Available via license: CC BY 4.0
Content may be subject to copyright.
Citation: Li, Q.; Maher, S. Deriving
an Electric Wave Equation from
Weber’s Electrodynamics.
Foundations 2023,3, 323–334.
https://doi.org/10.3390/
foundations3020024
Academic Editor: Eugene Oks
Received: 26 March 2023
Revised: 29 May 2023
Accepted: 1 June 2023
Published: 7 June 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Brief Report
Deriving an Electric Wave Equation from
Weber’s Electrodynamics
Qingsong Li 1, * and Simon Maher 2
1Independent Researcher, Sugar Land, TX 77479, USA
2Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, UK
*Correspondence: qingsong.li.geo@gmail.com
Abstract:
Weber’s electrodynamics presents an alternative theory to the widely accepted
Maxwell–Lorentz
electromagnetism. It is founded on the concept of direct action between par-
ticles, and has recently gained some momentum through theoretical and experimental advancements.
However, a major criticism remains: the lack of a comprehensive electromagnetic wave equation
for free space. Our motivation in this research article is to address this criticism, in some measure,
by deriving an electric wave equation from Weber’s electrodynamics based on the axiom of vac-
uum polarization. Although this assumption has limited experimental evidence and its validity
remains a topic of debate among researchers, it has been shown to be useful in the calculation of
various quantum mechanical phenomena. Based on this concept, and beginning with Weber’s force,
we derive an expression which resembles the familiar electric field wave equation derived from
Maxwell’s equations.
Keywords: electric wave equation; vacuum polarization; Weber’s electrodynamics
1. Introduction
Classic electromagnetism is based on Maxwell’s equations and the Lorentz force.
As an alternative to Maxwell–Lorentz electromagnetism, Weber’s electrodynamics can
also explain a range of electromagnetic phenomena, but it has received significantly less
attention [
1
,
2
]. In recent years, progress has been made in both experimental and theoretical
research on Weber electrodynamics. A series of new experiments, including electron beam
deflections [
3
,
4
], electron beam induction [
5
], the prediction of magnetic force direction
within a capacitor [
6
], and wave-particle duality investigations utilizing a Hertzian dipole
acting as a single quantum object [
7
], have provided evidence supporting the validity
of Weber’s electrodynamics. In addition, a six-component force representing Weber’s
electrodynamics has been introduced, making calculations simpler for cases involving large
numbers of particles [
8
]. Weber’s electrodynamics has also been extended for the regime of
high-velocity particles [
9
,
10
], and proposed as a more physically intuitive explanation of
Faraday’s paradox [11].
The success of Maxwell’s equations in deriving the electromagnetic wave equation
and explaining a range of electromagnetic phenomena is profound and has been well
documented. Whilst Maxwell’s equations have been influential in our understanding of
electromagnetic waves, it was Weber’s electrodynamics that first established the relation-
ship between the speed of light and electromagnetic waves [
2
]. Nevertheless, Weber’s
electrodynamics has faced criticism for its inability to predict electromagnetic waves in free
space, including its failure to explain electromagnetic wave phenomena, such as radar [
12
]
and cold plasma oscillations [
13
]. However, in the past few decades, researchers have
challenged this view and made progress in using Weber’s electrodynamics to explain
electromagnetic waves. In particular, Assis [
14
,
15
] employed Weber ’s electrodynamics to
derive a wave equation for signal propagation in conductors, while Wesley [16] derived a
Foundations 2023,3, 323–334. https://doi.org/10.3390/foundations3020024 https://www.mdpi.com/journal/foundations
Foundations 2023,3324
wave equation by introducing time retardation into Weber’s electrodynamics. Kühn [
17
]
developed an inhomogeneous wave equation from Maxwell’s equations to be compatible
with Weber ’s electrodynamics under certain conditions (observer rest frame, charge as a
function of velocity, and uniform charge velocity). Interestingly, there is another approach
in which the vacuum is not regarded as empty. It is based on the quantum mechanical
idea that, under the influence of an external electric field, the vacuum is polarized through
the creation of so-called virtual-particle/anti-particle pairs [
18
,
19
]. In 2003, Fukai [
20
]
put forward comparable ideas in which electrical signals propagating through a vacuum
were hypothesized to have an equivalent circuit consisting of a series of LC components
(i.e., inductors and capacitors).
Building on these ideas, herein we assume a polarized vacuum consisting of positive
and negative charges and, for the first time, successfully derive an electric wave equation
for free space based on the telegraphy ideas of Weber, Kirchoff, and Assis [
14
,
15
]. While
Maxwell’s equations have undoubtedly been successful in explaining electromagnetic
phenomena, recent advancements in Weber ’s electrodynamics can potentially offer an
intriguing and complementary insight regarding the behavior of electromagnetic waves.
