Journal of Modern Physics, 2013, 4, 899-903
http://dx.doi.org/10.4236/jmp.2013.47121 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Model of an Atom by Analogy with the Transmission Line
Milan Perkovac
The First Technical School Tesla, Zagreb, Croatia
Received March 24, 2013; revised April 27, 2013; accepted May 24, 2013
Copyright © 2013 Milan Perkovac. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Model of an atom by analogy with the transmission line is derived using Maxwell’s equations and Lorentz’ theory of
electrons. To be realistic such a model requires that the product of the structural coefficient of Lecher’s transmission
lines σ and atomic number Z is constant. It was calculated that this electromechanical constant is 8.27756, and we call it
structural constant. This constant builds the fine-structure constant 1/α = 137.036, and with permeability μ, permittivity
ε and elementary charge e builds Plank’s constant h. This suggests the electromagnetic character of Planck’s constant.
The relations of energy, frequency, wavelength and momentum of electromagnetic wave in an atom are also derived.
Finally, an equation, similar to Schrödinger’s equation, was derived, with a clear meaning of the wave function, which
represents the electric or magnetic field strength of the observed electromagnetic wave.
Keywords: Lecher Transmission Line; Lorentz’ Theory of Electrons; Maxwell’s Equations; Model of Atom; Planck
Constant; Structural Constant; Transverse Mass; Wave Equation
1. Introduction
A hundred years ago, classical physics, with Newton’s
mechanics and Maxwell’s theory, couldn’t explain de-
termined properties of atoms [1]. For that reason Max-
well’s equations are neglected in modern physics, despite
the fact that matter is composed of electrically charged
particles, and that static and current electricity are in
complete harmony with Maxwell’s equations. The main
motive of this paper is to show that the atom can be ex-
plored using Maxwell’s equations. Moreover, I want to
show that, in addition to Maxwell’s equations and Lor-
entz’ theory of electrons for the basic research of the
atom, nothing more is needed. By using Maxwell’s the-
ory differential equations of the electromagnetic wave in
any space are derived. This space could also be the space
within the atom. These differential equations have the
same form as the differential equations of wave on the
parallel-wire transmission line, on the so-called Lecher’s
line [2]. Therefore, this electromagnetic wave, with the
same differential equations, is treated in analogy with the
wave on Lecher’s line. Here I show a model of atom [3],
where structural coefficient of Lecher’s transmission line,
corresponding to a certain atom, multiplied by their
atomic number, appears as a structural constant 8.27756,
which is a universal constant. This constant builds the
fine structure constant, and with other constants builds
Planck’s constant [4]. The energy and momentum of the
electromagnetic wave can be determined using Max-
well’s and Lorentz theory. This paper makes it possible
to eliminate some disadvantages of classical physics
mentioned at the beginning.
The idea of this article is that it is not possible to make
the same thing in two distinctly different ways in nature.
Therefore an electromagnetic wave that originates in the
atom and the electromagnetic wave in the macro world
have common ground. Due to identical differential equ-
ations a linking of electromagnetic wave in atoms and
wave on the transmission lines is possible.
This linking of waves in atoms and waves on the trans-
mission line is carried out using the parameters of atoms
and transmission lines, as well as through their energy.
Researching electromagnetic energy from transmission
lines it was determined that this energy can be expressed
as linearly proportional to the frequency of oscillation of
its own LC circuit. Proportionality factor leads to struc-
tural constants and action constants. Structural constant
is introduced so as to make action constant independent
of LC circuit natural frequency. Here, in a unique way,
we can determine the value of the action constant. It’s
shown that action constant is equal to Planck’s constant
h.
Then we determine the frequency of the wave in the
atom, the wavelength, phase velocity and momentum of
C
opyright © 2013 SciRes. JMP

M. PERKOVAC 900
the wave.
Knowledge of these quantities allows determination of
the properties of space in which these phenomena take
place.
Finally, we can determine the wave equation accord-
ing to which all of these phenomena are governed by
investigating the atom.
2. The Atom and the Transmission Line
Using equations of Maxwell’s theory, [5], it is possible to
obtain the states of the electric field (vector E) and mag-
netic field (vector H) in any space, even within the atoms.
The mathematical description of these states are pre-
sented with two second-order linear partial differential
equations (wave equations) [1,6]:
22
2
2 2
em
1
0,
tt

 

