# XYLENE POWER LTD.

## SPHEROMAKS - INTRODUCTION

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

SPHEROMAK OVERVIEW:
This web page introduces basic spheromak concepts.

Spheromaks are natural electromagnetic structures that enable the existence of stable charged particles with rest mass. The rest mass is a local concentration of electromagnetic field energy. A spheromak is a stationary solution to Maxwells equations of electromagnetism.

Spheromaks are fundamental in nuclear and atomic particle physics. Spheromaks also have important roles in semi-stable plasmas, chemical binding, thermal radiation and gas spectroscopy.

Spheromaks cause the underlying mechanism of energy quantization in quantum mechanics.

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

Semi-stable spheromaks can be formed and photographed in plasmas.

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

It is likely 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.

SPHEROMAK GEOMETRY:
A spheromak is a stable distorted toroid shaped electromagnetic structure that naturally occurs as a result of a filament of current I continuously circulating around a closed path of length Lh at the speed of light C. The filament has a net charge Q that is uniformly distributed along the filament length.

The current path traces the surface of a distorted toroid. This surface, herein referred to as the spheromak wall, contains a blend of Np poloidal turns around the spheromak major axis of symmetry and Nt toroidal turns around the spheromak minor axis of symmetry. Experimental data indicates that Npo = 363 and Nto = 362.

The circulating current forms a purely toroidal magnetic field inside the spheromak wall and a purely poloidal magnetic field outside the spheromak wall.

The net electric charge causes a radial electric field outside the spheromak wall. Inside the spheromak wall the electric fields cancel so that the net electric field is zero.

At the center of the spheromak due to symmetry the electric fields also cancel to zero.

For a spheromak to be stable the integers Np and Nt must be correct. The integers Np = Npo and Nt = Nto cannot be equal and cannot have a common factor other than unity. As a result of Npo and Nto having no common factors the filament current path (winding) has only one layer with no cross overs. The filament only intersects itself at the current path closure point.

SPHEROMAK FIELDS:
Inside the spheromak wall there is a toroidal magnetic field but the net electric field is zero and the poloidal magnetic field is zero. Outside the spheromk wall there is both a poloidal magnetic field and a radial electric field but the toroidal magnetic field is zero. Hence a stable isolated spheromak has external radial electric, external poloidal magnetic and internal toroidal magnetic field energy components.

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 sometimes referred to as "spin".

The net charge and the charge motion along the closed current path cause the forces on the spheromak wall to net to zero. Note that inertial forces and relativistic phenomena apply to plasma spheromaks but do not affect quantum charge atomic particle spheromaks.

An isolated spheromak retains its size and shape due to its own electric and magnetic fields. 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 wall position is stable because the second derivative of total spheromak energy with respect to spheromak wall position is positive everywhere on the spheromak wall.

A spheromak forms a potential energy well because the field energy density inside the spheromak wall is less than the field energy density would be if the spheromak wall did not exist.

Spheromak geometry and the energy well are further discussed on the web page titled: THEORETICAL SPHEROMAK

SPHEROMAK APPLICATIONS:
Spheromaks occur as a result of electric charge circulation in atomic particles, as a result of electron circulation in atoms, molecules and crystals and as a result of electron and ion circulation in plasmas. Spheromaks are involved in well known particle physical phenomena that occur on application of an external magnetic field.

An isolated spheromak in its ground state has a unique energy state solution. An isolated spheromak subject to an external magnetic field has two discrete energy state solutions. Spheromaks can transfer between these two discrete energy states by absorption or emission of an electromagnetic photon of appropriate frequency.

This phenomena is variously known as nuclear magnetic resonance (NMR) and electron spin resonance. NMR of hydrogen atoms in body fluids is widely used to provide modern medical imaging.

In situations where there are multiple interacting spheromaks there is a spectrum of discrete energy state solutions.

Hot gas atoms also absorb or emit electromagnetic photons at various characteristic frequencies. That is the basis of gas spectroscopy. As a result of the underlying spheromak structure, there is a highly linear relationship between the photon energy Ep and photon frequency Fp of the form:
Ep = h Fp
where:
h is known as the Planck constant.

