(George Galeczki)

**I. Maxwell's Equations**

Freeman Dyson published twelve years ago [1] "Feynman's proof of the
Maxwell equations". He recalls that in 1948 Feynman showed him this "proof
assuming only Newton's law of motion and the commutation relation between
position and velocity for a single particle." Although formally obtaining
the two "vacuum equations" (*i.e.* those without source terms), the
claimed "proof" of the full Maxwell equations is wrong mathematically,
physically and conceptually. On top of all this, it expresses the - nowadays
common - arrogance of mathematical physicists giving priority to formalism
against empirical facts.

As a matter of fact, Maxwell's equations (ME) represent the mathematical expression of the experimentally discovered laws of Gauss, Ampère and Faraday and are widely used in physics and engineering. Several remarks and comments are in order, each of them being subsequently discussed in more detail:

1/ The basic formulation of ME - as derived from experiments - is in
**integral form**

pertaining to a finite, closed area or volume.

2/ The ME are formulated for **continuous fields** and are called,
therefore, **field equations**.

3/ ME hold for **closed circuits **only.

4/ The sources of the fields are **continuous charges **and **continuous
current densities**.

The** **discrete, quantized charges introduced in Maxwell's theory
are **foreign elements**.

5/ ME are **tautological** in the sense that they merely represent
relationships between **fields**

and their **sources. **One has to provide the initial charge distribution
in order to be able to

calculate the field distribution, or *vice versa*.

6/ ME is unable to describe the **interaction** between two discrete
charges.

7/ ME is unable to supply the **equation of motion** of one charge
in the field produced by all

others.

8/ ME are not suited for the description of **open circuits **like
**antennas**.

9/ ME are unable to prescribe the exact conditions under which a system
will **radiate**, or not.

The notorious example is Bohr's planetary model of the **hydrogen atom**.

10/ ME are unable to provide a **stable model** for the elementary
charge.

11/ ME are **generally covariant** and do not single out the Lorentz
transformation (LT) of the

"special" theory of relativity (STR).

12/ ME are formulated in terms of **independent, Eulerian coordinates**
x, y, z, t and **partial**

**derivatives ¶
**/¶
x , ¶ /¶
y , ¶ /¶
z , ¶ /¶
t .

ME in their original, **integral form** and in modern notation are:

**E.**d**s **= -(¶
**B**/¶
t)**.**d**a B.**d**a
= **0
(1)

**D.**d**a **= r
dV **H.**d**s
= **(**j + ¶
D**/¶ t)**.**d**a
**(2)

Contained in the above is the **equation of continuity**:

**j.**d**a** = -r
dV
(3)

where r and **j** denote charge and current
density, respectively.

In all cases the regions of integration are assumed to be **stationary
**and** mechanically rigid**.

ME in **differential form **, as derived from (1), (2) and (3) by
means of Stokes' and Gauss' theorem, are:

Ñ
**´
H**
= **j + ¶ D**/¶
t Ñ
**´
E**
= -¶
**B**/¶
t
(4)

Ñ
**. D** = r
Ñ
**. B** = 0
(5)

and the corresponding equation of continuity:

Ñ
**. j** = -¶
r
/¶ t
(6)

ME in differential form express the relationship that must exist between
the four field vectors **E**, **D**, **H** and **B** at any
point within a **continuous medium** (?). In this form, because they
involve space derivatives, they cannot be expected to yield information
at points of discontinuity in the medium. However, the integral form can
always be used to determine what happpens at the boundary surface between
different media. It follows then, that the **tangential** components
of **E **and **H **(except perfect conductors) and the **normal**
components of **B** and **D** (if no surface charges are present)
have to be continuous at the interface.

**II. Maxwell's Equations and "Special" Relativity**

As already said, besides generalizing Ampère's circuital law
by introducing the *"displacement current"*, Maxwell's achievment
was to express the experimental laws of Coulomb, Gauss, Ampère,
Faraday in mathematical terms. The modern, vector form of ME was introduced
by Gibbs. Einstein's "special" relativity of 1905 has built heavily upon
electromagnetism and, assuming the validity of ME in all **inertial frames
of reference** (IFR's) introduced the incomprehensible "postulate of
light velocity invariance". This constancy is not that refering to the
light source - a wellknown fact in classical wave theory - but to the independence
of the velocity of light from the uniform velocity of the observer/detector
relative to the source.