2. Weber’s Electrodynamics
Compared to Maxwell–Lorentz electromagnetism, Weber’s electrodynamics gives a
much simpler form for particle–particle interaction forces. For two particles interacting
with each other, the force exerted by one particle on the other is given by [21]
*
F=Qqˆ
r
4πε0r2[1+1
c2(
*
v·
*
v−3
2(ˆ
r·
*
v)2+rˆ
r·
*
a)] (1)
where
Q
and
q
are two electrical charges,
*
F
is the force that charge
Q
exerted on charge
q
,
r
is the distance between the two charges,
ˆ
r
is the unit vector pointing from charge
Q
to
charge
q
,
*
v
and
*
a
are the velocity and acceleration of charge
q
relative to charge
Q
,
ε0
is
the dielectric constant, and
c
is the speed of light. This force has been used in numerous
studies of Weber ’s electrodynamics before, e.g., [
1
–
11
,
14
–
17
,
21
]. The symbols used in this
paper are listed in Table 1below.
Table 1. List of symbols used in this paper.
Symbol Meaning Symbol Meaning
*
FTotal force. Q,qElectrical charges.
*
FcIntegrated static (Coulomb) force. ε0Dielectric constant.
*
FvIntegrated velocity related force. cSpeed of light.
*
FaIntegrated acceleration related force. ˆ
rUnit vector.
*
fForce between two parcels of charges. rDistance between the two charges.
*
fcStatic (Coulomb) force.
*
vRelative velocity.
*
fvVelocity related force.
*
aRelative acceleration.
*
faAcceleration related force.
*
EExternal electric field.
*
D+,
*
D−Displacement of positive and negative charges. g() Relationship between the electric field
and the displacement field.
*
v+,
*
v−Velocity of positive and negative charges. ∇· Divergence operator.
*
a+,
*
a−Acceleration of positive and negative charges. O(
*
r2)Higher-order terms.
Foundations 2023,3325
Table 1. Cont.
Symbol Meaning Symbol Meaning
ρ+,ρ−
Absolute density of positive and negative charges.
·Dot product of vector.
ρAverage density of positive and negative charges. ∇Gradient operator.
ρ−(
*
r)Density of negative charge at location
*
r. Similar
symbol for positive charge. dV Small parcel at origin.
*
v−(
*
r)Velocity of negative charge at location
*
r.d´
VSmall parcel at location
*
ron the shell.
*
a−(
*
r)Acceleration of negative charge at location
*
r.ds Small area on the shell.
ρ−(
*
0)Density of negative charge at origin. dr Thickness of the shell.
*
v−(
*
0)Velocity of negative charge at origin. ∂
∂tEulerian derivative.
*
a−(
*
0)Acceleration of negative charge at origin. d
dt Lagrangian derivative.
χConstant scalar. ∇× Curl operator.
ΦScalar potential. ∇2Laplacian operator.
The type of force described in Weber’s electrodynamics is often classified as action
at a distance (or direct action). The force is exerted instantaneously, regardless of the
distance between the interacting particles. Moreover, the force expression (as given in
Equation (1)) is relational and can be used to describe the interaction between particles
within any reference system.
3. Vacuum Polarization
In this article, in order to derive an electric wave equation from Weber’s force, we
postulate that the vacuum behaves similar to a polarizable material. With this assumption,
one can then speculate that the vacuum consists of positive and negative charges overlap-
ping each other (Figure 1). These charges can oscillate relative to each other, resulting in a
displacement represented by
*
D+
and
*
D−
for positive and negative charges, respectively
(Equation (2)). When there is no external electric field, the displacement of both positive
and negative charges is zero, and the vacuum remains electrically neutral (non-polarized).
However, when an external electric field is present, the displacement field becomes non-
zero (Equation (2)). As a result, the vacuum may become polarized due to the continuous
variation in charge density, which is related to the divergence of the displacement field.
*
D+=−
*
D−
*
v+=−
*
v−
*
a+=−
*
a−
*
D+=−
*
D−=g(
*
E)
ρ+=ρ(1− ∇·
*
D+)ρ−=ρ(1− ∇·
*
D−)ρ++ρ−=2ρ
(2)
where
*
D+
,
*
v+
,
*
a+
, and
ρ+
are displacement, velocity, acceleration, and density of positive
charge, respectively;
*
D−
,
*
v−
,
*
a−
, and
ρ−
are displacement, velocity, acceleration, and
density of negative charge, respectively;
*
E
is the external electric field;
g()
represents a
relationship between the electric field and the displacement field; and
ρ
is the average
density of positive and negative charges. When the density change is small, the formula
ρ+=ρ(1− ∇·
*
D+)
is approximated by
ρ+=ρ−ρ+∇·
*
D+
, which is equivalent to the
volumetric strain formula in continuum mechanics. Here, the rest frame of the media
(positive–negative charge pairs in the vacuum) is used.
