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2.2 Electromagnetic force in quantino theory

2.2.1 The force between two slowly uniformly moving point charges

Figure 2.2.1.1
As in Introduction is written, the quantino theory assumes that the electric charge is relative and its effect via the electric force depends on the relative velocity $\vec{u}-\vec{v}$. The equation for the force of a uniformly moving point charge $q_s$ on another uniformly moving point charge $q_d$ is as follows
$$\vec{F}_{R}(\vec{r},\vec{u}-\vec{v}) = \zeta_R(\vec{u}-\vec{v},\vec{r})\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\vec{r}$$ (2.2.1.1)
with
$$\zeta_R(\vec{w},\vec{r}) = \frac{\zeta_M(\vec{w},\vec{r})}{\sqrt{1-\frac{w^2}{c^2}}} = \frac{\sqrt{1-\frac{w^2}{c^2}}}{\sqrt{\left(1 - \frac{w^2}{c^2}\right) + \left(\frac{\vec{r}}{r}\,\frac{\vec{w}}{c}\right)^2}^{3}}.$$ (2.2.1.2)
By inserting the pre-factor (2.1.1.8), which has already occurred in Maxwell's electrodynamics before, and a series development with respect to $\vec{w}$ follows
$$\zeta_R(\vec{w},\vec{r}) = 1 + \frac{w^2}{c^2} - \frac{3\,\left(\vec{r}\,\vec{w}\right)^2}{2\,c^2\,r^2} + \mathcal{O}(w^3).$$ (2.2.1.3)
As will be shown later, $\zeta_R$ can be derived from a Force-Carrier Model. By using equation (2.2.1.3) in equation (2.2.1.1), neglecting all terms of order $\mathcal{O}(w^3)$ and above, follows the formula
$$\vec{F}_{R}(\vec{r},\vec{w}) \approx \left(1 + \frac{w^2}{c^2} - \frac{3}{2} \left(\frac{\vec{r}}{r}\,\frac{\vec{w}}{c}\right)^2\right)\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0}\,\frac{\vec{r}}{r^3}$$ (2.2.1.4)
for the electric force. This force formula (2.2.1.1) is similar to formula (2.1.1.8), which follows from the Maxwell equations, but it is not identical and cannot be derived from something known in standard physics. It is only possible to show that it makes sense. For example, by demonstrating that this formula can be used to calculate the correct magnetic fields of any current loop or by showing that for this force the law of momentum conservation, the law of angular momentum conservation and the law of energy conservation holds. It is important to note, that formula (2.2.1.4) applies only to very low speeds.

Do we compare equation (2.2.1.1) with equation (2.1.1.8), it is noticeable that the force formula of the electric force is here symmetrical, because the following applies
$$\vec{F}_{R}(\vec{r},\vec{u}-\vec{v}) = -\vec{F}_{R}(- \vec{r},\vec{v}-\vec{u}).$$ (2.2.1.5)
In quantino theory, a point charge thus experiences a force from another point charge which, conversely, is the same as the force that it itself exerts on the other point charge. Therefore, it is force = counterforce. This is the essential condition for fulfilling the law of momentum conservation.

In Maxwell's electrodynamics, on the other hand, the force is asymmetrical, since in the force formula (2.1.1.8) the pre-factor $\zeta_M$ depends only on the absolute velocity $\vec{u}$ of the source. Therefore, the conservation of momentum is violated here, unless one argues that the electromagnetic field carries the missing momentum (see also [Timm2016], section 4.2). However, this is not quite plausible for uniform motions. Moreover, this asymmetry also leads to real paradoxes, like the Lorentz force paradox.

