MATHEMATICS & NATURE
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Article
.
Physics
Dongfang Modified Equations of Electromagnetic Wave
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
Systematically show the correct solution process of the initial value problem of Maxwell equations
in rectangular coordinate system and cylindrical co ordinate system, and clearly point out that the
transverse electromagnetic wave equation described in the classical electromagnetic theory is the
result of incorrect understanding and treatment of Maxwell equations: the definite solution conditions
of magnetic field and electric field in the same phase that do not conform to the natural law are
assumed in advance, and then the plane transverse electromagnetic wave mode is determined through
the general solutions of the second-order Maxwell equations, and the conclusion violates the law of
energy conservation, In particular, the mathematical process does not conform to the solving rules
of differential equations. Through this simple example, I clarify my clear position: since the era of
Maxwell’s electromagnetic theory, theoretical physics has fallen into a strange circle in which the
wrong calculation of celebrities is defined as standard methods and the assumption that celebrities
cannot prove is defined as basic principles, resulting in a large number of wrong logic and wrong
conclusions in physical theory, even some branches of theoretical physics are completely wrong from
b eginning to end, and all these errors are covered up by formal mathematical operations, the formal
mathematics seriously distorts the laws of nature. However, all correctable and uncorrectable errors
have not been found and recognized by mainstream physicists, and especially the pointing out of
these errors has been rejected by mainstream scholars. A serious question arises: Can the future
mainstream physics change this academic abnormal state?
Keywords: Maxwell Equations; Optimal differential equation; Initial conditions; Plane transverse
electromagnetic wave; Cylindrical transverse electromagnetic wave.
PACS number(s): 03.50.De—Classical electromagnetism, Maxwell equations; 41.20.-q—Applied
classical electromagnetism; 41.20.Jb—Electromagnetic wave propagation; radiowave propagation.
1 Introduction
Maxwell equations
[1, 2]
point out that the changing
electric field and magnetic field can excite each other
and propagate in space to form electromagnetic waves.
However, Maxwell equations fail to indicate the time and
space interval of the causal effect of the magnetic field
excited by the changing electric field or the electric field
excited by the changing magnetic field. Is the electric
field near a point far away from the radiation source
propagated by the electric field of the radiation source
at the previous time, or is it excited by the magnetic
field changed in the surrounding space at which time?
The answer is unknown. It can be seen that the descrip-
tion of causality of an electromagnetic wave in Maxwell
theory is actually very vague.
Reviewing the research on plane electromagnetic wave
in classical electromagnetic wave theory: the expres-
sion forms of monochromatic plane electromagnetic wave
with the same phase of magnetic field and electric field
are predetermined in advance
[3-8]
, E (x, t) = E (x) e
−iωt
and B (x, t) = B (x) e
−iωt
, so as to determine that the
relationship between electric field and magnetic field is
B (x, t) =
√
µεn ×E, where n is the unit vector of prop-
agation direction. The main formal conclusions are as
follows: 1) the electromagnetic wave is shear wave, and
both E and B are perpendicular to the propagation di-
rection; 2)E and B are perpendicular to each other, and
E ×B is along the wave propagation direction; 3) E and
B are in phase, and the ratio of amplitudes in vacuum is
equal to the speed of light. The electric field is in phase
with the magnetic field, which breaks the law of con-
servation of energy. This kind of plane electromagnetic
wave mode in which the electric field and magnetic field
are in phase everywhere can not be generated or exist.
The expression of electric field and magnetic field
characterizing electromagnetic wave mode naturally
needs to meet Maxwell equations, but the expression of
Citation: Dongfang, X. D. Dongfang Modified Equations of Electromagnetic Wave. Mathematics & Nature 1, 202108 (2021).
*The name of the author’s ID card is Rui Chen, and the signature of his academic articles is X. D. Dongfang.
†This article integrates the regular solutions of plane transverse electromagnetic wave modes of Maxwell equations published in Chinese
in 2000 and the regular solutions of cylindrical transverse electromagnetic wave modes of Maxwell equations published in English in 2011.
Signatures of Chen in these two articles are my early signatures.