Foundations 2023,3326
Foundations 2023, 3, FOR PEER REVIEW 4
Figure 1. Sketch of vacuum postulated as a polarizable material. The positive–negative charge pairs
are not dipoles since the charges can fully overlap each other when
.
4. Derivation of Electric Wave Equation
Consider a homogeneous vacuum with positive and negative charges. The negative
charge density is , and the velocity and acceleration of negative charge are
and , respectively (Figure 2). The density , velocity , and acceleration
of positive charge are simply related with those of negative charge (Equation (2)).
Now, apply a Taylor expansion to the quantities of negative and positive charges
around the origin,
(Equation (3)). Note that the same expansion applies to positive
charge too. The higher-order terms will be dropped in the subsequent equation
derivations.
(3)
Figure 1.
Sketch of vacuum postulated as a polarizable material. The positive–negative charge pairs
are not dipoles since the charges can fully overlap each other when
*
D+=−
*
D−=0.
4. Derivation of Electric Wave Equation
Consider a homogeneous vacuum with positive and negative charges. The negative
charge density is
ρ−(
*
r)
, and the velocity and acceleration of negative charge are
*
v−(
*
r)
and
*
a−(
*
r)
, respectively (Figure 2). The density
ρ+(
*
r)
, velocity
*
v+(
*
r)
, and acceleration
*
a+(
*
r)of positive charge are simply related with those of negative charge (Equation (2)).
Foundations 2023, 3, FOR PEER REVIEW 5
Figure 2. Sketch of physical quantities at the origin,
, and at .
Both positive charge and negative charge on the shell exert force
on the negative charge
at the origin (Figure 3). Using Weber’s electrodynamics
(Equation (1)) and the Taylor expansion of charge quantities (Equation (3)), the combined
force can be wrien as
(4)
Equation (4) can be broadly separated into two parts, with the rst item of the above
equation being the force exerted by the negative charge . The second item is the
force exerted by the positive charge . Overall, the force consists of a static term
(Coulomb force), a velocity-related term, and an acceleration-related term.
Figure 2. Sketch of physical quantities at the origin,
*
0 , and at
*
r.
Now, apply a Taylor expansion to the quantities of negative and positive charges
around the origin,
*
0
(Equation (3)). Note that the same expansion applies to positive charge
Foundations 2023,3327
too. The higher-order terms
O(
*
r2)
will be dropped in the subsequent equation derivations.
ρ−(
*
r) = ρ−(
*
0) +
*
r·∇ρ−(
*
0) + O(
*
r2)
*
v−(
*
r) =
*
v−(
*
0) +
*
r·∇
*
v−(
*
0) + O(
*
r2)
*
a−(
*
r) =
*
a−(
*
0) +
*
r·∇
*
a−(
*
0) + O(
*
r2)
(3)
Both positive charge
ρ+(
*
r)d´
V
and negative charge
ρ−(
*
r)d´
V
on the shell exert force
on the negative charge
ρ−(
*
0)dV
at the origin (Figure 3). Using Weber’s electrodynamics
(Equation (1)) and the Taylor expansion of charge quantities (Equation (3)), the combined
force
*
fcan be written as
*
f=−
*
r
4πε0r3ρ−(
*
0)dVρ−(
*
r)d´
V(1+1
c2((
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)−
*
v−(
*
0))·(
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)−
*
v−(
*
0))
−3
2r2(
*
r·(
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)−
*
v−(
*
0)))2+
*
r·(
*
a−(
*
0) +
*
r·∇
*
a−(
*
0)−
*
a−(
*
0))))
+
*
r
4πε0r3ρ−(
*
0)dVρ+(
*
r)d´
V(1+1
c2((
*
v+(
*
0) +
*
r·∇
*
v+(
*
0)−
*
v−(
*
0))·(
*
v+(
*
0) +
*
r·∇
*
v+(
*
0)−
*
v−(
*
0))
−3
2r2(
*
r·(
*
v+(
*
0) +
*
r·∇
*
v+(
*
0)−
*
v−(
*
0)))2+
*
r·(
*
a+(
*
0) +
*
r·∇
*
a+(
*
0)−
*
a−(
*
0))))
(4)
Foundations 2023, 3, FOR PEER REVIEW 6
Figure 3. Sketch of charges in the volume element on the spherical shell, and charges in the
volume element at the origin. is an area on the shell and is the thickness of the shell.
First, consider the static term (Coulomb force) and use Equations (2) and (3)
(5)
Now, integrate around the shell
(6)
(N.B., further details supporting the integrations carried out in this derivation are
included in the Supplementary Information).
Second, consider the velocity-related term
(7)
Figure 3.
Sketch of charges in the volume element
d´
V
on the spherical shell, and charges in the
volume element dV at the origin. ds is an area on the shell and dr is the thickness of the shell.
