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

This web page introduces basic spheromak concepts.

Spheromaks are stable electromagnetic structures used by nature to store energy in rest mass. Spheromaks enable the existence of charged atomic particles. Spheromaks lie at the foundation quantum mechanics.

A spheromak is a stable toroidal shaped structure with a slightly elliptical cross section. The surface of this toroid forms the spheromak wall between the region inside the wall and the region outside the wall.

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

The region inside the wall has a cylindrical field energy density proportional to:
(1 / R)^2
which exists within a larger outside region which in a vacuum has a field energy density proportional to:
[1 /(Ro^2 + (A R)^2 + Z^2)]^2
where A is a constant of the order of unity.

At the spheromak wall the energy density of the inside region equals the energy density of the outside region.

A = 1
the elliptical cross section of the spheromak wall is circular.

In a quantum charge electromagnetic spheromak the spheromak wall consists of a filament composed of opposite types of quantized charge that follow a closed spiral path within the wall.

A single quantum charge spheromak 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-linear non-propagating solutions to electromagnetic equations.

The spheromak fields decay rapidly with increasing distance from the spheromak center but the fields extend to infinity.

Due to the mathematical structure of spheromaks involving coincidences between a real number Pi and a rational number (ratio of poloidal to toroidal turns), in the absence of external fields spheromaks exhibit a constant geometry. Application of an external magnetic field causes real but discrete energy state solutions known as magnetic resonance.

Spheromaks account for experimentally observed quantum energy exchanges between matter and electromagnetic radiation.

Confined photons, which account for most of a particle's rest mass, form standing waves within a spheromak.

Semi-stable spheromaks can also form in plasmas.

A plasma spheromak involves electrons and ions spiraling around a toroidal shaped closed path.

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

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 arises from the geometry of a stable quantum charge spheromak.

A multi-quantum charge spheromak can exist. However in isolation a neutral spheromak is unstable.

In the presence of a suitable external electric or magnetic field a neutral spheromak, such as a neutron, can have long term stability.

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

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 so that the net poloidal magnetic field is minimized.

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

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

Each spheromak has Np poloidal turns and Nt toroidal turns.

Spheromaks have an energy minimum that potentially permits quantum state (Np / Nt) values of the form:
(Np / Nt) = Np / [P - 2 Np]
P = prime number. eg For P = 757 some permitted (Np / Nt) values are:
223 / 311, 222 / 313, 221 / 315, 220 / 317, etc.

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

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

This author anticipates that with further investigation the spheromak model will precisely predict the electron and proton rest masses in terms of their magnetic resonance frequencies. Other atomic particle properties will also be revealed.

Our experience of time is defined by spheromak behaviour. The Planck constant provides the link between energy and time.

It is speculated that long range spheromak interactions might be responsible for gravitation.

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

A quantum charged particle spheromak results from quanta of uniformly distributed line charge moving along a stable three dimensional closed path. This path traces the shape of a toroidal surface known as a "spheromak wall". In the region inside the spheromak wall the magnetic field is toroidal and the electric field is cylindrically radial and self cancelling. The energy density of the inside field is proportional to (1 / R^2). In the region outside the spheromak wall the magnetic field is poloidal and the electric field is spherically radial. In a vacuum the total field energy density in this region is proportional to [Ro^2 / (Ro^2 + (A R)^2 + Z^2]^2.

The closed charge motion path has both poloidal and toroidal components. The charge motion path forms the spheromak wall. In free space the resulting toroid cross section is slightly elliptical. In a laboratory plasma spheromak, due to the proximity of vacuum chamber enclosure walls, the shape of a spheromak may be slightly distorted.

When a spheromak first forms its ratio of inside radius Rc to 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 it is in radiation balance with its environment. In a low radiation environment that is the spheromak minimum energy state.

Due to a spheromak's toroidal shape and the uniform charge distribution along the current path the surface charge density at the outer perimeter of the spheromak wall is less than the surface charge density at the inner perimeter of the spheromak wall.

Atomic particle spheromaks exhibit stable geometry. That stable geometry is now the basis for world metric unit standards for mass and energy measurements.

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.

Basic electromagnetic theory indicates that parallel electric currents flowing in the same direction magnetically attract each other. If these parallel currents are composed of charge strings that have the same net charge per unit length the charge strings electrically repel each other. In circumstances when the electric and magnetic forces on the charge strings 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 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, magnetic and inertial forces at the spheromak wall to net to zero. Note that inertial forces and relativistic phenomena apply to electrons forming plasma spheromaks but do not constrain atomic particle spheromaks.

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

For a spheromak to be stable the spheromak geometry must correspond to force balance on the spheromak walls, a spheromak total energy relative minimum and an integer ratio between the number of spheromak current path poloidal turns Np and the number of spheromak current path toroidal turns Nt. 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.

An important property of a 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.

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.

A single quantum charge spheromak such as an electron or proton results from sub-quanta of charge forming a uniform charge string. This charge string provides a stable three dimensional closed path along which net current moves at the speed of light. This charge motion path forms a dividing wall in the shape of a toroidal surface with an elliptical cross section. This wall is referred to as the "spheromak wall". The charge motion path has both poloidal and toroidal motion components.

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 typically:
3 < So^2 < 4.5

A quantum charged spheromak can only take discrete energies, which in the language of conventional quantum mechanics are referred to as Eigenvalues. However, the Eigenvalues can be modified by changing the spheromak's environment by application of an external magnetic field or by surrounding the spheromak with an array of other spheromaks such as in a silicon crystal.

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 toroidal surface with a slightly elliptical cross section. This sheet, 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) = 4.2

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 equally distributed amongst the particles. Hence presence of neutral particles in the same space as the plasma spheromak eventually leads to a spheromak plasma becoming 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 concentrated 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.

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, spheromak confined photons, 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 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.

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.

A spheromak exists because, except right at the spheromak wall, the energy density inside the spheromak wall is less than it would be if the energy density function applicable to the regioon outside the spheromak wall extended through the region inside the spheromak wall. The lower energy density inside the spheromak wall forms a stable potential energy well.

In a spheromak there is a toroidal shaped spheromak wall with a slightly elliptical cross section. Inside the spheromak wall 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 nearly spherically radial. The spheromak wall is located at the locus of points where the 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 quantum charged particles such as electrons and protons the spheromak walls confine energetic photons which contain most of the particle rest mass energy.

On this web site spheromak energy density functions are developed in terms of spheromak geometrical size, winding, 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 May 21, 2019.

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