2 Comparison of electrodynamics theories

2.1 Maxwell's electrodynamics

Maxwell's electrodynamics is currently the standard electrodynamics and comprises (for vacuum) four differential equations

$$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0},$$

(2.1.1)

$$\nabla\cdot\vec{B} = 0,$$

(2.1.2)

$$\nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t},$$

(2.1.3)

$$\nabla\times\vec{B} = \mu_0\,\vec{j} + \frac{1}{c^2}\,\frac{\partial\vec{E}}{\partial t}$$

(2.1.4)

and the supplementary formula

$$\vec{F} = q_d\,\vec{E} + q_d\,\vec{v}\times\vec{B}.$$

(2.1.5)

The formula (2.1.5) is known as the Lorentz force. The equation provides the force $\vec{F}(\vec{r},\vec{v},t)$, which is caused by the electric field $\vec{E}(\vec{r},t)$ and the magnetic field $\vec{B}(\vec{r},t)$ on a point-like test charge $q_d$, which is located at time $t$ at location $\vec{r}$. The fields $\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ are quantities that can be calculated by inserting the charge density $\rho(\vec{r},t)$ and current density $\vec{j}(\vec{r},t)$ and solving the differential equation system (2.1.1) to (2.1.4). However, solving systems of partial differential equations requires considerable experience and mathematical skills and is usually only possible for simple problems.

The velocity $\vec{v}$ is of particular importance. Usually, one uses for $\vec{v}$ the velocity of the test charge $q_d$ in the reference frame of the laboratory, i.e. in the same frame in which the field-generating device is located. Since electrical currents in metallic conductors consist of many charge carriers that only move extremely slowly, $\vec{v}$ in electrical engineering is almost always identical to the relative speed between the test charge $q_d$ and the field-generating charges in the wires and metallic conductors.

The fields $\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ are auxiliary quantities, since in the end only the force $\vec{F}(\vec{r},\vec{v},t)$ can be measured. It is a convention that the quantity $\vec{F}(\vec{r},\vec{v} = \vec{0},t)/q_d$ is referred to as the electric field strength $\vec{E}(\vec{r},t)$. The remaining part of the Lorentz force (2.1.5) is the magnetic component.

Note that the magnetic field $\vec{B}$ is a quantity that was introduced to describe the force between permanent magnets. However, Oersted and Ampere discovered at the beginning of the 19th century that these magnetic forces are caused by electric currents [16]. This also applies to permanent magnets, as they contain numerous small circular currents. The magnetic field was therefore actually already an outdated concept around 1850. Unfortunately, it was reintegrated into electrodynamics due to the success of Maxwell's equations in describing electromagnetic waves.

2.2 Weber electrodynamics

Weber electrodynamics is a very old electrodynamics from the middle of the 19th century [3] [4] [5] [6], i.e. from a time before the discovery of electromagnetic waves. Weber electrodynamics is a very compact and elegant representation of the scientific knowledge of that time in terms of a single equation known as the Weber force.

According to the state of knowledge at that time, it was believed that the electromagnetic force generated by a point charge $q_s$ onto another point charge $q_d$ could be fully described by the formula

$$\vec{F} = \frac{q_s\,q_d}{4\,\pi\,\varepsilon_0}\,\left(1 + \frac{v^2}{c^2} - \frac{3}{2}\left(\frac{\vec{r}}{r}\cdot\frac{\vec{v}}{c}\right)^2\right)\,\frac{\vec{r}}{r^3},$$

(2.2.1)

with $\vec{r}$ being the distance vector

$$\vec{r} := \vec{r}_d(t) - \vec{r}_s(t)$$

(2.2.2)

of the point charges $q_s$ and $q_d$ with the trajectories $\vec{r}_d(t)$ and $\vec{r}_s(t)$. In contrast to the Coulomb force, which is well known even today, the Weber force (2.2.1) also depends on the relative velocity

$$\vec{v} := \dot{\vec{r}}_d(t) - \dot{\vec{r}}_s(t).$$

(2.2.3)

It is interesting and important to emphasize that the Weber force works excellently in the case of direct currents and low-frequency alternating currents and makes it possible to explain some effects that are difficult to understand on the basis of Maxwell's equations alone [17] [18] [19]. Obviously, a magnetic field is not needed for many problems in electrical engineering. Instead, the Weber force shows that the Lorentz force is a multi-particle effect and is created by different relative velocities of the charge carriers in a line current. The Weber force is therefore more than just a mathematical description, as it also represents a compression of knowledge and an interpretative explanation of magnetism.

Unfortunately, the Weber force cannot describe electromagnetic waves, at least not directly and without a transmission medium. This can be recognized immediately by the fact that the Weber force $\vec{F}(\vec{r},\vec{v},t)$ depends only on the locations and velocities of the point charges at time $t$. The Weber force is therefore an instantaneous force that propagates from $q_s$ to $q_d$ without any delay. This contradicts numerous important effects, which unfortunately renders it largely useless for electrical engineering.

2.3 Weber-Maxwell electrodynamics

Weber-Maxwell electrodynamics is - as far as electrical engineering is concerned - equivalent to Maxwell's electrodynamics [20] [21] [2] [22]. However, it is not based on charge and current densities, but works with pairs of point charges, similar to Coulomb's law or the Weber force (2.2.1), both of which are included as special cases.

