The Importance of Weber–Maxwell Electrodynamics in Electrical Engineering

Weber–Maxwell electrodynamics combines classical Weber electrodynamics and Maxwell’s equations, including all four field equations and the Lorentz force, into a single de facto equivalent 3-D wave equation. From classical Weber electrodynamics, Weber–Maxwell electrodynamics inherits properties in which the concept of the magnetic field is unnecessary, and Newton’s third law is satisfied under all circumstances. From Maxwell’s electrodynamics, Weber–Maxwell electrodynamics inherits the ability to be compatible with electromagnetic waves. This article shows that in Weber–Maxwell electrodynamics, all conservation laws are satisfied, and that electromagnetic waves in an isolated system do not possess energy and momentum, but only mediate them between particles of matter. Furthermore, the article shows that the modern formulation of Weber electrodynamics is clearly superior to standard electrodynamics in electrical engineering, because it not only eliminates internal contradictions, but also represents considerable simplification and compression.


I. INTRODUCTION
W EBER electrodynamics is a theory of electromagnetism from before the time when Maxwell's equations and special relativity were developed; it dates back to the works of A.-M. Ampère, W. Weber and C. F. Gauss in the middle of the 19th century [1], [2], [3]. Weber electrodynamics is now considered obsolete, although it has some remarkable and highly attractive features that are absent in Lorentz-Einstein electrodynamics.
The most important of these features is that, in Weber electrodynamics, the concept of magnetic field is obsolete, because Weber electrodynamics generalizes the electrostatic Coulomb force to a velocity-dependent electrodynamic force. A highly important and essential aspect of the Weber force is that, despite this modification, it remains a central force that satisfies Newton's third law.
In contrast, the Liénard-Schwarzschild force [4,Eq. (26)], which follows from the canonical application of Maxwell's equations and Lorentz force to two point charges, violates Newton's third law, even at nonrelativistic velocities. Consequently, at everyday speeds, a formal violation of the law of conservation of momentum occurs. Wesley [5], for instance, has addressed this subject: It should be remarked that a failure to obey Newton's third law is a very serious matter; as it implies drastic consequences, such as the violation of the conservation of energy, the ability to propel a space craft using only forces internal to the space craft itself, and the ability to lift oneself by one's own boot straps. Later, J. P. Wesley showed that Lorentz-Einstein electrodynamics allows for absurd conclusions, even for standard engineering problems. Slightly sarcastically, he writes: Depending upon how one chooses portions 1 and 2, one can obtain a nonvanishing force with any value at all (within limits). Such a loop would be very convenient to drive an automobile or propel a spaceship.
One could obtain the desired magnitude of the force without having to change anything physically; one need only alter the mathematical labels. Experienced electrical engineers know that Lorentz-Einstein electrodynamics must be used with caution, and they instinctively know the limits of the theory. However, from a scientific point of view, this situation is unsatisfactory, especially because Lorentz-Einstein electrodynamics repeatedly tempts inexperienced practitioners to think about perpetual motion machines or to research propulsion devices that violate the conservation of momentum.
Physicists have also long studied the intrinsic contradictions of Lorentz-Einstein electrodynamics. For example, the missing momentum has often been argued to be carried by the radiation field [6]. However, good arguments in the literature have shown that this explanation is not sufficient [7], [8], [9]. Furthermore, experiments have investigated these aspects and demonstrated that, in the low-energy domain, Weber electrodynamics not only explains the behavior of nature in a simpler and more satisfactory manner, but also provides better predictions [10], [11], [12], [13], [14].
Unfortunately, Weber electrodynamics has two critical drawbacks that severely limit its practical value in electrical engineering and modern physics. One is the lack of connection with Maxwell's equations and the absence of a field concept, thus disqualifying Weber electrodynamics from use in all fields of electrical engineering involving electromagnetic waves and their propagation. In addition, classical Weber electrodynamics cannot be applied at relativistic velocities, thereby rendering it largely useless for particle physics, although it would actually be predestined for work with point charges. In addition, its value in atomic and plasma physics is low, although some interesting approaches have been reported [15], [16], [17], [18].