In an appropriate experimental apparatus knownas a Kibble Balance the Planck constant h is constant and is reproducible to many significant figures, so today this experimentally measured constant is used to connect international standard units of energy (mass) and time (frequency).

SPHEROMAK WALL:
The focus of the spheromak mathematical model developed on this web page 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.

The spheromak wall separates the region inside the spheromak wall and the region outside the spheromak wall.

The field energy density outside the spheromak wall is given by:
Up = Uo {Ro^2 / [(K Ro - R)^2 + Z^2]}^2

The field energy density inside the spheromak wall is given by:
Ut = Uto [K Ro / R}^2

At the spheromak wall within which current I circulates is at:
Up = Ut

Hence, at the spheromak wall:
Uo {Ro^2 / [(K Ro - R)^2 + Z^2]}^2 = Uto [K Ro / R}^2
or
[Uo / Uto] = [K Ro / R}^2 {[(K Ro - R)^2 + Z^2] / Ro^2}^2

SPHEROMAK GEOMETRY:
A spheromak has cylindrical symmetry. The major axis of symmetry is the Z axis where radius R = 0. The spheromak wall position is the locus of points defined by the equation:
[Uo / Uto]^0.5 = [K Ro / Rw][(K Ro - Rw)^2 + Zw^2] / Ro^2
= [K / (Rw Ro)][(K Ro - Rw)^2 + Zw^2]
where:
[Uo / Uto]^0.5 = constant
K = constant slightly greater than (1 / 2)
R = 0 is the major axis of symmetry of the spheromak;
Z = 0 is th equatorial plane;
R = 0, Z = 0 is the origin of the cylindrical co-ordinate system at the center of the spheromak;
R = Rc, Z = 0 is the smallest value of R on the spheromak wall where that wall intersects the equatorial plane;
R = Rs is the largest value of R on the spheromak wall where that wall intersects the equatorial plane;
R = Ro, Z = 0 defines the nominal minor axis of the spheromak which is also the equivalent poloidal current path.

Z = Ho, R = Ro is a circle on the spheromak wall above the equatorial plane;
Z = Hm, R = Rm is a circle on the spheromak wall above the equatorial plane where the distance between the spheromak wall and the equatorial plane is maximum;
Z = - Ho, R = Ro is a circle on the spheromak wall below the equatorial plane;
Z = - Hm, R = Rm is a circle on the spheromak wall below the equatorial plane where the distance between the spheromak wall and the equatorial plane is maximum;

SPHEROMAK FIELDS:
Inside the spheromak wall there is a toroidal magnetic field but the net electric field is zero and the poloidal magnetic field is zero. Outside the spheromak wall there is both a poloidal magnetic field and a radial electric field but the toroidal magnetic field is zero. Hence a stable isolated spheromak has external radial electric, external poloidal magnetic and internal toroidal magnetic field energy components.

On this web site spheromak energy density functions are developed in terms of spheromak geometrical size, poloidal and figure eight 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.

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, 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.

SPHEROMAK STABILITY:
Spheromak stability arises from prime number constraints that affect Npo and Nto as well as formation of an energy well inside the spheromak wall.

The optimum spheromak geometry is stable over a wide range of spheromak energy. The spheromak wall can be geometrically characterized by the parameters Ro and Ho. For a stable spheromak the ratio: (Ho / Ro) is constant independent of the spheromak's energy.

SPHEROMAK WINDING:
The spheromak winding has Npo poloidal turns and Nto toroidal turns. The numbers Npo and Nto are positive integers with various special mathematical properties.

The winding length Lh is given by:
Lh^2 = (Npo Lp)^2 + (Nto Lt)^2
= [Npo 2 Pi Ro]^2 + [Nto Lt]^2
where:
Lpo = 2 Pi Ro
is the length of a single poloidal filament winding turn and Lto is the length of a toroidal filament winding turn that encircles the toroidal magnetic flux once.

