International Journal of Magnetics and Electromagnetism
(ISSN: 2631-5068)
Volume 9, Issue 1
Research Article
DOI: 10.35840/2631-5068/6542
Electric Wave Equation Derived from Weber's Electrodynamics
Qingsong Li*
References
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Author Details
Qingsong Li*
Independent Researcher, Sugar Land, Texas, USA
Corresponding author
Qingsong Li, Sugar Land, Texas, USA.
Accepted: April 04, 2023 | Published Online: April 06, 2023
Citation: Li Q (2023) Electric Wave Equation Derived from Weber's Electrodynamics. Int J Magnetics Electromagnetism 9:042.
Copyright: © 2023 Li Q. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Abstract
Weber's electrodynamics, as an alternative to Maxwell-Lorentz electromagnetism, has maintained continual attention among the physics community. Recently it got some exiting advancements both experimentally and theoretically. However, one major criticism to Weber's electrodynamics still stands: It did not develop a general electromagnetic wave equation for free space. In this article, we try to derive an electric wave equation from Weber's electrodynamics, using a postulate of vacuum polarization.
Introduction
The classic electromagnetism is mainly built with Maxwell equations and Lorentz force axiom. As an alternative to Maxwell-Lorentz electromagnetism, Weber's electrodynamics can also explain a lot electromagnetic phenomena, but much less popular [1,2]. Recently, researchers have made exciting advancements both experimentally and theoretically. Some new experiments, including the electron beam deflection experiment [3,4], electron beam induction experiment [5], and the experiment of magnetic force direction within capacitors [6] support Weber's electrodynamics. A 6-component electric field was introduced to represent Weber's electrodynamics, which simplify calculations for cases of large number of particles [7]. Weber's Electrodynamics was extended for the regime of high velocity particles [8]. Weber's Electrodynamics has been seen as a better and natural solution to the Faraday's Paradox [9].
Maxwell equations have been successfully used to derive electromagnetic wave equations, explaining electromagnetic wave phenomena. On the other hand, Weber's electrodynamics was struggling to give an electromagnetic wave equation for free space. This was one major limitation of Weber's electrodynamics [2]. The previous efforts of extending Weber's electrodynamics to wave equation are mainly in two fronts. One is to use Weber's electrodynamics to derive wave equations for signal propagation in conductors [10,11]. The other is to introduce time retardation to Weber's electrodynamics [12,13] or to transform Maxwell inhomogeneous wave equation to be compatible with Weber's electrodynamics under certain conditions (observer rest frame, charge as a function of velocity, etc.) [14].
When using Maxwell equations to derive the wave equation, the vacuum can be seen as fully empty. The electric and magnetic fields propagate by themselves. However, it is hard to derive a wave equation with Weber's electrodynamics if there is no media in vacuum. Aether in vacuum was suggested as a media for electric wave propagation in 19^{th} century, but became unpopular in early 20^{th} century and thereafter [15]. However, the modern physics suggests that vacuum may not be fully empty. Quantum mechanics state that an external electric field can cause vacuum polarization and form virtual particle-antiparticle pairs [16]. And dark matter theory suggests that invisible materials exist in galaxies [17].
In this article, we postulate that vacuum is not fully empty, instead composed of positive and negative charges, and that vacuum experience polarization under external electric field. Fukai postulated similar idea of polarizable space before [18]. Using this vacuum postulate, we try to derive a wave equation from Weber's electrodynamics. The method used in this article originates from the telegraphy idea of Weber, Kirchhoff and Assis [10,11].