Figure 2.2.1.2: The blue arrows show the direction and approximate strength of the force between two charges when there is a relative speed of $\vec{u}$. The form of the force this time is the same in every inertial frame of reference.
It is also possible to construct further thought experiments which demonstrate the contradictions in Maxwell's electrodynamics. The force formula of Maxwell electrodynamics (2.1.1.8) has yet another unpleasant feature. And this consists in the fact that the force described by it does not represent a central force. Most obvious is this problem for the force of a current element (2.1.2.3), which does not necessarily point to the cause of the force and can even be orthogonal. Something like this does not occur in the force formula (2.2.1.1) of the quantino theory, since the pre-factor $\zeta_R$ is scalar and therefore cannot influence the direction of the force. This means that the equation (2.2.1.1) is a formula that always describes a central force. So it is already clear that the law of angular momentum conservation must hold.

The force field is the same as shown in Figure 2.1.1.2.1 or 2.2.1.2. The difference, however, is that the field also retains its shape if the observer moves uniformly with respect to the charges, i. e. beholds the situation from a different frame of reference.

The most important property of formula (2.2.1.1) is that it is possible to derive the Lorentz force with it. This is not self-evident, because - as mentioned above - the force (2.2.1.1) is a central force and the Lorentz force can, for example, also be aligned parallel to a current-carrying wire. The following shows that it is nevertheless possible.

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

As already mentioned in section 2.1.2, it is in quantino-theory possible as well to construct a current element from two oppositely charged and oppositely moved point charges at the same position. So we imagine a charge $q_s$ moving at the speed $\vec{u}/2$, while a second negative charge $-q_s$ is moving at the same position with speed $-\vec{u}/2$. The force $F_{RC}$ of these two charges on a test charge $q_d$ with the speed $\vec{v}$ is because of equation (2.2.1.1)
$$\vec{F}_{RC} = \zeta_R\left(\frac{\vec{u}}{2}-\vec{v},\vec{r}\right)\,\frac{(+q_s)\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\vec{r} + \zeta_R\left(-\frac{\vec{u}}{2}-\vec{v},\vec{r}\right)\,\frac{(-q_s)\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\vec{r}.$$ (2.2.2.1)
This can be simplified to
$$\vec{F}_{RC} = \left[\zeta_R\left(\frac{\vec{u}}{2}-\vec{v},\vec{r}\right) - \zeta_R\left(\frac{\vec{u}}{2}+\vec{v},\vec{r}\right)\right]\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\vec{r}$$ (2.2.2.2)
by using $\zeta_R(\vec{w},\vec{r}) = \zeta_R(-\vec{w},\vec{r})$. Next we use the definition (2.2.1.2) and get
$$\left[\zeta_R\left(\frac{\vec{u}}{2}-\vec{v},\vec{r}\right) - \zeta_R\left(\frac{\vec{u}}{2}+\vec{v},\vec{r}\right)\right] = \frac{3\,\left(\vec{r}\,\vec{u}\right)\left(\vec{r}\,\vec{v}\right)-2\,r^2\,\vec{u}\,\vec{v}}{r^2\,c^2},$$ (2.2.2.3)
what leads to
$$\vec{F}_{RC}(\vec{r},\vec{u},\vec{v},q_s,q_d) = \left(3\,\left(\frac{\vec{r}}{r}\,\vec{u}\right)\left(\frac{\vec{r}}{r}\,\vec{v}\right)-2\,\vec{u}\,\vec{v}\right)\,\frac{\mu_0\,q_s\,q_d}{4\,\pi}\,\frac{\vec{r}}{r^3}$$ (2.2.2.4)
by insertion of (2.2.2.3) into equation (2.2.2.2) and by using of $\varepsilon_0\,c^2 = 1/\mu_0$. The figure 2.2.2.1 shows that the force of a current element in quantino theory is fundamentally different to the force of the current element in classical electrodynamics (see figure 2.1.2.2).
Figure 2.2.2.1: The blue arrows show the direction and approximate strength of the force that a test charge through a current element at the coordinate origin would experience if it were at the corresponding point. On the left-hand side the test charge moves upwards, on the right-hand side it moves to the right. You can see that it is a central force. The Lorentz force, which would be created by a whole current thread, is not yet existent.

2.2.3 The field of a long straight wire in quantino theory

A reader with a well-developed geometrical imaginative power will probably recognize it already by looking at Figure 2.2.2.1 that the superposition of the fields of many current elements on the x-axis yield in the end the Lorentz force. In fact, three current elements - as shown in Figure 2.2.3.1 - already provide a fairly good approximation.