The two aforementioned drawbacks are substantial and are clearly the reason why Weber electrodynamics, despite its elegant concept and superior properties, never became accepted in the low-energy domain and nearly fell into obscurity. However, the first drawback exists only if Weber electrodynamics is used in its original form from the middle of the 19th century, i.e., before the introduction of Maxwell's equations. Indeed, Weber electrodynamics can also be generalized to Weber-Maxwell electrodynamics and can be formulated as a field theory in which Maxwell's equations are compressed and summarized into a single wave equation [19]. That this is possible demonstrates that Weber electrodynamics is as closely related to Maxwell's equations, as is canonical Lorentz-Einstein electrodynamics. Furthermore, the fusion of Weber electrodynamics with Maxwell's equations eliminates one of the two fundamental disadvantages of Weber electrodynamics.
This article is primarily aimed at showing that the conservation laws hold in Weber-Maxwell electrodynamics. In particular, in Weber-Maxwell electrodynamics, the radiation field is demonstrated to have neither energy nor momentum, and the electromagnetic field serves only as a mediator. Why this finding does not represent an experimental contradiction is also explained.
In conclusion, Weber-Maxwell electrodynamics is shown to be a theory that is clearly superior to standard electrodynamics in the nonrelativistic domain, because the concept of the magnetic field is no longer necessary, and the conservation laws are guaranteed to be valid even in the presence of electromagnetic waves. Moreover, from a purely practical point of view, Weber-Maxwell electrodynamics has considerable advantages because it allows engineering problems to be solved quickly and easily through numerically integrating the solutions for point charges or point antennas along arbitrary conductor paths or antenna structures. 1 Therefore, Weber-Maxwell electrodynamics possesses all the properties expected from a good theory in electrical engineering.

II. MODERN FORMULATION OF WEBER
ELECTRODYNAMICS In contrast to standard electrodynamics, Weber electrodynamics is a theory describing the force between two moving point charges. Terms such as current density, charge density, and magnetic or electric field strength do not appear. Instead, a formula is given for the electromagnetic force between a force-emitting point charge and a force-absorbing point charge. The original formulation of Weber electrodynamics is not yet based on a differential equation; consequently, only oscillations, but no wave phenomena, can be represented by classical Weber electrodynamics.
However, classical Weber electrodynamics can be combined with Maxwell's equations and expressed with the wave equation [19] □ F(r, t) = G(r + r ds (t),ṙ ds (t)) with and F ds (t) = F(0, t).
In words, (1), (2) and (3) describe the relation of the force F ds exerted by the point charge q s on another point charge q d by expressing how the force depends on the time-varying distance vector r ds . The Lorentz factor is defined as usual by δ(s) is the Dirac delta function in three dimensions. The vector r ds represents the difference or distance vector of the two trajectories r d and r s The time derivative of the difference vector of the two trajectories is consequently the difference velocity or relative velocityṙ The general solution of (1) is well known (e.g., [19,Eq. (33)]) and reads, since r is zero Note that F ds (t) has no explicit dependence on the location r, but can nevertheless represent a field. To understand this, one needs to realize that an electromagnetic field represents the force that a test charge q d would perceive if it were at the corresponding location. The field in the conventional sense at time t = 0 is therefore obtained, for example, by varying the location r ds (t = 0) := r 0 and setting velocityṙ ds (t = 0) := v to a specific value. One can also plot such fields. In the following, we will make use of this option several times.
Interpretatively new and unfamiliar about Weber-Maxwell electrodynamics is that the electromagnetic field is no longer thought of as something that exists independently of the involved point charges. Instead, each receiver perceives the same field in a different way according to its relative velocity.
(1) Summarizes all four Maxwell equations and the Lorentz force. In the form (1), it differs from the wave equation of Lorentz-Einstein electrodynamics in the rest frame of the receiver by only an additional Lorentz factor. However, this factor has no effect on the time behavior of the electromagnetic waves. The purpose of the additional Lorentz factor is only to re-normalize the force such that the concept of the magnetic field becomes obsolete. Compared with canonical electrodynamics, with its 4-D electromagnetic field tensor, this equation represents an immense simplification, which should not be underestimated.
From (1), both of Einstein's postulates are clearly fulfilled, because the force F ds depends on only the difference vector r ds and the difference velocityṙ ds . Both are relative quantities, and the velocity of an observer does not enter the wave equation. Consequently, the principle of relativity is concluded to be satisfied, and the force has exactly the same value for any inertial observer. Furthermore, the d'Alembert operator shows that the force between the point charges q s and q d propagates with velocity c independently of their relative velocityṙ ds .