The spheromak inner wall and the spheromak outer wall have different radii with respect to the main axis of symmetry. Hence, at the inner wall the filament to filament distance is less than at the outer wall. Since the net charge is evenly distributed along the filament this issue causes the surface charge per unit area at the outer wall to be less than the surface charge per unit area at the inner wall.

The current filament turns occur in a single layer which contains no cross overs. The current path only intersects itself at the current path closure point. The spheromak must remain stable in the presence of small geometric disturbances that can cause either Np or Nt to increment or decrament from their nominal Npo and Nto values. Hence, in a stable spheromak, Npo and Nto do not share common factors. Similarly there is no common factor sharing between: Npo + 2, Nto - 1; between Npo - 2 and Nto + 1, between Npo + 4 and Nto -2 and between Npo - 4 and Nto + 2.

In a spheromak Npo cannot equal Nto .

Any point on the spheromak winding can be identified by its Theta and Phi values. Theta is the angle about the spheromak minor axis measured with respct to a radial line from the minor axis to the orgin. Phi is the angle about the major axis of symmetry measured with respect to the same radial line. Hence at the current path closure point:
Theta = 0, Phi = 0 or Theta = 2 Pi Nt, Phi = 2 Pi Np

The range of Theta is:
0 < Theta < 2 Pi Nto radians;
The range of Phi is:
0 < Phi < 2 Pi Npo radians

The average value of dTheta / dPhi is given by:
dTheta / dPhi = 2 Pi Nto / 2 Pi Npo
= Nto / Npo.
However, in a spheromak dTheta / dPhi varies along the winding and is a function of R.

Symmetry indicates that at the center of the spheromak core the net electric field is zero and the net toroidal magnetic field is zero.

At R = Rc, Z = 0 the distance between adjacent filaments measured in the equatorial plane is:
2 Pi Rc / Nto = dPhi Rc
or:
dPhi = 2 Pi / Nto

At R = Rc, Z = 0 the distance between adjacent filaments measured perpendicular to the equatorial plane is:
[(Ro - Rc)] dTheta = [Ro - Rc)][(dTheta / dPhi)|Rc] dPhi = ((Ro - Rc))[(dTheta / dPhi)|Rc](2 Pi / Nto)
= (Rs - Rc) Pi [(dTheta / dPhi)|Rc] / Nto

At R = Ro , Z = H the distance between adjacent filaments measured parallel to the minor axis is:
2 Pi [Ro] / Nto = [Ro] dPhi
or: dPhi = 2 Pi / Nto

At R = Ro, Z = H the distance between adjacent filaments measured along a radial line is:
H dTheta = H [(dTheta / dPhi)|Ro] dPhi
= H [(dTheta / dPhi)|Ro] (2 Pi / Nto)

At R = Rs, Z = 0 the distance between adjacent filaments measured along the equatorial plane is:
Rs dPhi = 2 Pi Rs / Nto
or
dPhi = 2 Pi / Nto

CHECK FROM HERE

At R = Rs, Z = 0 the distance between filaments measured parallel to the Z axis is:
[(Rs- Rc) / 2] dTheta = [(Rs - Rc) / 2] [(dTheta / dPhi)|Rs] dPhi
= [(Rs - Rc) / 2][(dTheta / dPhi)|Rs][2 Pi / Nto]
= [(Rs - Rc)][(dTheta / dPhi)|Rs][Pi / Nto]

The above calculated distances can be used to compute the shortest separation between adjacent filaments at each of the three specified locations. The (shortest separation) X (Q / Lh) = charge / unit area The surface electric field Es = (charge / unit area) / Epsilono

Let Dh = horizontal distance between adjacent filaments
Let Dv = vertical distance between adjacent filaments
Let X = shortest distance betwen adjacent filaments.

