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Journal of High Energy Physics, Gravitation and Cosmology, 2020, 6, 340-352

https://www.scirp.org/journal/jhepgc

ISSN Online: 2380-4335

ISSN Print: 2380-4327

DOI:

10.4236/jhepgc.2020.63027 Jul. 2, 2020 340 Journal of High Energy Physics, Gravitation

and Cosmology

Electron Shape and Structure:

A New Vortex Theory

Nader Butto

Petah Tikva, Israel

Abstract

Along with all other quantum objects, an electron is partly a wave and partly

a particle. The corpuscular properties of a particle are demonstrated when it

is shown to have a localized position in space along its trajectory at any given

moment. When an electron looks more like a particle i

t has no shape, “point

particle”, according to the Standard Model, meaning that it interacts as if it is

entirely located at a single point in space and does not spread out to fill a

three-dimensional volume. Therefore, in the sense of particle-like interac-

tions, an electron has no shape.

In this paper, a new theory is proposed in

which the electron has a structure and a shape. The central idea is that an

electron is a frictionless vortex with conserved momentum made out of con-

densed vacuum generated in the Big Bang from massless virtual p

hotons that

acquire mass when moving in the vortex at the speed of light. Using Hydro-

dynamics laws and applying them on the superfluid vacuum the basic proper-

ties of the electron are described here forth. This study provides mathematical

models to calculat

e the mass, kinetic energy, density, volume, time, charge,

and particle-wave duality. Such mathematical formulations are presented to

confirm the theory. We conclude that the shape of the electron is accessible to

human imagination, knowing its shape helps

to determine its properties and

shed a light on how matter is made and to explain the interactions of sub-

atomic

particles.

Keywords

Electron Structure, Vortex, Hydrodynamics Laws, Vacuum Density

1. Introduction

The electron is a fundamental particle of nature, is the essential constituent of

electric currents, and together with protons and neutrons, is the most important

How to cite this paper:

Butto, N. (2020

)

Electron Shape and Structure: A New Vo

r-

tex Theory

.

Journal of High Energy Phy

s-

ics

,

Gravitation and Cosmology

,

6

, 340-352.

https://doi.org/10.4236/jhepgc.2020.63027

Received:

May 19, 2020

Accepted:

June 29, 2020

Published:

July 2, 2020

Copyright © 20

20 by author(s) and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

N. Butto

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10.4236/jhepgc.2020.63027 341 Journal of High Energy Physics, G

ravitation and Cosmology

element of an atom’s structure. It is negatively charged and may be free or

bound to an atom in electronic orbits. The electrons are subatomic particles,

play an essential role in numerous physical phenomena, such as electricity,

magnetism, chemistry and thermal conductivity, and they also participate in

gravitational, electromagnetic and weak interactions.

According to current experiments and theories, the electron is a structure-less,

point-like object, made out of nothing [1], and its entire mass is concentrated in

its extension-less center.

An electron looks like a particle when it interacts with other objects in certain

ways (such as in high-speed collisions) and is not point-like, as stated by quan-

tum mechanics.

In quantum mechanics, depending on the observation point of an electron, it

can appear to be a particle or it can appear to be a wave. As a

wave

, one can im-

agine “clouds” of electron orbitals around an atom, which are not physical

things

but rather representations of probabilities.

Conversely, electrons display properties that normally result from an extended

structure, namely, angular momentum (spin), a magnetic moment, and some

sort of internal oscillation.

The point-like depiction of elementary particles is so unsatisfactory that it has

spawned new theories of matter known as M theories and quantum gravity.

In 1928, when Paul Dirac presented the wave function of the electron in the

Dirac equation [2], it became obvious that there must be not only an internal os-

cillation but also some type of internal motion at the speed of light. Therefore,

the seemingly empty space that surrounds electrons is made up of “virtual par-

ticles,” and electrons are inseparable from the clouds of virtual particles sur-

rounding them.

Moreover, there is no theory that quantifies particles in a meaningful way us-

ing appropriate calculations. This implies that quantum mechanics actually have

no need for a particle as a concept because all the calculations are similar wheth-

er or not hard particles exist.

Subsequently, physicists attributed this intrinsic contradiction between the

electron’s different properties to the common-sense view that the electron is

subject to quantum mechanics and, therefore, is not accessible to human imagi-

nation.

