surplus of electrons on the negative plate. Within the metal conductor, there must be an overall balance of charge, so that the electron gas behaves a bit like an incompressible fluid and the result is a surplus of electrons on the negative plate of the capacitor and a deficit of electrons on the positive plate of capacitor. Energy is transferred through the system by the simple process of force acting through a distance as the electrons in the conduction band repel one another. At the surface of the capacitor plate, the electrons are no longer within the metal lattice and the attractions of the positive atomic nuclei are no longer balanced. Work is done again the attraction of the lattice. The same happens at the positive plate of the capacitor, but in reverse, electrons in the conduction band are pulled away from the surface layer of atoms so that the excess of fixed positive charge is no longer balanced by the presence of conduction band electrons. This results in a net attraction against which work must be done pulling the surface electrons into the body of the conductor. (This somewhat simple picture ignores the fact that the conduction band electrons are in violent thermal motion and that the presence of each electron on the surface is just a temporary presence. The surface population is constantly changing and the process is one of dynamic equilibrium.)

It would seem then that as the capacitor is charged, the work is being done in the region of the surface layers of atoms. But our understanding of dielectric theory says that if we place a layer of say glass between the positive and negative plates, the capacitor is able to store more energy. This is because the electron shells of the glass atoms are distorted. If we look at the surface of the negative capacitor plate we find that distortion of the electron shells of the adjacent glass atoms results in them producing a net force of attraction helping to pull the conduction band electrons onto the surface of the metal. If we could put a test charge in the space between the glass atoms, we would find that the electric field would be less than it was when the space was filled with air. The amazing thing about the dielectric is that it is its weakness which enables the capacitor to store more energy. The more easily the atoms can be distorted, the more help it will give to electron movement onto the surface of the negative capacitor plate and away from the surface of the positive plate. This is the opposite to our experience of mechanical systems. If we want to increase the energy a clockwork motor can store then we make its spring stiffer. The dielectric is able to function in this way because it is corpuscular in nature. (I use the word corpuscular to include atoms, molecules and possibly assemblies of molecules) Electric charge distributions within each corpuscle are altered creating an external electric field (which is in the opposite direction to the field which is causing the polarisation). An electron is influenced by the fields of nearby corpuscles. This differs from the polarization of space in that space is continuous and pervades everything.

It seems reasonable to suppose that polarising glass requires work to be done and that energy is therefore stored in the glass. But does this imply that energy is stored in space in the absence of the glass. As the capacitor is charged and more electrons migrate onto the surface of the negative plate and away from the surface of the positive plate, do their electric fields combine. Doubling the charge results in a fourfold increase in the energy stored in the capacitor. When the charge is doubled, does the increase in energy result from an electric field between the plates with four times the energy density, or does it result from their being twice as many displaced electrons on the surface, which are now under the influence of twice the force from the lattice. There is no way of telling. We are in one of those situations where the two interpretations are mathematically linked. They are mathematically equivalent in that whatever reasoning process we go through in one theory is in one to one correspondence with the reasoning process we would go through in the other. We could consider moving the plate closer together in the hope that the two theories would produce different results, but this is not the case.

lf to the question of what happens when we move the capacitor plates closer together, I found myself in difficulty because I still had the classical concepts of electric field and potential difference firmly embedded in my mind. If we are to explain electric fields in terms of the combined effect of the distribution of electrons, we need to understand that the electric field in a conductor is not zero. This is because conduction band electrons are the fastest thing on earth with velocities in the region of 100,000 metres per second. The atoms of a metal are in a state of violent thermal agitation and being more massive than the electrons, every time an electron and a vibrating atom have a head on collision, the effect is the same as hitting a tennis ball with a base ball bat. The velocity of the bat is only slight reduced, but the ball rebounds with a speed almost equal to its speed before the impact plus twice the velocity of the bat. Energy is transferred from the bat to the ball. Only when the oncoming ball has the same kinetic energy as the bat is equilibrium reached.

