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Mathematics of the Pure Charge Theory

This is one section from the Mathematics of the Pure Charge Theory. It is located on this separate page to speed downloading times.

Magnetic fields

In a universe containing only two charges, the movement of one polarisation field relative to the other generates two magnetic fields associated with the two polarisation fields. What we see depends upon our sense of scale and wether the charges are attracting or repelling each other. Zoom in and look closely at one charge, it will have its own well defined magnetic field travelling along with it. In this sense, there are two magnetic fields. In general, if we zoom out until we can no longer distinguish one charge from the other and we will see a single magnetic field, in the case of two equal charges in an empty universe, the field will be zero at the plane equidistant from them and in opposite senses on either side of it. While this is technically a single field, we can see two distinct fields.

Looking closely at one charge, the magnetic field surrounding it is generated by the relative movement of the two electric energy density fields. Its energy density is determined by the relative velocity and by the electric energy density of its electric polarisation field. Let us look closely at one of the charges and then create a third charge, a forth charge and so on. The energy density of the magnetic field surrounding our charge is unaffected by the number of other charges. It does not suddenly double when we create the third charge. But now our charge's polarisation field is permeated by the polarisation fields of two other charges and a distinct velocity is associated with each. If one of the other two charges is nearby and the other some distance away, the former must have far more influence on the magnetic field than the latter. What we must do is to calculate a weighted average of the two relative velocities taking into account the relative strengths of the two polarisation fields in the vicinity of our charge. As the third and subsequent charges are created, we must include them in the calculation of our weighted average. This weighted average velocity is the velocity of our charge relative to stasis. That is to say that it is the absolute velocity of our charge at that point.

In a universe containing much matter, it is better to think of one charge moving through a background of the polarisation fields of all the other charges in the universe. Each of these has its own intensity and velocity in the region of the charge we are observing and it is possible to calculate a weighted velocity of this background through which our charge is moving. This velocity determines the magnitude of the magnetic field generated by the movement of our particle.

The magnetic fields we are familiar with are generated by the movement of millions of charges moving in a fairly co-ordinated fashion amidst billions moving with random orbital and thermal motion. ( I use the terms millions and billions figuratively.) For each charge, we can define a magnetic field intensity

This is one of the few times when it is useful to use this form of the Biot Savart law. But remember that it applies to a single charge. It must be remembered that is only a mathematical artefact representing the effect of the two physical realities which are the polarisation field of the charge and its velocity. A second mathematical artefact can now be calculated. A magnetic energy density field will now attempt to form according to . We can define the magnetic induction as a property of the magnetic energy density field, but we must understand that the magnetic energy density field is a real entity and has a life of its own. This is because is not constant. In the language of electronics, it is noisy. At any one point, it varies far too quickly for the magnetic energy density to keep up with it. It is further complicated by the fact that magnetic energy density fields are large and within them the magnetic intensity has to obey strict geometrical laws. We may write the equation
but it is not absolutely obeyed. It is a desirable state which the magnetic field attempts to satisfy subject to other far more pressing factors. The magnetic field which forms is a magnetic energy density field and should be denoted by its magnetic energy density vector . The field moves with the velocity of its geometry. The magnetic energy density flux is not stationary relative to the moving charge(s) which generated it or to stasis; the background weighted mean velocity of the other electric fields against which it was generated. (Unless the geometry of the field is stationary relative to either.)

When we are close to a moving charge, the magnitude of its own magnetic intensity overwhelms the contributions of other neighbouring charges and we find the charge surrounded by its own private magnetic field. The energy content of this magnetic field is considerable and largely or wholly accounts for the property we call inertia. There is no need to assume there is such a thing as mass! We can account for inertia entirely as an electromagnetic interaction. We call the magnetic field surrounding a moving charge the "field of motion" of the charge.

The behaviour of magnetic fields in the regions between the the loops of wire in a coil indicates that magnetic energy density flux has no resistance to shear. We are used to seeing diagrams of magnetic fields depicted by their lines of force which represent the magnetic induction flux . We are told that a tension exists in these lines of force. We are told that they are continuous and do not cross. These are good mental pictures in many ways but for a time I was deceived by this image into thinking that magnetic flux is "continuous". That word is often used to convey the fact that the divergence is everywhere zero. My problem was that my internalised concept of continuous is best encapsulated by the image of a rubber band. Magnetic lines of force are not like rubber bands.

As the current in the coil is increased, loops of magnetic flux emerge from the wire and move outwards, but in the region between the wire loops they thin out and eventually part joining onto the parted loops of magnetic flux of the neighbouring wire loops. In that process, the flux is always free of divergence. If we go back to a mental image of rubber bands, what is happening is that individual bands are continually being cut and rejoined with super-superglue to neighbouring bands which are also cut to facilitate the process. This is what I mean by shear. In a metal crystal, lattice shear is the process by which two halves of a crystal slip past each other along the lattice plane. The crystal still remains in one piece, but the individual atoms along the shear plane are now bonded to different atoms.

