Mathematics of the Pure Charge Theory
Keywords
space, charge, electron, quark, superimposition, energy, energy density field, electric field, magnetic field, field interaction equations, stasis, stasis theory, acceleration, acceleration of a charge, kinetic energy, atoms, mass, inertial forces, inertia, Stern-Gerlach experiment
Around the end of the nineteenth century and beginning of the twentieth century some notable scientists made attempts to understand matter as a purely electromagnetic phenomena, but their understanding of the laws of electricity and magnetism broke down at atomic scales and the venture was doomed to failure. By re-thinking those laws, it has been possible to complete this work. This web-page takes us through the mathematics of the complete theory of pure charges and is of necessity rather long.
The Pure Charge model is dependent on a new understanding of the nature of the universe. This is not simply a neat package which sits alongside our current understanding of the nature of matter and of "Space Time", but the core of an alternative way of explaining natural phenomena. So before we can get into the detailed mathematics, we need to deepen our understanding of the nature of space and the way in which electric and magnetic fields behave.
Contents
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The nature of space
Empty space is the starting point for understanding the nature of the universe. It has physical properties which allow particles of mater to exist in it and to interact with one another. Our current understanding of matter as the real substance from which the universe is constructed is wrong. Rutherford discovered that atoms are not solid, but are composed of very very small electrons moving about a very small nucleus. Modern physics has shown, first, the nucleus to be composed of protons and neutrons and second, the protons and neutrons to be composed of quarks. I take this one stage further showing that electrons and quarks are simply bundles of electromagnetic energy. Space can be polarised, and these particles are no more than a particular polarisation of space with distinguishing geometrical properties. Mater then looses its final claim to solidness in the theories which I have developed. An interesting consequence of this is that these theories are not incompatible with the idea that the universe is nothing more than a giant mathematical model. Matter does not have any real existence. It is only the information about its existence which is real. Space has properties which allow the information to be written on it and then enables that information to interact. This understanding is metaphysically open ended compared with the way in which the solid deterministic universe of the Age of Enlightenment tended to crowd God out of the picture.
Light from stars billions of light years away is able to travel through empty space and reach our eyes. The light travels in bundles of energy existing in the form of interacting electric and magnetic fields. Empty space has properties which allow these electric and magnetic fields to exist and to interact. The primary property of space is its ability to be electrically polarised. Empty space consists of positiveness and negativeness which can be stretched apart. Matter is built up from discrete electric charges. An electric charge may be formed by polarising space towards a point. In this way we can form any one of the three constituent types of charge from which matter is built.
The presence of matter within the universe gives to every point in space a property which I call stasis. This is in effect an absolute zero velocity for each point in space. This is not a constant, but a weighted average of the velocities of all the particles in the universe. This concept is rooted in my new understanding of electromagnetism.
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Superimposition
An electric current in a wire is able to generate a magnetic field extending beyond the wire because the electric fields of the electrons and the quarks of the wire all have a separate existence. If they simply added together, the effect of the movement of the electrons within the wire could not be experienced beyond it because their electric fields would add with the net positive fields of the protons to give a zero field outside the wire.
We will see in the Pure Charge Model how an electron like charge can be created from empty space. While it is not yet proven that electrons are simply a polarisation of space, such a model mimics the electric properties of an electron, so I will use the word electron to describe a pure charge of equivalent charge and mass. Imagine we have one electron in an empty universe. When we create a second electron, the polarisation field of each electron is unaffected by the other. Each polarisation field has a separate existence and they coexist in the same space without distorting each other. The layer of charge which forms the inner surface of the polarisation field of each electron experiences a force exerted on it by the polarisation field of the other electron. We no longer have to ask how an electron can exert a force at a distance because the electron consists of its polarisation field and the charge which is manifested on its inner surface. The electron is infinite in size and its presence fills the whole universe.
This is true for every charge in the universe. Each charge is a polarisation field extending to infinity and has an inner surface of charge. The region of that inner surface of charge is permeated by the presence of the polarisation field of every other charge in the universe and each exerts a force on that charge. These forces add by the natural process of vector addition.
- Electric fields do not add up! They coexist.
- Electric fields coexist and the forces which they exert add up.