Figure 2.2.3.1: The superposition of the fields of only three current elements already results in a good approximation of the Lorentz force.
It is not only possible to make it plausible, but also to prove it mathematically. The simplest model case is the so-called infinitely long straight wire. It is obtained by integrating along of all current elements:
$$\vec{F}_{RCT}(\vec{r}) = \int\limits_{-\infty}^{+\infty} \vec{F}_{RC}(\vec{r}-\vec{s},\vec{u},\vec{v},\lambda,q_d)\,\mathrm{d}\vec{s}.$$ (2.2.3.1)
$\lambda$ is hereby the number of current elements per unit of length. It is clear that doubling the number of these current elements leads to a doubling of the current. The current $I$ is therefore proportional to $\lambda$. Furthermore, it is proportional to the speed $u$, i. e. the relative speed of the charges in the current element. A doubling of this speed doubles also the number of charge per unit which passes a surface across the direction of movement of the charge carriers. The current is thus definable by equation
$$\vec{I} = \lambda\, \vec{u}.$$ (2.2.3.2)

Without loss of generality it is possible to let the current flow only along the x-axis. With this, $\vec{u}=(u_x,0,0)$ and $\vec{s}=(s_x,0,0)$ follows. Furthermore, it is useful to place the sample charge $q_d$ on the z-axis. Thus is $\vec{r} = (0,0, r_z)$. If this is inserted into formula (2.2.2.4), equation
$$\vec{F}_{RC}(\vec{r}-\vec{s},\vec{u},\vec{v},\lambda,q_d) = \frac{\lambda\,q_d\,\mu_0}{4\,\pi}\,\frac{\left(2\,r_z^2\,v_x - s_x^2\,v_x + 3\,r_z\,s_x\,v_z\right)\, u_x}{\sqrt{s_x^2 + r_z^2}^5}\,\left(\begin{matrix}s_x \\ 0 \\ -r_z\end{matrix}\right)$$ (2.2.3.3)
follows. This can easily be inserted into the integral. This brings us to
$$\vec{F}_{RCT}(\vec{r}) = \frac{\lambda\,q_d\,\mu_0}{4\,\pi}\, \int\limits_{-\infty}^{+\infty} \frac{\left(2\,r_z^2\,v_x - s_x^2\,v_x + 3\,r_z\,s_x\,v_z\right)\, u_x}{\sqrt{s_x^2 + r_z^2}^5}\,\left(\begin{matrix}s_x \\ 0 \\ -r_z\end{matrix}\right) \mathrm{d}s_x.$$ (2.2.3.4)
The calculation of the integral results in
$$\vec{F}_{RCT}(\vec{r}) = \frac{q_d\,\mu_0\,I}{2\,\pi\,r_z}\,\left(\begin{matrix}v_z \\ 0 \\ -v_x\end{matrix}\right),$$ (2.2.3.5)
whereby formula (2.2.3.2) was used for the current $I$. The same result would we obtain if we would made conventional calculations using the Biot-Savart law and the Lorentz force formula.

2.2.4 The field of arbitrary conductor loops in quantino theory

Figure 2.2.4.1: conductor loop
It is not only possible to show that the field of an infinitely long wire matches the results to be expected experimentally, but it can also be proven that any closed conductor loop generates the known magnetic field. For this purpose, it is examined here which force a current $I$ in a small rectangular conductor loop, as shown in Figure 2.2.4.1, effects on a test charge $q_d$.