Equation (4) can be broadly separated into two parts, with the first item of the above
equation being the force exerted by the negative charge
ρ−(
*
r)d´
V
. The second item is the
force exerted by the positive charge
ρ+(
*
r)d´
V
. Overall, the force consists of a static term
(Coulomb force), a velocity-related term, and an acceleration-related term.
Foundations 2023,3328
First, consider the static term (Coulomb force) and use Equations (2) and (3)
*
fc=−
*
r
4πε0r3ρ−(
*
0)dVρ−(
*
r)d´
V+
*
r
4πε0r3ρ−(
*
0)dVρ+(
*
r)d´
V
=−
*
r
4πε0r3ρ−(
*
0)dV(2ρ−(
*
0) + 2
*
r·∇ρ−(
*
0)−2ρ)d´
V
(5)
Now, integrate around the shell
*
Fc=−1
4πε0r3ρ−(
*
0)dVZ(2ρ−(
*
0) + 2
*
r·∇ρ−(
*
0)−2ρ)
*
r dsdr =−2r
3ε0
ρ−(
*
0)dV∇ρ−(
*
0)dr (6)
(N.B., further details supporting the integrations carried out in this derivation are
included in the Supplementary Information).
Second, consider the velocity-related term
*
fv=−
*
r
4πε0r3ρ−(
*
0)dVρ−(
*
r)d´
V1
c2((
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)−
*
v−(
*
0))·(
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)−
*
v−(
*
0))
−3
2r2(
*
r·(
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)−
*
v−(
*
0)))2)
+
*
r
4πε0r3ρ−(
*
0)dVρ+(
*
r)d´
V1
c2((
*
v+(
*
0) +
*
r·∇
*
v+(
*
0)−
*
v−(
*
0))(
*
v+(
*
0) +
*
r·∇
*
v+(
*
0)−
*
v−(
*
0))
−3
2r2(
*
r·(
*
v+(
*
0) +
*
r·∇
*
v+(
*
0)−
*
v−(
*
0)))2)
(7)
From Equations (2) and (3), we can have
ρ−(
*
r)≈ρ−(
*
0) +
*
r·∇ρ−(
*
0)
ρ+(
*
r) = 2ρ−ρ−(
*
r)≈2ρ−ρ−(
*
0)−
*
r·∇ρ−(
*
0)
*
v+(
*
0) = −
*
v−(
*
0)∇
*
v+(
*
0) = −∇
*
v−(
*
0)
(8)
Inserting Equation (8) into Equation (7), with some simplification we obtain
*
fv=−
*
r
4πε0r3ρ−(
*
0)dV(ρ−(
*
0) +
*
r·∇ρ−(
*
0))d´
V1
c2((
*
r·∇
*
v−(
*
0))·(
*
r·∇
*
v−(
*
0)) −3
2r2(
*
r·(
*
r·∇
*
v−(
*
0)))2)
+
*
r
4πε0r3ρ−(
*
0)dV(2ρ−ρ−(
*
0)−
*
r·∇ρ−(
*
0))d´
V1
c2((2
*
v−(
*
0) +
*
r·∇
*
v−(
*
0))
·(2
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)) −3
2r2(
*
r·(2
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)))2)
(9)
We can then integrate around the shell and drop those higher-order terms which
consist of a product of
*
r·∇ρ−(
*
0)and
*
r·∇
*
v−(
*
0). Thus, we can obtain Equation (10),
*
Fv=−1
4πε0r3ρ−(
*
0)dV R1
c2((
*
r·∇
*
v−(
*
0))·(
*
r·∇
*
v−(
*
0))
−3
2r2(
*
r·(
*
r·∇
*
v−(
*
0)))2)(ρ−(
*
0) +
*
r·∇ρ−(
*
0))
*
r dsdr
+1
4πε0r3ρ−(
*
0)dV R1
c2((2
*
v−(
*
0) +
*
r·∇
*
v−(
*
0))·(2
*
v−(
*
0) +
*
r·∇
*
v−(
*
0))
−3
2r2(
*
r·(2
*
v−(
*
0) +
*
r·∇
*
v−(
*
0)))2)(2ρ−ρ−(
*
0)−
*
r·∇ρ−(
*
0))
*
r dsdr
=−14r
15ε0ρ−(
*
0)dV 1
c2(
*
v−(
*
0)·
*
v−(
*
0))∇ρ−(
*
0)dr +4r
5ε0ρ−(
*
0)dV 1
c2(
*
v−(
*
0)·∇ρ−(
*
0))
*
v−(
*
0)dr
(10)
Third, consider the acceleration-related term
*
fa=−
*
r
4πε0r3ρ−(
*
0)dVρ−(
*
r)d´
V1
c2
*
r·(
*
a−(
*
0) +
*
r·∇
*
a−(
*
0)−
*
a−(
*
0))
+
*
r
4πε0r3ρ−(
*
0)dVρ+(
*
r)d´
V1
c2
*
r·(
*
a+(
*
0) +
*
r·∇
*
a+(
*
0)−
*
a−(
*
0)) (11)
From Equations (2) and (3), we obtain
*
a+(
*
0) = −
*
a−(
*
0)∇
*
a+(
*
0) = −∇
*
a−(
*
0)(12)
Foundations 2023,3329
Inserting Equation (12) into Equation (11), we obtain
*
fa=−
*
r
4πε0r3ρ−(
*
0)dV2ρd´
V1
c2
*
r·(
*
r·∇
*
a−(
*
0)) −
*
r
4πε0r3ρ−(
*
0)dV(ρ+(
*
0) +
*
r·∇ρ+(
*
0))d´
V1
c2
*
r·2
*
a−(
*
0)(13)
Again, we can integrate around the shell and drop the higher-order terms which have
a product of
*
r·∇ρ+(
*
0)and
*
r·2
*
a−(
*
0). We can also use the approximation ρ+(0)≈ρ.