Weber's electrodynamics
Comparing to Maxwell-Lorentz electromagnetism, Weber's electrodynamics gives a much simpler form of particle-particle interaction force. For two particles interacting with each other, the force exerted by one particle to the other particle is [19]:
$$\stackrel{\rightharpoonup}{F}=\frac{Qq\widehat{r}}{4\pi {\epsilon}_{0}{r}^{\text{2}}}\left[1+\frac{1}{{c}^{2}}\left(\stackrel{\rightharpoonup}{v}\cdot \stackrel{\rightharpoonup}{v}-\frac{3}{2}{\left(\widehat{r}\cdot \stackrel{\rightharpoonup}{v}\right)}^{2}+r\widehat{r}\cdot \stackrel{\rightharpoonup}{a}\right)\right]\text{(1)}$$
Where Q and q are two electrical charges, $\stackrel{\rightharpoonup}{F}$ is the force that charge Q exerted on charge q, r is the distance between the two charges, $\widehat{r}$ is the unit vector pointing from charge Q to charge q, $\stackrel{\rightharpoonup}{v}$ and $\stackrel{\rightharpoonup}{a}$ are velocity and acceleration of charge q relative to charge Q, ${\epsilon}_{0}$ is the dielectric constant, c is the speed of light.
This force is of the type action-at-a-distance. Intuitively, the force is exerted instantaneously.
Vacuum polarization
In this article, we postulate thatthe vacuum behaves similarly to a polarization material. In detail, the vacuum consists of positive and negative charges, overlapping each other (Figure 1). The positive charges and negative charges can oscillate relative to each other. When there is no external electric field, the displacement of negative charge relative to positive charge is zero, and the vacuum stays neutral. However, when there is an external electric field, there exists relative displacement between negative charge and positive charge. The displacement may cause the vacuum to become non-neutral. The charge density varies with the divergence of displacement field (equation 2).
$${\stackrel{\rightharpoonup}{D}}_{+}=-\stackrel{\rightharpoonup}{D}\_{\stackrel{\rightharpoonup}{v}}_{+}=-\stackrel{\rightharpoonup}{v}\_{\stackrel{\rightharpoonup}{a}}_{+}=-\stackrel{\rightharpoonup}{a}\_$$
$${\rho}_{+\text{}}=\text{}\rho \left(1-\nabla \cdot {\stackrel{\rightharpoonup}{D}}_{+}\right){\rho}_{-\text{}}=\text{}\rho \left(1-\nabla \cdot {\stackrel{\rightharpoonup}{D}}_{-}\right){\rho}_{+\text{}}+\text{}{\rho}_{-\text{}}=\text{2}\rho \text{(2)}$$
Where, ${\stackrel{\rightharpoonup}{D}}_{+}$, ${\stackrel{\rightharpoonup}{v}}_{+}$, ${\stackrel{\rightharpoonup}{a}}_{+}$, and ${\rho}_{+}$ are displacement, velocity, acceleration, and density of positive charge, ${\stackrel{\rightharpoonup}{D}}_{-}$, ${\stackrel{\rightharpoonup}{v}}_{-},\text{\hspace{0.33em}}\stackrel{\rightharpoonup}{a}\_$, and $\rho \_$ are displacement, velocity, acceleration, and density of negative charge, and ρ is the average density of positive and negative charges.
Derivation of electric wave equation
Let's consider a homogeneous vacuum with positive and negative charges. The negative charge density is ${\rho}_{-}\left(\stackrel{\rightharpoonup}{r}\right)$, the velocity and acceleration of negative charge are ${\stackrel{\rightharpoonup}{v}}_{-}\left(\stackrel{\rightharpoonup}{r}\right)$ and ${\stackrel{\rightharpoonup}{a}}_{-}\left(\stackrel{\rightharpoonup}{r}\right)$, respectively (Figure 2). The density ${\rho}_{\text{+}}\left(\stackrel{\rightharpoonup}{r}\right)$, velocity ${\stackrel{\rightharpoonup}{v}}_{+}\left(\stackrel{\rightharpoonup}{r}\right)$ and acceleration ${\stackrel{\rightharpoonup}{a}}_{+}\text{}\left(\stackrel{\rightharpoonup}{r}\right)$ of positive charge are simply related with those of negative charge (equation 2).