EH
H
2

2
22
em
1
0; E
uu
(1)
where is del-squared,
222
222
x
yz





u
, t is time
and em is phase velocity of the electromagnetic wave,
which dependents on the medium, i.e., em 1

 u;
r0

  is permittivity, r

is relative permittivity, 0

is permittivity of free space, r0

  is permeability,
r

is relative permeability and 0

is permeability of
free space.
The same form of differential equations as previous,
but only in one dimension, [2], is also present on the par-
allel-wire transmission line, called Lecher’s line, con-
sisting of a pair of ideal conducting parallel wires of ra-
dius

, separated by

, wherein the ratio



:
22
22
0 ;
uu
LC
zt




22
22
0,
i i
LC
zt




(2)
where

ln1 4π   L is inductance of Lecher’s
line per unit length and

2
ln22 1C'  

 


u
dz
dz
is its capacitance per unit length, is voltage at the
entrance to the element of Lecher’s line, and i is
electrical current at the entrance to the element of
Lecher’s line [2,4].
The same form of differential equations for example
those in the atom and on the Lecher’s line, allows finding
unknown solutions in the atom using known solutions on
Lecher’s line. In this case, the voltage u on the Lecher’s
line is analogous to the electric field E, while the electric
current i on the Lecher’s line is analogous to the mag-
netic field H. Therefore, the voltage and current on
Lecher’s line behaves the same way as the electromag-
netic wave in the atom. The analogy between the wave
on the transmission line and electromagnetic wave will
be completed when we put that energy into transmission
line, then it is equal to the energy of the electromagnetic
wave.
On one hand, such transmission line can be regarded
as a limiting case of an LC network with infinitely small
capacitors and inductors [7]. If all small capacitors of the
network are put on the open end (C), and all small in-
ductances are put on the shot-circuited end (L) of
Lecher’s line, then the natural frequency of such os-


cillatory circuit is 12 LC  
E
, [8].
3. Electromagnetic Energy in an Atom and
on the Transmission Line
We can determine the amount em of electromagnetic
energy of a wave and bring that energy to the LC circuit
formed in the manner described by Lecher’s line, [9]:
2
ˆ
12EQCC
ˆ
em C, where Q is a maximum charge on
the said capacitor C.
On the other hand, however, in force equation accord-
ing to Newton’s second law,
F
ma, we substitute ace-
leration a from 2
arv, F from Coulomb’s law,
2
4qQ r

, and m from transverse mass of the electron
according to Lorentz’ theory. Because the acceleration is
at right angles with respect to the velocity, this transverse
2
1m

mass of the electron is , [10]. Therefore we
obtain:
2
2
222
2
1
;
44
1
qQ qQ
mr
rmc
r








v

qe
, (3)
where r is the radius of the circular orbit of the electron
in an atom, q is the charge of the electron

, Q is
the charge of the nucleus


QZe, Z is atomic number,
m is the electron rest mass, c is the speed of light in vac-
uum, c


v v, where is the electron velocity.
The kinetic energy of electron is
222
1
K
mc mc  , [9]. Using Equation (3) and
noting that an electron is of opposite charge of the nu-
cleus, than the potential energy of electron is
22 2
41UqQr mc

. The total mechanical
energy of the electron

22
T11EKU mc,

 
according to the law of conservation of energy is equal to
the negative emitted electromagnetic energy,


22
em 11EmceV
 
Tem
EKU EeV
, [4]; here V is the potential
difference, which passes an electron from the point of
reference potential, to the potential of the point at which
an electron is currently located, i.e.,
 .



22
em 11Emc
According to

 we can write:

M. PERKOVAC 901
2em em
22
11;2
EE
mc mc


 em
2
1 .
2
E
mc




(4)
Now, the radius r in Equation (3) we write:

22
em
12E mc



em em
18r qQEmcE   , that is
22
C
2
ˆ
1
.
2
c Q
CEmc

ˆ
em
em
em
1
1
24 12
qQ Em
Er


 (5)
This single Equation (5) has two unknown sizes, i.e.
parameter C and variable C Q. With the help of Dio-
phantine equations we obtain one of the solutions:
4Cr

 , and

2
Cem
22
em
11
2QqQEmcE mc 
ˆ,
[4].
We can transform now the Equation (5) of the elec-
tromagnetic energy in LC circuit [4]:
22
CC
em
2em
LC CLC
em
ˆˆ
11
22
1
ˆ
12
QQ
LL
ECC
CC L
2
C
2
2
ˆ
2
,
Q
LC
Emc
Z
QZqQ A
Emc






 


 
(6)
where
2
em
2
em
1
12
Emc
Emc


LC
AZ
qQ (7)
shall be called the action of the electromagnetic oscillator,
[6,11,12], and