For approximately uniformly moving point charges, (1) has the solution [19, Eq. (54)] For ∥v∥ ≪ c, F W becomes the classical Weber force (27), as will be shown below, and for ∥v∥ = 0, the Coulomb force is obtained.
For illustration, Fig. 1 shows the shape of the force for r d (t) = r 0 + v t and r s (t) = 0 at time t = 0. Therefore, r ds (0) = r 0 andṙ ds (0) = v. Fig. 1 demonstrates that the force (9), although it has no explicit dependence on the location r, is nonetheless a field in terms of the parameter r 0 . Fig. 1 indicates furthermore that the force is elliptically deformed but nevertheless remains a central force. Fig. 2 shows the force component that results when the Coulomb force is removed from the force in Fig. 1. Because the Coulomb force has no time dependence, the remainder is the velocity-dependent part and hence the component correlated with the magnetic force. However, splitting into magnetic and electric components is not usually performed in Weber electrodynamics, since that is only necessary when the forces between magnetic materials and electric charge carriers or the forces between permanent magnets have to be calculated.
Note that in Weber electrodynamics an elementary permanent magnet can be modeled as a small circular current with infinitesimal loop diameter. It can be shown that such an infinitesimal current loop corresponds to a magnetic dipole. A magnetic dipole, in turn, can be interpreted as being composed of two virtual, oppositely charged magnetic monopoles. The force exerted by such a virtual magnetic monopole on another likewise resting magnetic monopole formally corresponds to a force law that resembles Coulomb's law. This means that one can also define a magnetic induction B in Weber electrodynamics, and it turns out to be exactly the same as that found in Maxwell's equations. Unfortunately, showing this in detail is beyond the scope of this article.
(1) Provides not only the classical Weber force as a solution but also enables analysis of considerably more complex problems. In particular, the electromagnetic field F ds of a point antenna, which may also move with high relative velocity, can be calculated. The calculated solutions are consistent with the solutions in textbooks, provided that the point antenna and Fig. 1. Electromagnetic force of a point charge located at the coordinate origin is shown from the perspective of a test charge moving with relative velocity v = c/2 e x . As indicated, the force is elliptically deformed with respect to the Coulomb force, but it is nonetheless a central force. Fig. 1 is shown, but all force components not depending on the velocity (Coulomb force) have been removed. The residual shown is that part which is correlated with the magnetic force in Weber electrodynamics. receiver are at rest relative to each other. For a fast-moving point antenna, solutions are automatically obtained that clearly indicate that the force between the point charges propagates at the speed of light. The Lorentz transformation is not required. In a later section, the field of the point antenna is discussed in more detail.

Fig. 2. Same situation as in
As can easily be seen, the force (10) satisfies Newton's third law F W (q d , q s , r, v) = −F W (q s , q d , −r, −v), because when the source and the receiver are swapped, the force only changes its sign. It is very important that this aspect is true even in general, i.e., when electromagnetic waves are present, or when the two-point charges are far from each other, so that the finite propagation velocity of the force is relevant. This can be proven by exchanging the source and recipient of the force in (1), because with aid of the relations ∇ δ(r ds ) = −∇ δ(r sd ), δ(r ds ) = δ(r sd ), γ(ṙ ds ) = γ(ṙ sd ) andṙ ds = −ṙ sd , it follows that: That Newton's third law (12) is satisfied even in the presence of electromagnetic waves is highly important for the validity of the conservation laws. In particular, it can be shown that in Weber-Maxwell electrodynamics, electromagnetic waves have no energy and no momentum, but mediate both. This strongly distinguishes Weber-Maxwell electrodynamics interpretationally from Lorentz-Einstein electrodynamics, wherein energy and momentum must be assigned to the waves themselves. This is true even if these waves have traveled very long distances. Importantly, in Weber-Maxwell electrodynamics, the total energy and total momentum of all particles in the universe are conserved quantities without the electromagnetic field. However, if only a small part of the system, consisting only of the receiver and the incident electromagnetic wave without the transmitter, is considered, one must of course also assign energy and momentum to the wave, because the system without the transmitter does not represent an isolated system. This can also be seen in a scenario where an electric charge at rest is suddenly accelerated for a short instant. It is immediately obvious that the radiated electromagnetic field cannot have any effect on another charge located far away. One could now argue that only the electromagnetic field can have absorbed the change of energy and momentum, since the state of motion has changed for the first charge, but not for the second. This ostensible paradox can be solved by realizing that in an isolated system electric charges do not accelerate without reason, but can only change their state of motion due to internal forces. In other words, the whole system is in a dynamic equilibrium and a charged particle changes its state of motion only because somewhere else and at some time in the past other charges have changed their state of motion.