2 X. D. Dongfang Dongfang Modified Equations of Electromagnetic Wave
the electric field and magnetic field satisfying Maxwell
equations may not constitute an electromagnetic wave
mode that can be realized or exist. Because Maxwell
equations are differential and integral equations, the so-
lution of differential and integral equations depends not
only on the form of differential equations, but also on the
definite solution conditions
[9]
. Different definite solution
conditions will bring completely different solutions to
the same differential and integral equation. The definite
solution condition of Maxwell equations in unbounded
space is the initial condition, that is, to solve the initial
value problem. However, the process of determining the
plane electromagnetic wave mode according to Maxwell
equations in classical theory seems to indicate that the
definite solution of Maxwell equations has nothing to do
with the initial condition
[10, 11]
, which is obviously a mis-
take. On the other hand, the changing electric field gen-
erates a magnetic field, and the changing magnetic field
generates an electric field. If E and B are in phase, it will
be difficult at zero time! Usually, the starting point of
time is the time when the electric field or magnetic field
begins to oscillate, that is, zero time. In the LC oscilla-
tion circuit, after the parallel plate capacitor is charged
stably, the electric field in any region of the space is
constant. At this time, the electric field does not excite
the magnetic field. The electric field is maximum every-
where and the magnetic field is zero everywhere. On the
contrary, if the magnetic field in any area of space does
not change and the magnetic field does not excite the
electric field, the magnetic field is maximum everywhere
and the electric field is zero everywhere. It can be seen
that E and B cannot always reach the maximum at the
same time or disappear at the same time. The classical
theory of electromagnetic wave with the same phase of
the electric field and magnetic field leads to the difficulty
of zero time. The two equations E (x, t) = E (x) e
−iωt
and B (x, t) = B (x) e
−iωt
used to describe plane electro-
magnetic wave mode in classical theory can not represent
an objective or possible electromagnetic wave mode.
As a system of differential equations, the solution of
Maxwell’s equations obviously depends on the initial val-
ue or boundary conditions of electromagnetic field
[12, 13]
and the determination of plane electromagnetic wave
mode and cylindrical electromagnetic wave mode can be
attributed to the initial value problem of Maxwell’s e-
quations. According to different initial conditions, the
plane electromagnetic wave modes or cylindrical electro-
magnetic wave modes
[14-18]
obtained by solving Maxwell
equations are different. Here, reasonable solutions of
plane transverse electromagnetic wave in Chinese ver-
sion in 2000
[19]
and cylindrical electromagnetic wave in
an English version in 2011
[20]
are integrated, and the
correct solution method of determining electromagnetic
wave mode according to Maxwell equations is empha-
sized. The consistent conclusion is that both plane elec-
tromagnetic wave and cylindrical electromagnetic wave
are transverse waves. The amplitude of the magnetic
field excited by the changing electric field, or the ampli-
tude of the electric field excited by the changing magnet-
ic field is different at different positions, and the phase
difference between the electric field and the magnetic
field is also different.
2 Regular solution of plane transverse elec-
tromagnetic wave modes
In 2000, the author published a Chinese article on the
regular solutions of the plane transverse electromagnet-
ic waves of Maxwell equations with the signature Chen
Rui
[19]
, and gave the plane transverse electromagnetic
wave which can be realized theoretically and can ex-
ist. However, this most basic and important discovery
seems to have not attracted the attention of mainstream
scholars. College education has always advocated the
wrong electromagnetic wave mode of electric field and
magnetic field in phase. This part is introduced again
in English. One first determines the initial condition-
s of plane electromagnetic wave, then gives the general
plane wave solution of Maxwell equations, and finally
determines the special solution of Maxwell equations ac-
cording to the initial conditions to obtain a correct plane
transverse electromagnetic wave mode.
2.1 Physical background of generating plane
electromagnetic wave
The determination of initial conditions depends on the
specific physical background. Although the electromag-
netic wave can be independent of the moving charge or
current, the existence of the electromagnetic wave in the
surrounding space depends on the accelerating charge or
changing current of the emission source. Without the
normal operation of the TV transmission tower, there
will be no electromagnetic wave of a predetermined fre-
quency in the nearby space. The charge variation law
of the emission source determines the initial conditions
of Maxwell equations in free space and the mode of an
electromagnetic wave
[21]
.
What is a plane electromagnetic wave? As shown in
Figure 1, it is assumed that two rectangular metal plates
P and M with finite width and sufficient length are paral-
lel to each other and close to each other, the plane of the
rectangular plate is perpendicular to the y-axis, the left
end is at the co ordinate origin, and a power supply with
output voltage varying with time is connected. There-
fore, the changing electric field propagates from left to
right in the metal, resulting in the continuous change of
charge distribution at any position on the two parallel
metal plates. The changing electric field generated be-
tween the two plates is parallel to the y-axis, and the
electric field propagates along the positive direction of
the x-axis. When the rectangular plate is close togeth-
er, it can be considered that the electric field is evenly
Mathematics & Nature (2021) Vol. 1 3
distributed on the y-axis; Because the electric field prop-
agates at the speed of light, the distribution change of
the electric field to the finite width on the z-axis can be
ignored; Considering the case of superconductivity, the
energy of electromagnetic wave in the conduction process
is not converted into Joule heat and lost. Therefore, the
electric field between two parallel plates constitutes an
ideal plane electric field wave mode, that is, plane free
electric field wave.
