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Space Interpretation of Maxwell’s Equations
... Coming back to its own source...
R. Bekbudov
IEEE member /Canada/, E-mail: radbek@ieee.org
Abstract
The development of Maxwell’s equations based on the space interactions as the matter is presented.
The generalized equation for the transformations of vector fields is given. The extensions to Maxwell’s
equations are obtained, and the interpretation of it as a fundamental property of matter is provided.
I. Introduction
Maxwell’s equations are the base of classical electrodynamics. They describe the interaction of two
fundamental types of matter and their mutual transformation. The equations are presented in different
forms depend on the system of units, environment and mathematical tools [1]. These equations were
formulated by Maxwell more than 150 years ago and were modified by Heaviside later on. However,
since that time and until now they are unchangeable; even the physical understanding of matter was
changed. During this time, it became clear that sometimes equations give approximate results and it
was discovered that existing discrepancies cannot be explained by unaccounted factors [2]. Moreover,
there are some exceptions for the one of these equations such as well-known Faraday’s law or “flux
rule”, so the formal description is powerless for these cases [3].
In this paper, a different way to obtain of Maxwell’s equations is considered, and the presentation form
[3] for the case of isotropic and homogenous environment without dispersion such as the space is
selected
E 

0
B  0
(1a)
 E  
(1c)
 B 
B
t
J
1 E
 2
c  0
c t
2
(1b)
(1d)
(1)
The proposed approach is based on the concept that the space is a matter; therefore a philosophical
aspect would be appropriate in this paper.
The whole world around us is the moving matter in its infinitely varied forms and manifestations, with
all its properties, connections and relationships. According to Lenin's definition, "matter is a
philosophical category denoting the objective reality which is given to human being by his sensations,
and which is copied, photographed and reflected by our sensations, while existing independently of
them" [4]. This definition of matter covers the space as a concept, so the space could be endued by the
next properties:
1. The space is a universal matter that includes physical objects. The space is undisturbed if there
is no objects there. The undisturbed space is dynamically neutral, isotropic and homogeneous,
i.e. undisturbed space is in a dynamic balance.
2. The space disturbance is the perturbation of dynamic balance. The physical objects may be
considered as the space disturbances (distortions), and the space reacts on it by creating the
field. Any change of energy level of physical object is the space excitement.
3. The field is the reaction of space caused by a local disturbance and it may be considered as a
transition between the undisturbed and disturbed areas in a local space. The field has the same
nature as a disturbance. The permeability measure of disturbance is defined via its constants
such as electric permeability ε0, magnetic permeability μ0, and gravitational permeability γ0.
4. The moving field as a moving matter creates a secondary field with another nature in local
space around the object. The secondary field is moving as a primary one. If the primary field is
excited, the secondary field is excited too.
5. The space tries to keep its unexcited state by forming the compensating disturbance as a
negative feedback. The compensating disturbance is creating through the excited secondary
field. The excited secondary field can create in its turn the compensating field with same nature
as a primary one and it makes an impact on the excited object.
In the author’s opinion, the acceptance of this approach will allow us to expand our imagination about
the mutual transformation of matter and give better correspondence to reality.
II. Generalized Equation
It is well known that there are three types of vector fields such as gradient, vortex and hybrid fields.
According to Helmholtz decomposition [5], any sufficiently smooth, rapidly decaying vector field in
three dimensions can be resolved into the sum of the vector fields: irrotational (curl-free) field and a
solenoidal (divergence-free) field.
For analysis generality, it is assumed that initial vector field h is hybrid, i.e. it may be presented as the
combination of gradient g and vortex r fields:
h gr
(2)
Let’s assume that vector field (2) satisfies to the well-known transformation [6]
  (  h)  (  h)  h
(3)
where  is the vector laplasian.
It can be accepted that as the result of transformation (3) the initial hybrid field h creates the secondary
vortex field r’ of an another nature
r'  h
(4)
Taking into account (2) and the property of vector operator   r  0 the formula (3) may be
rewritten
  r '  (  g )  h
(5)
The vector laplasian in (5) may be presented by another way [6]
h  (2hx )i  (2hy ) j  (2hz ) k
(6)
where  is the scalar laplasian; hx , hy , hz are the scalar components of the vector field h.
2
The mathematical expression (6) is the formal description of operation with the abstract vector fields
that may cover the physical fields too. The physical fields usually have the inherent restrictions caused
by their nature. These restrictions can be expressed through the properties of (6), i.e. it looks
reasonable to change his properties to provide some restrictions for the transformation of physical
fields.
Let’s assume that properties of physical fields in (6) satisfy the next conditions
 2 hi
0
j 2 i  j
(7)
This restriction leads to the degeneracy of vector laplasian (6)
 2 hy
 2 hx
 2 hz
h 
i 
j
k
x 2
y 2
z 2
(8)
Taking into account (8) the generalized equation (5) may be changed to the next
  r '  (  g )  h
(9)
The formula (9) describes the transformation of hybrid field to the vortex one as the transformation of
a right-hand triple vectors. This equation may be considered as the generalized equation of field
transformation whereas it was obtained without reference to the nature of fields. The correctness of (9)
may be proved by its accordance with reality in passing from the abstract fields to the physical ones.
III. Development of Maxwell’s equations
To connect the generalized equation (9) to the real fields it will be enough to make the substitution of
abstract vector fields by its physical analogs. Let’s assume that hybrid field h is presented by an
electric field Eh = h that has the same structure (2), i.e.
 r '  (  E g )  E h
(10)
The right part of (10) is described by operators that corresponding to the movement of field E in the
space, i.e. to the movement of matter. According to the property 4 it indicates that another moving
field with different nature is appeared, i.e. the dynamic field should be in the left side of (10). Let’s
present field r’ by another way to take into account the next substitution
h
1 h

