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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

$$\vec{F}_{R}(\vec{r},\vec{u}-\vec{v}) = \zeta_R(\vec{u}-\vec{v},\vec{r})\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}\,\vec{r}$$ | (2.2.1.1) |

$$\zeta_R(\vec{w},\vec{r}) = 1 + \frac{w^2}{c^2} - \frac{3\,\left(\vec{r}\,\vec{w}\right)^2}{2\,c^2\,r^2} + \mathcal{O}(w^3).$$ | (2.2.1.3) |

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) |

**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.

**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) 2.2.1.2) and get 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.

$$\vec{F}_{RCT}(\vec{r}) = \int\limits_{-\infty}^{+\infty} \vec{F}_{RC}(\vec{r}-\vec{s},\vec{u},\vec{v},\lambda,q_d)\,\mathrm{d}\vec{s}.$$ | (2.2.3.1) |

$$\vec{I} = \lambda\, \vec{u}.$$ | (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

$$\vec{F}_{RCT}(\vec{r}) = \frac{q_d\,\mu_0\,I}{2\,\pi\,r_z}\,\left(\begin{matrix}v_z \\ 0 \\ -v_x\end{matrix}\right),$$ | (2.2.3.5) |

### 2.2.4 The field of arbitrary conductor loops in quantino theory

- Figure 2.2.4.1: conductor loop

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

$$\int\limits_{-L}^{+L} f(x,L)\,\mathrm{d}x \approx 2\,L\,f(0,0) + 2\,L^2\,\left.\frac{\partial}{\partial L}f(0,L)\right|_{L=0}.$$ | (2.2.4.2) |

$$\vec{F}_{RCT}(\vec{r}) = \frac{L^2\,\lambda\,q_d\,\mu_0\,u}{\pi}\,\left(\vec{v} \times \frac{\vec{e}_z\,r^2 - 3\,\vec{r}\,(\vec{e}_z\,\vec{r})}{r^5} \right).$$ | (2.2.4.5) |

$$\vec{F}_{RCT}(\vec{r}) = q_d\,\left(\vec{v} \times \frac{\mu_0}{4\,\pi}\,\frac{3\,\vec{r}\,(\vec{\mu}\,\vec{r}) - \vec{\mu}\,r^2}{r^5}\right).$$ | (2.2.4.6) |

$$\vec{B}(\vec{r}) = \frac{\mu_0}{4\,\pi}\,\frac{3\,\vec{r}\,(\vec{\mu}\,\vec{r}) - \vec{\mu}\,r^2}{r^5}$$ | (2.2.4.7) |

$$\vec{F}_{RCT}(\vec{r}) = q_d\,\left(\vec{v} \times \vec{B}\right).$$ | (2.2.4.8) |

- 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.

$$\nabla\vec{B} = 0$$ | (2.2.4.9) |

$$\nabla\times\vec{B} = \mu_0\,\vec{j}$$ | (2.2.4.10) |

### 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.