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

This web page introduces basic spheromak concepts.

A spheromak is a stable toroidal shaped structure consisting of circulating quantized charge that locally concentrates electric and magnetic field energy. A spheromak can exist in isolation independent of external fields.

Stable charged atomic particles with rest mass, such as protons and electrons, are spheromaks.

The Planck constant is fundamental to quantum mechanics. The spheromak mathematical model precisely predicts the experimentally measured value of the Planck constant.

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

Plasma spheromaks are used for energy injection in some nuclear fusion processes.

A spheromak results from uniform 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. In the region outside the spheromak wall the magnetic field is poloidal and the electric field is spherically radial. Between the toroidal and poloidal magnetic field regions is the spheromak wall formed by the closed circulating charge path.

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

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

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 the charge strings electrically repel each other. In circumstances when the electric and magnetic forces on the moving charges are in balance a spheromak can exist. This existence requirement is developed on the web page titled CHARGE HOSE THEORY.

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 apply to plasma spheromaks but do not apply to atomic particle spheromaks.

For a spheromak to be stable the spheromak geometry must correspond to both a spheromak total energy relative minimum and an integer relationship between the lengths of the toroidal and poloidal current path components. In a stable spheromak the charge moves through exaactly 222 poloidal turns while moving through exactly 305 toroidal turns. This mathematical relationship is developed on the web page titled: ELECTROMAGNETIC SPHEROMAK. This mathematical relationship causes the Planck constant.

The Planck constant, which is fundamental to quantum mechanics, is not an independent physical constant. It is shown on this web site that the Planck constant h is given by:
h = (Constant) (Mu Q^2 C) where:
Mu = permiability of free space;
Q = proton charge;
C = speed of light
Constant = a geometrical constant that arises from coincidence of a ratio of integers having no common factors and a function of the irrational number Pi

The mathematical model of a spheromak leads to the Planck constant. The theoretical calculation of the Planck constant is developed on the web pages titled:SPHEROMAK ENERGY and PLANCK CONSTANT.

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

An proton spheromak results from quanta of charge known as quarks forming a uniform charge string. This charge string provides a stable three dimensional closed path along which current moves at the speed of light, even though individual quarks move at only 60% of the speed of light. This charge motion path forms a charge sheet in the shape of a toroidal surface. The charge motion path has both poloidal and toroidal motion components. In free space a spheromak has the approximate shape of a round toroid. The shape of a spheromak may be slightly altered by the presence of external electric and magnetic fields.

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 charge sheet in the shape of a toroidal surface. This charge sheet, known as the spheromak wall, has a net charge. The current path has both poloidal and toroidal motion components. Plasma spheromaks have been generated and photographed in a laboratory. An ideal plasma spheromak in free space has the shape of a nearly round toroid. The shape of a real plasma spheromak may be distorted due to external electric and magnetic fields or due to the proximity of an enclosing cylindrical 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:
(Rs / Rc) = 4.2
This experimenatally observed (Rs / Rc) radius ratio is consistent with the spheromak mathematical model developed on this web site.

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 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 a plasma sheet with the radial electric field that provides spheromak stability.

Over time interactions between the participating charged particles and non-participating 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. 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 vacuum chamber, especially neutral particle species that have a 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.

A plasma spheromak stores concentrated electric and magnetic field energy, which is required for initiation of some nuclear fusion processes. The first step in realizing controlled deuterium-tritium nuclear fusion is often 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, spheromak magnetic field strength, spheromak electric field strength, spheromak total field energy, plasma spheromak free 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 closed form solutions to problems that would otherwise likely require extensive computing power.

The utility of this mathematical model is demonstrated by comparison of predictions from the spheromak mathematical model to experimental data.

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

A spheromak exists because, except at its toroidal walls, the energy density within the toroidal region confined by the spheromak wall is less than it would be if the external energy density function applicable to the poloidal region extended through the toroidal region. The lower energy density in the toroidal region with respect to the adjacent surrounding poloidal region forms a potential energy well that provides stability to the spheromak.

In a spheromak there is a toroidal shaped wall, a toroidal field energy density function inside the spheromak wall and a poloidal field energy density function outside the spheromak wall. The spheromak wall is located at the locus of points where the field energy densities on both sides of the 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.

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 toroidal spheromaks with known electric and magnetic field energy content. Hence the total spheromak electric and magnetic field energy is expressed in terms of measureable parameters. It is shown that quantum mechanical properties, such as the Planck constant, arise from the spheromak structure.

This web page last updated September 12, 2016.

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