Since in the special case of *vacuum-as-continuous-medium* the
ME displayed **covariance** (not **invariance**!) under the so called
Lorentz transformations (LT), ME and STR became indissolubly tied together,
one implying so to say the other. This strategy proved itself very useful,
since every criticism of STR was authomatically seen as criticism of ME,
thus contributing to the survival of the contradiction ridden STR.

The said bi-univocal correspondence between ME and STR is manifested,
however, only if one has in mind the differential form of ME. The reason
is that STR is a **local**, **point-event theory, **with **local
simultaneity **and **position **and** velocity dependent time! **Due
to this feature of STR, only **local conservation laws of energy and momentum**
are compatible with STR. While valid in the *hydrodynamic approximation*
- a *continuum theory* - **the local, differential conservation laws
fail in the case of discrete, extended systems**. The **global time
**required in this case, independent of position and velocity, is anathema
to STR. Farady's law of induction assumes tacitely such a global time and
distant simultaneity, thus allowing the definition of inductance and self-inductance
for macroscopic, closed circuits. Since the stationary circuits appearing
in the integral form of ME are incompatible with STR, it is quite understandable
why the STR "philosophy" gave almost exclusive prominence to ME in differential
form and eliminated the integral form from physics textbooks and monographies.
As a matter of fact, this form - more rich in physical information - is
actively in use in engineering books on electromagnetism [2].

**III. General covariance of Maxwell's equations**

It was repeatedly and forcefully pointed out by Post [3] that **Maxwell's
equations are general covariant.** Post quotes that Kottler, Cartan and
van Dantzig (KCD), quite independently of one another concluded on the
**natural invariance** of ME, **independent of any metric or linear
connection.** Post strongly emphasized the clean functional separation
obtainable between the **constitutive equations **{ **D**(**E**),
**B**(**H**), **J**(**E**) } and the **field equations. **In
this approach **the constitutive equations instead of ME carry all the
metric information**, while the field equations (4) and (5) are **covariant
**under all possible space and time coordinate transformations: Galilean,
Lorentzian, conform and so on. The use of (**E**, **D**, **B**,
**H**) rather than two field vectors eliminates the cgs free-space field
identification. The latter tied the cgs situation irrevocably to an inertial
reference frame. A free-space inertial situation is defined by an explicit
constitutive relation:

**D** = e
_{0}**E**
,
**B** = µ_{0}**H**
(7)

Which can be proven to be invariant under the Lorentz group, as well
as under scale changes of the conformal group. Here is the place to mention
the qualitative difference between **invariance** and **covariance**:
a physical law is said to be invariant under a coordinate transformation
when the vectors/tensors entering the law remain unchanged, while in the
case of covariance the components of the vectors/tensors are transformed,
or in Thomas Phipps' words "scrambled", according to the same rules as
the coordinates,. Only after performing this scrambling the equations in
question remain *form invariant*.

The superiority of the KCD approach is that the field equations retain
their form when a transition is made from an **inertial** to a **non-inertial**
frame, in particular a **rotating frame**. (N.B. Jan Evert Post was
the chief theoretician of the *ring laser gyro* project, which produced
the most sensible detector of rotation, *i.e. *of the degree of non-inertiality).
Moreover, the form invariance of ME is independent of whether the fields
exist in free space or in matter. The information about the reference frame
and the state of motion of the matter therein is conveyed exclusively by
the nature of the constitutive equations. The general form of these equations
is *tensorial *and the applications to specific problems was intensively
investigated by Post.