Let's do a Tylor expansion around the origin for quantities of negative charge (equation 3). The same expansion applies to positive charge too. The higher order terms will be dropped in later equation derivation.
$$\rho \_(\stackrel{\rightharpoonup}{r})\text{=}\rho \_(0)\text{+}\stackrel{\rightharpoonup}{r}\text{}\text{.}\nabla \rho \_(0)+O({\stackrel{\rightharpoonup}{r}}^{2})$$
$${\stackrel{\rightharpoonup}{v}}_{-}\left(\stackrel{\rightharpoonup}{r}\right)\text{=}{\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)+O\left(\stackrel{\rightharpoonup}{r}\right)$$
$${\stackrel{\rightharpoonup}{a}}_{-}\left(\stackrel{\rightharpoonup}{r}\right)\text{=}{\stackrel{\rightharpoonup}{a}}_{-}\left(0\right)+O\left(\stackrel{\rightharpoonup}{r}\right)\text{(3)}$$
The positive charge and negative charge in $d{V}^{\prime}$ on the shell of radius r expert force on the negative charge in dV at origin (Figure 3). With Weber's electrodynamics, the force can be written as:
$$\begin{array}{l}\text{}\stackrel{\rightharpoonup}{f}\text{=}\frac{-\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}\rho \_\left(0\right)dV{\rho}_{-}\left(\stackrel{\rightharpoonup}{r}\right)d{V}^{\prime}\text{(4)}\\ \text{}\text{+}\frac{-\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}{\rho}_{-}\left(0\right)dV{\rho}_{+}\left(\stackrel{\rightharpoonup}{r}\right)d{V}^{\prime}\\ \text{}\left(1\text{}+\text{}\frac{1}{{c}^{2}}\text{}\left(\left({\stackrel{\rightharpoonup}{v}}_{+}\text{}\left(0\right)\text{}-\text{}{\stackrel{\rightharpoonup}{v}}_{-}\text{}\left(0\right)\right)\text{}.\text{}\left({\stackrel{\rightharpoonup}{v}}_{+}\text{}\left(0\right)\text{}-\text{}{\stackrel{\rightharpoonup}{v}}_{-}\text{}\left(0\right)\right)\text{}-\text{}\frac{3}{2{r}^{2}}\text{}{\left(\stackrel{\rightharpoonup}{r}\text{\hspace{0.17em}}.\text{}\left({\stackrel{\rightharpoonup}{v}}_{+}\text{}\left(0\right)\text{}-\text{}{\stackrel{\rightharpoonup}{v}}_{-}\text{}\left(0\right)\right)\right)}^{2}+\text{}\stackrel{\rightharpoonup}{r}.\text{}\left({\stackrel{\rightharpoonup}{a}}_{+}\text{}\left(0\right)\text{}-\text{}{\stackrel{\rightharpoonup}{a}}_{-}\left(0\right)\right)\right)\right)\\ \end{array}$$
This force consists of static term (Coulomb force), velocity related term and acceleration related term.