Ultimately, and this is really remarkable, total energy, total momentum, and total angular momentum of all charges are already conserved quantities without the electromagnetic field. This will be shown in Section III.

III. PROOF OF THE CONSERVATION LAWS
The following section formally proves that in Weber-Maxwell electrodynamics in isolated systems, energy, momentum, and angular momentum are conserved quantities. This proof is presented for several reasons. The first reason is to show that J. C. Maxwell and H. Helmholtz [20, p. 396-397] were incorrect in their conjecture that Weber electrodynamics violates the conservation of energy. The second reason is to clarify that conservative forces may also depend on the relative velocity and relative acceleration, and that the concept of potential energy can easily be applied to such generalized conservative forces. Finally, the requirements to guarantee compliance with the conservation laws are identified. This aspect is important, because whether and how Weber-Maxwell electrodynamics is suitable for the relativistic domain, and how the formulas for kinetic energy and momentum must be adapted are not yet clear.

A. Potential Energy in Weber Electrodynamics
We start with the total energy E of a two-particle system where U ds = U sd is the potential energy possessed by the point charge q d in the field of the point charge q s . T s and T d are the kinetic energies of the two-point charges. Of note, for the kinetic energy, the relative velocityṙ ds is not used; instead, the velocities from the perspective of a laboratory system are used. Consequently, the kinetic energies as well as the total energy E usually have different values in each frame of reference. This aspect is not a property of Weber electrodynamics but is a basic characteristic of Newtonian mechanics. If the energy E is to be a conserved quantity, then the time derivative of (13) must disappear. For this reason, we insert (14) and calculate the derivative. In this way, we obtain the following equation: A comparison with Newton's second law shows that (15) is equivalent to Because in Weber-Maxwell electrodynamics, Newton's third law (12) is always valid, it follows that: which in turn can be rewritten as because of the definition (5). This equation is fulfilled if the relation is valid, as can be easily verified by inserting it into (19).
Therefore, the conservation of energy in the two-particle system is guaranteed if the force (9) can be expressed with (20) based on potential energy U ds . In fact, such a formula exists with where Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
is the potential energy of a point charge q d at rest in the field of another resting point charge q s . Note that in Weber-Maxwell electrodynamics, potential energy and the scalar potential must be clearly distinguished. That (21) together with (22) in fact provide a valid definition of the potential energy can be verified through substitution into (20) and calculation of the derivative. This process gives the force with κ := γ(ṙ ds ) 4 c 4 r ds ·ṙ ds r ds ·ṙ ds + γ(ṙ ds ) 2 c 2r ds · r ds (25) which forr ds ≈ 0 corresponds to the solution (10) of the wave equation for uniformly moving point charges. Furthermore, the force (24) is for small relative velocities ∥ṙ ds ∥ ≪ c equivalent to the Weber force (27). To provide a demonstration, the substitutionṙ ds → vṙ ds is performed, and the resulting equation with respect to the auxiliary scalar variable v is developed into a Taylor series of second order.
If v = 1 is set, the following is obtained: Because the term with the factor 1/c 4 can be neglected, this corresponds exactly to the classical Weber formula, which in vector notation reads The previous derivations show that a potential energy (22) can be defined for which, together with the kinetic energy (14) and Newton's second law (16), the law of conservation of energy is satisfied. Furthermore, the potential energy (22) is clearly compatible with classical Weber electrodynamics and (1) obtained from Maxwell's equations. The latter is not the case for alternative definitions of the potential energy in Weber electrodynamics [21]. Moreover, the potential energy (22), like the force (24), is recognized to be a purely relative quantity that does not depend on the velocity of a uniformly moving observer.