- Each electric field is superimposed upon the others. They coexist in the same space without distorting each other. (This is not exactly true because the energy in their electric energy density polarisation fields has to be braced against the fabric of space. As charges are brought together, a very small amount of energy is drained from their fields in gravitational effects.)
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Energy
It is important to realise that energy is the most fundamental constituent of the universe. The most fundamental laws are those dealing with the interaction of the various forms of energy. There are four forms of energy:
- Energy in the form of an electric energy density field.
- Energy in the form of a magnetic energy density field.
- Potential energy stored by virtue of relative position.
- Mechanical energy stored in the fabric of space as it braces the internal tensions of electric and magnetic energy density fields.
This is at once a most surprising list because of the omission of kinetic energy and any reference to the equivalence of energy and mass. Why is heat missing? The answer is that all other forms of energy are manifestations of the four fundamental forms listed above.
Energy is conserved. It can never be created or dissipated. It can only be transformed from one form to another by the process of a mechanical force acting through a distance.
Electric and magnetic fields consist of energy and it is necessary to treat them as such if we are to understand their behaviour. The concepts of electric field strength and magnetic induction are useful, but they do not represent the fundamental nature of their fields. To make it clear to the reader that they must abandon their original understanding, I will use the phrase "energy density field" instead of the word "field".
Moving electric energy density fields generate magnetic energy density fields and moving magnetic energy density fields generate electric energy density fields. But the nature of these interactions has been misunderstood. This is not surprising when we remember that the theory currently taught was worked out well before the discovery of the electron.
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Old fields and new fields
The historically accepted descriptions of electric and magnetic fields are misleading in that these descriptors are not fundamental in nature, but are in fact mathematical artefacts.
We have described electric fields in terms of the two vectors 
and 
. The electric field strength can be found by measuring the force exerted on a test charge. This would be fine if the process of exerting the force were a simple one, but it is not. An electron wondering past a metal object cast adrift in deep space is passing through space which is permeated by the polarisation fields of all the constituent charges of the atoms of the metal object. Since the energy density of a charge's polarisation field is inversely proportional to the forth power of the distance from that charge, we need only take into account the polarisation fields of the charges of the nearby metal object and may ignore the polarisation fields of the rest of the universe. If the metal object is electrically charged, it will have either a surplus or a deficit of electrons in its surface region. This means that there will be an imbalance between the polarisation fields of the negative and the positive charges. Each polarisation field exerts a force on the electron and these forces are unbalanced causing a net force.
Every electric energy density field exists in its own right and they are superimposed on one another without distorting each other. The vector is a good descriptor of what is happening within a single polarisation field. It represents both the geometric process of polarisation and the ability of the polarisation field to exert a force on any charge within it. There is a fundamental relationship between and the energy density of the polarisation field.
Where 
is a property of empty space determining the ease with which a polarisation field may be created.
The polarisation field constituting the extended presence of a charge is an electric energy density field. Since energy density fields have a directional property, it is convenient to describe them by a vector. The electric energy density vector.
Magnetic fields are more complex and it is even more imperative that we realise that the historic descriptors 
and 
do not describe their fundamental nature.
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Magnetic fields
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The field interaction equations
Moving magnetic energy density fields generate electric energy density fields and moving electric energy density fields generate magnetic energy density fields. These processes are described by the two field interaction equations which are the equivalent of the laws of Biot-Savart and Faraday, but are expressed in terms of energy density.
These equations contain two completely new concepts, the vocacity, 
(Greek letter upsilon) the and vector Kross product 
. The vector represents the fundamental property of motion proportional to the Kinetic Energy per unit of mass. I have named this vocasity. It is calculated from the velocity as 
and while the name comes from "V Over C All Squared + ity", it is also another word for speed in the English language. The vector operation ; (which I call the Kross product ) is similar to the cross product, but has the 
term replaced by 
and is defined as
(where 
is a unit vector perpendicular to the plane containing 
and 
and in the sense that a right hand thread would advance if rotated from to ).
These field interaction equations are the fundamental laws of electromagnetism. A fundamental law corresponds to an actual primary mechanism of nature. It cannot be explained or broken down further. On the other hand, we discover further laws empirically which can be mathematically derived from the fundamental laws. An example would be Newton's laws of motion which can be derived from the field interaction equations. On the other hand, the field interaction equations cannot be derived from Newton's laws of motion.