**IV. The force of Lorentz and Maxwell's equations**

It is clear from what has been said till now that the Maxwell field
equations (4) and (5) expresses a law of nature and will retain their validity
so long their limitation to closed circuits is assured. The **tautological
nature of ME** follows from the absence of **"detector charges"**
in the rhs, since only the **source charges** and their distribution
is considered. It means that the ME are intrinsically unable to provide
either a force law between discrete charges, or an equation of motion for
individual charge within a system of charges. Here I can mention another
fundamental difficulty connected with the discrete *vs. *continuum
dichotomy, reflected in the use of two different kinds of coordinates:
the already mentioned **four independent, Eulerian** x, y, z, t and
the **three time dependent Lagrangean** coordinates x(t), y(t), z(t).
The former are suited for field theories, while the later for particle
dynamics. Fields are functions of x, y, z, t, meaning that they have different
values at the points of a 4D continuum. They don't "propagate" in the 4D
continuum. The solutions x(t), y(t), z(t) of the dynamical equations of
motion, on the other hand, are 3D vectors and the coordinates "move - so
to say - with the particle". This already shows that STR is at least formally
compatible with pure field theories, but incompatible with (discrete) particle
dynamics [4]. Anyway, quite independently of STR, in order to brake the
tautology, ME have to be completed by a **force law** and an **equation
of motion** for discrete charges, the very program of Lorentz who introduced
the quantized electrical charges into Maxwell's field theory. The ME supplemented
by the Lorentz force-law (LF) is called Maxwell-Lorentz electrodynamics
(MLE). Lorentz himself remained unsatisfied with his force-law:

**F _{L}** = q (

In his own words [5]: "It is got by generalizing the results of electromagnetic
experiments. The first term represents the force acting on an electron
in an **electrostatic field** [**F**_{1}=q**E**] . On
the other hand, the part of the force expressed by the second term may
be derived from the law according to which an element of a wire carrying
a currect is acted on by a **magnetic field** [d**F**_{2}
= (q**v x B , **assuming Jd**s **= q**v **] . After having been
led in one particular case to the existence of the force [**F**_{1}
= q**E**] and in another to that of the force [**F**_{2 }**=
**q**v x B** ] , __we now combine the two in the way shown in the
equation, going beyond the direct result of experiments by the assumption
that in general the two forces exist at the same time."__

(a) The two 'particular cases' here 'combined' are, however, quite incompatible.
In one case we have a charge **at rest**, in the other the charges are
**moving**.

(b) Experiments with 'a wire carrying current' have to do with **neutral
**currents, yet the derivation contradicts this neutrality. The discovery
of the Hall effect, formally described as a "modified Ohm's law":

**j** = s
**E** + k(**E ´
B**)
(9)

where s is the conductivity and k a constant,
seemingly supports (8), but everybody familiar with the experimental set-up
used in the Hall effect studies will agree that ** E and B above
belong to different systems: a dc - or ac - source for E and a completely
separated permanent or electro-magnet for B. Maxwell's theory requires,
however that E and B belong to the same system of charges and currents.**
As shown elsewhere [6],

**F _{L} = **q (

*i.e. *a **phenomenological external force, rather than fundamental
force acting on a charge belonging to the same system, as implied by ME.
**The upper indexes ^{(1)} and ^{(2) }indicate that the
electric, respective magnetic field belong to different systems, therefore
**ME and LF do not form a coherent Maxwell-Lorentz theory as claimed in
present day textbooks and monographies! The reason for this persisting
mess is the seeming compliance with SRT's LT.** This belief is, however,
totally wrong, since: (a) The LT apply only to **E** and **B** belonging
to the same system and (b) The velocity **v **in most applications is
a **non-uniform velocity** **between magnets an current carrying wires**,
while the velocity entering the LT is the **uniform relative velocity
between two inertial frames of reference**. This confusion goes back
to Einstein's failure to distinguish between his theory involving *schesic
velocities* referred to abstract "reference frames" and relative velocities
between moving masses, as implied by Mach's program. **For this reason
STR is not a true relativity theory**! Using a somewhat different terminology
- '**principle of relativity** *vs.* '**principle of relative
motion**' - this point was discussed in a paper by Bartocci and Capria
[7], too. This explains also why the aged Ernst Mach unmistakable declined
the rôle of spiritual father of the "special" (very special, indeed!)
theory of the young Einstein.