*
Fa=−1
4πε0r3ρ−(
*
0)dVR2ρ
c2
*
r·(
*
r·∇
*
a−(
*
0))
*
r dsdr
−1
4πε0r3ρ−(
*
0)dVR2
c2(ρ+(
*
0) +
*
r·∇ρ+(
*
0))
*
r·
*
a−(
*
0)
*
r dsdr =−2r
3ε0ρ−(
*
0)dV ρ
c2
*
a−(
*
0)dr
(14)
The total force on negative charge in dV exerted by the shell is
*
F=
*
Fc+
*
Fv+
*
Fa(15)
Inserting Equations (6), (10), and (14) into Equation (15), we obtain
*
F=−2r
3ε0ρ−(
*
0)dV∇ρ−(
*
0)dr −14r
15ε0ρ−(
*
0)dV 1
c2(
*
v−(
*
0)·
*
v−(
*
0))∇ρ−(
*
0)dr
+4r
5ε0ρ−(
*
0)dV 1
c2(
*
v−(
*
0)·∇ρ−(
*
0))
*
v−(
*
0)dr −2r
3ε0ρ−(
*
0)dV ρ
c2
*
a−(
*
0)dr
(16)
Since the mass of positive–negative charges in vacuum is likely much smaller com-
pared to that of physical particles, we may choose to ignore their mass and acceleration force
in the volume element
dV
. Additionally, the polarization force within a
positive–negative
charge pair may also be neglected since it is likely small compared to the force exerted
by the nearby volume of charges. Equation (16) is valid for a small radius
r
, as we used
a Taylor expansion (Equation (3)). Without integrating along the radius
r
, we can set the
total force in Equation (16) equal to zero due to force balance. With some simplification,
we obtain
∇ρ−(
*
0) + 7
5
1
c2(
*
v−(
*
0)·
*
v−(
*
0))∇ρ−(
*
0)−6
5
1
c2(
*
v−(
*
0)·∇ρ−(
*
0))
*
v−(
*
0) + ρ
c2
*
a−(
*
0) = 0 (17)
When velocity
*
v−(
*
0)
is much less than
c
, we can neglect the velocity-related term. Thus
∇ρ−(
*
0) + ρ
c2
*
a−(
*
0) = 0 (18)
The above equation can also be written as
∇ρ−(
*
0) + ρ
c2
d
*
v−(
*
0)
dt =0 (19)
This expression holds for points other than the origin. Additionally, since
*
v−(
*
0)
is
small, the Lagrangian derivative can be approximated as a Eulerian derivative. Thus, we
can write
∇ρ−+ρ
c2
∂
*
v−
∂t=0 (20)
By applying divergence to the above equation
∇·∇ρ−+ρ
c2
∂(∇·
*
v−)
∂t=0 (21)
From Equation (2), we have
ρ−=ρ(1− ∇·
*
D−)
. Applying a time derivative to it,
we obtain
∂ρ−
∂t=−ρ∇· ∂
*
D−
∂t=−ρ∇·
*
v−(22)
Foundations 2023,3330
By inserting Equation (22) into Equation (21), we obtain
∇·∇ρ−−1
c2
∂2ρ−
∂t2=0 (23)
which represents the wave propagation equation for negative charge density.
