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2 Theoretical basics

2.1 The electromagnetic force in Maxwell's electrodynamics

The Maxwell equations contain well hidden contradictions. These only become obvious when the electromagnetic forces of very slow, uniformly moving point charges are subjected to a precise investigation. Since this analysis will finally show how the other forces can be traced back to the electrical force, the theoretical derivation of quantino theory begins by showing the points of classical electrostatics and magnetostatics that have not yet been thought through to the end.

2.1.1 The force between two slowly uniformly moving point charges

The electromagnetic fields of uniformly moving point charges can be determined from the Liénard Wiechert potentials ([Lehner2004], Page 618).
Figure 2.1.1.1
For potential $\varphi$, the equation
$$\varphi = \frac{c\,q_s}{4\,\pi\,\varepsilon_0\sqrt{\left(c^2\,t-\vec{u}\,\vec{r}\right)^2 + \left(c^2-u^2\right)\left(r^2-c^2\,t^2\right)}}$$ (2.1.1.1)
applies if the point charge $q_s$ is at the origin at time point $t=0$ and is uniformly moving away from there with speed $\vec{u}$. The vector potential $\vec{A}$ is
$$\vec{A} = \frac{\vec{u}}{c^2}\,\varphi$$ (2.1.1.2)
under these circumstances. The electrical and magnetic field is generally given by the equations ([Lehner2004], page 451)
$$\vec{E} = -\nabla\varphi - \frac{\partial\vec{A}}{\partial t}$$ (2.1.1.3)
and
$$\vec{B} = \nabla\times\vec{A}.$$ (2.1.1.4)
For the special case of the uniformly moving point charge follows from these formulas (2.1.1.1) and (2.1.1.2) the electrical field
$$\vec{E} = \frac{c\,q_s\,\left(c^2-u^2\right)\left(\vec{r}-\vec{u}\,t\right)}{4\,\pi\,\varepsilon_0\sqrt{ \left(r^2 - c^2\,t^2\right)\,\left(c^2 - u^2\right) + \left(c^2\,t - \vec{r}\,\vec{u}\right)^2 }^{3}}$$ (2.1.1.5)
and the magnetic field
$$\vec{B} = \frac{\vec{u}}{c^2}\times\vec{E}.$$ (2.1.1.6)
The aim of the further is to calculate the force of the electric point charge onto another electric point charge at an arbitrary point in time. To do this, we set in equation (2.1.1.5) without loss of generality the time of the consideration to $t=0$ and obtain
$$\vec{E} = \frac{c\,q_s\,\left(c^2-u^2\right)\,\vec{r}}{4\,\pi\,\varepsilon_0\sqrt{r^2\,\left(c^2 - u^2\right) + \left(\vec{r}\,\vec{u}\right)^2}^{3}}.$$ (2.1.1.7)
The formula (2.1.1.6) remains unaffected by this.

The fields $\vec{E}$ and $\vec{B}$ are not actually measurable, but only their force effects on test charges. In order to calculate the force $\vec{F}$ of the charge $q_s$ onto another point charge $q_d$, we need its velocity $\vec{v}$ and the formula of the Lorentz force $\vec{F} = q_d\,\vec{E} + q_d\,\vec{v}\times\vec{B}$. The Maxwell equations thus finally provide the formula
$$\vec{F}_{M}(\vec{r},\vec{u},\vec{v}) = \zeta_M(\vec{u},\vec{r})\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\left(\vec{r} + \frac{1}{c^2}\,\vec{r}\times\vec{u}\times\vec{v}\right)$$ (2.1.1.8)
for the force of a uniformly moved ideal point charge $q_s$ onto another uniformly moved ideal point charge $q_d$. The unitless, scalar pre-factor $\zeta_M$ is defined by
$$\zeta_M(\vec{u},\vec{r}) := \frac{c\,\left(c^2-u^2\right)}{\sqrt{\left(c^2 - u^2\right) + \left(\frac{\vec{r}}{r}\,\vec{u}\right)^2}^{3}} = 1 + \frac{u^2}{2\,c^2} - \frac{3\,\left(\vec{r}\,\vec{u}\right)^2}{2\,c^2\,r^2} + \mathcal{O}(u^3).$$ (2.1.1.9)
We now want to examine formula (2.1.1.8) in more detail and analyze the influence of the speeds $\vec{u}$ and $\vec{v}$. Furthermore, we want to understand if and how the force depends on the direction of the speeds.

