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By Charles Rhodes, P.Eng., Ph.D.

This web page introduces basic spheromak concepts.

A spheromak is a naturally occurring stable electromagnetic structure that stores electric and magnetic field energy and hence enables the existence of stable charged particles. Spheromaks form the foundation of quantum mechanics.

A spheromak has two cylindrically symmetric field energy density functions of the form:
Ut(R, Z)
Up(R, Z)

The spheromak has two regions which are separated by the thin wall closed wall. Everywhere on the wall:
Ut(R, Z) = Up (R, Z).

Everywhere inside the wall:
Ut(R, Z) < Up(R, Z)
and everywhere outside the wall:
Ut(R, Z) > Up(R, Z)

Ut(R, Z) is known as the toroidal energy density function and Up(R, Z) is known as the poloidal energy density function.

The following diagram shows the approximate cross sectional shape of a spheromak wall in free space.

The spheromak outside radius on the equatorial plane is Rs and the spheromak inside readius on the equatorial plane is Rc.

An isolated spheromak is a quasi-toroidal shaped electromagnetic structure with a nearly elliptical cross section. The surface of the toroid forms a separating wall between the region enclosed by the wall and the region outside the wall.

A spheromak tends to adopt the lowest stable energy which implies that the spheromak local energy density is U(R, Z) where:
inside the spheromak wall:
U = Ut(R, Z)
and outside the spheromak wall:
U = Up(R, Z)

The spheromak total field energy given by:
Ett = Integral over all space of:
U(R, Z) 2 Pi R dR dZ
= finite constant

An isolated spheromak is a quasi-toroidal shaped electromagnetic structure with a nearly elliptical cross section. The surface of the toroid forms a separating wall between the region inside the wall and the region outside the wall.

Spheromaks exhibit mirror symmetry about the equatorial plane. Hence:
U(R, Z) = U(R, - Z)

Ett = Integral from Z = 0 to Z = infinity of:
Integral from R = 0 to R = infinity of:
U(R, Z) 4 Pi R dR dZ

The spheromak wall forms due to a natural tendancy for free charge to form a filament with a net charge that locally circulates around a closed path at the speed of light. The circulating net charge Q forms a uniform current I which causes a magnetic field configuration that confines the spheromaks net charge against the net repulsive electrostatic force. Detailed analysis shows that spheromaks are stable due to formation of the relative potential energy well that exists inside the spheromak wall.

The spheromak geometry is stable over a wide range of spheromak energy. A stable spheromak contains Np poloidal filament turns and Nt toroidal filament turns. The numbers Np and Nt are positive integers. The spheromak will potentially collapse if Np and Nt contain a common integer factor.
Np = Npo
Nt = Nto
be stable low energy values of Np and Nt. Then for good stability the spheromak should not collapse if:
Np = Npo + 1
Np = Npo - 1
and should not collapse if:
Nt = Nto + 1
Nt = Nto - 1

Nto and Npo are both prime numbers.

Spheromaks have a characteristic relationship that if:
Ett = electromagnetic field energy associated with the spheromak
F = spheromak natural frequency
dEtt / dF = h,
h = Planck Constant

The spheromak natural frequency F is:
F = C / Lh

C = speed of light
Lh = [Charge circulation path length]

Let Lt = length of a toroidal current path turn
Let Lp = length of a poloidal current path turn

Lh^2 ~ (Np Lp)^2 + (Nt Lt)^2

Spheromaks gain or lose energy by absorption or emission of electromagnetic radiation.

Spheromaks are used by nature to store electromagnetic field energy. We commonly refer to this conceentrated stored electromagnetic field energy as rest mass. Spheromaks enable the existence of charged atomic particles, and are instrumental in nuclear and atomic particle interactions. Spheromaks also have important roles in semi-stable plasmas, chemical binding and thermal radiation. The potential energy wells contained in spheromaks make quantum charged particles such as electrons and protons highly stable.