LC
d
d
ln1 4ln2
LLzL
ZCCzC







 




24 1


(8)
is the characteristic impedance of Lecher’s line, [5],
while
 


2
+4 1  
em
E
ln+1 4ln2    (9)
shall be called the structural coefficient of Lecher’s line
[4].
The solution of Equation (6) now reads:


2
2
22
.mc
EA
em LC LC
EZ qQmcZqQ   (10)
4. Structural Constant and Action Constant
Now, with regard to Equation (6), em

, the action
of electromagnetic oscillator can be written as:
Constant part of this solution, which does not depend
on natural frequency

, we denote as action constant
0LC
A
ZqQ . Each oscillator has its own action con-
stant. Equation (11) is then:

2
222
00
,AA mcAmc    (12)
and also we can now write Equations (6) and (10) as (see
Planck-Einstein equation):






22
em 0emem
2
2
22
em 002
0
0
112,
0
,
EA EmcEmc
EAmcAmc
Amc
A

 

 
 

EeV
(13)
and with em

the first part of the Equation (13)
gives (see the test of Duane-Hunt’s law [4]):
2
2
2
00
12
.
0
1
eVeVmceV
eV mc

2
2
2. mc
Q




2
LC LC
mc
AZ qQ Zq

   (11)
A
AeV mc





0
(14)
A
can now be written using Equation (8): Also,


2
0LC .ZqQeZeZe




 

A
(15)
One specific Lecher’s line is dedicated to each atomic
element. We can choose Lecher’s line, which represents
an atom of atomic number Z, arbitrarily. If we choose it
so that the product of structural coefficient



 and
atomic number Z, i.e.


Z


in Equation (15), is con-
stant, so called structural constant, 0
s
Z  
,
, then,
providing rr



action constant 0
A
will be the
same for all atoms, i.e. it will be a universal constant for
all atomic oscillators: 22 22
00000
A
se se   
0
.
The only unknown quantity in this expression is
s
.
Let’s see how we shall determine it.


2, 2 
Namely, below


2
, Lecher’s line does
not exist, because in that case two conductors become
just one guide. Therefore, in the region below


,
Lecher’s line cannot represent a single atom. However,
the limit 2


can be used to determine the structural
constant 0
s
. For example, I estimate, [4], that only ten
percent increase of


,

.. 1.122.2ie   , is not big
enough to include all of about forty of unstable elements
in this region. An increase of twenty percent of

,


1.2 22.4  
294 Uuo 118Z
207
82 Pb 82Z

, however, should then include almost
40 elements, exactly from 118 , , to the first
always-stable atom of lead, , . Therefore,

02.4820.837 828.28sZ
   . Now,
one can calculate 2234
00
6.63 10JsAse 

00

 
2
2 137.11s (it is like Planck’s h), and 0 (like fine-struc-
ture constant 1

). The best agreement with the fine-
structure constant gives: 0. Then the action
constant of the atomic oscillator is always the same and it
8.277 56s

M. PERKOVAC 902
is equal to Planck’s constant h. This is a realistic and
proven solution.
5. The Wavelength and Momentum of the
Electromagnetic Wave in an Atom
The momentum of a limited plane electromagnetic wave,
like the momentum of a photon, is related to its phase
velocity by

em emem em
pE E

 u, where

is
the wavelength of the electromagnetic wave, and on the
other hand, in accordance with the law of conservation of
momentum, the linear momentum of the electron, [10], is
equal to the momentum of the electromagnetic wave,
(see Compton effect [9]),
em
22
11
Emm c
.






v
EeV
(16)
From Equation (16), by using Equations (4), (14) and
em , we obtain (see de Broglie wavelength; in the
case of low-energy is 22 ,2K mveV mceV):


2
2
3
2
1
212
eV mc
00
2
.
0
A
A
m

v
meV eV mc
eV mc



(17)
Phase velocity,

 , of electromagnetic wave from
Equations (14) and (17) is
2
em 2
1
212
.
2
eVeV mc
meV mc


 

u
v
20eV mc 
em
p
em
EeV
(18)
The momentum we obtain from Equations (13),
(17) and (18), with :
em 0
em
em 2
0
12
2
EA
peV eV mc
m
A





u20eV mc (19)
6. Properties of Space in the Model of Atom
As I have already stated at the beginning, that phase ve-
locity is also em 1

 u, while it is now clear that the
wave impedance

 , [which is an integral part of the
characteristic impedance of Lecher’s line in Equation (8),
and hence integral part of the action constant 0
A
in
Equation (15)], should therefore remain unchanged, i.e.,
we get a system of two equations:

00
2
,
11eVeV mcm
 



with two unknowns, μ and ε. The solutions of these equa-
tions are [6]:
2
0
r0
2
0
2
0
r0
2
0
2
rr 2
00
12 2,
1
12
2,
1
12
12 .
1
eV mc
m
eVeV mc
eV mc
m
eVeV mc
eV mc
m
eVeV mc





2
212,eV mc (20)



 







 

(21)
7. Equation Like Schrödinger’s Equation
A current value of linearly polarized standing wave reads
[3,4]:

 
x0
0
y
2
,sin cos,
2
,cossin,
z
Ezt Et
Ez
zt t


H









 

E

,Ezt
(22)
where 0 is the maximum value, i.e., the amplitude of
electric field strength E, x is the x-component of
the electric field strength dependent on the z-axis and the
time t, and


,
y
H
zt is the y-component of the magnetic
field strength H dependent on the z-axis and the time t.
All that we shall continue to write for the y-component of
the magnetic field


,
y
H
zt shall be applied in the exact
same way to the x-component of the electric field


,Ezt
x. If we use the second derivative of the previous
equation,


y,
H
zt

, with respect to z, we get:

2
22
yy
,2π,0Hzt zHzt 

 . After the inclu-
sion of wavelength

, from Equation (17), we obtain:





3
22
2
y
y
24
22
0
812
,,0.
1
meVeV mc
Hzt Hzt
zAeVmc

 (23)



In the case of low-energy


2
T
,eV mceVKEU , we have:


22
y
Ty
22
0
,8,0, Hzt mEUHzt
zA





(24)
which in this form resembles the non-relativistic Schrö-
dinger’s equation for a single particle moving in an elec-
tric field, [1,2]:

2
2
T
2
8
,,,,,, 0,
m
xyztE Uxyzt
h



  (25)


,,,
x
where yzt 
hA
is the state function, and h is Planck
constant, 0.

This similarity of Equations (24) and (25) is not sur-
prising, because Schrödinger’s equation is based on the
C opyright © 2013 SciRes. JMP

M. PERKOVAC 903
following assumptions [1]:
 For a micro-object there exists a state function, whose
complex amplitude satisfies the same equation as the
complex amplitude of the electromagnetic wave.
 Between the energy of the object T E and the fre-
quency of the wave

exists relation Eh

.
T
 The phase velocity of the wave corresponds to the
Equation (19).
At low energies, all these assumptions are met for the
electron and for the fields E and H. Therefore Equation
(24) actually represents Schrödinger’s equation.
In the same way, using the second derivative with re-
spect to time t, with 2

 , we obtain the following
from the current values


,
y
H
zt of linearly polarized
standing wave:
 
2
y
,,0Hzt
22
y
Hzt t 

 
T
eV E
; or, us-
ing Equation (14) (in addition to ),
 
Ty
,0 .Hzt
mc


2
22
yT
22
0T
,1+ 22
1+
Hzt Emc
E
A tE





(26)
Then in the case of low-energy:
 
2
2
y
2
0
,2 Hzt EU
At






Ty
,0 .Hzt (27)
8. Conclusion
Using Maxwell’s equations we determined that the volt-
age and current on the transmission line as well as an
electromagnetic wave in the atom are described using the
same differential equations. Electromagnetic energy in
the transmission line is linearly proportional to the natu-
ral frequency of oscillation of the LC circuit that belongs
to this line. By analogy, the same is true for the electro-
magnetic energy of the atom. Factor of proportionality,
which we call the action of electromagnetic oscillator,
however, is not constant. Because of Lorentz’ theory of
electrons this factor depends on the energy, or frequency,
of electromagnetic wave. The part of the factor of pro-
portionality, that is not dependent on the frequency,
which we denoted as action constant, completely coin-
cides with Planck’s constant. For low-energy the wave-
length of the wave in the atom corresponds to de Bro-
glie’s equation, and the frequency corresponds to Duane-
Hunt’s law. We found that the electromagnetic wave in
the atom is described by Schrödinger’s equation. All of
this indicates that atoms can be well described only with
the help of Maxwell’s equations and Lorentz’ theory of
electrons. All this provides a deeper entry into the matter
and reveals a different view of the atom.
9. Acknowledgements
The Wolfram Research, Inc. Mathematica software is
used by courtesy of Systemcom Ltd., Zagreb, Croatia.
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