The source charge rests, the target charge moves

First we set $\vec{u}=0$, i. e. we examine a case where the source charge $q_s$ rests and the target charge moves at an arbitrary speed $\vec{v}$. Equation (2.1.1.8) thus simplifies to
$$\vec{F}_{M}(\vec{r},\vec{0},\vec{v}) = \zeta_M(\vec{0},\vec{r})\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\vec{r} = \frac{q_s\,q_d}{4\,\pi\,\varepsilon_0}\,\frac{\vec{r}}{r^3},$$ (2.1.1.1.1)
which is easily recognizable as the Coulomb law. If the source charge $q_s$ has the same sign as the target charge $q_d$, the resulting force acts exactly in the direction of the distance vector $\vec{r}$ otherwise in the opposite direction (see figure 2.1.1.1).

It is remarkable that the force $\vec{F}_M$ for a resting source charge does not depend at all on the speed of $\vec{v}$ of the target charge! Here, the Lorentz invariance of the electrical charge in classical electrodynamics is clearly apparent.

The source charge moves, the target charge rests

Figure 2.1.1.2.1: The blue arrows show the direction and the approximate strength of the force that a fast moving charge at the origin would exert on a test charge resting at the cooresponding point.
For the case that the source charge $q_s$ moves, but the target charge $q_d$ rests, it follows
$$\vec{F}_{M}(\vec{r},\vec{u},\vec{0}) = \zeta_M(\vec{u},\vec{r})\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\vec{r},$$ (2.1.1.2.1)
which this time does not correspond to the Coulomb force, because $\zeta_M(\vec{u},\vec{r})$ is usually not one. But also in this case is the force $\vec{F}_M$ always aligned parallel to the distance vector $\vec{r}$, because the scalar prefactor $\zeta_M$ cannot affect the direction but only the strength of the force.

If we analyze the prefactor $\zeta_M (\vec{u}, \vec{r})$, we find that the numerical value depends on the angle between the distance vector $\vec{r}$ and the speed $\vec{u}$. For $\vec{u}\,\bot\,\vec{r}$ is
$$\zeta_{M}^{\bot} = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} = \gamma(u) \quad (>1),$$ (2.1.1.2.2)
where $\gamma$ is the Lorentz factor. For $\vec{u}\,\parallel\,\vec{r}$, however,
$$\zeta_{M}^{\parallel} = 1 - \frac{u^2}{c^2} = \gamma(u)^{-2} \quad (<1)$$ (2.1.1.2.3)
applies. Such a field distortion is besides also known for gravity. This effect was once called longitudinal and transverse mass.

It is now clear that in Maxwell's electrodynamics it is a difference whether the source or the receiver of the force moves. As long as the source is at rest, a moving point-charge always perceives only the normal coulomb force. Conversely, however, it is more complicated because the strength of the force exerted by a moving point-charge onto a resting test-charge depends on whether the point-charge moves sideways or in a direct line to the resting test-charge. A charge passing sideways generates a force that is increased in comparison to the coulomb force. On the other hand, a charge that is approaching or leaving on a straight line is weakened in its effect.

It is clear from the occurrence of Lorentz factors that this is a relativistic effect. However, the asymmetry is irritating. The situation becomes even more confusing when all speeds are not zero, because in this case the cross product term in equation (2.1.1.8) becomes effective additionally to the $\zeta_M$-factor. It is left up to the reader to figure out in which directions the force is deflected. It is only mentioned here that the force usually does not point to the source and that in Maxwell's electrodynamics the electromagnetic force is not a central force.

2.1.2 The force of a current-element on a slow moving electric point charge

We now want to use the formula (2.1.1.8) to calculate the force that electric currents exert on single moving point charges. In contrast to the forces that single point charges generate on other point charges, the forces of DC currents can be measured easily. As it will turn out, the strange formula (2.1.1.8) - as far as currents are concerned - is in agreement with the experiment.

For the calculation we require an intermediate step. This consists of imagining a positive electric charge $q_s$ moving at speed $\vec{u}/2$, while a second negative charge $-q_s$ of equal magnitude moves at the same place with speed $-\vec{u}/2$. We will refer to this structure as a current-element.