The introduction of exponential complex number to
express physical quantities often brings great conve-
nience. However, this does not mean that the complex
number has an inevitable causal relationship with phys-
ical theory. Some quantum force scholars have made
mistakes in their understanding of this problem. The
measurability of physical quantity originated in the field
of real numbers and finally ended in the field of real num-
bers. Here, the changing electric field between parallel
plates is expressed by a real function,
E = e
y
E
y
(x, t) (1)
Where e
y
represents the unit vector along the posi-
tive direction of the y-axis. The output voltage of
the alternating power supply between the two parallel
plates changes according to the sinusoidal law u (t) =
U
m
sin ωt. when t = 0, the magnetic field between t-
wo parallel plates as well as the surrounding space is
zero. If the edge effect of parallel plates is not consid-
ered, this physical background gives the definite solution
condition, that is, the initial condition is
B (t = 0) = 0
E (x, t) = e
y
E
m
sin ω
t −
x
c
(2)
The non-uniform changing electric field and magnetic
field can excite each other. Different definite solution
conditions determine the different modes of magnetic
field generated by the changing electric field between
parallel plates. Similarly, the changing magnetic field
can be used as the electromagnetic wave source to de-
termine a certain plane electromagnetic wave mode. In
unbounded space, no matter whether the wave source is
evacuated later or not, the electromagnetic field will not
turn back, and the initial conditions cannot be changed.
y
P
x
M
z
o
Figure 1 plane electromagnetic wave modes propagating along the positive direction of x-axis between sufficiently long parallel plate
capacitors
2.2 General solution of plane wave modes for
Maxwell equations
In a vacuum, electromagnetic waves travel at the
speed of light. Maxwell’s electromagnetic theory hold-
s that in the process of electromagnetic wave propaga-
tion, the changing electric field wave and the changing
magnetic field wave exist at the same time. In order to
obtain the plane electromagnetic wave mode satisfying
the initial condition (2), the Maxwell equations need to
be solved first to determine the general magnetic field
solution of the plane wave mode. The vector form of
Maxwell equations is
∇ × E = −
∂B
∂t
∇ × B =
1
c
2
∂E
∂t
∇ · E = 0
∇ · B = 0
(3)
From this, the second-order wave equations for the elec-
tric field E and the magnetic field B can be derived
∇
2
E −
1
c
2
∂
2
E
∂t
2
= 0
∇
2
B −
1
c
2
∂
2
B
∂t
2
= 0
(4)
(3) and (4) are vector differential equations, which can
be transformed into optimization differential equations.
The mainstream literature directly writes the general so-
lutions, E (x, t) = E (x) e
−iωt
and B (x, t) = B (x) e
−iωt
,
of the time harmonic electromagnetic waves of the two e-
quations, and then writes the special solutions according
to the shear wave conditions. This solution completely
ignores the initial conditions of the wave equation and
violates the solution rules of the wave equation. Strict-
ly speaking, it lacks the mathematical basis for solving
the differential equation, and the conclusion is naturally
untenable.
⊗ Reference [9] uses the circulation integral to “prove” the result that is inconsistent with the initial condition: H
z
= ε
0
E
y
v (page 613).
Generally speaking, the magnitude of the electromagnetic field vector can be solved by flux integral or circulation integral only under the
condition of spherical symmetry or axial symmetry, but plane electromagnetic wave does not have such symmetry.
