Weber-Maxwell electrodynamics

Weber-Maxwell electrodynamics is a representation of classical electrodynamics expressed in terms of a generalized Coulomb law that can also be applied to moving and accelerated non-relativistic point charges.

Weber-Maxwell electrodynamics is based on exactly the same field equations as Maxwell's electrodynamics. In contrast to Maxwell's theory of electrodynamics, Weber-Maxwell electrodynamics does not define the Lorentz force with the Lorentz force law, but explains Lorentz force and magnetism by means of a hypothesis by Carl Friedrich Gauss.

Field of the electromagnetic force of an accelerated negative point charge from the perspective of a resting positive test charge in Weber-Maxwell or Maxwell electrodynamics.

Weber-Maxwell electrodynamics is largely equivalent to Maxwell's electrodynamics, as it uses exactly the same fields and only describes the effect of the electromagnetic fields on matter slightly differently. For small relative velocities and negligibly small accelerations, it is compatible with Weber electrodynamics. As a result, Weber-Maxwell electrodynamics is also compatible with André-Marie Ampère's original force law for small relative velocities. In contrast to Weber electrodynamics, Weber-Maxwell electrodynamics is also suitable for describing electromagnetic waves in a vacuum, providing practically identical predictions as Maxwell's electrodynamics.

Weber-Maxwell electrodynamics is not suitable for applications in which charged particles move at almost the speed of light. In such cases, it is necessary to use relativistic mechanics.

History

In the years around 1820, André-Marie Ampère carried out numerous experiments with direct current. He summarized his results by means ofAmpère's force law, which describes the force that a short segment of direct current exerts on another short segment of direct current. Since conductor loops with direct current must always be closed and therefore no isolated segments with direct currents exist, Ampere's original force law is only one of several possibilities to express the forces between conductor loops with direct current. For example, the Biot-Savart law, together with Ampère's force law in modern form, is also an equivalent representation for conductor loops with direct current.

Around 1835, Carl Friedrich Gauss noticed that Ampère's original force law can be explained by modifying Coulomb's law. He assumed that Coulomb's law is incomplete and, in addition to its dependence on distance, also contains a dependence on relative velocity. Based on this modification, Wilhelm Eduard Weber developed Weber electrodynamics in the following years. Gauss thus generalized a law that was only applicable to direct currents as point charges, which marked the beginning of the development of a first, still incomplete, electrodynamics.

Another generalization of quasistatics was developed by James Clerk Maxwell. In 1864, he studied whether the Biot-Savart law could also be applied to conductor loops that contain gaps. He found that, in this case, inconsistencies occur. As a solution, he proposed an additional term, which is known today as displacement current. A remarkable consequence of his extension was the prediction of electromagnetic waves in a vacuum.

At first glance, Gauss' and Maxwell's generalizations of quasistatics are incompatible concepts, as they are each based on different force laws for direct current elements. As the electrodynamics of Gauss and Weber around 1865 was still unable to describe electromagnetic waves in a vacuum, research in the following decade focused increasingly on Maxwell's electrodynamics and detection of the postulated electromagnetic waves. Heinrich Rudolf Hertz succeeded in demonstrating the existence of electromagnetic waves in several experiments and articles between 1886 and 1889. This soon resulted in Weber's electrodynamics being considered obsolete.

Several years later, experiments such as the Michelson-Morley experiment began to indicate that Maxwell's equations in combination with the Lorentz force law are not completely correct, as they describe wave propagation in a propagation medium. However, detection of this propagation medium failed, which led to development of the Lorentz transformation, which culminated in 1905 with Albert Einstein's special theory of relativity. Weber electrodynamics was no longer taken into account in these developments.

Ampère's law in its original form, Gauss's interpretation of magnetism, and Weber's electrodynamics saw a minor renaissance around 1990 due to the work of J. P. Wesley and A. K. T. Assis. A complicating aspect was that many works by Ampère, Coulomb, Gauss, and Weber had never been translated into English. This has only been achieved in recent years by A. K. T. Assis and others.