The entire current in it consists of four partial currents. This means that this time we have to integrate four times. The total force is therefore given by
$$ \begin{align} \vec{F}_{RCT}(\vec{r}) \; = \; & \int\limits_{-L}^{+L}\,\vec{F}_{RC}(\vec{r} - (x \,\vec{e}_x + L \,\vec{e}_y), u\,\vec{e}_x,\vec{v},\lambda,q_d)\,\mathrm{d}x + \\ & \int\limits_{-L}^{+L}\,\vec{F}_{RC}(\vec{r} - (L \,\vec{e}_x - y \,\vec{e}_y), -u\,\vec{e}_y, \vec{v},\lambda,q_d)\,\mathrm{d}x + \\ & \int\limits_{-L}^{+L}\,\vec{F}_{RC}(\vec{r} - (-x \,\vec{e}_x - L \,\vec{e}_y), -u\,\vec{e}_x,\vec{v}, \lambda,q_d)\,\mathrm{d}x + \\ & \int\limits_{-L}^{+L}\,\vec{F}_{RC}(\vec{r} - (-L \,\vec{e}_x + y \,\vec{e}_y), u\,\vec{e}_y, \vec{v},\lambda,q_d)\,\mathrm{d}x. \end{align}$$ (2.2.4.1)
This integral is very difficult to solve for a $L$ which is not very small. However, the calculation can be considerably simplified by developing expression $\int_{-L}^{+L} f(x,L)\,\mathrm{d}x$ with respect to $L$ at $0$ in a Taylor series and breaking it after the second order. It follows the equation
$$\int\limits_{-L}^{+L} f(x,L)\,\mathrm{d}x \approx 2\,L\,f(0,0) + 2\,L^2\,\left.\frac{\partial}{\partial L}f(0,L)\right|_{L=0}.$$ (2.2.4.2)
With the help of equation (2.2.4.2), Integral (2.2.4.1) can be considerably simplified and we get
$$ \begin{align} \vec{F}_{RCT}(\vec{r}) \; \approx \; & \,2\,L\,\vec{F}_{RC}(\vec{r}, u\,\vec{e}_x, \lambda) + \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{C}(\vec{r} - L \,\vec{e}_y, u\,\vec{e}_x, \vec{v},\lambda,q_d)\right|_{L=0} \\ & 2\,L\,\vec{F}_{RC}(\vec{r}, -u\,\vec{e}_y, \lambda) + \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{C}(\vec{r} - L \,\vec{e}_x , -u\,\vec{e}_y, \vec{v},\lambda,q_d)\right|_{L=0} \\ & 2\,L\,\vec{F}_{RC}(\vec{r}, -u\,\vec{e}_x, \lambda) + \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{C}(\vec{r} + L \,\vec{e}_y, -u\,\vec{e}_x, \vec{v},\lambda,q_d)\right|_{L=0} \\ & 2\,L\,\vec{F}_{RC}(\vec{r}, u\,\vec{e}_y, \lambda) + \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{C}(\vec{r} + L \,\vec{e}_x, u\,\vec{e}_y, \vec{v},\lambda,q_d)\right|_{L=0}. \end{align} $$ (2.2.4.3)
With the help of equations $\vec{F}_{RC}(\vec{r}, u\,\vec{e}_x,\vec{v}, \lambda,q_d) = -\vec{F}_{RC}(\vec{r}, -u\,\vec{e}_x, \vec{v}, \lambda,q_d)$ and $\vec{F}_{RC}(\vec{r}, u\,\vec{e}_y,\vec{v}, \lambda,q_d) = -\vec{F}_{RC}(\vec{r}, -u\,\vec{e}_y,\vec{v}, \lambda,q_d)$ we get
$$ \begin{align} \vec{F}_{RCT}(\vec{r})\; = \; & \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{RC}(\vec{r} - L \,\vec{e}_y, u\,\vec{e}_x,\vec{v}, \lambda,q_d)\right|_{L=0} + \\ & \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{RC}(\vec{r} - L \,\vec{e}_x , -u\,\vec{e}_y,\vec{v}, \lambda,q_d)\right|_{L=0} + \\ & \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{RC}(\vec{r} + L \,\vec{e}_y, -u\,\vec{e}_x,\vec{v}, \lambda,q_d)\right|_{L=0} + \\ & \left.2\,L^2\,\frac{\partial}{\partial L}\vec{F}_{RC}(\vec{r} + L \,\vec{e}_x, u\,\vec{e}_y,\vec{v}, \lambda,q_d)\right|_{L=0}. \end{align} $$ (2.2.4.4)
Now, into the equation (2.2.4.4) the formula for the current element (2.2.2.4) is inserted. The calculation results in
$$\vec{F}_{RCT}(\vec{r}) = \frac{L^2\,\lambda\,q_d\,\mu_0\,u}{\pi}\,\left(\vec{v} \times \frac{\vec{e}_z\,r^2 - 3\,\vec{r}\,(\vec{e}_z\,\vec{r})}{r^5} \right).$$ (2.2.4.5)
The expression $\lambda\,u$ corresponds here again to the current $I$ in the conductor loop. If the magnetic dipole moment $\vec{\mu} = I\,(2\,L)^2\,(-\vec{e}_z)$ is, as usually common, is defined as the product of the current and the area enclosed by it, then we get
$$\vec{F}_{RCT}(\vec{r}) = q_d\,\left(\vec{v} \times \frac{\mu_0}{4\,\pi}\,\frac{3\,\vec{r}\,(\vec{\mu}\,\vec{r}) - \vec{\mu}\,r^2}{r^5}\right).$$ (2.2.4.6)
Term
$$\vec{B}(\vec{r}) = \frac{\mu_0}{4\,\pi}\,\frac{3\,\vec{r}\,(\vec{\mu}\,\vec{r}) - \vec{\mu}\,r^2}{r^5}$$ (2.2.4.7)
is known as the magnetic flux density of the magnetic dipole, i. e.
$$\vec{F}_{RCT}(\vec{r}) = q_d\,\left(\vec{v} \times \vec{B}\right).$$ (2.2.4.8)
is valid. This is clearly recognizable as Lorentz force. The result is exactly the same as that which can be obtained in Maxwell's electrodynamics.