**V. Magnetic field, vector potential and induction**

In the spirit of the ME in their integral form, **B ^{(}2)**
in (10) has

**B**^{(2)}(**r**) = I' (d**s**'
´ **R**)/R^{3}
(11)

where **R** = **r** - **r**' and the integral is performed
around the closed current loop. Attempts to generalize the Biot-Savart
law for **time-variable magnetic fields** have been made by Jefimenco
[8] in the form:

**B** = (µ_{0}/4p ){[**j**]/r²
+ (1/rc)¶ [**j**]/¶
t} ´ (**r**/r)_{ }dV'
(12)

where [..] denotes the **retardation symbol **indicating that the
quantities between the square brackets are to be evaluated for t' = t -
r/c , where t is the moment for which **B** is calculated. It is interesting
to note that that Eq. (12) **does not** contain displacement currents,
thus indicating that although time-dependent magnetic fields and displacement
currents are coupled together, **displacement currents are not sources
of magnetic fields in the conventional sense**.

**Definition** (11) **of the magnetic field** **B **- rightly
called magnetic flux density in older books - **is incompatible** **with
the "Lorentz transformed E field" definition of B in "special" relativistic
electromagnetism**:

**B** = **V** ´ **E**
(13)

valid for **uniform **velocity **V** only! This incompatibility
brings us to the most important issue of (electromagnetic) **induction
**and the status of **Faraday's** "flux rule". According to textbook
(and also monography) knowledge, electromagnetic induction were always
due to a time variable magnetic flux crossing a closed conducting loop.
Although Faraday discovered both this so called **transformer induction
**as well as the **motional induction**, only the first is embedded
in the integral form of ME formulated for **stationary integration regions**.
This deficiency of the integral ME is, of course, transferred to the differential
form of the Maxwellian law of induction:

Ñ ´
**E** = -¶
**B**/¶
t
(4)

The correct expression for the induced electromotive force (emf), in
terms of the **vector potential A**, follows from the integral form:

emf = **E.**d**s
**- (d/dt)**B.**d**a**
= -(d/dt)(Ñ
´ **A**)**.**d**a** = -(d/dt)**A.**d**s**
(14)

which provides the formula:

**E**_{ind} = -d**A**/dt
(15)

for the induced electric field **E**_{ind}. Keeping the integration
region stationary, one gets the 'transformer field':

**E**_{ind} = -¶
**A**/¶
t
(16)

The difference between Eq. (15) containing the **total time derivative
**d/dt and Eq. (16) containing the **partial time derivative ¶
**/¶
t is huge and has fatal consequences for "special" relativity! This is
obvious, since in ME the four partial derivatives ¶
/¶ x , ¶
/¶ y , ¶
/¶ z , ¶
/¶ t are on equal footing (see, for example,
Eq.(6)) and obey the LT. **The presence of the total time derivative,
by giving to the time derivative a distinct status, destroys the Lorentz
covariance of ME**!

Here is the place to mention the incompleteness of the traditional formula for the total derivative of a vector field:

d**A**/dt = ¶
**A**/¶
t + (**v.Ñ
**)**A**
(17)

and the time rate of change "seen" by a point moving with velocity **v
**in a vector field **A **[9]:

d**A**/dt = ¶
**A/¶
**t
+ (**v.Ñ
**)**A **+ (**A.Ñ
**)**v**
(18)

Although the vector identity:

(**v.Ñ
**)**A** + (**A.Ñ
**)**v**
+ **A ´ (Ñ
´ v**) - Ñ
**(v.A)**
= -**v** ´ (Ñ
´
**A**) = -**v** ´ **B**
(19)

for **v.A** = **const.** and Ñ
´ **v** = 0 leads to the 150 years
old formula of Neumann:

**E**_{ind }= -¶
**A**/¶
t + **v** ´ **B**
(20)

equation (18) covers **all** known situation of electromagnetic induction,
including those where Eq. (19) fails. Eq. (20) is still in exclusive use,
although it has never been rigorously justified. Neumann - just like later
Lorentz with his force law (8) - just juxtaposed Faraday's and Maxwell's
**transformer field** and the empirical field found in the unipolar
induction experiments of Faraday and in the (then) recently discovered
Hall effect, called **motional induction field**. Wesley derived for
the first time (!) the most general law of induction which includes (20)
as a particular case. The surprising result is that the law based upon
(18) is able to describe phenomena governed by the term (**v.Ñ
**)**A**
like the Aharonov**-**Bohm (AB) effect and the** **Marinov motor
[10]. The demystification of the "strange quantum-mechanical (AB) effect"
[10] and its explanation in the framework of electrodynamics has been a
real *tour de force. *The term (**A.Ñ
**)**v
**is presently insufficiently investigated, but preliminary results seem
to support its explaining the interaction between two toroidal magnets
(closed magnetic field configurations) [11], which, according to Maxwell's
electromagnetism should not interact. The **local** form of the correct
law of induction, involving the total derivative (18), puts an end to the
perennial disputes between the supporters of **fields **and **potentials**,
respectively. It has to be clear that the description by means of **A
**is more general than the usual by means of **B**, since it provides
an induced electric field even if Ñ
F
= 0 (F denotes here the scalar potential), ¶
**A**/¶
t = 0 and **B** = Ñ
**´
A**
= 0 .

One is tempted to say that Maxwellian electrodynamics overcame all difficulties
and retained its original form since, after all, the use of the truncated
form (17) for the total derivative was not Maxwell's fault. The painful
fact for STR-supporters is, however, that **Eqs. (4) and (5) do** **not
cover all experimental** **situation and** - acutely painful **-**
that** they do not remain Lorentz covariant if one replaces the partial
time derivative with the total one**!!

**VI. Beyond the Lorentz force law**

The force law of Lorentz (8) applies only in situations where the fields
**E **and **B **are static, or quasistatic, when radiation could
safely be neglected. In such situations, however, Eqs. (4) and (5) decouple
in two pairs of electrostatic and magnetostatic equations, respectively:

Ñ ´
**E** = 0
;
Ñ
**D** = r
(21)

and

Ñ
**´
B** = **j**
;
Ñ
**B** = 0
(22)

This explains the upper indexes appearing in Eq. (10), indicating that
the sources of **E**^{(1) }and **B**^{(2)} are different.
Moreover, as already pointed out, the field **B** has to be produced
by a closed current loop. It follows then, that **the force of Lorentz
can by no means be applied to a system of two charges, so that charge ^{(1)
}moves in the field B^{(2) }and vice-versa**:

d(m_{1}**v**_{1})/dt = q_{1}(**E**^{(2)}
+ **v**_{1} ´
**B**^{(2)})
; d(m_{2}**v**_{2})/dt
= q_{2}(**E**^{(1)}** + v**_{2}** ´
B**^{(1)})
(23)

No wonder that **this two-body problem would violate the linear momentum
conservation law**, since the sum of internal forces would be different
from zero! The replacement of particle linear momentum **p **by (**p
- **q.**A**)** - **as suggested by the "operator formalism" of
quantum mechanics - doesn't save the conservation law.

The inability of Lorentz force to describe the simplest system of two interacting charges is presented in textbooks and monographies as due to the fact that "at least one charge path has to be closed", which is obviously false! In a dense plasma, for example, even in external magnetic fields, where charges are permanently colliding with each other, there may well be no closed paths at all. The fact that the external magnetic field is produced by the closed circuits of the electromagnets is irrelevant for the plasma system!

There exists a rich experimental evidence for the failure of Maxwell-Lorentz
electrodynamics at **low velocities** (v/c << 1), which is lethal
to MLE, since the laws of Gauss, Ampère, Faraday were all formulated
according to the experimental results obtained for low velocities. Moreover,
the interactions between electric currents and magnets were all investigated
by using **metallic conductors**. The microscopic nature of currents
in metallic conductors remained undecided, till Weber introduced atomism,
*i.e.* quantized electric charges in physics. He assumed that electric
currents consists of a stream of electrons and made first the identification:

Id**s** = q**v**
(24)

With this, Weber succeeded to derive Ampère's law between metallic current elements:

d²**F**_{A} = (I.I'**R**/R^{3})[-2d**s.**d**s**'
+ 3(**R.**d**s**)(**R.**d**s**')/R²]
(25)