First, let's consider the static term (Coulomb force):
$$\begin{array}{l}{\stackrel{\rightharpoonup}{f}}_{c}\text{=}\frac{-\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}{\rho}_{-}\left(0\right)dV{\rho}_{-}\left(\stackrel{\rightharpoonup}{r}\right)d{V}^{\prime}+\frac{\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}{\rho}_{-}\left(0\right)dV{\rho}_{+}\left(\stackrel{\rightharpoonup}{r}\right)d{V}^{\prime}\text{(5)}\\ \text{=}\frac{-\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}{\rho}_{-}\left(0\right)\text{}dV\left(2{\rho}_{-}\text{}\left(0\right)\text{}+\text{}2\stackrel{\rightharpoonup}{r}\text{}.\text{}\nabla {\rho}_{-}\text{}\left(0\right)\text{}-\text{}2\rho \right)d{V}^{\prime}\end{array}$$
Let's integrate around the shell. Due to the symmetry, after integration, the net force of several terms becomes zero. The remaining terms can be written as:
$${\stackrel{\rightharpoonup}{f}}_{c}\text{=}-\frac{1}{4\prod {\epsilon}_{0}{r}^{3}}\rho \_\left(0\right)dV\text{\hspace{0.33em}}{\displaystyle \int \left(2\rho \_\text{}\left(0\right)\text{}+\text{}2\stackrel{\rightharpoonup}{r}\text{}\cdot \text{}\nabla \rho \text{\_}\left(0\right)\text{}-\text{}2\rho \right)\stackrel{\rightharpoonup}{r}dsdr\text{}=-\frac{2r}{3{\epsilon}_{0}}\rho \_\text{}\left(0\right)dV\nabla \rho \text{\_}\left(0\right)dr}\text{\hspace{0.33em}}$$ (6)
Second, let's consider the velocity related term:
$${\stackrel{\rightharpoonup}{f}}_{v}\text{=}\frac{\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}{\rho}_{-}\left(0\right)dV{\rho}_{+}\left(\stackrel{\rightharpoonup}{r}\right)d{V}^{\prime}\frac{1}{{c}^{2}}\text{\hspace{0.33em}}\left(\left({\stackrel{\rightharpoonup}{v}}_{+}\left(0\right)-{\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)\right)\cdot \left({\stackrel{\rightharpoonup}{v}}_{+}\left(0\right)-{\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)\right)-\frac{3}{2{r}^{2}}{\left(\stackrel{\rightharpoonup}{r}\cdot \left({\stackrel{\rightharpoonup}{v}}_{+}\left(0\right)-{\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)\right)\right)}^{2}\right)\text{(7)}$$
From equation (2, 3), we can have:
$${\rho}_{+}\left(\stackrel{\rightharpoonup}{r}\right)\text{=2}\rho \text{}-\text{}{\rho}_{-}\left(\stackrel{\rightharpoonup}{r}\right)\approx 2\rho \text{}-\text{}{\rho}_{-}\left(0\right)-\stackrel{\rightharpoonup}{r}\cdot \nabla {\rho}_{-}\left(0\right)$$
$$\stackrel{\rightharpoonup}{v}+\left(0\right)\text{=}-\text{}\stackrel{\rightharpoonup}{v}\_\text{}\left(0\right)\text{(8)}$$
Inserting equation (8) into equation (7), we get
$${\stackrel{\rightharpoonup}{f}}_{v\text{}}=\text{}\frac{\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}\rho \_\left(0\right)dV\left(2\rho -\rho \_\left(0\right)-\stackrel{\rightharpoonup}{r}\cdot \nabla \rho \_\left(0\right)\right)d{V}^{\prime}\frac{1}{{c}^{2}}\left(2{\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)\cdot 2\stackrel{\rightharpoonup}{v}-\left(0\right)\right.\text{\hspace{0.33em}}-\frac{3}{2{r}^{2}}{\left(\stackrel{\rightharpoonup}{r}\cdot \text{2}\stackrel{\rightharpoonup}{v}\_\left(0\right)\right)}^{2}\text{(9)}$$
