8. Force

It is shown in this section that the interaction energy of two fields must indicate a corresponding force. In fact it indicates two forces, one for the interaction of the electric fields and one for the interaction of the magnetic fields. These are claimed to be the correct fundamental laws of force in electro-magnetism. Then these forces are used to successfully derive the Lorentz force law. However, here is an important difference. The Lorentz force law is not an invariant, but is only valid at the non-relativistic limit. A universally valid force law must always be derived from the inter-field force equations given here. While an explanation is given for why two solitons should accelerate when their fields overlap, thus acting as if there were a force, the results of this section are actually true regardless of any specific model, since it is required to remain consistent with the energy content of the fields.

Note that the interaction energy is (from Eq.(7-6)) given by

(1)


There is also an interaction energy for the magnetic field:

(2)


These represent the potential energy between two charges and thus ultimately relates to the force. This suggests that there should exist an associated force which depends directly on the interaction of the fields. If the dot products are zero, no energy is available to apply a force (do work) on the charges1. However, the fields so far described do not exert any forces. Since there are no bodies present, no definition of force exists. However, imagine two particle fields in proximity. The field from one would overlap the other. This alters the symmetry of the fields and thus changes the velocity of propagation2. This interaction mimics a force when considering the inertial content of the fields (the inertial content to be determined in the next section). Coulomb's law is not the basis of this stationary electromagnetic theory, but a secondary result. It does not occur for every infinitesimal piece of charge, but only in relation to a complete charge field. This explains how various continuous charge distributions survive (which we are then labeling as charge fields). To then derive the relevant force laws in general, it is only necessary to utilize the appropriate interaction energies of the fields of two sources. The two equations are thus,

(3)


and

(4)


This is the force law required to remain consistent with the energy content of the electric and magnetic fields. The Lorentz Force law can then be derived from these. First it will be useful to note that for a point charge:

(5)


where u is a distance in some dimension and r is perpendicular to u (cylindrical coordinates aligned along u). This point charge will eventually be put in a uniform electric field which needs to be confined between two infinite plates divided by a distance b to avoid an infinite energy. The charge will be at a distance s from one of the plates. So only this limited volume is considered in the integral.

(6)


The electric force on a charge in a uniform electric field is given by:

(7)


where E is the uniform field and Eq is the electric field of the charge. Since E is uniform and in the direction of s, it can be brought out of the integral and we obtain

(8)


Thus by Eq.(6) the electric force is

(9)


There is also an analogous magnetic force based on the direct interaction of the magnetic fields (Eq.(2)). In a similar manner the overlapping magnetic fields affect the soliton propagation. The magnetic force on a particle travelling through a uniform magnetic field is

(10)


where B is the uniform field and Bq is the magnetic field of the charge. From Eq.(5-9) we have

(11)


where Eq is the electric field of the charge. By an identity

(12)


Since v is constant and B is uniform, they can be taken out of the integral to obtain

(13)


where s is in the direction of v×B. The magnetic force is thus

(14)


where v is the absolute velocity. Now imagine that a single charge is traveling through a uniform electric field and a uniform magnetic field (from a current with no net charge) at an absolute velocity v2. The sources of the electric and magnetic fields have an absolute velocity v1. Because of Eq.(4-6) the magnetic field contributes an additional electric field given by:

(15)


since there is no net charge and thus Ñ×E=0. The force on the charge is then the electric force plus the magnetic force such that

(16)


which reduces to

(17)


This is the Lorentz force law3. Normally, this is used as a postulate in classical electromagnetism, but here the postulate is that the electric and magnetic fields contain energy by the amount described in Poynting's theorem (Eq.(1-5)). The Lorentz law is then derived from this. The switching of causality in this case turns out to be significant. One reason, of course, is that the particle-less space could hardly use a force as a primary postulate. Another reason is that this approach maintains a consistent representation of absolute velocity, a further requirement of a space where only fields exist (that have no velocity associated with them). The causality here, however, is not completely reciprocal in that the Lorentz Force Law is only a specific case of the more general laws described by Eq.(3) and Eq.(4). If the charge fields are at a high velocity, then the Lorentz Force Law is not the same. Eq.(3) and Eq.(4) are not completely general however. They can only be used if one of the fields is based on a single charge field and are used as an indirect method of determining the soliton interaction (a direct method not being known at this time). Once this is done, the Lorentz force can be used generally for non-relativistic velocities. Later, we will use this same method to determine the soliton interaction when relativistic velocities are involved.

As a pure magnetic force is not recognized, it is assumed that in relation to Eq.(4-6) that

(18)


giving all the credit to the electric field for the force. Thus we have

(19)


where v is the relative velocity4.

1. This is a bit misleading for the magnetic field as the magnetic field does no work. It can be considered here as a virtual work which can be utilized to derive a force, which as seen in Eq.(17) fails to actually do any work.

2. If simple superposition were involved, this would not be the case, but again (as in footnote 2, Section 3), we are assuming that an additional process is involved with a divergence based non-linearity.

3. Griffiths1 198 or Jackson2 2.

4. This corresponds to Jackson2 212 as discussed in Section 4.
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