B. Conservation of Energy
We now prove that in Weber-Maxwell electrodynamics, the total energy is a conserved quantity even if the system consists of n particles, where n may be arbitrarily large. To provide a demonstration, we write the total energy E of the system as a sum of all kinetic and potential energies The n equations of motion (16) are for k = 1, . . . , n n i=1 F ki = m kr k .
We define the force F ii of the ith particle on itself as zero to avoid handling exceptions in the summation. Multiplication byṙ k yields The time derivative of the kinetic energy of the kth particle is due to (14) i.e., (30) can be rewritten as The sum of all n equations of motion gives By formally manipulating the summation indices, we obtain Because of Newton's third law (12), F ik = −F ki applies. Using this and the relationṙ ki =ṙ k −ṙ i , (34) can be simplified to But because of (20), it follows that: This allows (35) to be written as However, because of the symmetry of the potential energy U ki = U ik , the equation However, this corresponds to the time derivative of the total energy, as can be recognized by comparison with (28). Clearly, E = 0 applies, thus indicating that the total energy E is a conserved quantity.

C. Conservation of Momentum
For an isolated system of n point charges, the temporal change in the momentum p k of the kth point charge is equal to the sum of all forces acting on this point particle. Thereforė provided that F ii := 0 is defined. The temporal change in the total momentum p of the system is the sum of the temporal momentum changes of all point particles. For this reasoṅ Again, Newton's third law (12) is ensured to be valid under all circumstances, and it follows that: By formally renaming the summation indices, the following is then obtained: (43) which can be true only ifṗ is exactly zero. Therefore, the total momentum p must be a conserved quantity.
Furthermore, the sum of all forces acting on the matter particles disappears in an isolated system. Consequently, electromagnetic waves themselves do not have any momentum. Instead, the momentum is a property that only particles of matter can possess. The electromagnetic force, in contrast, is only the mediator. Nevertheless, in a subsystem of an electromagnetic wave and a receiver of the force without the source, the electromagnetic wave seems to possess momentum, because it causes a momentum change at the receiver. However, one must not forget that in another location, the source of the wave also simultaneously experiences a compensating momentum change. Electromagnetic waves therefore have momentum only in non-isolated systems.

D. Conservation of Angular Momentum
The conservation of angular momentum L can also be demonstrated. The total angular momentum of a system of n point charges is defined by The derivative with respect to time yieldṡ Because the momentum p k is always parallel to the velocitẏ r k , it follows thatṙ k × p k = 0. Thus, (45) becomeṡ Substituting (40) giveṡ This can be further rewritten aṡ Because of Newton's third law (12), it follows that: i.e.,L applies. However, the Weber force (10) is a central force, and r ki × F ki = 0 holds. Thus, we obtaiṅ This shows that the total angular momentum of an isolated system must also be a conserved quantity.

E. Necessary Features of Valid Dynamics
The previous sections show which features are needed for which steps in the proofs of the conservation laws. Of note, Newton's second law and the kinetic energy could also be defined such that they would converge to the classical versions for small relative velocities, on the one hand, and would become compatible with the empirical results of high-energy physics, on the other hand. We must only ensure that the presented proofs retain their validity. Therefore, certain conditions must be met, which should therefore be identified and summarized.
The following features are central to the proofs of the conservation laws.
1) Newton's third law (12): this property is necessary for the proof of all three conservation laws. 2) The time derivative of the kinetic energy must be equal to the scalar product of time derivative of momentum and velocity: this property is needed in the step from (15) to (17), i.e., to prove the conservation of energy.
3) The momentum must be parallel to the velocity: this is required in the proof of conservation of angular momentum. 4) The force formula between two point charges must be a central force: this is needed in the proof of conservation of angular momentum. As becomes clear, the last two features are needed only in the proof of the conservation of angular momentum. The property 4) in Weber electrodynamics is satisfied only if the two-point charges are so close to each other that the finite propagation speed of the force is unimportant. The conservation of angular momentum is therefore the weakest of the three conservation laws. Conservation of momentum, in contrast, is always guaranteed in every case in Weber-Maxwell electrodynamics. Somewhat less clear is the conservation of energy, because the possibility of defining a potential energy is required.