However, because subatomic particles can’t yet be directly observed, scientists

learn about the objects through indirect evidence. By observing what happens in

the vacuum around negatively charged electrons—thought to be swarming with

clouds of as-yet-unseen particles—researchers can create models of particle be-

havior.

The Standard Model predicts that particles surrounding electrons do affect an

electron’s shape, and according to this framework, the electron should be close

to perfectly spherical. But at such an infinitesimal scale as to be pretty much un-

detectable using existing technology. One framework to explain physics beyond

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the Standard Model is known as supersymmetry. However, this theory predicts

that the electron has a more distorted shape than that suggested by the Standard

Model. According to this idea, the electron is predicted to be slightly aspheric

[3].

In particle physics, the fundamental blocks of matter are continuous fluid-like

substances known as “quantum fields” that permeate the whole space around us.

With this article, the vacuum is proposed to behave as a superfluid and a vor-

tex shape of the electron is proposed as a condensation of the vacuum.

Hydrodynamics laws are applied to the superfluid vacuum to describe the ba-

sic properties of the electron. Hydrodynamic formulas are provided to calculate

its mass, density, volume, time, constant angular momentum (spin), and electric

charge.

2. Nature of the Vacuum

Both dark matter and new subatomic particles that were not predicted by the

Standard Model are yet to be directly spotted; still, a growing amount of compel-

ling evidence suggests that these phenomena do exist.

Vacuum density is generally viewed as a fundamental property of the cosmos

whose magnitude should not depend on whether we choose subatomic, astro-

nomical, or cosmological methods to assess its value.

According to quantum field theory, even in the absence of real particles, the

vacuum is always filled by pairs of created and annihilated virtual particles.

Therefore, the physical vacuum is assumed to be a non-trivial medium to which

one can associate a certain energy and density. Therefore, quantum theory re-

quires additional attributes to the vacuum. For instance, vacuum is not empty as

previously considered but rather filled with quantum mechanical zero point

energy.

The simulation of gravity (as far as Newton’s universal law of gravitation is

concerned) run through a computational fluid dynamic (CFD) approach is proven

successful [4] and it seems that such an approach (space quanta absorption af-

fected by massive particles, described as vortices of the same quanta [4]) is also

able to describe any other effect related to general relativity’s space topology.

In superfluid vacuum theory, the physical vacuum is described as a quantum

superfluid and is characterized to behave like a frictionless fluid with density and

extremely high thermal conductivity. The vacuum extends everywhere, has no

size, shape, center, direction, time, or extent, and is immovable.

Superfluid vacuum theory proposes a mass generation mechanism that may

replace or supplement the electroweak Higgs theory. It has been shown that the

masses of elementary particles could be a result of interactions with a superfluid

vacuum, similar to the gap generation mechanism in superconductors [5] [6].

Therefore, vacuum energy has real physically observable consequences, and its

properties can be observed as real physical effects [7] [8].

The quantization of gravity indicates that there is an

elementary quantum of

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ravitation and Cosmology

matter

, indivisible, whose virtual imaginary mass is

()

73

0 min 3.9 10 kg

i

m−

=±×

[9].

These elementary quanta of matter should fill all the space in the Universe,

forming a continuous and stationary quantum fluid. The density of the universal

quantum fluid is clearly not uniform

throughout the Universe because it can be

strongly compressed in several regions (e.g., galaxies, stars, black holes, and pla-

nets). In the normal state (free space), the above-mentioned fluid is invisible.

In the

super-compressed

state, it can become visible in the form of known

matter, since matter, as we have seen, is

quantized

and is consequently formed

by an integer number

of imaginary elementary quanta of matter with imaginary

mass

( )

0 mini

m

. This means that there are no particles in the Universe with masses

smaller than the minimum mass and that all bodies are formed by a whole

number of these particles [10].

3. Internal Structure of the Electron

The Standard Model describes most of the interactions between all of matter’s

building blocks, as well as the forces that act on those particles. For decades, this

theory has successfully predicted how matter behaves, however it does not pre-

dict the structure of the electron.

The angular momentum (spin) of the electron indicates that there is an inter-

nal rotation that confers upon it its rest mass. According to Higgs theory, the

interaction between particles and the Higgs field is continuously maintained and

renewed, converting the amorphous potential energy of the field into individual

structures. The seemingly empty space that surrounds the electron is teeming

with pairs of particles and antiparticles that fleet in and out of existence; these

are called “virtual particles”.