The atomic lattice spacing of copper is 1.28 x 10-10 metres and a conduction band electron can cover this distance in less than 2 x 10-15 seconds. If we look into a wire of an electric circuit at these scales of time and distance, we will find electric fields of a few volts per metre in operation. Far from being a static situation in which the electric field is zero, this is a dynamic situation in which electrons are all the time being accelerated and decelerated by ever changing local electric fields. We define voltage, the measure of potential difference as the line integral of electric field strength along the lines of force of the field. I had thought of this as being a mathematical artefact, but the picture I have just painted of all this seething electron activity make potential a very real thing. If we connect a wire between the terminals of a battery (please don't actually do it), the conduction band electrons which are all the time doing work on each other and having work done on them simply have their behaviour modified a little giving them an average drift along the wire from the negative terminal to the positive terminal. At any instance, the potential difference of the battery is split between a milliard of discontinuities of the electron distribution as electrons which have hit atoms rebounded and are again accelerated in the direction of the drift. Every collision causes energy to be transferred generating heat. (Since an average car battery can deliver several hundred amps at about 6 volts, heating effects in six inches of copper wire can be explosive. Really good thick wire will explode the battery.)

When we connect a capacitor across the battery, electrons flow into the negative plate and out of the positive plate of the capacitor. Equilibrium is quickly reached. The voltage of the battery is a push-me-pull-you sort of a thing. In the gap between the capacitor plates, the electric field strength has two equal components, one from the electrons on the negative plate pulling and one from the excess of positive charge in the positive plate pushing from the opposite direction. When we look at the electrons in the surface layer of the lattice of the positive plate, we find that they are being pushed from behind and pulled from in front. The push from behind is determined by the battery voltage, but the pull from in front is determined by the reduction in the electron presence in the surface layers of the lattice of the positive plate. This situation results in the charge density on the plates being a function of the battery voltage and of the distance between the plates. But how?

## Potential difference

When we consider individual charges, they each have nice spherically symmetric electric fields and it is easy to see how the geometry of these fields might convey the properties of the field. When we come to the situation created by the geometry of a parallel plate capacitor, the situation changes. The electric field strength depends not on the distance between the plates, but on the charge on the plates. Imagine a capacitor made of two large flat surfaces which can be moved in and out allowing the gap between them to be varied from a tenth of a millimetre to say ten centimetres. With the plates 1mm apart, connect the plates to a 10 volt battery and charge them. Now connect an almost perfect digital voltmeter with a very high input impedance between the plates and disconnect the battery. As the hours pass, the capacitor will slowly discharge and we will see the recorded voltage dropping, but if we are quick, we can move the plates in and out and see that the voltage changes. At 0.1 mm apart, the voltage is reduced to 1 volt. Move the plates away from each other and the voltage increases. In theory it reaches 100 V at 1cm apart and 1000 V at 10 cm. Now, I must admit that I have never performed this experiment because there are a few practical difficulties like the cost of very high impedance voltmeter, but I do accept the theory. However we get the result, it posses a question. How do the plates know how far apart they are?

The problem is that when we consider the electric field close to a flat surface of a charge on that surface, the geometry of the situation produces an unexpected result. The electric field strength as we would measure with a test charge does not fall of with the inverse square law, but remains constant. There is a very simple reason for this which might be illustrated by imagining that we are in a large dark room with a torch. We shine the light on one wall and walk towards it. The patch of light on the wall becomes smaller and brighter. The brightness varies according to the inverse square law, but the area of the patch of light is proportional to the square of the distance. When we calculate how much light falls on the wall, the two cancel each other and the amount remains constant. Now consider a charged flat surface, the force form individual charges is subject to the inverse square law, but as we move further from the wall, the number of charges making a significant contribution to the force increases with the square of the distance and the two effects cancel.

Now we have said that the effect of the charge on the opposite plates of the capacitor is to help to pull electrons out of the negative plate and into the positive plate. But the geometry of the situation determines that force exerted by each surface charge on the other depends only on the charge and not on the distance between the plates because the field strength close to a flat surface is independent of the distance from it.