4 X. D. Dongfang Dongfang Modified Equations of Electromagnetic Wave
If limited to the dogma of standard literature, it may
involuntarily produce the sophistry of deviating from the
inevitable logic and choosing the formal logic to maintain
the wrong classical logic and conclusion. At this time,
you might as well directly solve the first-order Maxwell
equations to obtain the correct plane transverse elec-
tromagnetic wave mode, and all problems will b ecome
suddenly clear. The component form of the first-order
Maxwell equations (3) is
e
x
∂
∂x
+ e
y
∂
∂y
+ e
z
∂
∂z
× (e
x
E
x
+ e
y
E
y
+ e
z
E
z
) = −
∂
∂t
(e
x
B
x
+ e
y
B
y
+ e
z
B
z
)
e
x
∂
∂x
+ e
y
∂
∂y
+ e
z
∂
∂z
× (e
x
B
x
+ e
y
B
y
+ e
z
B
z
) =
1
c
2
∂
∂t
(e
x
E
x
+ e
y
E
y
+ e
z
E
z
)
e
x
∂
∂x
+ e
y
∂
∂y
+ e
z
∂
∂z
· (e
x
E
x
+ e
y
E
y
+ e
z
E
z
) = 0
e
x
∂
∂x
+ e
y
∂
∂y
+ e
z
∂
∂z
· (e
x
B
x
+ e
y
B
y
+ e
z
B
z
) = 0
(5)
The plane wave modes satisfying equation (1) are implied in the following equation
E
x
= E
z
= 0,
∂E
y
(x, t)
∂z
= 0 (6)
It can be seen that the third formula of equation (5) is valid, and the other three formulas are simplified to
e
z
∂E
y
∂x
= −e
x
∂B
x
∂t
− e
y
∂B
y
∂t
− e
z
∂B
z
∂t
e
x
∂B
z
∂y
−
∂B
y
∂z
+ e
y
∂B
x
∂z
−
∂B
z
∂x
+ e
z
∂B
y
∂x
−
∂B
x
∂y
= e
y
1
c
2
∂E
y
∂t
∂B
x
∂x
+
∂B
y
∂y
+
∂B
z
∂z
= 0
(7)
therefore
∂B
x
∂t
= 0,
∂B
y
∂t
= 0,
∂E
y
(x, t)
∂x
= −
∂B
z
∂t
∂B
z
∂y
−
∂B
y
∂z
= 0,
∂B
x
∂z
−
∂B
z
∂x
=
1
c
2
∂E
y
(x, t)
∂t
,
∂B
y
∂x
−
∂B
x
∂y
= 0
∂B
x
∂x
+
∂B
y
∂y
+
∂B
z
∂z
= 0
(8)
The general solutions of the three equations in the first line of (8) are
B
x
= B
x
(x, y, z) , B
y
= B
y
(x, y, z) , B
z
= a (x, y, z) −
∂E
y
(x, t)
∂x
dt (9)
Where a (x, y, z) is the partial integral ”constant”. So the expression of the magnetic field is
B = e
x
B
x
(x, y, z) + e
y
B
y
(x, y, z) + e
z
a (x, y, z) −
∂E
y
(x, t)
∂x
dt
(10)
Of course, the expression of magnetic field B must satisfy the third line equation of (8). It can be seen from the
definite solution condition (2)
B
x
(y, z) = 0; B
y
(x, z) = 0 (11)
Therefore, the general expression for the magnetic field wave mode is obtained from equation (10),
B = e
z
a (x, y, z) −
∂E
y
(x, t)
∂x
dt
(12)
It should be noted that using equation (11) to integrate the second equation in the second line of (8), one gets
B
z
= b (y, z, t) −
1
c
2
∂E
y
(x, t)
∂t
dx +
∂B
x
∂z
dx
Mathematics & Nature (2021) Vol. 1 5
Where b (y, z, t) is also the ”constant” of the partial integral. The expression for the magnetic field can be expressed
as
B = e
x
B
x
(y, z) + e
y
B
y
(x, z) + e
z
b (x, y, z) −
1
c
2
∂E
y
(x, t)
∂t
dx +
∂B
x
∂z
dx
(13)
It is also obtained from B
x
(y, z) = 0 and B
y
(x, z) = 0
given in (11)
B = e
z
b (y, z, t) −
1
c
2
∂E
y
(x, t)
∂t
dx
(14)
(12) and (14) are equivalent and represent the general
solution of plane electromagnetic wave modes of Maxwell
equations.
The first-order Maxwell differential equations or inte-
gral equations are sufficient to give the correct solution of
the electromagnetic wave mode, while transforming the
first-order differential equations into second-order differ-
ential equations will produce additional roots and unrea-
sonable roots, which are mathematical common sense.
E⊥v has been explained earlier, and these two results
show that B⊥E and B⊥v, so the plane electromagnetic
wave is a transverse wave. According to (8), the scalar
form of the electric and magnetic field wave equation
can also be derived, and the solution of an initial value
problem in unbounded space is usually d’Alembert solu-
tion. Similarly, if the plane magnetic field wave mode is
given, the integral expression of the plane electric field
wave mode can be obtained.
2.3 Special solution of plane wave mode for
Maxwell equations
The definite solution condition (2) and equation (12)
determine a plane transverse electromagnetic wave mod-
e:
E = e
y
E
m
sin ω (t − x/c)
B = e
z
a
z
(x, y, t) −
∂E
y
(x, t)
∂x
dt
B (t = 0) = 0
(15)
The solution of the above second equation is
B = e
z
E
m
c
sin
ω x
c
+ sin ω
t −
x
c
It can be seen that the amplitude of the time harmon-
ic magnetic field excited by the time harmonic electric
field is different at different positions in space. The plane
transverse electromagnetic wave mode is
E = e
y
E
m
sin ω
t −
x
c
B = e
z
E
m
c
sin
ω x
c
+ sin ω
t −
x
c
(16)
The phase difference between electric field and magnetic
field is also a function of position.