Although precisely measuring this cloud is beyond the capabilities of modern

methods, the current model predicts that electrons are slightly aspheric, with a

distortion characterized by the electric dipole moment. However, no experiment

so far has detected this deviation [11]. Thus, the electric dipole moment remains

an elusive (and unproven) phenomenon, in which an electron’s spherical shape

appears deformed—“dented on one end and bulged on the other” [12].

We propose that the electron is an irrotational circular vortex of frictionless

superfluid space with concentric streamlines that was created from the primor-

dial vacuum during the Big Bang. The rate of rotation of the fluid is greatest at

the center and decreases progressively with distance from the center until there

is no gradient pressure on the boundaries of the vortex where the flow is laminar

and the friction is null. In such a case, the absence of friction would make it im-

possible to create or destroy the vortex motion. If the negative suction point vo-

lume in the center of the vortex does not have enough energy to drag the virtual

photons to the speed of light, then a stable situation cannot occur [13].

If we take a deeper look at the spiral arms of the electron vortex, we notice

that the currents are made of smaller vortices that correspond to the Higgs par-

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ravitation and Cosmology

ticles. Higgs particles have no mass when they are created but acquire mass

when they travel through space. The interaction between the whirlpool-shaped

particles and the Higgs field (the vacuum) is continuously maintained and re-

newed, converting the amorphous potential energy of the field into Higgs par-

ticle vortices.

The superfluid accommodates the rotation by forming a lattice of quantized

vortices in which the vortex core, typically singular, breaks the topological con-

straint against rotational motion.

This new viewpoint of particles allows us to see them as a composite web

structure and at the same time as energy motion processes (Figure 1).

A spinning system along an axis with angular momentum has a torque when

the force is directed toward the center of gravity; this is known as the Coriolis

effect.

The flow to the center of the vortex due to the Coriolis effect results in vortex

tubes, which are always composed of the same virtual particles that rotate at the

speed of light; they remain

unbroken

, so they are ring-like (Figure 2).

If the velocity of the space circulation reaches the limiting speed,

c

, which is

the speed of light in the absolute vacuum, and the velocity field gradient around

the center of the vortex becomes the postulated limiting angular rotation,

ω

, the

Figure 1. Artistic presentation of an elementary particle that has a vortex structure made

by mini vortices of Higgs bosons.

Figure 2. Two vortex tube constituents of an electron vortex.

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ravitation and Cosmology

space breaks down, creating a spherical void, which is defined as a field-less,

energy-less, and space-less volume of vacuum at the vortex center.

These maximums occur at the point where the centrifugal force and the radial

force are equalized. The inflowing medium and the free surface dip sharply and

the inflowing medium turns at 90˚ near the axis line, with the depth and velocity

inversely proportional to

r

2, to form a concave paraboloid. This is where the

highest vortex energy exists [14].

Maxwell worked out a theory of electromagnetism assuming that every mag-

netic tube of force was a vortex with an axis of rotation coinciding with the di-

rection of the force. Several properties have been mathematically proved for a

perfect frictionless fluid [15].

The magnitude of the vorticity in a vortex line increases proportionally as the

vortex line is stretched. Consider a very thin vortex tube around the vortex line,

so thin that the vorticity is practically constant over its width. As the vortex tube

stretches, the cross-sectional area decreases by the same factor; thus, the vorticity

must increase proportionally for the flux across the cross section to remain con-

stant.

Therefore, the electron is a vortex characterized by its power and volume, with

the specific magnitude depending on the rest energy of the electron.

4. Mass of the Electron Vortex

The angular momentum (spin) indicates that there is an internal rotation that

determines the rest mass. The mass of an electron is the amount of fluid-like

virtual photons with a certain density that passes in 1 s. Therefore, the mass of

the electron is calculated to be density times volume.

In hydrodynamics, the force

F

that moves the vortex is directly related to the

pressure that creates the vortex, known as the dynamic pressure

Pd

, and the area

A

according to the formula

d

F PA=

.

(1)

The dynamic pressure (

Pd

) representing the fluid kinetic energy is expressed

as

2

1

2

d

Pv

ρ

=

, (2)

where

ρ

is the density of the fluid and

v = c

is the

velocity.