The electric field strength depends only on the charge density, but the voltage is a measure the work which needs to be done adding charge to the capacitor. Now if the charge had to be dragged between the capacitor plates removing electrons from the positive plate and moving them across the gap between the plates, the link between field strength, distance between the plates and potential would be obvious. But we do not do that. At the start, we bring the extra electrons in the back door by way of the capacitor's leads. Then when we vary the distance between the plates, there is no movement of charge. There is of cause a force between the plates and we do work against this force pulling the plates apart and we let the plates do work against their mounting as we bring them closer together. Where does the energy go to and come from? One easy answer is that it is stored in the space between the plates. When we translate that answer from its pre-electron-discovery form to its post-electron discovery form, it seems to be a good proof that the electric fields of individual charges combine rather than coexist together. But if they combine, then how does a current generate a magnetic field beyond its wire.

Within the wires, the movement of the conduction band electrons transports energy. In the static position where the capacitor is charged to a constant voltage, the electron gas in the wires of the circuit and the plates of the capacitor is all the time transporting energy to and fro. Work is being done and undone all the time. While the statistical time average is zero, the situation is very dynamic. But what is happening away from the electron gas in the conduction band of the wires and the capacitor plates: what is happening in the gap between the plates. The answer is that all of the electric fields of the individual charges are on the move. Each consists of a stretched apart-ness of the positiveness and negativeness of space and as the polarisation fields of two charges move through each other, equal and opposite amounts of work are done on each other. The dynamic situation of the conduction band electrons has its counterpart in the constant relative motion of the superimposed electric energy density fields. As we approach the surface of an individual electron, we find that the raw edge of its polarisation field which is its charge also participates in this dance. Thus there is a dynamic continuum of the doing to, and being done to, of work which conveys potential through the electric field.

That explanation needs repeating because it is again conceptually difficult. The space between the plates of the capacitor is not in a static condition: it is full of the moving polarisation fields of billions upon billions of electrons. The polarisation fields all share the same space and they exist within each other. Consider the action of just one polarisation field on another. The one being acted upon has both positiveness and negativeness. An element of the positiveness finds itself within the polarisation of the other and is pulled from one side and pushed from the other. An element of the negativeness likewise finds itself within the polarisation field of the other charge and it is pulled from one side and pushed from the other, but in the opposite direction. The push and the pull on the positiveness combine into a single force. If the polarisation fields are in relative motion, then that does work. The push and the pull on the negativeness combine into a single force and that does an opposite amount of work. Equal and opposite amounts of energy are continually being exchanged. There is a continuum of dancing energy and it is this which transmits the potential difference between the plates as surely as if we were moving charge between the two. This process terminates at the actual surface of the charge because we meet the raw edge of the polarisation field and there are no longer two balancing forces. Here the surface charge acts as an interface between the energy dance within the conductor and one between the plates. As the capacitor plates are moved apart, the electric field between them remains constant, but the distance over which this continual two way of exchange of energy takes place is increased. Thus the voltage we measure between the now disconnected leads of the capacitor increases.

## Of words and understanding

We happily teach electric theory to students at all levels and I do not suppose that more than one lecturer in a thousand seeks to address this problem or that more than one student in a million asks the question "How do the plates of a capacitor know how far apart they are?" The physicist would do well to cast a sceptical eye at the biological sciences to see just how often understanding is limited to naming things. The Latin words sound good, but an arts wallah could tell us that the wonderful sounding Latin translates into "thick brown layer". It is not so obvious in physics, we are better at formulating our explanations in ways which are not so transparent to passing Latin scholars.

In trying to put a new theory together, much time is spent asking every possible question and agonising over the answers. The more detailed the theory becomes, the more awkward questions it is possible to ask. But there are times when I wonder if the old theories actually address the awkward questions. To what extent do they just give names to things. How does potential equate to electric field in the absence of the movement of charge. Is the question answered merely by defining a mathematical relationship in the form of an integral.

As the capacitor is charged from the battery, the electric polarisation energy density field of each electron and of each quark remains unaltered. The only thing that changes is their positions. This change in positions creates a measurable effect on a test charge placed in the region of the electric field. The test charge experiences a separate force from every one of the superimposed electric energy density fields of the surrounding charges. These forces sum vectorially to produce a resultant force which is what we measure. Our mistake is to think that the resultant electric field, which is the sum of all the electric fields of the individual charges, has a real existence. It does not, it is only a mathematical artifact.