Obviously, the above plane electromagnetic wave is a
transverse wave. Although the amplitude of the elec-
tric field is the same everywhere, the amplitude of the
magnetic field is different at different positions. Al-
though the frequency of the electric field and magnet-
ic field oscillation is the same, they are not in phase
everywhere. The ratio of electric field to the magnetic
field is not a constant, and the maximum ratio of elec-
tric field amplitude to magnetic field amplitude is c and
the minimum is c/2. It can be seen that the plane elec-
tromagnetic wave modes of E (x, t) = E (x) e
−iω t
and
B (x, t) = B (x) e
−iωt
described by the classical theory
is not tenable. The discussion of plane electromagnetic
wave should be based on the objectively existing wave
modes.
3 Regular solution of cylindrical transverse
electromagnetic wave mode
In 2011, the author cooperated with Xijun Li and pub-
lished an English article on regular solutions of cylindri-
cal transverse electromagnetic modes
[20]
of Maxwell e-
quations with the signature Rui Chen. Here, the relevant
mathematical process and conclusion are introduced a-
gain to emphasize the correct treatment of the initial
value problem of Maxwell equations. Firstly, the gen-
eral solution of vector Maxwell equations is solved in
cylindrical coordinate system, and then the cylindrical
transverse electromagnetic wave mode is determined ac-
cording to the initial value conditions. The results show
that the phase of electric field and magnetic field of cylin-
drical transverse electromagnetic mode is not the same
everywhere, and the ratio of amplitude is also a function
of time and space, which has the same characteristics
as the plane transverse electromagnetic wave mode that
can be generated and existing in practice.
3.1 Initial conditions of cylindrical transverse
electromagnetic wave
Theoretically, the coaxial transmission line device
composed of a hollow conductor tube and core wire as
shown in Figure 2 can realize cylindrical transverse elec-
tromagnetic wave. Let the radius of the core wire be
R
1
and the radius of the coaxial hollow conductor tube
be R
2
. By using the distribution formula of electrostat-
ic field, when the core wire and hollow conductor tube
have the equal amounts of different kinds of charges re-
spectively, let the charged amount on the unit length of
the surface along the axis be λ
m
, and the electrostatic
field in the transmission line can be obtained by using
6 X. D. Dongfang Dongfang Modified Equations of Electromagnetic Wave
Gauss theorem
E =
λ
m
2πε
0
r
e
r
(R
1
6 r 6 R
2
) (17)
Among them, e
r
is the radial unit vector, and r is the
distance from a certain point in the vacuum of the coax-
ial transmission line to the axis.
When the p eriodically changing power supply is con-
nected at the left end so as to the periodically changing
charge distribution state is excited to propagate from left
to right, a periodically changing electric field is generat-
ed in the coaxial transmission line and propagates from
left to right to form an electric field wave. The prop-
agation direction is along the positive direction of the
z-axis and perpendicular to the direction of the electric
field Assuming that the hollow conductor tube is close
to the core wire and the distribution of electric field to
the radial direction is stable, the form of electric field
wave can be deduced according to the electrostatic field
distribution formula (17). This result can also be deter-
mined by Maxwell’s equations. Naturally, they are the
same, and the specific form is
x
y
z
o
Figure 2 realization of cylindrical transverse electromagnetic wave with coaxial transmission line
E = e
r
λ
m
2πε
0
r
f (z, t) (R
1
6 r 6 R
2
)
B (t = 0) = 0
(18)
f (z, t) can be a sine function or a cosine function, which
depends on the change of charge distribution on the con-
ductor surface, and the corresponding wave mode is ex-
pressed in a complex numb er as
E = e
r
λ
m
2πε
0
r
e
i ω
(
t−
z
c
)
(R
1
6 r 6 R
2
)
B (t = 0) = 0
(19)
At zero time, the electric field in the transmission line is
zero, so the magnetic field is zero everywhere, which is
the initial condition.