An ideal isolated spheromak has a unique energy state solution. An ideal isolated spheromak in an external magnetic field has two energy state solutions. However, when there are multiple interacting spheromaks there is a spectrum of discrete state solutions.

A problem with existing spheromak mathematical models is failure to correctly predict the experimentally observed inertial mass of the particle. The issue of inertial mass and gravitational field energy within and around a spheromak's structure needs further study. It has been speculated that most of a particle's inertial/gravitational mass is a result of a Higgs field.

Within the spheromak wall is a single layer filament current path which has Np poloidal turns around the main axis of spheromak symmetry and Nt toroidal turns. The current path only intersects itself at the current path closure point. There is distributed net charge along the current path. Along the current path the current is constant and propagates at the speed of light C. Since the inner spheromak wall and the outer spheromak wall have different radii there is some spheromak cross sectional shape distortion. For an isolated spheromak in a vacuum the electric field inside the spheromak wall is zero. However, for an electron spheromak around a central positive charged nucleus the field situation may be more complicated.

For an isolated spheromak the field energy density outside the wall is the sum of the energy densities caused by the surface electric field of the spheromak and by the polodial magnetic field. The field energy density inside the wall is caused by the toroidal magnetic field.

Thus, from basic electromagnetic theory inside the spheromak wall the toroidal magnetic field is given by:
Bt = Muo Nt I / 2 Pi R so inside the spheromak wall the toroidal magnetic field energy density is given by:
Ut = Bt^2 / 2 Muo
= [Muo Nt I / 2 Pi R]^2 / 2 Muo
= Muo Nt^2 I^2 / 8 Pi^2 R^2
Muo = permiability of free space
Pi = 3.14159265
Nt = number of toroidal turns
I = circulating current
R = the radial distance from the main axis of spheromak symmetry, herein referred to as the Z axis.

This region inside the spheromak wall exists within a larger region outside the spheromak wall where the magnetic and electric field energy densities are approximately equal to the field produced by a thin ring located at R = Ro, Z = 0 carrying a current (Np I) and having a net charge Q.

For R >> Ro and Z >> Ro the electric field outside the spheromak wall caused by a spheromak is the same as for an ideal point particle with charge Q surrounded by a radial electric field.

Due to spheromak symmetry, for R < Rc on the equatorial plane the electric field Z components all cancel. Hence on this plane the poloidal magnetic field energy density plus the radial electric field energy density at the inner spheromak wall equals the toroidal magnetic field energy density at the inner spheromak wall. Thus:
[Bp|(R = Rc)]^2 / 2 Muo + (Epsilono / 2) [Er|(R = Rc)]^2
= [Bt|(R = Rc)]^2 / 2 Muo
= (Uo / 8) [Nt I / Pi Rc]^2

Similarly, at the outer wall of the spheromak on the equatorial plane the sum of the poloidal magnetic field energy density and the radial electric field energy density equals the toroidal magnetic field energy density inside the spheromak wall. Thus:
[Bp|(R = Rs)]^2 (1 / 2 Muo) + (Epsilono / 2) [Er|(R = Rs)]^2
= Uo / 8) [Nt I / Pi Rs]^2

This condition is especially important because balancing the electric field energy density outside the spheromak demands a minimum magnetic field energy density inside the spheromak. However, the spheromak seeks an overall minimum energy configuration. Hence the boundary condition at the outside wall in effect sets the spheromak overall contained energy. That boundary condition effectively sets the ratio of poloidal turns to toroidal turns in the spheromak. For spheromak stability those numbers of turns must be integers with special properties.

All around the surface of the spheromak just outside the spheromak wall there is a blend of electric and poloidal magnetic fields that together provide a field energy density equal to the toroidal magnetic field energy density just inside the spheromak wall.

The spheromak wall consists of a long filament of charge that forms a complex closed spiral current path. This current path contains Nt toroidal turns and Np poloidal turns. Electric current flows along this path at the speed of light. The length of this current path and the speed of light give the spheromak a characteristic frequency Fh. The numbers Np and Nt have no common factors so the current path never crosses itself. This issue is crucial in spheromak stability and hence in determination of the Fine Structure Constant and the Planck Constant.