Next, we can insert equation
ρ−=ρ(1− ∇·
*
D−)
into Equation (20), assuming a
homogenous vacuum and constant charge density, ρ. This gives us
−ρ∇∇·
*
D−+ρ
c2
∂
*
v−
∂t=0 (24)
which can be simplified to
−∇∇·
*
D−+1
c2
∂2
*
D−
∂t2=0 (25)
Using the vector formula,
∇∇·
*
D−=∇×∇×
*
D−+∇2
*
D−
, we can simplify
Equation (25) further to obtain
∇ × ∇ ×
*
D−+∇2
*
D−=1
c2
∂2
*
D−
∂t2(26)
At first glance, this equation appears a little complex. Therefore, considering a simple
scenario of irrotational field
*
D−(∇ × ∇ ×
*
D−=0), we obtain
∇2
*
D−=1
c2
∂2
*
D−
∂t2(27)
Further, we can assume the vacuum is a homogeneous, linear, non-dispersive, and
isotropic dielectric medium. We may use a simple expression of vacuum polarization
*
D+=−
*
D−=χε0
*
E
, where
χ
is a constant scalar; this expression is similar to that of a
typical isotropic dielectric medium [22]. Finally, we obtain
∇2
*
E=1
c2
∂2
*
E
∂t2(28)
The equation presented above (28) bears remarkable resemblance to the widely recog-
nized electric field wave equation derived from Maxwell’s equations. For the derivation
herein, we start with the balance of Weber’s force, which leads us to the wave equation of
the charge displacement field, and, subsequently, we arrive at the wave equation of the
electric field. It is important to note that Weber’s force, which arises from the interactions
among charges, determines the displacement field. Furthermore, this force can generally
be represented by an electric field [
8
]. In this context, it is worth highlighting that the
displacement field is merely a manifestation or an indicator of the electric field.
5. Longitudinal Electric Wave
Longitudinal electric waves travel in the same direction as the electric field. While they
have been observed in plasmas [
23
] and focused beams [
24
], their existence in free space has
been a topic of debate. According to Gauss’s law, the divergence of the electric field in free
space must be zero, which implies that the plane or spherical longitudinal waves cannot
exist in free space [
25
]. However, there have been reports of longitudinal electric waves in
free space, such as during the eclipse of the sun by the moon [
26
] and in experiments with
spherical antennas [
27
,
28
]. To explain these observations, Monstein and Wesley proposed
Foundations 2023,3331
an inhomogeneous wave equation for the scalar potential using Coulomb’s law and time
retardation [27]. For clarity, the wave equation is transcribed below (Equation (29)):
∇2Φ−∂2Φ2
∂2t2=−4πρ (29)
where
Φ
is the scalar potential and
ρ
is the source charge density. Since introducing the
time retardation term
∂2Φ2
∂2t2
, the above equation does not obey Gauss’s law in free space. A
similar theory was used to explain longitudinal electric waves in vacuum radiated by an
electric dipole [29].
The wave equation derived in this paper does not rely on Gauss’s law. Instead, it
is based on the non-zero divergence of electric displacement (and/or electric field). If
the divergence is zero, the field will be static and there will be no wave propagation
(Equation (25)). Therefore, the theory developed in this paper is compatible with the
phenomenon of longitudinal waves and with the theory proposed by Monstein and Wesley.
6. Discussion
This paper posits that the vacuum is not ‘empty’, but rather filled with
positive–negative
charge pairs. This postulation is not without precedent, as it is somewhat in line with
the concept of vacuum polarization in quantum mechanics [
18
,
19
], and the Casimir effect
of the void [
30
]. However, it is important to note that there is a fundamental difference
between the vacuum postulate in this paper, which has speculated regarding the existence
of physical charges in the vacuum, and the quantum mechanical assumption that virtual
particle–anti-particle pairs are created in the vacuum.
The Michelson–Morley experiment has long been cited as evidence against the exis-
tence of aether or any other free-space medium. However, in this paper, a hypothetical situ-
ation is described whereby the vacuum serves as a medium, consisting of
positive–negative
charge pairs. By considering the rest frame of this medium and assuming small charge
velocity, the Lagrangian derivative is approximated as a Eulerian derivative, as shown
in Equations (19) and (20). Notably, the wave equation derived herein does not include
any sources. Further research should investigate how to incorporate sources and consider
cases with varying source and medium velocities. It is also possible that the wave equation
derived here may only be valid for stationary media, and further work is needed to explore
its limitations and applicability in more general cases.
A criticism that has long been levelled at Weber ’s electrodynamics is its supposed
incompatibility with fields and electromagnetic waves, since Weber ’s force is based on
direct action and thus is not conceptually dependent upon fields. Some, such as O’Rahilly,
consider that the field is a merely a ‘metaphor’ and that force formulae are the ultimate
element of ‘scientific description’ [
2
]. Nevertheless, even though the literature is heavily
weighted towards the idea of fields (electric and magnetic fields), with little consideration
for the more fundamental entity of force, Weber’s electrodynamics is remarkably compatible
with field theory [
17
]. It has been shown that the force law of Weber is consistent with
Maxwell’s equations [
2
], and it can be generalized in such a way that its interpretation is
based on the concept of fields [2].