3.2 General solution of cylindrical wave for
Maxwell equations
In order to obtain the cylindrical electromagnetic
wave mode satisfying the initial condition (18), the gen-
eral magnetic field solution of Maxwell equations in
cylindrical coordinate system
[22]
should be solved. The
component form
[23]
of Maxwell equations (3) in the cylin-
drical coordinate system is
1
r
∂E
z
∂θ
−
∂E
θ
∂z
e
r
+
∂E
r
∂z
−
∂E
z
∂ r
e
θ
+
1
r
∂
∂ r
(rE
θ
) −
1
r
∂E
r
∂θ
e
z
= −
∂B
∂ t
1
r
∂B
z
∂θ
−
∂B
θ
∂z
e
r
+
∂B
r
∂z
−
∂B
z
∂ r
e
θ
+
1
r
∂
∂ r
(rB
θ
) −
1
r
∂B
r
∂θ
e
z
=
1
c
2
∂E
∂ t
1
r
∂
∂ r
(rE
r
) +
1
r
∂E
θ
∂θ
+
∂E
z
∂z
= 0
1
r
∂
∂ r
(rB
r
) +
1
r
∂B
θ
∂θ
+
∂B
z
∂z
= 0
(20)
For the electric field wave mode (18), because E
r
= E
r
(r, z, t), E
θ
= 0 and E
z
= 0, the system of equations (20) is
reduced to
∂E
r
∂z
e
θ
= −e
r
∂B
r
∂ t
− e
θ
∂B
θ
∂ t
− e
z
∂B
z
∂ t
1
r
∂B
z
∂θ
−
∂B
θ
∂z
e
r
+
∂B
r
∂z
−
∂B
z
∂ r
e
θ
+
1
r
∂
∂ r
(rB
θ
) −
1
r
∂B
r
∂θ
e
z
= e
r
1
c
2
∂E
r
∂ t
1
r
∂
∂ r
(rE
r
) = 0
1
r
∂
∂ r
(rB
r
) +
1
r
∂B
θ
∂θ
+
∂B
z
∂z
= 0
(21)
Mathematics & Nature (2021) Vol. 1 7
According to the third line of equation (21), the electric field under the condition of axis-symmetry must satisfy the
equation rE
r
= g (z, t), so
E
r
=
g (z, t)
r
(22)
Consistent with the result inferred from the electrostatic field (18), it can be seen that the third line of (21) is true.
The components on both sides of the vector equation are equal, so the first, second and fourth equations in (21)
are reduced to
∂B
r
∂ t
= 0,
∂B
θ
∂ t
= −
∂E
r
∂z
,
∂B
z
∂ t
= 0
1
r
∂B
z
∂θ
−
∂B
θ
∂z
=
1
c
2
∂E
r
∂ t
,
∂B
r
∂z
−
∂B
z
∂ r
= 0,
1
r
∂
∂ r
(rB
θ
) −
1
r
∂B
r
∂θ
= 0
1
r
∂
∂ r
(rB
r
) +
1
r
∂B
θ
∂θ
+
∂B
z
∂z
= 0
(23)
According to the three equations in the first row, one gets
B
r
= B
r
(r, θ, z) , B
θ
= −
∂E
r
∂z
dt + a (r, θ, z) , B
z
= B
z
(r, θ, z) (24)
Of course, this result must satisfy the last two equations in (23). Thus, the expression of the magnetic field is
B = e
r
B
r
(r, θ, z) + e
θ
−
∂E
r
∂z
dt + a (r, θ, z)
+ e
z
B
z
(r, θ, z) (25)
From the initial value condition B (t = 0) = 0, it can
be seen that
B
r
(r, θ, z) = 0; B
z
(r, θ, z) = 0 (26)
Replace the corresponding magnetic field component in
(25) with the above results, so there is
B = e
θ
−
∂E
r
∂z
dt + a (r, θ, z)
(27)
On the other hand, using equation (26), the integration
result of the first equation in the second line of (23) is
B = e
θ
−
1
c
2
∂E
r
∂ t
dz + b (r, θ, t)
(28)
The general expression of cylindrical electromagnetic
wave mode given by synthesizing (18) and (27) is
E = e
r
λ
m
2πε
0
r
f (z, t) (R
1
6 r 6 R
2
)
B = e
θ
−
∂E
r
∂z
dt + a (r, θ, z)
B (t = 0) = 0
(29)
The general expression of cylindrical electromagnetic
wave mode given by synthesizing (18) and (28) is
E = e
r
λ
m
2πε
0
r
f (z, t) (R
1
6 r 6 R
2
)
B = e
θ
−
1
c
2
∂E
r
∂ t
dz + b (r, θ, t)
B (t = 0) = 0
(30)
Equations (29) and (30) are equivalent. Within the
framework of Maxwell’s electromagnetic theory that the
transformed electric field produces a changed magnetic
field and the changed magnetic field produces a trans-
formed electric field, equations (29) and (30) represent
the general solution of Maxwell’s equations of cylindrical
magnetic field excited by a radial changing electric field.