The change in the total field energy content of a spheromak is proportional to its change in frequency.

A single quantum charge spheromak is a stable entity that can exist in isolation in a vacuum with no external fields.

Particles with rest mass are stable packets of charge and energy that are non-propagating solutions to electromagnetic equations. Hence atoms and charged atomic particles embody spheromaks.

The spheromak's poloidal magnetic field energy density decays rapidly with increasing distance from the spheromak center in proportion to (1 / R^6) but the spheromak's electric field energy density extends to infinity decaying in proportion to (1 / R^4) for R >> Ro.

Due to the mathematical structure of spheromaks involving a coincidence between a real number Pi and a rational number (ratio of integer Np to interger Nt), in stable spheromaks exhibit a constant geometry.

Application of an external magnetic field causes splitting of spheromak energy states corresponding to a change in the orientation of the spheromak main axis of symmetry with respect to the applied magnetic field. This phenomena accounts for experimentally observed quantum energy exchanges between matter and electromagnetic radiation, known as particle magnetic resonance.

Semi-stable spheromaks can also form in plasmas.

A semi-stable plasma spheromak forms when there are plasma electrons spiraling around a quasi-toroidal shaped closed path.

The spheromak mathematical model accurately predicts the experimentally observed geometry of plasma spheromaks.

A spheromak has a quasi-toroidal shaped wall with a nearly elliptical cross section. Inside the wall of an isolated spheromak the magnetic field is toroidal and the electric field is zero. Outside the spheromak wall the magnetic field is poloidal and the electric field is normal to the spheromak surface. However, on the equatorial plane the electic field components normal to the equitorial plane cancel. The spheromak wall is located at the locus of points where the total field energy densities on both sides of the spheromak wall are exactly equal. The spheromak forms a potential energy well. The spheromak shape is stable because the second derivative of total spheromak energy with respect to spheromak wall position is positive everywhere on the spheromak wall.

In certain circumstances a stable atomic particle spheromak can absorb quantum amounts of electromagnetic radiation (photons) with energy Ep = h Fp where:
Ep = photon energy
h = Planck constant
Fp = photon frequency

The PLANCK CONSTANT h is a combination of physical constants that arise from the geometry of a quantum charged spheromak.

Multi-quantum charge spheromaks can exist. A free neutron is a spheromak assembly with no net charge. In a reactor a free neutron will eventually spontaneously decay into a proton, an electron and a neutrino. However, in an atomic nucleus, a neutron is often stable.

Spheromaks can bind together and/or merge to form atomic assemblies with larger rest energy.

Nucleons tend to arrange themselves so that their constituant poloidal magnetic fields cancel.

The quantum electron charges around an atomic nucleus form multi-particle spheromaks. About half of these quantum electron charges move poloidally opposite to the other half to minimize the net poloidal magnetic field.

The spheromak structure of atomic electrons explains experimentally observed atomic ionization energies and chemical bonding.

Spheromaks have an electromagnetic energy structure that potentially permits quantum state (Np / Nt) values of the form:
(Np / Nt) = Np / [P - 2 Np]
P = prime number.

The lowest stable energy state is known as the ground state.

These Np, Nt integer numbers have the mathematical property that they do not share any common factors, which is one of the criteria required for the existence of a spheromak and hence the Fine Structure Constant and the Planck constant. Notice that between each adjacent number pair:
dNt = - 2 dNp.
Hence the energy differences between these quantum states are very small.

Spheromak stability is enhanced if a small error in either Nt or P caused by an external disturbance does not lead to spheromak collapse. This objective is automatically realized if both Np and Nt are prime numbers. Another way of achieving this objective is for Np = (Npr)^2 and Nt = (Ntr)^2 where Npr and Ntr are different prime numbers.