Let's integrate around the shell:
$$\begin{array}{l}{\stackrel{\rightharpoonup}{F}}_{v}\text{=}\text{\hspace{0.33em}}-\frac{1}{4\pi {\epsilon}_{0}{r}^{3}}\rho \_\left(0\right)dV{\displaystyle \int \frac{1}{{c}^{2}}}\left(2\stackrel{\rightharpoonup}{v}\_\left(0\right)\cdot 2\stackrel{\rightharpoonup}{v}\_\left(0\right)-\frac{3}{2{r}^{2}}{\left(\stackrel{\rightharpoonup}{r}\cdot \text{2}\stackrel{\rightharpoonup}{v}\_\left(0\right)\right)}^{2}\right)\stackrel{\rightharpoonup}{r}\cdot \nabla \rho \_\left(0\right)\stackrel{\rightharpoonup}{r}dsdr\text{(10)}\\ \\ \text{}\text{}=-\frac{14r}{15\epsilon 0}\rho \_\left(0\right)dV\frac{1}{{c}^{2}}\left(\stackrel{\rightharpoonup}{v}\_\left(0\right)\cdot {\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)\right)\nabla {\rho}_{-}\left(0\right)dr\\ \\ \text{}\text{}+\frac{4r}{5{\epsilon}_{0}}{\rho}_{-}\left(0\right)dV\frac{1}{{c}^{2}}\left({\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)\cdot \nabla {\rho}_{-}\left(0\right)\right)\stackrel{\rightharpoonup}{v}\_\left(0\right)dr\end{array}$$
Third, let's consider the acceleration related term:
$${\stackrel{\rightharpoonup}{f}}_{\alpha}\text{=}\frac{\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}\rho \_\left(0\right)dV{\rho}_{+}\left(\stackrel{\rightharpoonup}{r}\right)d{V}^{\prime}\frac{1}{{c}^{2}}\stackrel{\rightharpoonup}{r}\cdot \left({\stackrel{\rightharpoonup}{a}}_{+}\left(0\right)-\stackrel{\rightharpoonup}{a}\_\left(0\right)\right)\text{(11)}$$
From equation (2, 3) and using an approximation, we can have:
$$\stackrel{\rightharpoonup}{a}+\left(0\right)\text{=}-\stackrel{\rightharpoonup}{a}\_\left(0\right)$$
$${\rho}_{+}\left(\stackrel{\rightharpoonup}{r}\right)\approx {\rho}_{+}\left(0\right)\approx \text{}\rho \text{(12)}$$
Inserting equation (12) into equation (11), we get
$${\stackrel{\rightharpoonup}{f}}_{a}=\text{}-\frac{\stackrel{\rightharpoonup}{r}}{4\pi {\epsilon}_{0}{r}^{3}}\rho \_\left(0\right)dV\rho d{V}^{\prime}\frac{1}{{c}^{2}}\stackrel{\rightharpoonup}{r}\cdot 2\stackrel{\rightharpoonup}{a}\_\left(0\right)\text{(13)}$$
Let's integrate around the shell:
$${\stackrel{\rightharpoonup}{F}}_{a}\text{=}-\frac{1}{4\pi {\epsilon}_{0}{r}^{3}}\rho \_\left(0\right)dV\int \frac{2\rho}{{c}^{2}}\stackrel{\rightharpoonup}{r}\cdot \stackrel{\rightharpoonup}{a}\_\left(0\right)\stackrel{\rightharpoonup}{r}dsdr\text{=}-\frac{2r}{3{\epsilon}_{0}}\rho \_\left(0\right)dV\frac{\rho}{{c}^{2}}\stackrel{\rightharpoonup}{a}\_\left(0\right)dr\text{(14)}$$
The total force on negative charge in dV exerted by the shell is:
$$\stackrel{\rightharpoonup}{F}\text{=}{\overrightarrow{F}}_{c}+\text{}{\overrightarrow{F}}_{v}\text{+}{\overrightarrow{F}}_{a}\text{(15)}$$
Inserting equations (6, 10, 14) into equation (15), we can get:
$$\begin{array}{l}\stackrel{\rightharpoonup}{F}\text{=}-\frac{2r}{3{\epsilon}_{0}}\rho \_\left(0\right)dV\nabla \rho \_\left(0\right)dr-\frac{14r}{15{\epsilon}_{0}}\rho \_\left(0\right)dV\frac{1}{{c}^{2}}\left(\stackrel{\rightharpoonup}{v}\_\text{}\left(0\right)\text{\hspace{0.33em}}\cdot \text{\hspace{0.33em}}\stackrel{\rightharpoonup}{v}\_\text{}\left(0\right)\right)\nabla \rho \_\left(0\right)dr\text{(16)}\\ \\ \text{}\text{}+\frac{4r}{5{\epsilon}_{0}}\rho \_\left(0\right)dV\frac{1}{{c}^{2}}\left(\stackrel{\rightharpoonup}{v}\_\text{}\left(0\right)\text{\hspace{0.33em}}\cdot \text{\hspace{0.33em}}\nabla \rho \_\text{}\left(0\right)\right)\stackrel{\rightharpoonup}{v}\_\left(0\right)dr\\ \\ \text{}\text{}-\frac{2r}{3{\epsilon}_{0}}\rho \_\left(0\right)dV\frac{\rho}{{c}^{2}}\stackrel{\rightharpoonup}{a}\_\left(0\right)dr\end{array}$$
We may choose to neglect the mass and acceleration force of negative charge in parcel dV and neglect the polarization force between negative and positive charges. Equation (16) holds for small radius r since Taylor expansion was used (equation 3). Without doing integration along radius r, we set the total force in equation (16) equal zero due to force balance. With some simplification, we can get:
$$\nabla \rho \_\left(0\right)+\frac{7}{5}\frac{1}{{c}^{2}}\left(\stackrel{\rightharpoonup}{v}\_\left(0\right)\cdot \stackrel{\rightharpoonup}{v}\_\left(0\right)\right)\nabla \rho \_\left(0\right)-\frac{6}{5}\frac{1}{{c}^{2}}\left(\stackrel{\rightharpoonup}{v}\_\left(0\right)\cdot \nabla \rho \_\left(0\right)\right)\stackrel{\rightharpoonup}{v}\_\left(0\right)+\frac{\rho}{{c}^{2}}\stackrel{\rightharpoonup}{a}\_\left(0\right)\text{=0(17)}$$
When velocity $\stackrel{\rightharpoonup}{v}\_\left(0\right)$ is much less than c, we can neglect the velocity related term. Thus
$$\nabla \rho \_\left(0\right)+\frac{\rho}{{c}^{2}}\stackrel{\rightharpoonup}{a}\_\left(0\right)\text{=0(18)}$$
The above equation can also be written as:
$$\nabla \rho \_\left(0\right)+\frac{\rho}{{c}^{2}}\frac{d\stackrel{\rightharpoonup}{v}-\left(0\right)}{dt}\text{=0(19)}$$
This expression holds for points other than the origin. And since the ${\stackrel{\rightharpoonup}{v}}_{-}\left(0\right)$ is small, the Lagrangian derivative can be approximated as Eulerian derivative. Thus we can write:
$$\nabla \rho \_+\frac{\rho}{{c}^{2}}\frac{\partial \stackrel{\rightharpoonup}{v}\_}{\partial t}\text{=0(20)}$$
Apply divergence to the above equation:
$$\nabla \cdot \nabla \rho \_\text{}+\frac{\rho}{{c}^{2}}\frac{\partial \stackrel{\rightharpoonup}{v}\left(\nabla \text{\hspace{0.33em}}\cdot \text{\hspace{0.33em}}\stackrel{\rightharpoonup}{v}\text{\_}\right)}{\partial t}\text{=0(21)}$$
From equation (2), we have From equation (2), we have $\rho \_\text{=}\rho \left(1-\nabla \cdot \stackrel{\rightharpoonup}{D}\_\right)$. Apply time derivative to it, we get
$$\frac{\partial {\rho}_{-}}{\partial t}\text{=}-\rho \nabla \text{}\cdot \text{\hspace{0.33em}}\frac{\partial {\stackrel{\rightharpoonup}{D}}_{-}}{\partial t}\text{=}-\rho \text{}\nabla \cdot \stackrel{\rightharpoonup}{v}\_\text{(22)}$$
Insert the equation (22) into equation (21), we can get
$$\nabla \cdot \nabla \rho \text{\_}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}{\rho}_{-}}{\partial {t}^{2}}\text{=0(23)}$$
This is the wave propagation equation for negative charge density.