Weber-Maxwell electrodynamics is a relatively new development and shows that Weber electrodynamics is fully compatible with Maxwell's equations. It integrates Gauss's interpretation of magnetism into Maxwell's electrodynamics by calibrating the solution of Maxwell's equations for arbitrarily moving point charges calculated by Oleg D. Jefimenko in such a way that it becomes compatible with Gauss's hypothesis.

Mathematical representation

Generalized Coulomb force

In Weber-Maxwell electrodynamics, the electromagnetic force $\mathbf {F}$ that a point charge $q_{s}$ with the trajectory $\mathbf {r} _{s}(t)$ exerts on another point charge $q_{d}$ with trajectory $\mathbf {r} _{d}(t)$ at time $t$ is given by

$$\mathbf {F} ={\frac {q_{d}\,q_{s}\,\gamma (v)}{4\,\pi \,\varepsilon _{0}}}\left({\frac {\left(\mathbf {r} \,c+r\,\mathbf {v} \right)\,\left(c^{2}-v^{2}-\mathbf {r} \cdot \mathbf {a} \right)}{\left(r\,c+\mathbf {r} \cdot \mathbf {v} \right)^{3}}}+{\frac {\mathbf {a} \,r}{\left(r\,c+\mathbf {r} \cdot \mathbf {v} \right)^{2}}}\right).$$ (1)

Here

$$\mathbf {r} :=\mathbf {r} _{d}(\tau )-\mathbf {r} _{s}(\tau ),$$ (2)

is the retarded distance vector,

$$\mathbf {v} :={\dot {\mathbf {r} }}_{d}(\tau )-{\dot {\mathbf {r} }}_{s}(\tau )$$ (3)

the retarded relative velocity and

$$\mathbf {a} :={\ddot {\mathbf {r} }}_{d}(\tau )-{\ddot {\mathbf {r} }}_{s}(\tau )$$ (4)

the retarded relative acceleration. $\gamma (v)$ is the Lorentz factor. Bold letters are vectors, regular letters represent the corresponding Euclidean distance, for example $r=\Vert \mathbf {r} \Vert$.

In addition to the force formula, the time $\tau$ is required, which can be calculated using

$$\tau :=t-{\frac {r}{c}}.$$ (5)

The time $\tau$ corresponds to the time when the force has left the charge $q_{s}$ in order to reach the charge $q_{d}$ at the time $t$. The equation shows that the electromagnetic force propagates with the speed of light in a vacuum $c=r/(t-\tau )$ in every inertial frame, although the equations are Galilean invariant.

Incidentally, it is important that in formulas (2), (3), and (4) only the past time $\tau$ is used but not the current time $t$. This follows from the derivation of formula (1) from Maxwell's equations and also ensures that the force for any point charge $q_{d}$ propagates at the speed of light regardless of its speed relative to the source of the force $q_{s}$.

Interpretation as a field theory

Formula (1) can be interpreted as a force between two point charges. However, formula (1) can also be used to calculate the force of a single point charge on an entire grid of other point charges. In this way, one generates what is usually referred to as a field. This grid may also move; for example, at velocity $\mathbf {v} '.$ If the velocity is constant, the trajectory of a grid particle is $\mathbf {r} _{d}(t)=\mathbf {r} '+\mathbf {v} '\,t$, where $\mathbf {r} '$ is the location of the particle at time $t=0.$ The force (1) can therefore also be interpreted as a location- and velocity-dependent field formula, just like the Lorentz force formula.

In contrast to the classical interpretation of Maxwell's equations, Weber-Maxwell electrodynamics emphasizes the role of the field-generating point charges, as their trajectories uniquely define the fields. In the standard interpretation of Maxwell's equations, however, fields are often interpreted as entities that exist independently of the point charges.

This different interpretation has the consequence that in Weber-Maxwell electrodynamics it is not necessary to know the fields everywhere in space and time in order to predict their development in the future by means of Maxwell's equations. Instead, it is only necessary to know the trajectories of all point charges. This can reduce the amount of information to be stored and the computational costs of in silico simulations.