These integer pairs and the spheromak mathematical model predict the experimentally measured Planck constant h, and the corresponding Fine Structure constant Alpha which are fundamental to quantum mechanics.

The spheromak model allows precise calculation of the contribution of electric and magnetic field energies to electron and proton rest masses.

Atomic free particle spheromaks exhibit stable geometry. That stable geometry is now the basis for world metric unit standards for mass and energy measurements. The Planck constant provides the link between standards of energy and time.

Plasma spheromaks are used for energy and fuel injection in some nuclear fusion processes. "Ball Lightning" is an occasionally observed form of plasma spheromak.

When a spheromak first forms its ratio of minimum inside radius Rc to maximum outside radius Rs may not conform to the spheromaks most stable minimum energy state. The spheromak will emit or absorb photons in order to reach its most stable energy state at which state it is in radiation balance with its environment. In a low radiation environment that is the spheromak minimum energy or ground state.

The electrons surrounding an atomic nucleus form spheromaks with nearly cancelling poloidal magnetic fields.

The toroidal magnetic field in a spheromak may be either clockwise (CW) or counter clockwise (CCW) with respect to the spheromak's poloidal magnetic field. This issue is referred to as "spin".

An important property of an isolated charged particle spheromak is that in its minimum energy state, also known as its ground state, the spheromak does not emit radiation. This property enables the existence of stable quantum charged particles, stable atomic nuclei and stable atoms.

Basic electromagnetic theory indicates that parallel electric currents flowing in the same direction magnetically attract each other. If these parallel currents have the same net charge per unit length the adjacent current filaments electrically repel each other. In circumstances when the electric and magnetic forces on the charge filaments are in balance a spheromak can exist. This existence requirement is developed on the web page titled CHARGE HOSE PROPERTIES.

A spheromak retains its size and shape due to its own electric and magnetic fields. The spheromak wall position is stable because at every point on the spheromak wall there is stable field energy density balance (and hence force balance) between the internal and external fields. The net charge and the charge motion along the closed charge motion path cause the electric and magnetic forces at the spheromak wall to net to zero. Note that inertial forces and relativistic phenomena also apply to electrons forming plasma spheromaks but do not affect the quantum charge which forms atomic particle spheromaks.

Spheromak geometry is discussed on the web page titled: THEORETICAL SPHEROMAK

For a spheromak to be stable the integer ratio between the number of spheromak current path poloidal turns Np and the number of spheromak current path toroidal turns Nt must be correct. The integers Np and Nt cannot have a common factor other than unity.

The mathematical model of a spheromak for discrete quantum charged particles leads to the Planck constant and the Fine Structure constant. The theoretical calculation of these constants is developed on the web pages titled:SPHEROMAK ENERGY, ELECTROMAGNETIC SPHEROMAK and PLANCK CONSTANT.

The Planck constant, which is fundamental to quantum mechanics, is not an independent physical constant. The Planck constant h is given by:
h = (Muo Q^2 C) / (2 Alpha)
Muo = permiability of free space;
Q = quantum proton charge;
C = speed of light
Alpha = a geometrical constant known as the "Fine Structure Constant" given by:
Alpha^-1 = 137.035999
that arises from spheromak theory.

Spheromaks involve concepts that can be difficult for uninitiated persons to grasp. The mathematical structure of spheromaks is complicated but the underlying physics is very basic. It is helpful for the reader to first grasp the electromagnetic principles set out on the web page titled CHARGE HOSE PROPERTIES before moving on to study the structure and energy content of a spheromak.

At its stable minimum energy state the ratio of the spheromak outside radius Rs to its inside radius Rc is given by:
So^2 = Rs / Rc
where for an isolated plasma spheromak:
So^2 ~ 4.0

The electromagnetic field energy associated with a quantum charged spheromak is proportional to the charge circulation frequency. Thus if there is a change in energy there is a proportional change in charge circulation frequency.

The change in energy:
Ep = Ea - Eb
causes a change in frequency:
Fp = Fa - Fb.