Let's insert equation $\rho \_\text{=}\rho \left(1-\nabla \cdot \stackrel{\rightharpoonup}{D}\text{\_}\right)$ into equation (20). Let's assume that the vacuum is homogeneous, and is a constant. Then we can get
$$-\rho \nabla \nabla \text{}\cdot \stackrel{\rightharpoonup}{D}\text{\_+}\frac{\rho}{{c}^{2}}\frac{\partial \stackrel{\rightharpoonup}{v}}{\partial t}\text{=0(24)}$$
The above equation can also be written as:
$$-\nabla \nabla .\stackrel{\rightharpoonup}{D}-\text{+}\frac{1}{{c}^{2}}\frac{{\partial}^{2}\stackrel{\rightharpoonup}{D}-}{\partial {t}^{2}}\text{=0(25)}$$
Using the vector formula $\nabla \nabla \cdot \stackrel{\rightharpoonup}{D}\text{\_=}\nabla \times \nabla \times {\stackrel{\rightharpoonup}{D}}_{-}+{\nabla}^{2}{\stackrel{\rightharpoonup}{D}}_{-}$, we can get
$$\text{}\nabla \times \nabla \times \stackrel{\rightharpoonup}{D}\_+{\nabla}^{2}\stackrel{\rightharpoonup}{D}\_\text{=}\frac{1}{{c}^{2}}\frac{{\partial}^{2}\stackrel{\rightharpoonup}{D}\_}{\partial {t}^{2}}\text{(26)}$$
This equation looks a little complicated. Let's consider a simple scenario when $\text{}\nabla \times \nabla \times \stackrel{\rightharpoonup}{D}\_\text{=0}$. Then we can get
$$\text{}{\nabla}^{2}\stackrel{\rightharpoonup}{D}\_=\text{}\frac{1}{{c}^{2}}\frac{{\partial}^{2}\stackrel{\rightharpoonup}{D}}{\partial {t}^{2}}\text{(27)}$$
Let's assume a simple relationship of vacuum polarization $\text{}{\stackrel{\rightharpoonup}{D}}_{+}\text{=}-{\stackrel{\rightharpoonup}{D}}_{-}=\frac{1}{2}{\epsilon}_{0}\stackrel{\rightharpoonup}{{\rm E}}$, we can get
$${\nabla}^{2}\stackrel{\rightharpoonup}{E}\text{=}\frac{1}{{c}^{2}}\frac{{\partial}^{2}\stackrel{\rightharpoonup}{{\rm E}}}{\partial {t}^{2}}\text{(28)}$$
The above equation is similar to the well-known electrical field wave equation derived from Maxwell equations.
Longitudinal electric wave
Longitudinal electric wave travels in the same direction as electric field. It has been observed in plasma [20] and focused beam [21]. However, the existence of longitudinal electric wave in free space has been a controversial topic. Maxwell equations do not allow plane or spherical longitudinal electric wave, which violates the Gauss's law (zero divergence of electric field in free space) [22]. On the other hand, there have been reports of longitudinal electric wave in free space. E.g., longitudinal electric wave was observed during the eclipse of the sun by the moon [23]. Also, the existence of spherical longitudinal electric wave was observed in an experiment of spherical antenna [24].The experiment was repeated later [25]. To explain this phenomenon, aninhomogeneous wave equation of scalar potential was proposed by Monstein& Wesley [24], using Coulomb's law and time retardation. The wave equation is copied below:
$${\nabla}^{2}\Phi -\frac{{\partial}^{2}{\Phi}^{2}}{{\partial}^{2}{\text{t}}^{2}}\text{=}-4\pi \rho \text{(29)}$$
Where φ is the scalar potential, ρ is the source charge density. Since introducing the time retardation term $\frac{{\partial}^{2}{\Phi}^{2}}{{\partial}^{2}{t}^{2}}$, the above equation does not obey Gauss's law in free space. Similar theory was used to explain longitudinal electric wave in vacuum radiated by electric dipole [26].