Derivation from Maxwell's equations

Starting point of the derivation are the Liénard-Wiechert potentials. The Liénard-Wiechert potentials describe the electric and magnetic potentials generated by a moving electric point charge $q_{s}$. It is possible to use the potentials $\phi$ and $\mathbf {A}$ to calculate the corresponding fields $\mathbf {E}$ and $\mathbf {B}$.

To obtain Weber-Maxwell electrodynamics, it can be exploited that a stationary test charge $q_{d}$ is only receptive to the electric field $\mathbf {E}$, but not to the magnetic field $\mathbf {B} .$ In this special case, $\mathbf {F} =q_{d}\,\mathbf {E}$ applies. Provided that the field-generating charge $q_{s}$ is much slower than the speed of light, it is allowed to apply a Galilean transformation to transform the force from the center-of-momentum frame of the test charge $q_{d}$ to another slowly moving inertial frame. Furthermore, it is permitted to apply Newtonian mechanics.

After performing the Galilean transformation, it becomes apparent that the resulting force formula is not completely correct. This is a consequence of the fact that Galilean transformation and Newton's laws are approximations. To correct the result, instead of the formula $\mathbf {F} =q_{d}\,\mathbf {E}$, the formula $\mathbf {F} =\gamma (v)\,q_{d}\,\mathbf {E}$ can be applied. For small $v$, $\gamma (v)$ is almost unity. Regardless of this, the additional factor is necessary to ensure that the resulting force formula produces experimentally correct predictions. As a consequence, by introducing this calibration, Weber-Maxwell electrodynamics becomes compatible with Gauss's hypothesis and Weber electrodynamics. This in turn has the consequence that the force (1) correctly reproduces the Lorentz force generated by a direct current or a low-frequency alternating current without taking the field $\mathbf {B}$ into account.

Characteristics

Constancy of the speed of light

In Weber-Maxwell electrodynamics, the propagation speed of the electromagnetic force for each point charge $q_{d}$ corresponds exactly to the speed of light in a vacuum, regardless of its speed relative to the source $q_{s}$ of the force. The reason for this is represented in equation (5).

Principle of relativity

Weber-Maxwell electrodynamics is Galilean invariant, since formulas (1) and (5) have the same form in any inertial frame and depend only on quantities that are invariant under a Galilean transformation.

Magnetism and Lorentz force

It can be demonstrated that the force (1) is equivalent to the Weber force for small relative velocities and small relative accelerations. This implies that Weber-Maxwell electrodynamics can correctly represent the magnetic effects of direct currents and low-frequency alternating currents. Further details on this topic can be found in the specialist literature on Weber electrodynamics.

Conservation of momentum

In Weber-Maxwell electrodynamics, Newton's third law also applies when charges are accelerated and emit electromagnetic waves. This is easy to show, as only the sign of the force changes when the two point charges and their trajectories are swapped in equations (1) to (5). Newton's third law is the only assumption required to prove that the total momentum of an isolated system consisting of any number of point charges is time-invariant.

Electromagnetic waves

Weber-Maxwell electrodynamics is capable of describing electromagnetic waves, because it is derived from Maxwell's equations with explicit inclusion of the displacement current.

Pseudo-instantaneous interactions

The force (1) often acts like an instantaneous action-at-a-distance force, which does not require any time for propagation, although it is evident from formula (5) that it always propagates at the speed of light. This applies if the trajectories are almost straight lines during the period of time in which the force moves from $q_{s}$ to $q_{d}$. Under these circumstances, $\mathbf {a} \approx \mathbf {0}$ and the relative velocity $\mathbf {v}$ becomes a time-independent constant. It can be shown that under these conditions the formula (1) is identical to the Weber force, which, like Coulomb's law, seems to describe an instantaneous action-at-a-distance interaction.

Practical applications

Dipole antenna

The field of the electromagnetic force calculated by means of Weber-Maxwell electrodynamics from the perspective of a grid of stationary test charges.