(I, I' denote currents and **R **= **r **- **r**') from his
**interaction law between discrete**, **moving electrons in a metallic
conductor**:

**F**_{W }= (qq'**R**/R^{3})[1 + V²/c²
- 3(**V.R**)²/2c²R² + **R.**d**V**/dt.c²]
(26)

vhere **V** = **v **- **v**' and d**V**/dt denote the **relative
velocity** and the **relative acceleration** between the moving charges.
This truly relativistic (Machian) and instantaneous force law explains
all known experiments at low velocities with metallic currents, including
**Ampère's moving bridge **one and the '**electromagnetic rail
gun'** used in frame of the SDI program [12] which both imply **longitudinal
forces** between **parallel** metallic current elements. It is notorious
that the Lorentz force - acting **perpendicular** on current elements
- is unable to account for these experiments. In spite of this, the **belief
**in uniqueness of the Grassmann-Biot-Savart-Lorentz (GBSL) force law
is so strong, that Ampère's law (25) - called by Maxwell "the cardinal
formula of electrodynamics" - is not even mentioned in the vast literature
on electrodynamics. Once again, this **belief** is motivated by the
seeming compliance of the Lorentz force (8) with the LT, *i.e.* with
STR. Wesley put his finger on the sorepoint of the interminable controversy
about Lorentz *vs.* Ampère force. The supporters of the LF
cramp to the equivalence of the two forces when two closed current loops
are involved. This is totally irrelevant as it is only a question of the
analysis of the mechanical forces between the two objects, the metallic
bridge and the remainder of the circuit as a mechanical object. However,
Grassmann's derivation of his law (equivalent with that of Lorentz) is
**only valid for mechanically rigid electrical circuits.** This means
that **the GBSL law cannot be applied to the electrical circuit involved
in the non-rigid Ampère bridge**!

Weber's law correctly describes the motion of electric charges in vacuum
- for example in the electron microscope - since in this case **E **and
**B **are external and **B** is produced by closed current loops.
The useful phenomenon of "self-focussing", or "pinch effect" wellknown
to electron microscopist, is also explainable within the traditional frame
of Maxwellian electrodynamics, as attraction between parallel currents.

Remarkably, Ampère's and Weber's laws comply with Newton's third
law (*actio = reaction*), since the **forces act instantaneously**
along the line joining the current elements, or the moving charges. **This
condition for law velocities is one of the requirements for a system being
non-radiating even for charges moving with high (**v** >> **c**)
velocities. **As a matter of fact, both the **hydrogen atom** and
the '**rotating ring** **electron model'** are **conservative**,
*i.e.* **non-radiating**, provided the forces are of Weber type!

**VII. Some comments on rapidly varying fields and radiation**

1/ The characteristic feature of Maxwell's equations is the presence
of the terms ¶
**D**/¶
t and ¶
**B**/¶
t which **couple** the electric and magnetic fields and lead to the
existence of **electromagnetic waves, **or** radiation**. The field
equations completed with the Lorentz force law (the MEL equations) are
therefore incoherent, since the fields in the LF expression are static,
or quasistatic, which means that radiation is neglected. No wonder that
the attempts of Dirac and others to add "radiation terms" to the equation
of motion of the electron leads to strange "runaway solutions" and other
unsolved difficulties.

2/ The 104 years old Liénard-Wichert formula [9] for the retarded
potentials corresponding to a point-charge moving with acceleration **a**
along the positive direction of the x-axis has been seriously questioned
by Chubykalo and Smirnov-Rueda [13] and independently by Wesley [9]. This
indicates that the "special" relativistic Maxwell-Lorentz electromagnetism
is **an unsatisfactory** **theory by itself**, although the reason
is hidden in the mathematics of d'Alembert's wave equation, rather than
in the ME themselves.