The formula for the electromagnetic force that a point charge $q_s$ with trajectory $\vec{r}_s(t)$ exerts on another point charge $q_d$ with trajectory $\vec{r}_d(t)$ is

$$\vec{F} = \frac{q_d\,q_s\,\gamma(v)\,\left(\left(\vec{r}\,c + r\,\vec{v}\right)\left(c^2 - v^2 - \vec{r}\cdot\vec{a}\right) + \vec{a}\,r\,\left(r\,c + \vec{r}\cdot\vec{v}\right)\right)}{4\,\pi\,\varepsilon_0\,\left(r\,c + \vec{r}\cdot\vec{v}\right)^3},$$

(2.3.1)

in Weber-Maxwell electrodynamics, whereby

$$\vec{r} := \vec{r}_d(\tau) - \vec{r}_s(\tau)$$

(2.3.2)

is the retarded distance vector

$$\vec{v} := \dot{\vec{r}}_d(\tau) - \dot{\vec{r}}_s(\tau)$$

(2.3.3)

the retarded difference velocity and

$$\vec{a} := \ddot{\vec{r}}_d(\tau) - \ddot{\vec{r}}_s(\tau)$$

(2.3.4)

the retarded difference acceleration. $\gamma(.)$ is the Lorentz factor. In addition, one needs the time $\tau$, which can be calculated iteratively by means of equation

$$\tau = t - \frac{\Vert\vec{r}_d(\tau) - \vec{r}_s(\tau)\Vert}{c}.$$

(2.3.5)

In order to find the value of $\tau$, any initial value (e.g. $\tau = t$) can be chosen and recursively inserted until $\tau$ does not change any longer. The fixed-point iteration converges always as long as the difference speed between the two point charges is less than the speed of light $c$. In somewhat simplified terms, $\tau$ corresponds to the time when the force has left the charge $q_s$ to reach the charge $q_d$ at time $t$.

As one can see, equation (2.3.1) is not a differential equation. If we know the trajectories $\vec{r}_s(t)$ and $\vec{r}_d(t)$ for all times less than or equal to $t$, we can easily calculate the force $\vec{F}(\vec{r},\vec{v},t)$ at time $t$. This can be achieved by means of a simple computer program in Python or C++. Knowledge of vector analysis and differential geometry is not required.

2.4 Relationship between the theories

Maxwell's electrodynamics and Weber-Maxwell electrodynamics are equivalent for non-relativistic velocities, because equation (2.3.1) is the solution of Maxwell's equations for point charges [1]. The proof can be found here. The restriction to the non-relativistic regime is necessary because, strictly speaking, the Weber-Maxwell force only represents the solution of the Maxwell equations for an arbitrarily moving $q_s$ but a resting $q_d$. The generalization to arbitrarily moving $q_d$ was achieved by means of a Galilean transformation. This is only an approved method in the non-relativistic regime. Note, however, that the equation (2.3.1) together with the equation (2.3.5) nevertheless fulfills Einstein's two postulates.

That the principle of relativity is fulfilled can be recognized by the fact that the equation (2.3.1) does not depend on the choice of the reference frame. This means that the formulas of Weber-Maxwell electrodynamics have the same form in every inertial frame and even in every non-inertial frame, because the coordinate transformations only affect $\vec{r}_d(t)$ and $\vec{r}_s(t)$. Furthermore, it can be seen that due to equation (2.3.5) the electromagnetic force cannot propagate faster than light in any frame of reference.

The fact that Einstein's two postulates are satisfied in Weber-Maxwell electrodynamics even without the Lorentz transformation is a major advantage. In contrast, Maxwell's electrodynamics can only be used with the Lorentz transformation, even at low difference velocities. In electrical engineering, this is practically always ignored, which can cause subtle problems, as the solutions are then only approximations that violate the conservation of momentum, for example.

In Weber-Maxwell electrodynamics, however, it is guaranteed that the conservation of momentum applies even at very high velocities. This can be recognized by the fact that in equation (2.3.1), an exchange of source charge and target charge only results in the sign being reversed. This means that Newton's third law is fulfilled.

It is also important that Weber-Maxwell electrodynamics is compatible with Weber electrodynamics, since the approximations $a \approx 0$ and $v \ll c$ lead to the Weber force (2.2.1). When demonstrating the equivalence, it should be noted that $\vec{r}$ is the non-retarded distance vector in Weber electrodynamics and the retarded distance vector in Weber-Maxwell electrodynamics. The proof can be found here.

As one can see, Weber-Maxwell electrodynamics combines the advantages of both philosophies (Maxwell's electrodynamics and Weber electrodynamics) and eliminates the individual drawbacks. Weber-Maxwell electrodynamics is also compatible with the Newtonian electrodynamics of Peter and Neal Graneau [23], as this forms the quasi-static limiting case of Weber-Maxwell electrodynamics. However, Weber-Maxwell electrodynamics is not suitable for speeds close to the speed of light, as it is still necessary to investigate how relativistic mechanics has to be applied. This is the subject of current research.

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