The field of the electromagnetic force calculated by means of Weber-Maxwell electrodynamics from the perspective of a grid of test charges that are moving leftwards at half the speed of light.

Weber-Maxwell electrodynamics is well suited for calculating fields that are generated by one or a few point charges. The Hertzian dipole can serve as an example. The Hertzian dipole consists of a negative point charge that oscillates up and down on the z-axis with angular frequency $\omega$ and amplitude $R$ and a positive point charge, which oscillates inversely.

We only calculate the field of one point charge $q_{s}$, since the calculation for the other point charge is identical. The trajectory of the oscillating point charge is $\mathbf {r} _{s}(t)=\mathbf {e} _{z}\,R\,\sin(\omega \,t)$. For the test charge, we assume the trajectory $\mathbf {r} _{d}(t)=\mathbf {r} '+\mathbf {v} '\,t$, where $\mathbf {r} '$ is the location of the test charge at time $t=0$. $\mathbf {v} '$ represents a constant velocity of the test charge.

The two trajectories $\mathbf {r} _{s}(t)$ and $\mathbf {r} _{d}(t)$ can now be inserted into equations (2), (3), and (4). This yields

$$\mathbf {r} =\mathbf {r} '+\mathbf {v} '\,\tau -\mathbf {e} _{z}\,R\,\sin(\omega \,\tau ),$$ (6)

$$\mathbf {v} =\mathbf {v} '-\mathbf {e} _{z}\,R\,\omega \,\cos(\omega \,\tau ),$$ (7)

and

$$\mathbf {a} =\mathbf {e} _{z}\,R\,\omega ^{2}\,\sin(\omega \,\tau ).$$ (8)

Now the time $\tau$ is required. This can be obtained by means of equation (5):

$$\tau =t-{\frac {\Vert \mathbf {r} '+\mathbf {v} '\,\tau -\mathbf {e} _{z}\,R\,\sin(\omega \,\tau )\Vert }{c}}\approx t-{\frac {\Vert \mathbf {r} '+\mathbf {v} '\,\tau \Vert }{c}}.$$ (9)

Equation (9) has two solutions, with only one of them satisfying the causality condition $\tau <t$. This solution is

$$\tau ={\frac {c^{2}t+\mathbf {r} '\cdot \mathbf {v} '-{\sqrt {\left(c^{2}-v'^{2}\right)\left(r'^{2}-c^{2}\,t^{2}\right)+\left(c^{2}t+\mathbf {r} '\cdot \mathbf {v} '\right)^{2}}}}{c^{2}-v'^{2}}}.$$ (10)

Equation (10) can now be inserted into equations (6), (7), and (8), and these in turn into equation (1).

To illustrate the solution, two examples are shown. When the test charge is moving, an additional Doppler effect becomes visible.

Computational electromagnetics

Modeling of diffraction using point charges

Modelling of interference at a double slit using point charges

The force formula (1) is well suited for computer simulation of multibody systems due to its great similarity to Coulomb's law, Newton's law of gravitation, and other types of forces, such as constraints or spring forces. In contrast to conventional field solvers of computational electromagnetics, Maxwell's equations do not have to be solved numerically. The challenge with Weber-Maxwell electrodynamics, on the other hand, is to model all involved charge quantities with point charges and to model their motions and constraints in a suitable manner.

However, many fundamental phenomena of electrodynamics can already be modelled with relatively few point charges. Two examples are shown in which a dipole antenna is simulated by means of several Hertzian dipoles (green). The radiated wave acts on a barrier of negative and positive point charges that are connected by spring forces (blue). The incident wave generates a force that sets these point charges into motion. After a certain time, the mechanical system is in its steady state and it can be seen how the secondary wave of the dipoles shown in blue weakens the incoming field. Depending on the arrangement of the dipoles, effects like wave interference or diffraction can be seen.

These models demonstrate that classical electrodynamics and even optics can also be interpreted as subdomains of Newtonian mechanics. The fact that the electromagnetic force does not propagate infinitely fast is not relevant in this context.