Thus if:
Ea / Fa = h
Eb / Fb = h
Ea - Eb = h (Fa - Fb)
Ep = h Fp
which is fundamental to quantum mechanics.

A plasma spheromak is also known as a toroidal plasma, a compact toroid or an electron spiral toroid.

Plasma spheromaks result from free electrons and ions following a stable three dimensional closed path that forms a sheet in the shape of a quasi-toroidal surface with a nearly elliptical cross section. This sheet, also known as the spheromak wall, has a net charge. The current path has both poloidal and toroidal motion components. The current makes Np revolutions around the main spheromak axis of symmetry and Nt revolutions around the toroidal axis before retracing its path. Plasma spheromaks have been generated and photographed in a laboratory. The shape of a laboratory plasma spheromak may be slightly distorted due to external electric and magnetic fields or due to the proximity of an enclosing vacuum chamber wall. The image below shows a plasma spheromak photograph made by General Fusion Inc.

This photograph shows that for this experimental spheromak the ratio of outside surface radius Rs to inside core radius Rc is about:
So^2 = (Rs / Rc) = 4.0

When a plasma spheromak is formed via ionization of a gas by an electric field the free electrons and ions initially have similar but opposite linear momenta. These electrons and ions move in opposite directions along almost the same closed path. However, the electrons have much more kinetic energy than the ions. A plasma spheromak relies on free electron and ion linear momentum balance to form the spheromak wall with the radial electric field that provides spheromak stability.

Over time collisions between the circulating charged particles and neutral particles cause particle energy, not linear momenta, to be become more equally distributed among the particles. Hence presence of neutral particles in the same space as the plasma spheromak eventually causes a plasma spheromak to become a random plasma. Thus a plasma spheromak is only semi-stable. The plasma spheromak lifetime, which is typically of the order of 100 to 500 microseconds, can be enhanced by minimizing the neutral particle concentration in the space, especially neutral particle species that have high electron impact ionization cross sections.

Understanding atomic particle spheromaks is key to understanding the existence and properties of stable charged particles such as electrons and protons. Understanding spheromaks also enhances understanding of quantum mechanics and chemical bonding.

A plasma spheromak stores electric and magnetic field energy, which energy may be used for initiation of some nuclear fusion processes. The first step in realizing controlled deuterium-tritium nuclear fusion may be formation of high energy deuterium plasma spheromaks.

Today the relative dimensional stability of spheromaks is relied upon to relate the standard of energy and mass to the standard of time.

In most introductory physics courses electricity and magnetism are taught from a point charge and force perspective. However, dealing with spheromaks from a point charge and force perspective is mathematically very difficult. It is mathematically much easier to recognize that a force is a change in total field energy with respect to position and deal with spheromaks from a field energy density perspective.

The focus of the spheromak mathematical model developed on this web site is on practical engineering issues such as relationships between spheromak linear size, spheromak shape, spheromak net charge, frequency, spheromak poloidal and toroidal magnetic field strengths, spheromak electric field strength, spheromak total field energy, plasma spheromak circulating electron kinetic energy, the number of free electrons in a plasma spheromak, the plasma spheromak enclosure size and plasma spheromak lifetime. The result is a practical mathematical model that gives relatively simple closed form solutions to problems that would otherwise likely require extensive computing power.

The utility of the speromak mathematical model is demonstrated by comparison of predictions from the spheromak mathematical model to experimental data. Spheromaks account for most experimentally observed quantum mechanical phenomena.

On this web site spheromak energy density functions are developed in terms of spheromak geometrical size, poloidal and toroidal turns, charge and current parameters. The spheromak energy density functions are shown to yield spheromaks with known static electric and magnetic field energy content. Hence the total spheromak static electric and magnetic field energy is expressed in terms of measureable parameters. It is shown that quantum mechanical properties, such as the Planck constant and Fine Structure constant, arise from these parameters.

This web page last updated December 15, 2021.

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