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When to use which equations
Application of the field interaction equations is complicated by the principle of superimposition. The field interaction equation is always valid because magnetic fields are not subject to superimposition. Only one magnetic field can be present in a region. In any situation where a single electric field is dominant, the field interaction equation is valid, but in the case of the magnetic field produced by an electric current in a circuit, we have a very large number of electric fields present and we have to sum the effect of them all and the equation 
should be used instead. Fortunately this equation is equivalent to the definition of magnetic induction derived from the Biot-Savart law. The interesting thing to note is that the velocities vi are measured relative to the wire. We could measure them relative to any reference frame, but we would then need to include every charge in the wire with the result that components of B, due to the velocity of wire, among the conduction band electrons would be exactly cancelled by the movement of the positive charge of the crystal lattice of the wire.
Faraday's law of induction provides a good model for calculating the induced emf in an electric circuit, but it does not describe what is actually happening because there are many situations in which the lines of flux do not actually cut the wires of the circuit. On a fundamental level, we must consider the movement of magnetic energy density flux into and out of the surface of each charge. Such a calculation is impossible because of the number of charges involved. In the situation where we want to calculate the force generated by the movement of energy density flux into or out of the surface of a charge, we can use Faraday's law as a short cut provided that we have calculated the velocity from the changes in the energy content of the magnetic field.
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Stasis theory
There is in fact a local absolute stationary-ness called stasis against which magnetic fields are generated by the movement of electric fields. This comes from the symmetry of the situation in which every charge interacts with every other charge. The magnetic intensity 
is generated by the relative velocity of the charge to each of the other charges in the universe. What happens is that the relative velocities average out so that the magnitude of the magnetic intensity is independent of the number of charges in the universe. Each charge has an effect proportional to the strength of its electric polarisation field. So we have to calculate a weighted average of the vi according to the magnitudes of the Ei. An observer may calculate the stasis vector
which is the velocity of stasis relative to their frame of reference. Fortunately, when we look closely at the problem of performing this calculation for the universe, we find that is rather similar to calculating the gravitational field in that both are dominated by the inverse square law. On earth, gravity pulls us down to the ground, so stasis equates to the earth's velocity. For a space probe on transit between the planets, stasis equates to the velocity of the solar system.
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The Pure Charge
The electric field surrounding a single charge is
We can calculate the total energy outside a spherical region of radius a with the same centre as our charge. For a small element of volume
And we can integrate
Where 
When an electric charge is in motion relative to stasis, a magnetic field is generated. This is called its field of motion and accounts for its kinetic energy and inertia. We can calculate the intensity of this magnetic energy density field. The magnetic field surrounding a single charge q moving at velocity v is
This result could alternatively be derived by use of the field interaction equation to transform 
into 
.
We know that 
so we substitute and cancel to give
The energy content of this field of motion can be calculated by integration.
We note that the energy content of the electric field depends both on the charge and on its radius. If we substitute the charge on the electron and its Bohr radius, the result is half the energy equivalence of its mass. We might at this stage suggest that we can model electrons, up quarks and down quarks simply by varying the charge and radius of the pure charge.
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Acceleration of a Pure Charge
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Kinetic energy
We now need to look at the properties of kinetic energy. The kinetic energy of a pure charge is the energy contained in the magnetic field generated by its motion. This magnetic field is absolute, it does not change in size and orientation to take into account an observers relative motion to the charge. It is a function of the absolute velocity of the charge, not a function of our attempts to observe the charge. However, within a local situation, observed relative to the motion of the locality, we will expect the work done in accelerating the charge to equal its gain in kinetic energy. The important thing to realise is that we are looking at mathematical properties of Newtonian mechanics, not at mysterious electromagnetic phenomena requiring complex integrations.
Consider a number of particles whose velocities in the centre of mass reference frame are vi and whose total mass is M. In this reference frame, the total kinetic energy is
The momentum sum of the Momentums is zero in the C of M frame.
