12. Relativistic1 Force
After 11 sections we are now finished with the electromagnetic model of particles and Newtonian mechanics. All these mechanical concepts can now be described purely in electromagnetic terms. Since this is a field theory with propagations which can be as fast as the speed of light, there is no reason to think that this theory should be different for solitons propagating at near light velocity. So in the next 3 sections (12-14) we will simply calculate some effects at high velocity (in the absolute frame of the electric and magnetic fields) and see the results. We will make no more postulates and we will not apply a single Lorentz transformation.
In this section the Lorentz force is calculated for relativistic velocities. There are two derivations. The first one is for a charge moving through an electric field where either of the absolute velocities can be large. The second one is for a charge moving through a magnetic field where again either of the velocities can be large. The result if the field sources are non-relativistic is F=F0g where g is the gamma factor using the speed of the particle through the fields. That is both fields apply more force to the particle by this factor.
We found in Section 8 that force is derived from the interaction energies of the associated electro-magnetic fields. However, as the velocity of the charge increases the electric and magnetic fields change and thus the energy content. The electric field of a charge at a constant velocity2 is given by:
(1)
where q is the angle between the direction of the electric field E and the velocity v of the charge. The magnetic field is then still determined by Eq.(5-9) and thus also changed by the factor in Eq.(1). It should be noted that by superposition these factors apply to any distribution of charge which is at constant and uniform velocity.
Now we can derive the Lorentz force again (as in Section 8). Imagine a charge at absolute velocity vq through a constant uniform electric field whose source is moving at absolute velocity vE. Since only the electric field of the charge that is parallel to the uniform electric field contributes to the interaction energy, the electric force is
(2)
where qq is angle between vq and the uniform electric field E; qE is the angle between vE and E. However, this is not the whole effect. Since both sources have an absolute velocity, they each create a magnetic field and there is thus a corresponding magnetic force (from Eq.(8-4)). Substituting the magnetic fields for charge distributions at uniform constant velocity yields
(3)
Utilizing an identity gives
(4)
where Eq is the electric field of the charge. The first term can be reduced similar to Eq.(8-7) hence,
(5)
To do the integral we will divide the velocity of the uniform field into two components, one parallel to the field and one perpendicular. z is aligned along the uniform electric field. Thus,
(6)
Note that only an additional x-component is given to vE. It could be only a y-component or any combination of the two, but it does not matter since the contribution is the same. In this case, it is cancelled out and the x-component does not contribute. Thus:
(7)
The total force is the sum of the electric force and the magnetic force which gives
(8)
Next imagine that there is a uniform magnetic field whose source is moving at absolute velocity vB and a charge q moving at absolute velocity vq. Before we can continue, it will be necessary to determine the effect of velocity on the magnetic field. We cannot use Eq.(5-9) directly because a uniform magnetic field is created by charge circulating, moving in all directions within a plane. In addition, we will want to reference the velocity to the direction of the magnetic field which is not the same as the associated electric field. Imagine a single current loop creating a magnetic field as in the following diagram.
Fig.5: Magnetic field and associated current.
If the absolute velocity of the loop is parallel to the magnetic field (at the center), then the velocity effects of the significant charge contributions are all based on the electric field which is perpendicular to the velocity. If the velocity is perpendicular to the magnetic field, then half of the significant charge contributions are based on the electric field that is perpendicular to the velocity and half of the charge contributions are based on the electric field that is parallel to the velocity. This equal division is possible because of the symmetry and because the force is dependent on an energy, which will divide out equally in correspondence to each dimension (in the plane). Thus the velocity dependent uniform magnetic field is given by:
(9)
where qB is the angle between the absolute velocity of the source of the magnetic field and the direction of the magnetic field itself. As in Eq.(8-11) we have
(10)
where Eq is the electric field of the charge. This can be simplified the same as with Eq.(8-11) resulting in
(11)
However, as before, this is not the whole effect. Since the source of the uniform magnetic field is moving, there is an associated electric field and therefore an electric force given by:
(12)
This can be simplified the same as Eq.(8-11) resulting in
(13)
The sum of this force and the magnetic force of Eq.(11) determines the total force on the charge which is
(14)
So we now have the equation of the Lorentz force law at any velocity in the sum of Eq.(8) and Eq.(13). In the case where the uniform fields are non-relativistic (vE=vB and both much less than c), this sum becomes
(15)
where qE is the angle between the velocity v of the charge and the direction of the electric field and qB is the angle between the velocity of the charge and the direction of the magnetic field.
Note that when qq = qE = p/2 Eq.(8) becomes
(16)
If vq=vE then F=qE, the same as at rest. Let vq=v+u and vE=v where v is the velocity of the lab and u is the observed (relative) velocity of the charge. Hence,
(17)
Assuming that v,u is much less than c we can expand by the binomial theorem to second order and obtain
(18)
By reversing the expansion this can be written as
(19)
though this is still only valid to second order. But we can see by its form that by ignoring the strictly magnetic force, one would obtain the equivalent force using the relative J in Eq.(6-4).
1. "Relativistic" in this context does not involve the actual theory of relativity at all, but is being used to indicate (absolute) velocities where the term v2/c2 becomes significant.
2. This can be found from the Liénard-Wiechert potentials as demonstrated in Griffiths1 425 or Jackson2 657.
Note that the transverse field of a moving charge is increased by the factor:
(a)
and the longitudinal field is reduced by:
(b)
giving a longitudinal to transverse ratio k of:
(c)
Now solve this for v to get:
(d)
Now compare this with Eq.(3-5). Note the similarity and the reduced effect of the longitudinal to transverse ratio on the velocity by an exponent of 2/3. There is apparently a kind of causal symmetry here. While the Liénard-Wiechert potentials describe the deformation of the electric field around the charge, it is, in fact, the deformation away from the symmetrical field that causes the soliton to propagate at the given velocity in the first place.
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