It is textbook knowledge [13] that the solutions of the wave equation (d'Alembertian) are:

F = [r
]**/**R.dV + F
_{0} ; **A** = [**j**]/R.dV
+ **A**_{0}
(27)

which are the retarded potentials. F
_{0
}and **A**_{0} denote the solutions of the homogeneous
wave equation. This is OK. From here one usually derives:

F = q/(R - **v.R**/c) ; **A** = q**v**/(c**R**
- **v.R**)
(28)

and the fields:

**E** = - q(1 v²/c²)(**R - v**R/c)/(R - **R.v**)^{3}
+ q**R ´
**{(**R **- **v**R/c)
´ **a**}/(R - **R.v**/c)^{3}c²
(29)

**B** = (**R** ´ **E**)/R

**Chubykalo and Smirnov-Rueda show that formula (29) does not satisfy
the d'Alembert equation along the x-axis at any time**. This follows
from the fact that the wave equation for E_{x}describes only transverse
modes and - on the other hand - the x-component according to (29) is different
from zero. Thus, the Liénard-Wichert potentials, as solutions of
the complete set of Maxwell equations, are inadequate for describing the
properties of electromagnetic field along the direction of an arbitrarily
moving charge. Whitney [14] found another inadequacy of the Liénard-Wichert
potentials for describing the properties of relativistic fields. Further,
it is easy to verify that the Poynting vector calculated with Eq. (29)
equals zero, *i.e.* no energy transport takes place along the x-axis,
while the energy conservation law requires both energy density and divergence
of the Poynting vector to be different from zero!

**The criticism of Wesley **[9]** is even more fundamental and relies
upon the fact that in the wave equation of a field theory, the four variables
x, y, z, t have to be independent (Eulerian) as explained also in **[4].
Despite this clear mathematical requirement that **r**' and t' be independent
variables, in the integral representation (28) of the retarded potentials
Liénard and Wiechert argued incorrectly that the **independent
**space variable **r**' is a **dependent **function of the time
variable t'.The change in the 'delta function' - which accounts for the
point-like nature of the charge - leads to the correct expressions for
the retarded Coulomb potential:

F
_{r} = q'/R_{r}
(30)

where R_{r} = R(t)/(1 - v/c) for an observer moving directly
away with v < c from the point charge.

3/ The ubiquitous presence of radiation, *i.e.* of electromagnetic
fields detached from their fields requires the existence of a unique, fundamental
frame of reference, relative to which the energy transmission velocity
is "c". This is a consequence of the fact that the velocity of light doesn't
obey either the hypotheses of Ritz ("ballistic propagation", or dependence
on the state of motion of the source), or the untenable second postulate
of "special" relativity which is discused in [15]. The existence of a fundamental
frame of reference which could be experimentally approached by successive
approximations, is in line with Maxwell theory, but disagrees with "special
relativistic electrodynamics" which states the validity of ME in **any**
inertial frame of reference.

**VIII. Conclusions**

- Maxwell's equations (ME) retain their validity for closed current loops.

- ME have to be completed with a force-law and a corresponding equation of motion.

- In all applications the force of Lorentz is a phenomenological external force, with different

sources for **E **and **B**, the latter being produced by closed
circuits.

- MLE fails to explain low-velocity experiments with non-rigid loops.

- Ampère and Weber's force law accounts for all electrodynamic phenomena in which

radiation can be neglected.

- MLE in its accepted form is unable to account for all induction phenomena.

- The correct law of induction is given by the total derivative of the vector potential. This is

compatible with ME, but destroys the Lorentz covariance of the theory.

- The vector potential is of primary importance and is uniquely defined for specific systems.

- There is no "gauge invariance".

- Low-velocity Weber electrodynamics is truly relativistic in the sense of Mach.

- The presence of radiation requires an absolute, fundamental frame
of reference.