In some other frame of reference, the velocity of the C of M of these particles is 
and the total kinetic energy is
But the vector dot product is distributive across addition and scalar multiplication is associative, so 
and we know that so
This is a very important result. Suppose we can can construct an atom like structure from pure charges. Each of the pure charges will be surrounded by a magnetic field generated by its motion. This gives the pure charge the property of inertia which we can for convenience call its inertial mass. This psudo-atom will have a centre of mass and relative to that we can calculate 
the sum of the kinetic energies of its constituent pure charges. If the psudo-atom were at absolute rest (relative to stasis), then would be equal to the sum of the energy in the magnetic fields of motion of the charges. If the psudo-atom is in motion, then we can calculate its kinetic energy 
as if it were a single particle. This motion of the psudo-atom alters the absolute velocity of every charge within the psudo-atom and the total energy stored in the magnetic fields of motion is now 
. According to the result we have just derived, the
kinetic energy of the psudo-atom is stored in the increased energy of the magnetic fields of motion of the pure charges it is formed from.
We can carry this analysis outwards and construct a crystal from psudo-atoms each of which is in thermal motion. If the crystal is at absolute rest, when we find the sum of the kinetic energies of all its constituent charges: lets call it 
then 
will contain the thermal energy. If the crystal is in motion, then its kinetic energy will be
Thus the magnetic fields generated by the motion of the pure charges forming the psudo-atoms of our crystal are able to store the thermal energy of the atoms and the kinetic energy of the whole crystal.
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Real atoms
We have seen that we can account for inertial mass and kinetic energy in a model of the universe constructed from pure charges. The question to be answered is whether or not such mechanisms can equally well explain these properties of the real universe. Real electrons are said to have magnetic moments. The magnetic fields associated with these magnetic moments are very strong. Their shape at a scale relative to the size of the electron is undetermined and even if it were known, it is very unlikely that it would yield to integration. We are thus unable to carry out any of the detailed mathematical analysis which we were able to carry out with pure charges.
We can account for the much greater mass of the neutrons and protons by the much smaller size of the two types of quark from which they are built. The particle formed from the three quarks acquires its mass from the sum of their masses and its size from their motion.
These then are the problems to be encountered in trying to account for the inertial mass of real particles in terms of electromagnetic theory. By far the most insurmountable is the supposed size of the magnetic field of the electron. I am forced to the conclusion that we are going to have to throw away some of the holly grails of modern physics.
Modern physics describes the electron as having zero size, a charge of 1.6×10-19 coulombs, a mass of 9.1×10-31 kg, an angular momentum of 3.3×10-34 Js, and a magnetic moment of 9.3×10-24 Am2 . It sends out quantum waves to advertise its presence to all the other charged particles in the universe and it detects the quantum waves which they have sent out. It senses the quantum waves sent by all the other charged particles in the universe and replies to each with a second type of quantum wave which travels backwards in time. When it receives each reply, it contains information about the size and the distance of the other charge. The electron then calculates the energy content of a virtual photon which it sends to the other charge, which adsorbs it and then sends back an identical virtual photon. Somehow these virtual photons are able to know whether they are to transmit an attractive or a repulsive force to the other charge.
There are some glaring problems with this picture. If electrons really could send quantum waves back in time, then every physicist worth their salt would be secretly working on producing a machine to send information back through time in order to win the national lottery. Here is the list of problems:
- The zero size would require an infinite energy to be stored in the electrostatic and magnetic fields surrounding the electron.
- The angular momentum of a point mass would be zero.
- Assuming that we adopt the classical radius, the magnetic moment is still too big by a factor of 137 to be generated by the rotation of the charge around the equator of the electron, but if the electron is a point mass, its charge would need to rotate with infinite angular velocity to posses such a magnetic moment.
- The quantum mechanical explanation of the transmission of force between charges is not a mechanism capable of doing the job, but a complex procedure in need of vastly complicated mechanism to execute it. It all seems just too far fetched, yet humans have a love for believing in the incredible. Cricket commentators believe fervently that the ball will turn when the sun is hidden by a passing cloud and academics will defend anything too hard to understand.
