4. Velocity Dependence
In the last section we proposed that a soliton wave (particle) moved throught the aether. This particle has a symmetrical field around it that looks like a field of a charge. Thus we have a charge moving through the rest frame of the electric and magnetic fields (aether) at some absolute (by definition) velocity. Now what is needed is to transform the initial postulates (Maxwell's equations in empty space) to a set of equations that describe the electric and magnetic fields when charge is present in the space (Maxwell's equations including charge and current). This is accomplished in these next three sections (4-6). There are several things that should be noted. At this point we are no longer concerned with a model of charge itself, but merely the result of charge being present, thus, the results of these following sections do not depend on the validity of the model described in Sections 2 and 3. The charge could be caused by anything and these results would be unchanged. Also one might wonder why we are rehashing Maxwell's equations. The reason has to do with continued consistency with the purely electro-magnetic field model proposed here. The displacement term of Maxwell's equations was included in the postulates because it is an effect based directly on the fields as opposed to the presence of charge (some inexplicable connection to a foreign object in the aether). We will show here that there is NO direct connection between current and the magnetic field, rather the current merely causes the electric field to change intensity in a region of space. This change in intensity then creates a magnetic field. In this section we use a convection derivative to show the contribution to the changing electric field by the motion of charge.
Now that a velocity has been defined, it can be determined how this velocity affects the fields.
Imagine a symmetric electric field (charge) propagating at a velocity vx in relation to a stationary observation point. As the field approaches the observation point, the field strength at the point increases such that
(1)
This can be generalized by:
(2)
(3)
Thus, with this type of propagation the initial field equations (Eq.(1-1) and Eq.(1-2)) can be rewritten as
(4)
(5)
which describe the relation between electric fields and magnetic fields when this velocity is present1. Note that there is not necessarily a particle moving which carries the field along with it. The field propagates of its own accord (provided that the longitudinal propagation problems can be resolved), but its propagation can mimic that of charged particle in motion.
We can apply an identity to these equations and obtain
(6)
(7)
These equations2 represent the contribution to the changing fields from the motion of a charge (or sub-light velocity of a "charge field").
1. This is essentially the convection derivative (and we will refer to it as that for future reference), except the sign is reversed.
2. This derivation can also be found in a footnote on page 212 of Jackson2 with sign reversed. This is due to the absolute velocity used here, whereas he refers to the relative velocity. This sign reversal relates to the nature of the Lorentz force which will be discussed in Section 8. The cross-product term comes in useful for deriving the voltage on a conductor moving in a magnetic field. It is surprising that he did not use the same derivation on the other curl equation as we have done here. This ultimately reveals the first-principles relationship between a magnetic field being created by a changing electric field and a magnetic field being created by a current.
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