It can be said that Maxwell’s equations possess a form of beauty, and the manifes-
tation of the speed of light from the electric and magnetic wave equations, in relation to
permittivity and permeability, is masterful. Yet, Maxwell introduced this concept with
regard to his displacement current, based on the electrodynamics of Weber. This quantity
was first hypothesized by Weber in his force law, which he was also the first to measure
(in collaboration Kohlrausch), and, furthermore, along with Kirchoff (independently and
at approximately the same time), he was the first to deduce that signals within an electri-
cal circuit would propagate at light velocity based on his force law [
14
]. Whilst Weber’s
electrodynamics has received comparably little/no attention, the derivation herein shows
that with certain assumptions, it is possible to obtain an expression that resembles the
Foundations 2023,3332
wave equation for the electric field in free space associated with Maxwell’s equations from
Weber ’s force.
It is noteworthy that the wave equation for an electric field derived herein (
Equation (28)
)
employs a simple polarization expression
*
D+=−
*
D−=χε0
*
E
. However, Weber’s electro-
dynamics has been shown to consist of six components [
8
]. Deriving the wave equation for
all six components would require further research and a more sophisticated polarization
expression that takes into account the relative displacement, velocity, and acceleration of
positive–negative charge pairs.
It should also be noted that the wave equation derived in this paper is based on further
assumptions and approximations additional to those already discussed. For instance, we
neglected higher-order terms, acceleration force, polarization force, and velocity terms,
among others. These assumptions and approximations determine the range of applicability
of our derivation. Our approach specifically applies to homogeneous vacuum and is most
suitable for scenarios involving low relative speeds of charge pairs. Future research is
needed to extend to cases when charged particles, such as electrons, exist in the vacuum,
and to cases when charge pairs have high relative speeds. Such investigations may require
more sophisticated theoretical and mathematical tools.
While the wave equation derived under the assumption of a zero curl (
Equation (26)
)
shares similarities with the widely recognized electric wave equation from Maxwell’s
equations, it remains uncertain whether the curl term is always zero. Most likely, it is not,
leading to differences with the electromagnetic wave equation associated with Maxwell’s
equations. Future research is needed to explore how this equation can be solved to explain
electric wave phenomena. Interestingly though, this equation has some similarity to fluid
dynamics equations. The curl term may indicate vortex structures in the electric field, akin
to those seen in fluids. For instance, ball lightning, a rare and unexplained luminescent
spherical object phenomenon [
31
], may be a potential candidate to study vortex structures
in electric fields.
7. Conclusions
A wave equation for the electric field has been successfully derived from Weber’s
force, which is also compatible with longitudinal waves. This demonstrates that Weber’s
electrodynamics can yield an electric field wave equation for free space, similar to that
derived from Maxwell’s equations. However, we speculate that the vacuum is not ‘empty’,
but rather it is filled with positive–negative charge pairs, providing the medium for wave
propagation. It is important to note that this derivation is based on certain assumptions
and approximations which determine its validity and range of applicability, in particular,
the hypothetical assumption of charged particle pairs in the vacuum. This idea has some
semblance to the concept of vacuum polarization, but the postulation in this paper is some-
what different from the quantum mechanical assumption that virtual particle–anti-particle
pairs are created in the vacuum. The electric wave equation derived from Weber ’s electro-
dynamics demonstrates the potential of a Weber-type theory in predicting electromagnetic
waves for free space and offers the advantage of dealing only with electrical forces between
charges, avoiding magnetic fields, displacement currents, and the like.
Supplementary Materials:
The following supporting information can be downloaded at: https://www.
mdpi.com/article/10.3390/foundations3020024/s1, further details relating to the integrations per-
formed in the main manuscript.
Author Contributions:
Conceptualization, Q.L. and S.M.; Methodology, Q.L.; writing—original
draft preparation, Q.L.; writing—review and editing, S.M. All authors have read and agreed to the
published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Foundations 2023,3333
Data Availability Statement: Not applicable.
Acknowledgments:
Q.L. thanks Steffen Kühn, Juan Manuel Montes Martos, and Mischa Moerkamp
for their constructive suggestions and feedback. S.M. thanks Christof Baumgärtel and Ray T. Smith
for fruitful discussions.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Tran, M. Evidence for Maxwell’s equations, fields, force laws and alternative theories of classical electrodynamics. Eur. J. Phys.
2018,39, 063001. [CrossRef]
2.
Baumgärtel, C.; Maher, S. Foundations of Electromagnetism: A Review of Wilhelm Weber ’s Electrodynamic Force Law. Foundations
2022,2, 949–980. [CrossRef]
3.
Baumgärtel, C.; Smith, R.T.; Maher, S. Accurately predicting electron beam deflections in fringing fields of a solenoid. Sci. Rep.
2020,10, 10903. [CrossRef] [PubMed]
4.
Baumgärtel, C.; Maher, S. A Charged Particle Model Based on Weber Electrodynamics for Electron Beam Trajectories in Coil and
Solenoid Elements. Prog. Electromagn. Res. C 2022,123, 151–166. [CrossRef]
5.