Then a little geometry shows that:
Dh^2 + Dv^2 = {[Dh^2 - X^2]^0.5 + [Dv^2 - X^2]^0.5}^2
or
Dh^2 + Dv^2 = Dh^2 - X^2 + Dv^2 - X^2 + 2{[Dh^2 - X^2]^0.5 [Dv^2 - X^2]^0.5}
or
2 X^2 = 2{[Dh^2 - X^2]^0.5 [Dv^2 - X^2]^0.5}
or
X^4 = [Dh^2 - X^2] [Dv^2 - X^2]
or
X^4 = Dh^2 Dv^2 + X^4 - X^2 (Dv^2 + Dh^2)
or
X^2 = Dh^2 Dv^2 / (Dv^2 + Dh^2)
or
X = Dh Dv / (Dv^2 + Dh^2)^0.5

SHORTEST DISTANCE BETWEEN ADJACENT FILAMENTS AT R = Rc, Z = 0:
Dh = 2 Pi Rc / Nto
Dv = (Rs - Rc) Pi [(dTheta / dPhi)|Rc] / Nto
Dh Dv = 2 Pi^2 Rc (Rs - Rc) [(dTheta / dPhi)|Rc] / Nto^2
Dh^2 = 4 Pi^2 Rc^2 / Nto^2
Dv^2 = (Rs - Rc) Pi^2 [(dTheta / dPhi)|Rc]^2 / Nto^2
Xc = Dh Dv / (Dv^2 + Dh^2)^0.5>
= [2 Pi^2 Rc (Rs - Rc)[(dTheta / dPhi)|Rc] / Nto^2] / [(4 Pi^2 Rc^2 / Nto^2) + ((Rs - Rc)^2 Pi^2 [(dTheta / dPhi)|Rc]^2 / Nto^2)]^0.5
= [2 Pi Rc (Rs - Rc)[(dTheta / dPhi)|Rc] / Nto] /[(4 Rc^2) + ((Rs - Rc)^2[(dTheta / dPhi)|Rc]^2)]^0.5

SHORTEST DISTANCE BETWEEN ADJACENT FILAMENTS AT R = (Rs + Rc) / 2, Z = H:
Dh = 2 Pi [(Rs + Rc) / 2] / Nto
Dv = H [(dTheta / dPhi)|Rm] (2 Pi / Nto)
Dh Dv = 2 Pi^2 H [Rs + Rc][(dTheta / dPhi)|Rm] / Nto^2
Dh^2 = Pi^2 (Rs + Rc)^2 / Nto^2
Dv^2 = 4 Pi^2 H^2 [(dTheta / dPhi)|Rm]^2 / Nto^2
Xm = Dh Dv / (Dv^2 + Dh^2)^0.5
= [2 Pi^2 H [Rs + Rc][(dTheta / dPhi)|Rm] / Nto^2] / [(Pi^2 (Rs + Rc)^2 / Nto^2) + (4 Pi^2 H^2 [(dTheta / dPhi)|Rm]^2 / Nto^2)]^0.5
= [2 Pi H [Rs + Rc][(dTheta / dPhi)|Rm]] / [((Rs + Rc)^2) + (4 H^2 [(dTheta / dPhi)|Rm]^2)]^0.5

SHORTEST DISTANCE BETWEEN ADJACENT FILAMENTS AT R = Rs, Z =0:
Dh = 2 Pi Rs / Nto
Dv = [(Rs - Rc)][(dTheta / dPhi)|Rs][Pi / Nto]
Dh Dv = 2 Pi^2 Rs (Rs - Rc)[(dTheta / dPhi)|Rs] / Nto^2
Dh^2 = 4 Pi^2 Rs^2 / Nto^2
Dv^2 = Pi^2 (Rs - Rc)^2 [(dTheta / dPhi)|Rs]^2 / Nto^2
Xs = Dh Dv / (Dv^2 + Dh^2)^0.5
= [2 Pi^2 Rs (Rs - Rc)[(dTheta / dPhi)|Rs] / Nto^2] / [(4 Pi^2 Rs^2 / Nto^2) + (Pi^2 (Rs - Rc)^2 [(dTheta / dPhi)|Rs]^2 / Nto^2)]^0.5
= [2 Pi Rs (Rs - Rc)[(dTheta / dPhi)|Rs] / Nto] / [(4 Rs^2) + ((Rs - Rc)^2 (dTheta / dPhi)|Rs]^2)]^0.5