Therefore,

2

1

2

F cA

ρ

=

. (3)

The area of the vortex is approximately a circle,

2

2Ar= π

, that interacts with

the adjacent vacuum from both sides; therefore,

22

F cr

ρ

= π

. (4)

If we multiply and divide the right side of the above equation by time

t

, we

obtain

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ravitation and Cosmology

2

F ct r c t

ρ

π=

.

(5)

However,

v/t

is equivalent to the acceleration

a

,

and

vt

is equivalent to the

length

L

.

According to Newtonian theory, the force

F

divided by the acceleration equals

the mass. Dividing both sides by the acceleration, we obtain

2

Fa m rL

ρ

π= =

. (6)

However, the area times the length is equal to the volume

Q.

Therefore,

2

Fa m rL

ρ

π= =

.

(7)

which corresponds to the inertial mass of the electron.

5. Minimum Time of the Electron Vortex

Because the electron has no shape in quantum physics, the time of the electron

has never been reported before. The electron as a vortex has a minimum time

less than which the electron converts into a virtual particle.

If

2

m rL

ρ

= π

and the length is the product of velocity times time,

L ct

=

,

then

2

m r tc

ρ

=π

.

(8)

The mass of the electron depends on the spin and the time. If the relative rota-

tion velocity of the vortex or time is zero, the mass will be zero; accordingly, the

mass will disappear and convert into an amorphous vacuum. The minimum

time of the electron, that needed to complete one rotation cycle 2π

re

, is

21

2 8.08586 10 s

e

t rc

−

= ×π=

, (9)

where

13

3.86 10 m

e

r

−

= ×

and

8

2.99792458 10 m sc= ×

.

6. Density and Volume of the Electron

Normally, electron density is a measure of the probability of an electron being

present at a specific location. However, what we mean by the density of an elec-

tron is the density of virtual photons of which the electron is made. Hydrody-

namically, vortex formation depends on the density of the medium in which it

occurs. In our case, the vacuum’s “energy density” is related to the number of

virtual particles that are continually created and annihilated in the vacuum;

however, the density of the electron is in a supercompressed state and is higher

than that of the surrounding vacuum field.

Knowing the time of the electron, the density of the electron can be calculated:

2 63

2.519470 10 kg mm r tc

ρ

π= = ×

,

(10)

where

2

r tcπ

is the volume. Knowing the density of the electron, the volume

Ve

can be calculated:

37 3

3.6118 10 kg m

e

Vm

ρ

−

= = ×

. (11)

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ravitation and Cosmology

7. Particle-Wave Relationship

Particle-wave duality is a central tenet of quantum physics, and an electron has

wave-like properties. The essence and physical relationship between the particle

and the wave remains an unresolved problem in physics. The vortex model of

the electron gives three different solutions:

1) Louis de Broglie developed a hypothesis [16] to relate the dual wave and

particle behavior that can be applied to electrons. The de Broglie wavelength

formula relates the wavelength

λ

to the momentum

mv

of a wave/particle: [17]

h p h mc

λ

= =

and

h mc

λ

=

.

In hydrodynamics, the velocity of the fluid element instantaneously passing

through a given point in space in the vortex with radius r is constant in time;

therefore, the circulation or the vorticity in the core of the vortex

2rc

π= Γ

is

constant. Γm is a conserved momentum; therefore,

2rcm

π

is constant, which

corresponds to the Planck constant.

If

2h rcm= π

, then

2r

λ

= π

, which is the circumference of the core of the

vortex.

If the radius of the core of the vortex electron,

13

2 3.86 10 mr h mc

−

= = ×π

where

m

is the mass of the electron, the value of the circumference is

12

2 2.42408 10 mr

−

= ×π

, which is similar to the CODATA 2014 [18] value for

the Compton wavelength of the electron, 2.4263102367 (11) × 10−12 m.

Vortices also have frequencies described in units of time (rotations per second). If

the time required to complete a cycle around the vortex is

21

2 8.08586 10 s

e

t rc

−

= ×π=

, then the frequency of the electron related to the

cycles of rotation of the vortex is

20

1 1.2367 10 Hz

e

ft= = ×

. In fact, the rota-

tional velocity of the vortex

2f cr

ω

= π =

; therefore,

2fc r= π

,

which gives

us the same frequency as Planck’s theory. If the invariant mass of an electron is

approximately 9.109 × 10−31 kilograms, then its energy would be

2 14

8.1981 10 JE mc

−

= = ×

And if

E hf=

where

h

is Planck constant = 6.62607004 × 10−34 m2∙kg/s

then

20

1.2372492 10 Hzf Eh

= = ×

.