Evidence of the magnitude of the magnetic moment comes from two sources, the hyperfine structure of atomic spectra and the Stern-Gerlach experiment. Both of these produce results remarkably similar to the magnetic moment of an orbiting electron. In explaining the hyperfine structure, the magnetic moment of the electron is said to interact with the magnetic moment of the nucleus. I am inclined to think that if there is a magnetic moment, it interacts with its own orbital magnetic field. In the Stern-Gerlach experiment, atoms move through a strong magnetic field varying in strength perpendicular to their line of motion. This exerts a force on a magnetic dipole or current loop and diverts the beam of atoms. It is found that the beam is split into two beams. The argument is that if the orbital magnetic fields were involved, they would not align with the powerful magnetic field of the apparatus, but would precess giving a different force on each atom and spreading the beam in a fan rather than splitting it in two. I am not convinced that this orbital precession would occur. There is no mention of orbital precession in the theory of ferro magnetism and I rather fancy that if it were, the manufactures of magnets would all be rather proud of the fact that they could make magnets which are stronger than is theoretically possible by a factor of 
2. I therefor believe that both of these pieces of experimental evidence are in fact measuring an orbital magnetic field.
All of this demands quite a change in our understanding of the workings of the atom. My belief is that quantum theory has so far given good results because it is homomorphic with a statistical treatment of the chaotic motion the electrons. Our only firm evidence for the actual behaviour of electrons within an atom is the structure of their atomic spectra. Atoms are actually quite reluctant to emit photons of light. About 99% of photons are emitted in response to the stimulation of a passing photon which is why light is coherent and we can produce interference patterns. In terms of the number of orbits an electron must make before emitting a photon and decaying into a lower orbit, the emitting of photons is quite improbable, it is just that we observe things on such a show time scale and observe such a vast number of atoms at a time that we see a few isolated improbable events as a large outpouring of light. From this point of view, it is easy to see where Bohr went wrong. He thought he saw evidence for the fact that only certain orbits are allowed. I would suggest that the reverse is the case. These are orbits which are not allowed. They are not allowed because they allow the electron to radiate electromagnetic energy and so to decay to a lower orbit from which they are quickly perturbed by their interaction with other electrons.
One of the problems facing early theoreticians was to explain why orbiting electrons did not rapidly radiate electromagnetic radiation, loose energy and decay into the nucleus of their atom. My explanation for this is that magnetic fields are formed and decay on a much slower time scale than that on which electrons orbit atoms. If we look at the magnetic field surrounding an electron on a scale commensurate with the size of the electron, then we see its field of motion formed in perfect circles symmetrically surrounding its line of motion. But if we look at on the scale of the atom, we find the magnetic intensity varying wildly at any point and changing at far too great a rate for the magnetic induction to follow it. Remember, and are very different animals. is a mathematical artefact describing the sum of the effects of the moving polarisation fields of all the charges at a point. 
. The magnetic induction on the other hand is generated throughout a region in such a way that it forms continuous loops which are free of any divergence. On the scale of the atom, the orbiting electron creates the kind of magnetic field we would expect a current loop to generate. On the scale of the atom, we do not see the particle like behaviour of its magnetic field. But what of the electron, it is moving through the magnetic field which it has created. It could be that its own magnetic moment interacts with this affecting energy levels of the disallowed orbits resulting in the hyperfine splitting of its spectral lines. This would only require the electron to have a magnetic moment which is 137 times smaller than the value currently quoted. All this is rather speculative.
To develop such new ideas might take generations, so I will make no apology for leaving the theory incomplete, what I wish to do is to demonstrate that there might be other ways of understanding the behaviour of electrons within atoms which do not demand us to give the electron such ridiculous and contradictory properties.
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Mass and inertia and forces
We have seen how matter constructed from pure charges will exhibit inertial properties and obey Newtons laws of motion. We have worked out all the equations and seen how the movement of the electric polarization fields of the charges generate magnetic fields surrounding the charge and that these contain an energy proportional to the square of the velocity. We have seen that work must be done to store energy in these magnetic fields and that an inertial force is generated against which that work is done. We have modified the laws of electromagnetism so that we get the correct force for the rate of increase in the energy stored in the magnetic field. We might even be happy with the idea that these inertial effects are the essence of what we call mass. We found that the kinetic energy of a moving crystal was stored in the energy of the magnetic fields generated by the motions of all the charges. All this would be true in a universe where the charges were held together by God's will.