x x t
(11)
where x is the speed of movement along the x-coordinate.
For simplicity, let’s consider the x- component of field r’
 Ezh
E yh
r' x  (  E ) x  

 y
z

h
h
 1 Ezh

1 E y 
i 

i


 v t
vz t 

 y

(12)
In case that the space is an isotropic, the above-all mentioned speeds are satisfied to the condition
vx  vy  vz  x  c , where с is the speed of light.
The sign of speed ν of electric field E depends on the sign of charge q, that is why it should be taken
into account to avoid any mess ( for example, the assigned direction of current is opposite to direction
of charge transfer by electrons)
(  E h ) x  sign( q )
1 
E zh  E yh  i

c t
(13)
Taking into account (13) and the above mentioned condition the right part of (12) may be presented as
the next
r '  sign(q )
 1 h
Ez  E yh  i   Exh  Ezh  j   E yh  E xh  k 


t c
(14)
The expression (14) testifies that dynamic vector’s field is under sign of derivation, and according to
above mentioned it may be considered as a vector field of different nature, i.e. as a magnetic field B
r '  sign(q)
B
t
(15)
One of the terms in the right side of (10) may be transformed by applying well known equation
 E   /  0 and the ratio J    as noticed in [7]
   1 J 1
(  E g )         
 J 
vx  v y  v z  c
  0   0     0c
where
(16)
 is a volume charge density; J is a current density.
By using the expended forms to (16) and degenerate laplasian in (10) together with the applying
substitution (11) the following can be obtained
(  E g ) 
1  J
J
J 
 J
i

j

k

sign
(
q
)
 0c  x
x
x 
t  0с2
 2 E yh
E yh
 2 Exh
 2 Ezh
 1  Exh
Ezh 
E 
i
j
k  sign(q)
i
j
k

x 2
y 2
z 2
t c  x
y
z 
(17)
h
(18)
The substitutions (16), (17) and 18) may be applied to (10) and together with elimination of derivation
procedure in both sides it gives the next
E yh
Ezh 
J
1  Exh
 B 
 
i 
j
k
 0 c 2 c  x
y
z


(19)
The transformation of magnetic field B into electric field E occurs in the similar way with one
difference only: there is no need for the applying of sign function due to the absence of heteropolar
magnetic charges. Taking it into account and that the ratio between fields E and B is saving all time,
i.e. |B| = |Eh |/c, the next is following
r'
1 E
c 2 t
(20)
It should be noted that due to nature of vortex field B the equation   B  0 is applied to (9), so
there is no gradient component of field there
B 
By
 1  Bx
Bz
i