2) The third solution is related to the energy of the particle. A particle of mass

m

has a rest energy of

2

E mc=

. The non-reduced Compton wavelength for this

particle is the wavelength of a photon of the same energy. According to Planck’s

theory, for photons of frequency

f

, the energy is given by

e

E hf=

.

(12)

The frequency of the electron is

e

f Eh=

,

(13)

where

7

0.511 MeV 8.1866 10 ergs

e

f Eh h h

−

= = = ×

. Therefore,

20

1.2355 10 cycles s

e

f= ×

.

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ravitation and Cosmology

where

h

is 6.626176 × 10−27 erg-seconds.

How is

E hf=

related to the vortex model?

The force that rotates the vortex is

22

F cr

ρ

= π

. (14)

If we multiply and divide the right side of Equation (3

.

5

.

3)

by time,

t

, we ob-

tain

2

F ct r c t

ρ

π=

,

(15)

where

ct

is equivalent to the distance

L

,

L

π

r

2 is equivalent to the volume

Q

,

ρQ

is

equivalent to the mass, and 1/

t

is equivalent to the frequency

f.

Therefore,

F mcf=

.

(16)

If

E

= force × distance and the distance that an electron moves in one cycle is

2rπ

, then

2E rmcf= π

, where

2rmcπ

is equivalent to

h.

Therefore,

E hf=

and

e

f Eh=

.

8. Electron Charge

Charge is a fundamental physical property of matter that is responsible for its

interactions with electromagnetic fields. An electron is a particle that possesses

this property, and experiments show that it possesses a negative charge.

The real nature and the essence of charge are unknown. In this section, a new

theory to describe the nature of electric charge is formulated based on the vortex

model of the electron.

In hydrodynamics, the rotation of a vortex creates a drag force that attracts

the vacuum to the center of the vortex. This force is directly related to the densi-

ty of the vacuum, the speed of rotation, and the area is inversely related to the

distance from the vortex center according to the equation

2

1

2

F cAr

ρ

=

.

(17)

Multiplying and dividing the right side of the above equation by time, we obtain

11

22

F ctAc rt Vc rt

ρρ

= =

. (18)

However, this force is reduced owing to interactions with the adjacent va-

cuum. If the density of the vacuum is

ρv

and the rotation speed of the vortex is

c

,

dividing the momentum by the length of the circumference of the vortex will

give the diminished momentum

P

for a unit of length:

v

Pc

ρλ

=

.

(19)

Therefore, the momentum of the vortex is reduced for every unit of length

according to the equation

2

1

2

v

F V c rt

ρρ λ

=

.

(20)

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If

2r

λ

= π

, then

2 22

12 42

2

vv

F Vcrtr Vc tr

ρρ ρρ

π= π=

. (21)

In hydrodynamics,

ρvc

2

is the elasticity of the vacuum. In fact, the velocity of a

particle in an elastic medium can be expressed by the formula

( )

12

c Ed=

, (22)

where

c

, the speed of light,

E

is the elasticity, and

d

=

ρv

, the density of the me-

dium.

Therefore, the elasticity,

E

, can be written as

2

v

Ec

ρ

=

, (23)

which is the inverse of the stiffness of the vacuum and has the same value as the

electric permittivity,

i.e.

,

21

0v

v

ρε

−

=

.

This makes sense in terms of dimensions because the elasticity modulus is

Newton × m−2 whereas the permittivity is Newton−1 × m−2 (C2/Nm2).

Then, the Equation (3

.

6

.

5)

becomes

2

0

4

v

F V tr

ρε

π

=

. (24)

The force density at a point in a fluid divided by the density is the acceleration

of the fluid at that point:

2

0

4F f Vt r

ρε

= π

=

. (25)

In fluid mechanics, the force density [19] is the negative gradient of the pres-

sure. It has physical dimensions of force per unit volume. The force density is a

vector field representing the flux density of the hydrostatic force within the bulk

of a fluid.

