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**SPHEROMAK OVERVIEW:**

This web page introduces basic spheromak concepts.

A spheromak is a naturally occurring electromagnetic structure that enables the existence of stable isolated packets of charge and energy. These packets of charge and energy may be single quantum charged particles, atomic nuclei, atomic ions or spheromak plasmas. A spheromak concentrates electric and magnetic field energy which gives spheromaks rest mass. Thus the spheromak structure enables the existence of atomic particles.

An isolated charged particle spheromak is a stable minimum energy toroidal shaped structure consisting of quantized string charge circulating around a closed path at the speed of light. A plasma spheromak involves electrons and ions circulating around a similar closed path at a small fraction of the speed of light.

A stable spheromak can absorb or emit photons with energy Ep = h Fp where:

Ep = photon energy

h = Planck constant

Fp = photon frequency

The Planck Constant h arises from the geometry of a stable minimum energypheromak.

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

A single quantum charge spheromak is stable and can exist in isolation.

Highly stable charged atomic particles with rest mass, such as protons and electrons, have spheromak like structures.

In the presence of an externally applied magnetic field quantum charged particle spheromaks exhibit magnetic resonance.

A multi-quantum charge spheromak can exist in the presence of a suitable external radial electric field.

Spheromaks can bind together and merge to form atomic particles.

The quantum electron charges around an atomic nucleus form spheromaks. About half of these quantum electron charges move poloidally opposite to the other half so that the net poloidal magnetic field is minimized.

The spheromak structure of atomic electrons explains experimentally observed co-valent chemical bonding.

The spheromak structure has an energy minimum at number of poloidal turns Np = 222 and number of toroidal turns Nt = 305. These two integers and the spheromak mathematical model precisely predict the experimentally measured Planck constant h, which is fundamental to quantum mechanics.

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

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

**SPHEROMAK CHARGE MOTION:**

An isolated charged particle spheromak results from a quantum uniformly distributed line charge moving at the speed of light 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 in the far field. Between the toroidal and poloidal magnetic field regions is the spheromak wall formed by the closed circulating charge path.

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

When a spheromak first forms its ratio of internal radius Rc to external radius Rs may not conform to the spheromaks minimum energy state. However, the spheromak spontaneously emits or absorbs particles and/or photons in order to reach its stable minimum energy state.

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.

The electrons around 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. In order for two atomic particle spheromaks to merge to form a new stable spheromak the toroidal fields of the two merging spheromaks must both be either CW or CCW.

**SPHEROMAK EXISTENCE:**

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 PROPERTIES.

**SPHEROMAK SHAPE:**

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 minimum energy charged particle spheromak the charge moves through exactly 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 results in the Planck constant.

**PLANCK CONSTANT:**

The mathematical model of a spheromak for discrete quantum charged particles leads to the Planck constant. The theoretical calculation of the Planck constant is developed on the web pages titled:SPHEROMAK ENERGY and 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 = quantum 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.

**NO RADIATION IN THE GROUND STATE:**

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.

**UNDERSTANDING SPHEROMAKS:**

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.

**SINGLE QUANTUM CHARGE SPHEROMAKS:**

A single quantum charge spheromak such as an electron or proton results from a quantum of charge forming a uniform charge string. This charge string provides a stable three dimensional closed path along which current moves at the speed of light. This charge motion path forms a dividing wall in the shape of a toroidal surface. This surface is referred to as the "spheromak wall". The charge motion path has both poloidal and toroidal motion components. In free space a spheromak has the shape of a round toroid. At its minimum energy state the ratio of the spheromak outside radius Rs to its inside radius Rc is given by:

Rs / Rc ~ 4.1

**PLASMA SPHEROMAKS:**

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. This sheet, known as the spheromak wall, has a net charge. The current path has both poloidal and toroidal motion components. The charge motion is complex. A charge makes 222 revolutions around the spheromak axis of symmetry and 305 revolutions around the toroidal axis before it retraces its previous path. Plasma spheromaks have been generated and photographed in a laboratory. An ideal plasma spheromak in free space has the shape of a round toroid. The shape of a laboratory plasma spheromak may be 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:

**(Rs / Rc) = 4.2**

This experimenatally observed (Rs / Rc) radius ratio is consistent with the spheromak mathematical model developed on this web site which indicates that for a stable spheromak:

(Rs / Rc) ~ 4.1.

**PLASMA SPHEROMAK LIFETIME:**

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 non-circulating 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.

**APPLICATIONS:**

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 energy is required 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.

**MATHEMATICAL MODEL OF A SPHEROMAK:**

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 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 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.

**SPHEROMAK ANALYSIS:**

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 potential energy with respect to position and deal with spheromaks from a field energy density perspective.

A spheromak exists because, except at the spheromak wall, the energy density within the toroidal region confined by the spheromak wall is less than it would be if the energy density function applicable to the external 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.

**SUMMARY:**

In a spheromak there is a toroidal shaped spheromak wall. Inside the spheromak wall the magnetic field is toroidal and the electric field is cylindrically radial. Outside the spheromak wall the magnetic field is poloidal and the electric field is spherically radial in the far field. 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.

**THIS WEB SITE:**

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 shape of the spheromak structure.

This web page last updated February 17, 2018.

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