**References**

[1] Dyson F. J. "Feynman's proof of Maxwell equations",
Am. J. Phys. **58** (1990) 209-211

[2] Jordan E. C. and Balmain K. G., "Electromagnetic Waves and Radiating Systems"

(Prentice-Hall, Inc., New Jersey, 1968) p. 103

[3] Post E. J. "Formal Structure of Electrodynamics" (North Holland Publ. Co.,Amsterdam,

1962); "Kottler-Cartan-van Dantzig (KCD) and Noninertial Systems", Found. of Physics,

**9** (1976) 619-640

[4] Galeczki G., "Minkowski Scalar Invariant Incompatible with any Equation of Motion"

(Proc. 2-nd Intrnational Workshop: "Physics as a Science", Koeln, 1997)

[5] O'Rahilly A., "Electromagnetic Theory: A Critical Examination of Fundamentals",Dover

Publ. Inc., New York, 1965) Vol. 2, p. 561

[6] Galeczki G., "What does the Lorentz force have to do with Maxwell's equations?",

Galilean Electrodynamics, **9** (1998) 95-98; Galeczki
G., "What does the Lorentz force

have to do with special relativity?", Galilean Electrodynamics 8 (1997) 1-4

[7] Bartocci U. and Mamone Capria M., "Symmetries and Asymmetries in Classical and Relativistic

Electrodynamics", Found. of Physics 7, (1991) 787-801

[8] Jefimenko O. D., "Comment on "On the equivalence of the laws of Biot-Savart and

Ampère", by T. A. Weber and D. J. Macomb [Am. J.
Phys. **57**, (1989) 57-59]", Am. J.

Phys. **58**, May 1990, 505

[9] Wesley J. P.,"Induction Produces Aharonov-Bohm Effect",
Apeiron, **5** (1998) 89-95;

[10] Selected Topics in SCIENTIFIC PHYSICS" (Benjamin Wesley, Blumberg, 2002)

[11] "Force between two identical coaxial toroidal solenoids" (in print)

[12] Graneau P. and Graneau N, "Newton versus Einstein: How Matter Interacts with Matter"

(Carlton Press, Inc., New York, 1993); "Wesley J. P., Selected Topics in Advanced

Fundamental Physics" (Benjamin Wesley, Blumberg, 1991

[13] Chubykalo A. E., "Action at a distance as a full-value solution of Maxwell equations:

The basis and application of the separated-potentials
method", Phys. Rev. E, **53**, (1996)

5373-5381

[14] Landau L. and Lifchitz E. M., "Théorie du Champ" (Éditions de la Paix, Moscou, 1965)

[15] Whitney C. K. "A quantum of light shed on classical potentials and fields", Apeiron (?)

[16] Galeczki G. and Marquardt P., "Requiem fuer die Spezielle Relativitaet" (Haag und

Herchen, Frankfurt a. M., 1997)

**Acknowledgement**

I am indebted to Paul Wesley, Thomas Phipps Jr., Jan Post
and Patrick Cornille for stimulating exchange of ideas during the years.

- - - - -

**George Galeczki*** received a Licence in Physics
from Bucharest University in 1968, M.Sc. (1975) and D.Sc. (1979) degrees
from The Technion - Israel Institute of Technology - in Haifa (Israel),
for works in the field of ordered magnetism. In 1979 he received the Michael
Landau for his research beyond his work toward a degree. After lecturing
three semesters at the Technion, he moved to the governmental research
center RAFAEL, where he did (mostly classified) work on HgCdTe-infrared
detectors. After cumulating two sabbatical years, he left Israel, responding
to an invitation from the University of Cologne (Germany). There he did
research on heterodyne HgCdTe-infrared detectors for astrophysical applications
and continued, in parallel, his critical work on fundamental physics started
in 1978 under the influence of Nathan Rosen ("the EPR one") and Marinov's
successful experiment to measure the absolute velocity of the Earth. He
published about 50 papers on magnetism, narrow-bandgap semiconductor physics,
nanoscopy, and about an equal number of papers criticizing "special" and
general relativities, Copenhagen quantum mechanics, and Big Bang theory.
He is the co-author (with Peter Marquardt) of *REQUIEM TO SPECIAL RELATIVITY*
(in German, published by Haag + Herchen, Frankfurt/Main, 1997) and organizer
(with P. Marquardt and J. P. Wesley) of three (1997, 2000, 2002)International
Workshops: *PHYSICS AS A SCIENCE*. He is presently an independent
science consultant, science writer, president of the Society for the Advancement
of Physics, R.S. and member of the Natural Philosophy Alliance.*

Society for the Advancement of Physics, R.S.