Figure 2.1.2.1: Electric current is the number of charge carriers that pass through the grey plane per unit of time.
It is obvious that two oppositely moving point charges are no longer at the same position after a short time. However, if we imagine many current elements arranged in a straight line, so it becomes clear that there are always two oppositely charged point charges at a certain position to every point in time, because neighbors always replace the missing point charges. Such a line represents a direct current, because electrical current is defined as the number of charge carriers that pass through a surface per unit of time.

Figure 2.1.2.1 illustrates this relationship in a sketch. The positive charge carriers penetrate the gray surface from the left, while the negative ones come from the right. Incidentally, the resulting current is exactly the same as the current which we would have if the positive charge carriers did not move at all and the negative charge carriers would move to the left at twice the speed. This will be important later.

Now we want to calculate the force $\vec{F}_{MC}$, which a single current element at the coordinate origin exerts on a moving point charge $q_d$ at position $\vec{r}$ with the velocity $\vec{v}$. Because of formula (2.1.1.8) is
$$\vec{F}_{MC} = \zeta_M\left(-\frac{\vec{u}}{2},\vec{r}\right)\,\frac{(-q_s)\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\left(\vec{r} - \frac{1}{2\,c^2}\,\vec{r}\times\vec{u}\times\vec{v}\right) + \zeta_M\left(+\frac{\vec{u}}{2},\vec{r}\right)\,\frac{(+q_s)\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\left(\vec{r} + \frac{1}{2\,c^2}\,\vec{r}\times\vec{u}\times\vec{v}\right)$$ (2.1.2.1)
With definition (2.1.1.9) we get $\zeta_M\left(-\vec{u}/2,\vec{r}\right) = \zeta_M\left(\vec{u}/2,\vec{r}\right)$ and we can simplify formula (2.1.2.1) to
$$\vec{F}_{MC} = \zeta_M\left(\frac{\vec{u}}{2},\vec{r}\right)\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\frac{1}{c^2}\,\vec{r}\times\vec{u}\times\vec{v}.$$ (2.1.2.2)

We reformulate the expression (2.1.2.2) a little bit more and write
$$\vec{F}_{MC} = \left[u\,\zeta_M\left(\frac{\vec{u}}{2},\vec{r}\right)\right]\cdot\left[\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\frac{1}{c^2}\,\vec{r}\times\frac{\vec{u}}{u}\times\vec{v}\right].$$ (2.1.2.3)
The second term of the product in formula (2.1.2.3) no longer depends on the magnitude $u$ of velocity $\vec{u}$. The first term can be derived for $u$ into a Taylor series and it becomes obvious that
$$u\,\zeta_M\left(\frac{\vec{u}}{2},\vec{r}\right) = u + \mathcal{O}(u^2)$$ (2.1.2.4)
applies. This gives us the second order approximation
$$\vec{F}_{MC} \approx \frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,c^2\,r^3}\,\vec{r}\times\vec{u}\times\vec{v},$$ (2.1.2.3)
which is completely sufficient, because at electric currents the charge carriers not only move much slower than at the speed of light, but also slowly in comparison to the usual everyday speeds. We can now replace the electric field constant $\varepsilon_0$ durch $1/(\mu_0\,c^2)$ and reshape something. Finally, we get to
$$\vec{F}_{MC} = q_d\,\vec{v}\times\,\vec{B}$$ (2.1.2.5)
with
$$\vec{B} = \frac{\mu_0}{4\,\pi}\,(q_s\,\vec{u})\times\frac{\vec{r}}{r^3}.$$ (2.1.2.6)
The reader will recognize immediately that equation (2.1.2.5) is the Lorentz force and (2.1.2.6) the Biot-Savart law.

What does this mean? Well, on the one hand this means that the formula (2.1.1.8) for any current loops leads to the correct experimental predictions. But this was not to be expected otherwise, because equation (2.1.1.8) was derived from the Maxwell equations. However, it is much more important to note that two individual electric point charges already behave like a small piece of wire that has been cut out of a conductor loop. For this reason, Maxwell's electrodynamics always make it possible to integrate along arbitrary conductor loops. The cross-product term automatically ensures that the result also fits after the integration. The integration itself, however, has no significance for the explanation of the magnetic force, since it only overlaps the individual partial magnetic fields to form a total magnetic field. Figure 2.1.2.2 shows the illogical force of a current element on two differently moved point charges.