j


t c  x
y
z

k

(21)
In case of magnetic field B, the applying of (20) and (21) into (9) gives the next
By
 B
B
  E  c  x i 
j z
y
z
 x

k

(22)
Thus, the substitution of abstract fields in generalized equation (9) by their physical analogs leads to
the following
By
 B
B 
  E  c  x i 
j  z k
y
z 
 x
E yh
Ezh 
J
1  Exh
 B 
 
i 
j
k
 0 c 2 c  x
y
z 
(23)
The equations (23) may be considered as a vector form of Maxwell’s equations that is obtained as a
result of their space interpretation. To present these equations in conventional form it is enough to
apply substitution (11) to the (23) with taking into account the sign of field E corresponding to the
transfer of charge by electrons

 Bx  By  Bz 
t 
J
1 
 Exh  E yh  Ezh 
 B 
 2
2
 0c
c t
 E  
(24)
The given representation is the extensions to the traditional Maxwell’s equations, and the transition to
them from (24) is occurred by limitation to one the all components of vector fields. The obtained
equations allow the taking into account the action of vector fields in their full space scale.
IV. CONCLUSION
The form of the right parts of (23) is very close to the gradient operator  , however the difference is
in their content presented by scalar components of vector field. Formally, the transition to the operator
 may be performed in case when these components are equal, i.e. it depends on the field properties.
The well-known models of physical field are not quite suitable for this role. However, there are no
obvious restrictions to use another model for it. For example, there is a conical model of vector field
presented in Fig.1 [8].
h
-h
Fig.1 The conical model of vector field
Depending on the coincidence or non-coincidence of rotation directions for the left and right sides, this
field may be both a vortex (   h  0 ) and an irrotational (   h  0 ). The space combination of these
fields may allow the give opportunity for the transition in (23) to the operator  .
It should be noted, that Maxwell’s equations are obtained based on the transition of vector’s laplasian
 to its degenerated form  . The obtained result as the extensions (24) shows the correctness of
this transition and there is no doubt in it. However, there is no practical confirmation of it yet. In the
case that the confirmation occurs, to avoid any confusion in the future, this operator  should be
fairly named as a maxwellian.
The equations (23) and (24) are principally different from the original one (1) due to the absence of
equations (1a) and (1c) there. There is no need to present them as the independent equations because
they are already used in the formation of structure of two main equations in formulas (17) and (21).
Moreover, the extensions of (23) and (24) to the full space representation open the possibilities not
only for the elimination of discrepancies between theory and practice, but for the opening of new
technologies in the field of electromagnetic energy transformation.
In the author’s opinion, taking into account all of the components of vector fields in case of Faraday’s
law leads to the new generation of electric machines that combine the features of induction and
synchronous machines. One of the best applications of this new type of electrical machines is the onboard applications such as electrical transportation including the trains with magnetic levitation. The
key role of these applications is the expected significant reduction of costs in combination with their
energy efficiency.
Acknowledgements
The author does not have any opportunity to perform experimental researches in this field, but he hopes
that other researchers will obtain experimental confirmation of obtained results and thanks them in
advance.
22/04/2015
References
[1] Maxwell’s equations.
http://en.wikipedia.org/wiki/Maxwell%27s_equations.
[2] The articles on alternative electrodynamics. (in Russ.).
http://alt-teoria.ucoz.ru/load/sekcija_ehlektrodinamiki/7
[3] R. Feynman, R. Leighton, and M. Sands. The Feynman Lectures on Physics.Vol.2.
Mainly electromagnetism and matter. Addison–Wesley, 1964.
[4] V. I. Lenin. Materialism and Empirio-Criticism. Collected Works, vol.14.
Progress Publishers, Moscow, 1972.
[5] Helmholtz decomposition.
http://en.wikipedia.org/wiki/Helmholtz_decomposition
[6] Bronstein I.N, Semendyaev K.A. Handbook of mathematics for engineers.
Nauka, 1981. (in Russ.)
[7] Ya. P. Terletsky, and Yu. P. Rybakov. Electrodynamics. Vysshaya Shcola (1990).
[8] R.Bekbudov. Space interpretation of Lorentz force.
http://www.sciteclibrary.ru/eng/catalog/pages/13335.html