SURFACE ELECTRIC FIELD:
The outward pointing surface electric field Es resulting from a surface charge per unit area is:
Es = Q / (Lh X Epsilono)
where:
Lh = [(Npo Lp)^2 + (Nto Lt)^2]^0.5
= [(Npo Pi (Rs + Rc))^2 + (Nto Lt)^2]^0.5

INTERIOR ELECTRIC FIELD CANCELATION:
Note that inside the spheromak wall, in order to have no net electric field on the equatorial plane, the radial electric field must be inversely proportional to X. Hence:
Ess / Esc = Rc / Rs

However:
Ess / Esc = [Xc / Xs]
= {[2 Pi Rc (Rs - Rc)[(dTheta / dPhi)|Rc] / Nto] /[(4 Rc^2) + ((Rs - Rc)^2 [(dTheta / dPhi)|Rc]^2)]^0.5}
/ [2 Pi Rs (Rs - Rc)[(dTheta / dPhi)|Rs] / Nto] / [(4 Rs^2) + ((Rs - Rc)^2 [(dTheta / dPhi)|Rs]^2)]^0.5

In order to have no net electric field inside the spheromak wall:
{[[(dTheta / dPhi)|Rc] / Nto] / [(4 Rc^2) + ((Rs - Rc)^2 [(dTheta / dPhi)|Rc]^2)]^0.5}
= [[(dTheta / dPhi)|Rs] / Nto] / [(4 Rs^2) + ((Rs - Rc)^2 [(dTheta / dPhi)|Rs]^2)]^0.5

Try for a general solution to this equation of the form:
[[(dTheta / dPhi)|R] / Nto] / [(4 R^2) + ((Rs - Rc)^2 [(dTheta / dPhi)|R]^2)]^0.5 = K = constant
which is valid for all R values.

Then:
[[(dTheta / dPhi)|R] / Nto]^2
= K^2 [(4 R^2) + ((Rs - Rc)^2 [(dTheta / dPhi)|R]^2)]
or
[(dTheta / dPhi)|R]^2 {(1 / Nto^2) - K^2 (Rs - Rc)^2} = 4 K^2 R^2
or
[(dTheta / dPhi)|R]^2 = 4 K^2 R^2 / {(1 / Nto^2) - K^2 (Rs - Rc)^2}
or
[(dTheta / dPhi)|R]^2 = 4 K^2 R^2 Nto^2 / {1 - Nto^2 K^2 (Rs - Rc)^2}
[(dTheta / dPhi)|R] = 2 K R Nto / {1 - Nto^2 K^2 (Rs - Rc)^2}^0.5

Recall that the average value of (dTheta / dPhi) = (Nto / Npo).
Hence integral from R = Rc to R = Rs of
[(dTheta / dPhi)|R]dR / (Rs - Rc)
= (Nto / Npo) or K (Rs^2 - Rc^2) Nto / {{1 - Nto^2 K^2 (Rs - Rc)^2}^0.5 (Rs - Rc)}
= (Nto / Npo) or
K (Rs + Rc) Nto / {1 - Nto^2 K^2 (Rs - Rc)^2}^0.5 = (Nto / Npo) or
K Npo (Rs + Rc) / {1 - Nto^2 K^2 (Rs - Rc)^2}^0.5 = 1
or
K^2 Npo^2 (Rs + Rc)^2 = 1 - Nto^2 K^2 (Rs - Rc)^2
or
K^2 {Npo^2 (Rs + Rc)^2 + Nto^2 (Rs - Rc)^2} = 1

This is an important equation because it relates constant K to filament length Lh for a spheromak. This equation arose from the requirement that the net radial electric field inside the spheromak wall be zero.

ELECTRIC FIELD COMMENTARY:
Ideally the shortest separation between adjacent filament turns on the ellipsoid surface should be proportional to (1 / R). Then the outward pointing electric field Es on the spheromak surface will be proportional to (1 / R). Then, due to cylindrical symmetry, inside the spheromak wall the net electric field will be zero.