How can we relate this to our real universe. The existence of magnetic moments of electrons and (presumably) of quarks does not in any way invalidate the discoveries we have made so far. A part of the inertia of matter can be accounted for by the magnetic fields generated by the motion of the charged particles from which it is made. The proportion is entirely determined by their actual size. One thing is certain, electrons would have infinite mass if they were the size of a point. Let us at this point put forward the proposal that the magnetic fields produced by the magnetic moments of electrons and quarks generate electric fields and that the storing of energy in these electric fields produces an inertial effect such that the two inertial effects, magnetic and electric, wholly account for the inertial mass of matter.
I am convinced that there is no such thing as mass in the accepted sense. I am convinced that matter is composed only of electric and magnetic fields and their effects. Because of the nature of electromagnetism, forces are generated opposing acceleration giving rise to the property of inertia.
One of the greatest achievements of the theories discussed in this work is to describe the way in which forces are able to act at a distance. We no longer need the virtual photon theory of quantum mechanics which sounds very good on paper, but falls many orders of magnitude short in predicting the size of the force between two charges. By accepting the principle of superimposition, force becomes something a charge experiences because of a property of space in its vicinity. Each charge is, from a non-magnetic point of view simply a spherically polarisation of space towards a point. The polarization extends outwards to infinity falling off in intensity according to the inverse square law as determined by the geometry of the situation. These polarisation fields coexist with each other in space, each existing in its own right throughout all space without being distorted by the presence of other charges or their polarisation fields. Thus space in the vicinity of a charge is pervaded by a milliard of these polarisation forces each causing it to feel a force acting on it. In the act of acting together on the same charge, these forces are combined to form a resultant force acting on the charge. Because forces have magnitude and direction, they are vectors and the process may be described mathematically by vector addition.
Electric forces are able to do work as they cause charges to accelerate. The principle law of physics is the law of conservation of energy. A force which does work must obey Newton's third law of motion if the law of conservation of energy is not to be violated. When we consider the force between two charges, we find that each polarises space in the region of the other. The polarisation is proportional to the magnitude of the charge divided by the square of the distance between them. The force experienced is proportional to the magnitude of the charge experiencing and therefore proportional to the product of the magnitudes of the two charges. Therefore the forces experienced by the two charges are equal and opposite. When we consider all the interactions between all the charges in the universe, the fact that the components of the forces come in equal and opposite pairs insures that for any system in which we can identify components and the forces acting on them, Newton's third law is obeyed.
Magnetic forces are experienced by a charge by virtue of its motion through a magnetic field. Magnetic fields are generated by the motion of charges through the background presence of polarisation fields. The velocity of a magnetic field is the velocity of its geometry. The force on a charge moving relative to a magnetic field is perpendicular to its velocity and does not change the magnitude of its velocity. It therefore does no work and is not required to obey Newton's third law. When charges move in some organised way as for instance in an electric circuit, the geometry of the situation ensures that when we add up all the individual magnetic forces on all the charges of the electric currents involved, we find that Newtons's third law is obeyed. When we have two current loops a few centimetres apart, they exert forces on each other which have the potential to pull them together or push them apart. They have the potential to do work and we should expect the forces to obey Newton's third law. We can in every instant calculate the forces by numerical integration and find them to be equal and opposite.
At the present time, I am not sure about Einstein's famous equation. Let us say that it expresses a principle, rather than an actual relationship applicable in every instance. The general principle is that mass is not something which exists in its own right, but is an inertial and a gravitational effect experienced by electric charges. Mass then is proportional to the total energy contained in the electric and magnetic fields associated with a charge. When an electron and a proton form a hydrogen atom, it is the sum of the magnetic intensities which produces the magnetic field of motion. Outside of the region of the atom, the components of the fields of motion will be of the same order of intensity, but of opposite direction due to the different signs of the charges. This results in the first order magnetic field of motion being less in total energy than the sum would be of the fields of two separate particles. Thus there is a reduction in the observed mass. The same effect is true for the much greater changes in mass associated with the nucleus.
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© Copyright Bruce Harvey 1997.