Figure 2.1.2.2: The blue arrows show the direction and approximate strength of the force that a test charge would perceive if it were at that point. On the left-hand side the test charge moves upwards, on the right-hand side it moves to the right. It can be clearly seen that the current element also generates a Lorentz force where it is not at all.
What is also noticeable is that the interesting, relativistic term (2.1.1.9) does not contribute anything to the explanation of the magnetic force, since it has already fallen out by the approximation of the second order in expression (2.1.2.4). As will become clear later on, the cross product term is a kind of "bug fix" to describe the magnetic force at currents correctly. Later sections will show how the magnetic force can be explained much better by a simple symmetric central force. But before we can come to this, an important paradox of classical magnetostatics has to be explained.

2.1.3 The Lorentz force paradox

In the context of figure 2.1.2.1 it has already been mentioned that the electrical current does not depend on the speed of the current elements per se, but only on the differential velocity of the positive and negative charge in a current element. Figure 2.1.3.1 illustrates this fact even more clearly.

Figure 2.1.3.1: The magnetic field of a current does not affect a resting test charge.
Figure 2.1.3.2: The same situation as before seen from a reference system that moves with the electrons.
In a resting, current-carrying metal wire drift the electrons with velocity $u$ and the metal ions rest. This corresponds exactly to the case shown in Figure 2.1.3.1. The number of negative charge carriers that pass the grey surface per unit of time corresponds to the current in the wire.

Figure 2.1.3.2 shows the wire and the test charge from the perspective of an observer moving to the left at speed $u$. It is now important to understand that the amount of charge moved per unit of time does not change. This means that formally the same current results. But now it consists of positive charge carriers which drift in the opposite direction.

According to the Biot-Savart law, a current in an infinitely long wire generates a magnetic field whose strength is directly proportional to the current in the wire. This means that both cases result in the same magnetic field. Since magnetic fields have no influence onto resting charges, in the first case there is no force acting on the test charge below the wire. In the second case, this does not apply because the test charge moves.

But this is a contradiction! Whether an object is accelerated or not cannot depend on the inertial system from which an uninvolved observer views the process.

As readers have correctly pointed out, this contradiction can be solved by using the Lorentz transformation instead of the Galilean transformation. In the special theory of relativity, a moving observer perceives a different current than a resting observer. At the same time, the wire is no longer electrically neutral. The change in magnetic force is precisely compensated by the added electrical force. This transformation from current to charge is discussed for this special case in [Lehner2004] on page 656 or in [Orear1979] on page 359.

Figure 2.1.3.3: An electric current flows in the x-direction. The positive test charge moving downwards experiences a force in the x-direction.
Figure 2.1.3.4: The current is still flowing only in the x-direction. The test charge rests, but the magnetic field is time-varying.
However, the Lorentz transformation cannot explain why a test charge experiences a force parallel to the wire when it moves perpendicular to it. Figure 2.1.3.3 shows an electric current in the x-direction and a test charge moving vertically to it. A transformation into the rest frame of the test charge gives the situation as shown in Figure 2.1.3.4. The current still flows only in x-direction, because of the symmetry the wire has to be electrically neutral even after the Lorentz transformation and therefore no charge transport in y-direction can occur. The charge is resting in Figure 2.1.3.4, but the magnetic field strength is changing in time because the wire is moving away from the test charge. As a result the force onto the test charge remains. However, this results only from the fundamental laws of electromagnetism and not from the Lorentz transformation.

Is the Lorentz force paradox now resolved? In principle yes, because mathematically everything is now without contradictions. But we have to accept that in the vertical case the magnetic force is simply present and we cannot explain its existence by relativity as it was possible in the parallel case. As a consequence, the magnetic field cannot be completely eliminated by the special theory of relativity and a four-dimensional field strength tensor is required to describe it.

In quantino theory, however, the magnetic field can be completely traced back to the relative speeds between all point charges. This is a considerable simplification and a clear gain in knowledge. Furthermore, this alone already suggests that quantino theory describes reality better than the formalism consisting of Lorentz transformation and Maxwell equations.