The wave equation derived in this paper does not involve the Gauss's law. Instead, it is depending on the non-zero divergence of electric displacement (electric field). If the divergence is zero, the field will be static and there is no wave propagation (equation 25). Thus the theory in this paper is compatible with the longitudinal wave phenomenon and the Monstein& Wesley's theory.
Discussion
In this paper's theory, we postulate that the vacuum is not fully empty, instead filled with positive-negative charge pairs. This postulate may not be unreasonable given that dark matter exists in galaxies [17], and that that quantum mechanics has the concept of vacuum polarization [16]. There is difference between the vacuum postulate in this paper and those of dark matter and quantum mechanics. Further research may be needed to address their relationship and differences.
Michelson-Morley experiment has been used as a proof that the free space media (aether) does not exist and/or is not necessary for electric wave propagation [15]. In this paper's theory, vacuum of positive-negative charge pairs is the media for wave propagation. Does this mean that our theory contradicts with Michelson-Morley experiment? The answer is no. If we carefully look at the derivation process in this paper, we shall notice that a uniform media velocity has no effect on interacting force (equation 4). This is due to the relativistic nature of Weber's electrodynamics, in which only relative velocity (and distance, acceleration between charges) matters. Thus a uniform media velocity has no effect on the wave equation (equation 23, 28), and does not impact the wave propagation speed. This seems contradicting intuitively, but actually compatible with Michelson-Morley experiment outcome.
In this paper, we came up with a wave equation of electric field (equation 28) using a simple polarization assumption $\text{}{\stackrel{\rightharpoonup}{D}}_{+}\text{=}-{\stackrel{\rightharpoonup}{D}}_{-}=\frac{1}{2}{\epsilon}_{0}\stackrel{\rightharpoonup}{E}$. However, the electric field from Weber's electrodynamics consists of 6 components [7]. The derivation of the wave equation for all 6 components needs further research, and potentially needs a more sophisticated polarization assumption between 6-component electric field and relative displacement, velocity and acceleration of positive-negative charge pairs.
In this paper, when we derive the wave equation, we used some approximations, e.g., neglecting higher order terms, acceleration force, polarization force, velocity term etc. These approximations set the applicable range of the theory/derivation of this paper. It is applicable to homogeneous vacuum only, and only applicable to cases of low relative speed of charge pairs. More research is needed for the cases when electric particles (such as electron) exist in the vacuum and/or charge pairs have high relative speed.
If we assume that the curl term equals zero (equation 26), we came to a wave equation similar to the popular electric wave equation derived from Maxwell equations. However, we can't prove that the curl term always equal zero. Most likely, they don't. In that case, the equation will be different from the popularly known electromagnetic wave equation. How the equation can be solved to explain electric wave phenomena is left for future researches. On the other hand, this equation has some similarity to fluid dynamics equations. The curl term may indicate vortex structure in electric field, similar to that in fluids. Ball lighting, a rare and unexplained luminescent spherical object phenomenon [27], may be a potential candidate to study vortex structure in electric field.
Conclusions
We postulate a non-empty vacuum of positive and negative charge pairs, which have polarization under external electric field. With this postulate, we derived an electric wave equation from Weber's electrodynamics. This wave equation may be applicable to cases of homogenous vacuum and low relative speed of charge pairs (low energy). It gives analternative way to study electric waves other than Maxwell equations.
This paper shows that even though Weber's electrodynamic force is exerted instantaneously, the propagation of electric field (or energy) has a limited speed. For a homogeneous vacuum, its speed happens to be light speed.