On the equatorial plane, due to cylindrical symmetry, for R < Rc the net electric field should be zero.

At R = Rc, Z = 0 the electric field inside the spheromak wall due to the inner wall pointing radially outward is proportional to (1 / Rc). when this field reaches the outer wall, due to cylindrical symmetry the electric field strength is proportional to:
(1 / Rc)(Rc / Rs) = (1 / Rs).

However, the electric field pointing inwards due to charge on the outer wall is proportional to (1 / Rs). Hence in the toroidal region the two electric fields cancel.

At R = Rm = (Rs + Rc) / 2, Z = H the surface electric field points along an outward pointing vector involving both Rm and H and should be proportional to [2 / (Rs + Rc)].

At R = Rs, Z = 0 the surface electric field points radially and should be proportional to (1 / Rs).

To achieve this electric field pattern objective the spacing between adjacent filament turns must change with R as set out above.

At R = Rs, Z = 0 the electric field pointing radially outwards is: Es = 2 (Surface charge / unit area)(1 / Epsilono)
= 2 (Q / Lh)[1 / X](1 / Epsilono)

Note that this electric field is a result of both the surface charge on the outer wall and the surface charge on the inner wall.

Note that in terms of angle about the major axis of symmetry the number of toroidal turn equatorial plane crossings per unit angle is the same for both the inner and the outer wall.

Inside the toroidal region the electric fields parallel to the Z axis emitted by the spheromak ends cancel.

It is tempting to attempt to simplify the situation by assuming that a spheromak winding everywhere follows:
dTheta / dPhi = Npo / Nto,
but while this equation is true on average making that assumption at each point on the winding does not lead to a net zero electric field inside the spheromak wall.

Instead, we found that to achieve a net zero electric field inside the spheromak wall:
[(dTheta / dPhi)|R] = 2 K R Nto / {1 - Nto^2 K^2 (Rs - Rc)^2}^0.5
where K is a constant.

After taking into account the required average value of [(dTheta / dPhi)average] = [Nto / Npo]
we found the important spheromak constraint that:
K^2 {Npo^2 (Rs + Rc)^2 + Nto^2 (Rs - Rc)^2} = 1

For a spheromak with a round cross section:
Lh^2 = (Npo Lp)^2 + (Nto Lt)^2
= (Npo 2 Pi (Rs + Rc) / 2)^2 + (Nto 2 Pi (Rs - Rc) / 2)^2
= (Npo Pi (Rs + Rc))^2 + (Nto Pi (Rs - Rc))^2
= Pi^2 (Npo (Rs + Rc))^2 + (Nto (Rs - Rc))^2

Hence:
K = Pi / Lh

Note that for a spheromak with a round cross section K is proportional to the spheromak's frequency and total energy.

At a particular spheromak energy:
(Pi / Lh){Npo^2 (Rs + Rc)^2 + Nto^2 (Rs - Rc)^2}^0.5 = 1
or
(Pi Rc / Lh){Npo^2 ((Rs / Rc) + 1)^2 + Nto^2 ((Rs / Rc) - 1)^2}^0.5 = 1
or
(Lh / Pi Rc)
= {Npo^2 [((Rs / Rc) + 1]^2 + Nto^2 [(Rs / Rc) - 1]^2}^0.5

In a geometrically stable spheromak the quantities (Lh / Pi Rc) and (Rs / Rc) are constant, independent of the spheromak's energy.

In order to prevent Npo and Nto having a common integer factor, for Npo odd:
Npo + 2 Nto = P
, where P is a fixed prime number, characteristic of the spheromak. Hence:
dNpo = - 2 dNto

We need to find the Npo and Nto values that result in minimum spheromak energy. Hence differentiating gives:
2 Npo dNpo [((Rs / Rc) + 1]^2 + 2 Nto dNto [(Rs / Rc) - 1]^2 = 0
or
2 Npo (- 2 dNto) [((Rs / Rc) + 1]^2 + 2 Nto dNto [(Rs / Rc) - 1]^2 = 0
or
Nto [(Rs / Rc) - 1]^2 = Npo (2) [((Rs / Rc) + 1]^2