Of note, a fourth conserved quantity exists in Newtonian mechanics, which is referred to as the Runge-Lenz vector. This conservation law is satisfied for only very simple force laws, such as Coulomb's law or Newton's law of universal gravitation. In fact, measurements show that the Runge-Lenz vector is often not a conserved quantity in real physical systems [22].

IV. MAGNETISM IN WEBER-MAXWELL ELECTRODYNAMICS
Because this article is also intended for readers unfamiliar with Weber electrodynamics, a brief discussion of the question of why the concept of magnetism is obsolete in Weber electrodynamics follows. We begin with a simple example and consider an infinitely long straight wire on the x-axis. Let an electric current I flow in this wire. Furthermore, somewhere on the z-axis at z, let there be a point-like test charge q moving with velocity v = v x e x + v y e y + v z e z .
In standard electrodynamics, an infinitely long wire in which a current I flows is surrounded by a magnetic field B = −(µ 0 /2 π z) I e y . The force on the test charge q is obtained using the Lorentz force F = q E + q v × B. Because the wire is electrically neutral, and therefore the electric field E vanishes, the force is The same result can be obtained with Weber electrodynamics. To provide a demonstration, we model the current I with resting metal ions +e and an equal number of moving electrons of charge −e, which move with an average drift velocity of u = −u e x . Because the electrons drift to the left, the technical direction of the current points to the right. The current strength of the resulting current is I = e n u, where n represents the number of moving electrons per meter of the conductor length.
In contrast to Lorentz-Einstein electrodynamics, magnetism in Weber electrodynamics is not a fundamental force but is a multi-particle effect, i.e., the net force of all individual forces between the test charge q and the charge carriers inside the wire. Expressed as an integral, the resulting force is where F W is the Weber force (10) in modern representation. This integral can be solved exactly, and the following approximation can be obtained: for very small electron drift velocities u ≪ c. This in turn becomes identical to (52) when e n u = I and c 2 ε 0 = 1/µ 0 is used, provided that equation (54) regarding ∥v∥ := v is expressed as a Taylor series of second order. Of note, the Fechner hypothesis is indeed not necessary [2, p. 87-88]. Moreover, it is irrelevant with which velocity an observer moves relative to the wire and to the test charge. This aspect is not the case in Lorentz-Einstein electrodynamics. The simple example above illustrates that magnetism is not a fundamental force from the viewpoint of Weber electrodynamics. In fact, the concept of the magnetic field is obsolete not only in the case of an infinitely long wire, but in general, since elementary permanent magnets, as already mentioned, can be modeled as infinitessimal current loops. The fact that magnetic fields are no longer needed at the elementary level represents a significant conceptual simplification, even though solving integrals of the type (53) may seem somewhat tedious.
However, this integration method allows every practical problem to be solved numerically. This aspect is also true for high-frequency alternating current, because the solution of (1) for the point antenna 2 can be used: In (55), s(t) is the time-dependent infinitesimal displacement of the charges +q and −q in a compound particle located at the coordinate origin. The time constant τ is given by thereby ensuring that the electromagnetic wave travels at speed of light c in the rest frame of any test charge q d moving with relative velocity v at location r 0 . Fig. 3 shows the field of the force of a point antenna polarized in the z-direction at the coordinate origin, from the perspective of a test charge q d resting relative to the point antenna somewhere in the x-z plane. As can be seen, the field of the force corresponds exactly to the electric field of the Hertzian dipole, as indicated in many textbooks.
The situation is different when the test charge is moving. Fig. 4 shows an example of the same field from the perspective of a point charge moving with speed c/3 to the right. As clearly indicated, a Doppler effect now occurs. If the test charge is located at the left side of the point antenna, a blueshift occurs, because the test charge moves toward the point antenna. On the right side of the point antenna, in contrast, a redshift is seen.
For basic understanding, an important and critical aspect is that this field is real only for a moving point charge. A stationary observer would perceive the same field as in Fig. 3. However, the observer could measure the acceleration of the test charge and indirectly infer the field that is actually perceived by a moving test charge, which would correspond   Fig. 3. Here, however, the test charge moves to the right with speed c/3. exactly to the field in Fig. 4 and not to the field measured by the observer.