If we could follow the path of a line of flux around the wire loop and in some way mark it with a dye, we would produce a visible rubber band in the flux. But if we now increase the current causing more flux to move outwards from the wire, the line of dye stain will no longer follow the line of the flux because it will have been subject to shear. The flux which was part of a loop is now distributed among many loops because magnetic flux does not resist shear. In fact it would be more accurate to say that it cannot experience shear when it occurs. The flux is still continuous in the sense that if we start at any dye marked point and trace the new flux path from that point, we will eventually come back to it, and that includes the possibility of going off on a ten to twenty thousand mile journey through the centre of the earth and back!

These properties of the magnetic field apply to the magnetic induction , but this does not represent the fundamental nature of the magnetic field. The lines of force represent a directional property of the magnetic energy density flux. The idea of continuity, in the sense that a rubber band is continuous, is not appropriate to what is simply a directional property. Magnetic energy density flux has a tensile property associated with its directional property. But the tensile property is unlike any which we meet in mechanics or materials science. It requires that tension is transmitted continuously in the same way that pressure in a fluid is transmitted throughout the fluid, but it is unlike a fluid which transmits pressure in all directions. The one accurate description of the way in which magnetic lines of force behave is that they are free of divergence. That means that the number of lines of force flowing into a region is always equal to the number flowing out.

The real continuity of magnetic fields lies in the flow of magnetic energy density flux. Magnetic energy density flux is generated in the surface of charges and it can be adsorbed back into them. These processes are both associated with the generation of force which acts as the means of energy transfer. Apart from these, magnetic energy density flux can neither be created or destroyed. When the movement of charges is such as to produce a magnetic intensity field, magnetic energy density flux is, sort of sucked out, of the charges. As the movement becomes more random again, the tension in the magnetic energy density flux causes it to shrink back into the charges. The energy content of the magnetic field is determined by the balance between these two factors; the pull of the magnetic intensity drawing energy out from the charges and the pull of the magnetic induction seeking to contract.

The movements of the electric fields of all charges combine to form a magnetic intensity field

this is a mathematical artefact describing the reality of the moving electric fields of the charges.

Magnetic energy density fields form such that

The sign is read "attempts to form according to".

The magnetic field is an energy density field and its energy density is given by

This equation can be written

If we now express the magnetic intensity as a sum of the effects of the movements of all charges.

The vector dot product is distributive over addition and so we can expand this

Thus the contribution of the motion of an individual charge to the energy density of the magnetic field at a point is

This is a most important result for it shows that the magnetic energy density at a point is the sum of the contributions of all of the charges contributing to the magnetic intensity at that point. Magnetic energy density flux belongs to individual charges. Since a dot product is involved, it can be seen that the contribution from charges whose motion adds to is positive and the energy contribution from those charges whose motion lessens is negative. This reflects the fact that magnetic energy density flux has directional properties.

If we consider what happens when the magnetic field changes in strength, we see that changes in magnitude. That means that changes in the magnetic induction at a point change the contribution of each charge to the magnetic energy density at that point. Magnetic energy density flux must move in out of all of the charges. The movement of magnetic energy density flux to accommodate a change in a magnetic field is the sum of the movements of the personal magnetic energy density flux of the charges affecting the magnetic field. It is this movement of magnetic energy density flux into and/or out of charges which generates the forces they feel as the result of the change in the magnetic field.

This results in a paradox for we have found a way of explaining the force on charges which is independent of the concept of induction as developed by Faraday. Can we account for the force with two quite separate mechanisms? In that case, we should have twice the force! The answer to this paradox is most unexpected. Movement of magnetic energy density flux generates an electric energy density field. There are actually two fields. One is the raw electric force generated by the movement of the magnetic energy density flux. This acts on the fabric of space and polarises it. If we now place a test charge in the electric field to measure it, the test charge feels the raw electric force produced by the movement of the magnetic field, but it also feels the force from the electric polarisation of space. These two electric forces are equal and opposite to each other and the test charge feels no net electric field.

At the surface of a charge, the situation is quite different. Here the polarisation of space comes to an end. Here is the raw edge of the polarisation manifesting itself as electric charge. The magnetic energy density flux generated by the motion of the charge moves in an out of the surface of the charge. The movement of that magnetic energy density flux generates an electric field, but here the scale is too small for it to produce a significant polarisation which could exert an equal and opposite force on the charge surface. So the charge surface feels the full effect of the force generated by the movement of its own magnetic energy density flux in and out of it.

Charges can feel forces generated by the movement of their own magnetic energy density flux out of and into their surface. They cannot feel any force from the movement of any other magnetic energy density flux.

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© Copyright Bruce Harvey 1997.