Figure 2.2.4.2: Every conductor loop can be interpreted as a superposition of an infinite number of infinitely small conductor loops, since all currents except those in the outer boundary compensate each other.
Since many of these conductor loops (Figure 2.2.4.2) can be used to form any arbitrarily shaped conductor loop, it means that the quantino theory corresponds in this aspect to the predictions of Maxwell's electro and magnetostatics. In particular, the two Maxwell equations of this special case apply.The first equation
$$\nabla\vec{B} = 0$$ (2.2.4.9)
states that the magnetic induction $\vec{B}$ has no sources, i. e. its field lines are always closed in themselves and there are no places anywhere where they begin or end. In other words, there is no such thing as magnetic monopolies. Instead, the magnetic field is created by the presence of a closed current path. The second Maxwell equation
$$\nabla\times\vec{B} = \mu_0\,\vec{j}$$ (2.2.4.10)
describes this fact by linking the magnetic induction $\vec{B}$ with the current density $\vec{j}$.

2.2.5 Summary of the section

Before we proceed, we briefly summarize what has been shown up to now. These were several things. On the one hand, it was shown that in Maxwell's electrodynamics serious logical problems are concealed. These only become obvious when we solve Maxwell's equations for point charges and calculate the force between two slowly and uniformly moving point charges from the $E$ and $B$ fields. A detailed analysis then reveals that the calculated force does not make any physical sense, as it contradicts the principle of relativity. Furthermore, it becomes clear that the magnetic force is not explained by Maxwell's electrodynamics, but only postulated because it already occurs at ideal point charges. Maxwell's electrodynamics does not provide a deeper reason for the occurrence of the magnetic force.

Quantino theory, on the other hand, assumes that the relativistic deformation of the electric field is the true cause of magnetism. Also in Maxwell's electrodynamics the potential for a real explanation of magnetism is already present. However, the important term describing the relativistic deformation of the electric force only comes into effect when we symmetrize the formula and remove the correction term, which became necessary to compensate the missing relativity. The price that has to be paid for this adjustment is that the electric charge is no longer relativistically invariant. That this is not necessarily a disadvantage will become clear in the following sections. However, first of all it should be proven that for the force (2.2.1.4) all laws of conservation are valid.