3.3 Special solutions of cylindrical waves for
Maxwell equations
According to (29) or (30), the expression of the mag-
netic field can be obtained from the expression of the
electric field, that is, the special solution of the cylindri-
cal wave of Maxwell equations. Substituting (19) into
(29) , one obtains
E = e
r
λ
m
2πε
0
r
e
i ω
(
t−
z
c
)
(R
1
6 r 6 R
2
)
B = e
θ
−
∂E
r
∂z
dt + a (r, θ, z)
B (t = 0) = 0
(31)
The partial derivative of electric field E is obtained from
the first equation above,
∂E
r
∂z
= −
iω
c
λ
m
2πε
0
r
e
i ω
(
t−
z
c
)
(32)
Substitute it into the second equation of (31) and inte-
grate it to obtain
B = e
θ
1
c
λ
m
2πε
0
r
e
i ω
(
t−
z
c
)
+ a (r, θ, z)
(R
1
6 r 6 R
2
)
(33)
8 X. D. Dongfang Dongfang Modified Equations of Electromagnetic Wave
Using the initial condition B (t = 0) = 0, it is deter-
mined that the result of the undetermined function is
a (r, θ , z) = −
1
c
λ
m
2πε
0
r
e
−i ω
z
c
(R
1
6 r 6 R
2
) (34)
Then substitute it into equation (33) to obtain
B = e
θ
λ
m
2πcε
0
r
e
i ω
(
t−
z
c
)
− e
−i ω
z
c
(R
1
6 r 6 R
2
)
(35)
Therefore, the specific form of (31) is
E = e
r
λ
m
2πε
0
r
e
i ω
(
t−
z
c
)
(R
1
6 r 6 R
2
)
B = e
θ
λ
m
2πcε
0
r
e
i ω
(
t−
z
c
)
− e
−i ω
z
c
(R
1
6 r 6 R
2
)
(36)
It can be verified that the expression (36) of the above
electromagnetic wave satisfies each equation of the com-
ponent form (7) of Maxwell’s equations, that is, it sat-
isfies Maxwell’s equations. (36) is a special solution of
Maxwell’s equations that meets the initial conditions.
It expresses a cylindrical time harmonic electromagnetic
wave mode that can exist in theory. Take the imaginary
part of formula (36) to obtain the cylindrical electro-
magnetic wave mode excited by sinusoidal electric field
wave
[24-28]
E = e
r
λ
m
2πε
0
r
sin ω
t −
z
c
(R
1
6 r 6 R
2
)
B = e
θ
λ
m
2πcε
0
r
sin ω
t −
z
c
+ sin
ω z
c
(R
1
6 r 6 R
2
)
(37)
Taking the real part of equation (36), the cylindrical
electromagnetic wave mode excited by cosine electric
field wave is obtained,
E = e
r
λ
m
2πε
0
r
cos ω
t −
z
c
(R
1
6 r 6 R
2
)
B = e
θ
ωλ
m
2πcε
0
r
cos ω
t −
z
c
− cos
ω z
c
(R
1
6 r 6 R
2
)
(38)
4 Conclusions and comments
Since the classical theory first preconcert the forms
of solutions before solving the Maxwell equations, the
obtained transverse electromagnetic wave mode shows
a most incredible characteristic that the electric field
and magnetic field are in phase everywhere. Now that
the periodic changing electric field generates periodically
changing magnetic field, the process of the transmission
of an electromagnetic wave is that of energy radiation,
and the electromagnetic energy is transmitted with the
electromagnetic waves. If the maximum or minimum
value of the electric field and magnetic field in an elec-
tromagnetic wave are always obtained simultaneously,
how do the electric energy and magnetic energy inter-
change? Consequently, it can be concluded that the clas-
sical electromagnetic wave in which the electric field and
magnetic field are in phase everywhere violates the law of
conservation of energy. In both mathematical and phys-
ical sense, the plane transverse electromagnetic wave de-
termined by the classical electromagnetic theory is not
tenable. Theoretically, the solution of the initial value
problem of Maxwell equations is the regularity of plane
or cylindrical transverse electromagnetic wave modes.
From Maxwell’ theory, if the electric field in space
changes, the magnetic field at the same place would
change and these changing electric field and magnetic
field will generate new changing electric field and mag-
netic field in farther space. Consequently, the changed
electric field and magnetic field are not confined to a
region but propagate from near site to farther places.