Furthermore, in fluid dynamics, the volume of a fluid that passes per unit of

time,

V/t

, is the volume flow rate, which is usually represented by the symbol

Q

.

Its SI unit is m3/s.

The force between two vortices is directly proportional to the magnitude of

the flow rate of the elementary density in each vortex

q

1

q

2 and inversely propor-

tional to the distance of the separation between their centers, r, diminished by

the stiffness of the vacuum represented by the electric permittivity per unit of

length:

2

12 0

4f qq r

ε

π=

. (26)

The origin and essence of the electric permittivity will be discussed in detail

separately in another paper.

9. Discussion

The nature and the essence of the electron was central to the development of

quantum theory early in the twentieth century, and remains at the frontier of

physics today. A century after Danish physicist Niels Bohr conceived of the elec-

tron as the proton’s satellite [20], our perception of the electron continues to

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ravitation and Cosmology

evolve and expand.

In particle physics, the fundamental blocks of matter are continuous fluid-like

substances known as “quantum fields” that permeate the whole space around us.

In this article, the electron is proposed to have frictionless vortex shape and

the hydrodynamic laws are applied.

The central idea is that an electron is a frictionless vortex with conserved

momentum made out of condensed vacuum generated in the Big Bang from

massless virtual photons that acquire mass when moving in the vortex at the

speed of light, as described by Higgs theory. Considering the vacuum density

and applying classical hydrodynamics, analytical formulations are applied to

calculate the mass, volume, density, time, and frequency of the vortex. We ob-

tained the properties of an electron and unified particle-wave duality using the

same vortex model.

We summarize the findings of this article in the following points:

1) The electron has a vortex shape with density, mass, radius, area, circumfe-

rence, volume, rotational velocity, minimal time, frequency and flow rate.

2) The force that rotates the vortex is directly proportional to the density of

the vortex, times the square of its rotation speed and to its area according to the

Equation (4)

22

F cr

ρ

= π

.

3) The electron vortex mass is directly proportional to its density times its

area and its length according to the Equation (6)

2

m rL

ρ

= π

.

4) The density of the electron is directly proportional to the vortex mass and

inversely proportional to its area, and to the time needed to complete one cycle

and rotation speed of the vortex according to the Equation (10)

2

m r tc

ρ

= π

.

5) The minimum time of the electron is the time needed to complete on rota-

tion cycle of the vortex which directly proportional to the vortex circumference

and inversely to the rotation speed according to the Equation (9)

2

e

t rc

= π

.

6) Therefore, the frequency of the electron is directly proportional to its rota-

tion speed and inversely related to its circumference according to the equation in

section (6.1)

2

e

t rc= π

.

7) Furthermore, the frequency of the electron vortex is directly related to its

energy and inversely related to its mass, circumference and rotation speed ac-

cording to the equation in section (6.2)

2f E rmc= π

.

8) The electric charge q is equal to flow rate of the elementary density V/t

from the periphery to the center of the electron vortex divided by the area in

both sides of the vortex diminished by the stiffness of the vacuum according to

the Equation (25)

22

00

44f Vt r q r

εε

ππ= =

.

9) The electric force between two charges is the force between two vortices

which is directly proportional to the magnitude of the flow rate of the elementa-

ry density in each vortex and inversely proportional to the distance of the sepa-

ration between their centers, diminished by the stiffness of the vacuum

represented by the electric permittivity per unit of length according to the Equa-

tion (26)

2

12 0

4f qq r

ε

π=

.

N. Butto

DOI:

10.4236/jhepgc.2020.63027 351 Journal of High Energy Physics, G

ravitation and Cosmology

We conclude that the shape of the electron is accessible to human imagina-

tion.

The electron properties can be accurately described using classical laws of hy-

drodynamics. Knowing its shape helps to determine its properties and shed a

light on how matter is made and to explain the interactions of sub-atomic par-

ticles.

This theory could overturn several alternative physics theories that attempt to

fill in the blanks about phenomena that the Standard Model can’t explain.

The electron wave-particle duality, electron charge and the origin of electron

spin will be discussed in detail in separate papers.

Future experimental studies are needed to confirm the vortex structure of the

electron.

Acknowledgements

The author would like to thank Enago (https://www.enago.com/) for the English

language review.

This research did not receive any specific grant from funding agencies in the

public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this pa-

per.

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