The Figs. 3 and 4 show the electromagnetic forces on a test charge in its entirety; consequently, the effects that would resemble the Lorentz force are difficult to recognize. However, the magnetic effects are also very real in Weber-Maxwell electrodynamics, even though the concept of the magnetic field is no longer needed. To illustrate this aspect, Figs. 5 and 6 show the force components that remain when all components of the electromagnetic force that do not depend on the velocity are subtracted. 3 In this way, the counterpart of the magnetic force q d v × B in Weber-Maxwell electrodynamics is obtained. Fig. 5. Only the velocity-dependent force components are shown. The test charge moves to the right with speed c/3. In the x-y plane, i.e., transverse to the axis of oscillation, the force acts perpendicularly to the direction of motion. Fig. 6. Only the velocity-dependent force components are shown. The test charge moves upward with speed c/3. In the x-y plane, i.e., transverse to the axis of oscillation, the force acts perpendicularly to the direction of motion. Fig. 5 shows this velocity-dependent force component for a test charge q d moving to the right. As can be seen, the force component in the x-y plane agrees exactly with the expectation according to Lorentz-Einstein electrodynamics, because the moving test charge experiences a force that is oriented exactly perpendicular to the direction of motion. The same is true for Fig. 6. Here, the test charge is moving in the z-direction. As can be seen, the force also acts in the x-y plane orthogonally to the direction of motion of the test charge.
However, the equivalence to the Lorentz force is present in only the x-y plane, i.e., transverse to the axis of oscillation. Above and below, in contrast, the predictions of Weber-Maxwell electrodynamics and Lorentz-Einstein electrodynamics are partly diametrically different. However, these differences often do not matter in practice, because an alternating current carrying wire can be interpreted as a chain of many point antennas in the direction of oscillation. Therefore, in the examples in Figs. (3)-(6), the fields of additional point antennas must be imagined on the z-axis. The fields of all point antennas superimpose, and the force components typical of Weber electrodynamics are canceled out because of the symmetry.
This symmetry is the reason why Lorentz-Einstein electrodynamics and Weber-Maxwell electrodynamics are de facto equivalent for closed conductor loops. However, in many applications in electrical engineering, this equivalence is not the case. For example, a capacitor represents a gap in a conductor loop. Indeed, forces between the plates can be calculated with Lorentz-Einstein electrodynamics, which cannot be real, because they violate the conservation laws. However, if the force effects are calculated with Weber-Maxwell electrodynamics, realistic forces are always obtained that satisfy the conservation laws and agree perfectly with the measurement results in reality.

V. SUMMARY AND CONCLUSION
The article has shown that the modern formulation of Weber electrodynamics by means of (1) is a theory of electromagnetism that, in the nonrelativistic domain, is clearly superior to standard electrodynamics, because the concept of the magnetic field becomes completely obsolete, and the conservation laws are ensured to hold even in the presence of electromagnetic waves. Furthermore, Weber-Maxwell electrodynamics is also considerably more compact and decreases the necessary number of fundamental equations to a single equation.
The last field of application in which Lorentz-Einstein electrodynamics seems necessary is when electric charges move with velocities close to c. This scenario is usually the case only in designing particle accelerators or nuclear fusion research. How Weber-Maxwell electrodynamics can be applied in such extreme applications is currently an open question and a subject of current research.
Importantly, in the field of electrical engineering, Weber-Maxwell electrodynamics allows many types of problems to be rapidly and easily solved; applications include electrical power engineering, radio frequency engineering, plasma physics, electronics, and computational electromagnetics. In particular, the basic solution for a uniformly moving point charge in the case of direct current, or a point antenna in the case of alternating current, can be numerically integrated along the current paths. Therefore, the need to solve Maxwell's equations, which not only requires experience in dealing with partial differential equations, but also can lead to paradoxical and unrealistic results, is eliminated. Weber-Maxwell electrodynamics, in contrast, ensures that the solutions always satisfy the conservation laws and are consequently reasonable and realistic. The dissertation addressed the feasibility of generating models in engineering when the available information is uncertain or the available data are noisy or unreliable. He is currently working in industry as an Electronics and Embedded Software Engineer. Furthermore, he is also a Developer of an economical speech-controlled home automation technology that functions without internet connectivity and thus cannot be remotely compromised. His current research interests include the physical foundations of electrical engineering.