Kühn, S. Experimental investigation of an unusual induction effect and its interpretation as a necessary consequence of Weber
electrodynamics. J. Electr. Eng. 2021,72, 366–373. [CrossRef]
6. Kühn, S. Experimental confirmation of Weber electrodynamics against Maxwell electrodynamics. Vixra 2019, 1–18.
7. Kühn, S. On the wave-particle duality of a Hertzian dipole. TechRxiv 2023, preprint. [CrossRef]
8. Li, Q. Electric Field Theory Based on Weber ’s Electrodynamics. Int. J. Magn. Electromagn. 2021,7, 039.
9. Li, Q. Extending Weber ’s Electrodynamics to High Velocity Particles. Int. J. Magn. Electromagn. 2022,8, 1–9.
10. Montes, J.M. On the Modernisation of Weber ’s Electrodynamics. Magnetism 2023,3, 102–120. [CrossRef]
11.
Baumgärtel, C.; Maher, S. Resolving the Paradox of Unipolar Induction—New Experimental Evidence on the Influence of the Test
Circuit. Sci. Rep. 2022,12, 16791. [CrossRef] [PubMed]
12. Sherwin, C.W. The Weber radar. Phys. Essays 1991,4, 417–419. [CrossRef]
13.
Clemente, R.A.; Cesar, R.G. Cold plasma oscillations according to Weber ’s law an unphysical result. Int. J. Theor. Phys.
1993
,32,
1257–1260. [CrossRef]
14.
Assis, A. The meaning of the constant c in Weber’s electrodynamics. In Proceeding of the International Conference Galileo Back in Italy
II; Monti, R., Ed.; SocietáEditrice Andromeda: Bologna, Italy, 2000; pp. 23–36.
15.
Assis, A.K.T.; Hernandes, J. Telegraphy equation from Weber ’s electrodynamics. IEEE Trans. Circuits Syst. II Express Briefs
2005
,
52, 289–292. [CrossRef]
16. Wesley, J. Weber electrodynamics extended to include radiation. Specul. Sci. Technol. 1987,10, 50–53.
17.
Kühn, S. Inhomogeneous wave equation, Liénard-Wiechert potentials, and Hertzian dipoles in Weber electrodynamics. Electro-
magnetics 2022,42, 571–593. [CrossRef]
18.
Weinberg, S. Foundations. In The Quantum Theory of Fields; Cambridge University Press: Cambridge, UK, 2002; Volume I,
ISBN 978-0-521-55001-7.
19. Karbstein, F. Probing Vacuum Polarization Effects with High-Intensity Lasers. Particles 2020,3, 39–61. [CrossRef]
20. Fukai, J. A Promenade along Electrodynamics; Vales Lake Publishing: Pueblo West, CO, USA, 2003.
21.
Assis, A.K.T.; Tajmar, M. Rotation of a superconductor due to electromagnetic induction using Weber’s electrodynamics. Ann.
Fond. Louis Broglie 2019,44, 111–123.
22.
Feynman, R.P.; Leighton, R.B.; Sands, M. Feynman Lectures on Physics; Addison-Wesley: Boston, MA, USA, 1964; Volume 2,
ISBN 0-201-02117-X.
23.
Gurnett, D.A. Introduction to Plasma Physics: With Space, Laboratory and Astrophysical Applications, 2nd ed.; Cambridge University
Press: Cambridge, UK, 2017; ISBN 9781107027374.
24.
Maluenda, D.; Aviñoá, M.; Ahmadi, K.; Martínez-Herrero, R.; Carnicer, A. Experimental estimation of the longitudinal component
of a highly focused electromagnetic field. Sci. Rep. 2021,11, 17992. [CrossRef]
25. Bruhn, G.W. Can longitudinal electromagnetic waves exist? J. Sci. Explor. 2002,16, 359–362.
26. Zaimidoroga, O. An electroscalar energy of the Sun: Observation and research. J. Mod. Phys. 2016,7, 806–818. [CrossRef]
27.
Monstein, C.; Wesley, J.P. Observation of scalar longitudinal electrodynamic waves. Europhys. Lett.
2002
,59, 514–520. [CrossRef]
28.
Butterworth, E.; Allison, C.; Cavazos, D.; Mullen, F. Longitudinal electromagnetic waves? The Monstein-Wesley experiment
reconstructed. J. Sci. Explor 2013,27, 13–23.
29.
Zhakatayev, A.; Tlebaldiyeva, L. Long-range longitudinal electric wave in vacuum radiated by electric dipole: Part I. Radio Sci.
2020,55, e2019RS006881. [CrossRef]
Foundations 2023,3334
30.
Esquivel Sirvent, R. The force of the void. The Casimir effect. Nano world. Interdiscip. J. Nanosci. Nanotechnol.
2011
,4, 1.
[CrossRef]
31.
Cen, J.; Yuan, P.; Xue, S. Observation of the Optical and Spectral Characteristics of Ball Lightning. Phys. Rev. Lett.
2014
,112, 035001.
[CrossRef]
Disclaimer/Publisher’s Note:
The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