The propagation of electromagnetic field forms electro-
magnetic waves. The characteristics of the electromag-
netic wave are actually described by the solutions of the
Maxwell equations. The electromagnetic wave is a trans-
verse wave, which is just determined by the transverse
wave conditions ∇·E = 0 and ∇·B = 0 of the Maxwell
equations. We strive to maintain the logic and conclu-
sion of classical electromagnetic theory, but this does
not mean that Maxwell’s electromagnetic wave theory is
perfect. The important problems mentioned at the be-
ginning of the article and the deeper problems hidden in
Maxwell’s electromagnetic field theory are easy to solve.
We will discuss them later. At present, the most im-
portant problem that should attract our attention is the
hidden problem of physics fame and wealth thought and
degree system, which is the fundamental reason for the
confusion of physics theory and a large number of errors.
Celebrities in theoretical physics often regard person-
al very random calculation as new mathematics, and are
good at explaining the contradictory logic as physical
mathematics different from pure mathematics. In fact,
it is a distorted understanding of mathematical rules and
natural laws due to the lack of due mathematical basis.
By systematically testing the logic and basic mathemat-
ical calculation of theoretical physics with the unitary
principle, it will be found that famous theoretical physi-
cists often make mistakes in physical concepts, physi-
cal logic and mathematical calculation
[29-35]
, especially
in the calculation of elementary calculus and elemen-
tary algebra. The electromagnetic wave mode solution
of Maxwell equations is only a simple example. Theoret-
ical physics is mixed with a large number of similar low-
level errors, which have not been found and recognized
by mainstream physics scholars, and non mainstream
scholars who find errors have been excluded, slandered
and denied. Since mainstream theoretical physics schol-
ars have made a large number of very serious logical
and computational errors on very simple problems for
a long time, and have been unaware of these errors for
more than 100 years, we should not firmly refuse to test
Mathematics & Nature (2021) Vol. 1 9
the logical and mathematical calculations of theoretical
physics with too much confidence, let alone ignore or
even slander the rigorous arguments of non mainstream
physics scholars, Unless we lose the awe of scientific truth
and abandon the dignity of human nature for personal
fame and wealth.
The reason why theoretical physics is limited to this
embarrassing situation is that the desire for fame and
wealth drives researchers to excessively pursue celebri-
ties and degree system, resulting in generations of stu-
dents having to clone textbooks. Theoretical physics
scholars who grew up in this environment have long lost
their basic consciousness and ability to independently
judge the right and wrong of physical logic. The deter-
mination of the transverse wave mode equation is an ini-
tial value problem of Maxwell equation, but it has been
completely misunderstoo d. Most of the wrong mathe-
matical operations in theoretical physics are unknown
to later scholars, which just reflected the serious defect-
s of the degree system in the past. Perhaps changing
the academic monopoly system of fame and wealth and
the degree system of cloning celebrities are the key to
the breakthrough and major development of theoretical
physics in the future.
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Latest findings
Nuclear Force Constants Mapped by Yukawa Potential
The nuclear meson theory evolved from Yukawa potential regards the observed mesons as its experimental test, which is far fetched. It goes without saying that there are no mesons in nuclei such as deuterons. However, it is the lack of scientific basis to construct the wave equation by using the quantum mechanical operator to act on the unknown function, so as to piece up the exact solution of the potential function. The relevant calculation results do not form an inevitable causal relationship with the existence of mesons. Even if a celebrity patchwork theory is maintained, meson theory should have at least one observable quantitative inference from the perspective of experimental test theory. Here, the ground state energy corresponding to the Yukawa potential expressed by the nuclear force constant and the mass of the PI meson is calculated by the variational method, and then the average binding energy of the atomic nucleus is connected with the ground state energy corresponding to the Yukawa potential to calculate the average nuclear force constant. From this result, it is estimated that the distribution range of the average nuclear force constant of 118 atomic nuclei in the periodic table of elements is from 1.093992 to 1.413981. The results also show that the estimated value of nuclear force constant up to 15 is unreliable. However, the nuclear force constant can not be predicted by other methods. Calculating the average nuclear force constant by using the ground state energy corresponding to Yukawa potential is only a dead cycle, and the desire to test meson theory through experimental observation can not be realized. This unexpected result shows that modern physical theory has long fallen into an unrealistically strange circle of formal logic shrouded in the halo of mathematics: although the conditions and conclusions of the theory are far from the true description of natural phenomena, the theory can be deduced infinitely, and is constantly beautified and advocated as a profound revelation of natural laws. The reason why this paper introduces the average nuclear force constant mapped by the unreasonable Yukawa potential is to disclose some unique calculation methods that need to be applied in establishing a reasonable nuclear force model in the future.