Recall that:
Npo + 2 Nto = P
or
Npo = (P - 2 Nto)
giving:
Nto [(Rs / Rc) - 1]^2 = (P - 2 Nto) (2) [((Rs / Rc) + 1]^2
or
2 P [((Rs / Rc) + 1]^2
= 4 Nto [((Rs / Rc) + 1]^2 + Nto [(Rs / Rc) - 1]^2
or
[P / Nto]
= {4 [((Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2} / {2 [((Rs / Rc) + 1]^2}
= 2 + {[(Rs / Rc) - 1]^2 / (2 [((Rs / Rc) + 1]^2)}

Thus, if we can accurately determine (Rs / Rc) we can use this equation together with a table of prime numbers to determine P and Nto.

Note that for stable spheromaks (Rs / Rc) is a relative geometric constant independent of spheromak energy.

Recall that:
Nto [(Rs / Rc) - 1]^2 = Npo (2) [((Rs / Rc) + 1]^2
or
2 Npo / Nto = {[(Rs / Rc) - 1]^2 / [((Rs / Rc) + 1]^2}
or
4 Npo^2 = Nto^2 {[(Rs / Rc) - 1]^4 / [((Rs / Rc) + 1]^4}
giving:
(Lh / Pi Rc)
= {Npo^2 [((Rs / Rc) + 1]^2 + Nto^2 [(Rs / Rc) - 1]^2}^0.5

= {(Nto^2 / 4) {[(Rs / Rc) - 1]^4 / [((Rs / Rc) + 1]^4}
[[(Rs / Rc) + 1]^2 + Nto^2 [(Rs / Rc) - 1]^2}^0.5
= {(Nto^2 / 4)[(Rs / Rc) - 1]^4 /[(Rs / Rc) + 1]^2 + Nto^2 [(Rs / Rc) - 1]^2}^0.5
= {(Nto^2 [(Rs / Rc) - 1]^2 [[(Rs / Rc) - 1]^2 / 4 [(Rs / Rc) + 1]^2] + 1}^0.5
= {(Nto^2 [(Rs / Rc) - 1]^2 [[(Rs / Rc) - 1]^2 + 4 [(Rs / Rc) + 1]^2] / 4 [(Rs / Rc) + 1]^2}^0.5
= {(Nto^2 [(Rs / Rc) - 1]^2 {[[(Rs / Rc)^2 - 2 (Rs / Rc) + 1] + 4 [(Rs / Rc)^2 + 2 (Rs / Rc) + 1] / 4 [(Rs / Rc) + 1]^2]}^0.5
= Nto [(Rs / Rc) - 1] {[[5 (Rs / Rc)^2 + 6 (Rs / Rc) + 5] / 4 [(Rs / Rc) + 1]^2]}^0.5
which expresses Nto in terms of (Lh / Pi Rc) which is a natural geometric constant.

Note that (Lh / Pi Rc) and Nto are both spheromak geometric constants that are independent of spheromak energy.

Nto is a prime number and (Lh / Pi Rc)^2 is an integer.

Recall that:
[P / Nto]
= {4 [((Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2} / {2 [(Rs / Rc) + 1]^2}
or
Nto
= P / ({4 [((Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2} / {2 [(Rs / Rc) + 1]^2})
= P {2 [(Rs / Rc) + 1]^2} / {4 [(Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2}
giving:
(Lh / Pi Rc)
= P {2 [(Rs / Rc) + 1]^2} [(Rs / Rc) - 1]
/ {2 [(Rs / Rc) + 1)] {4 [(Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2}^0.5}}
= P {[(Rs / Rc) + 1]} [(Rs / Rc) - 1]
/ {4 [(Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2}^0.5}

Note that stable spheromaks have a characteristic (Rs / Rc) value and hence a characteristic prime number P.

EXPLORATION OF POSSIBLE VALUES OF (Rs / Rc):
Recall that:
(Lh / Pi Rc) = P {[(Rs / Rc) + 1]} [(Rs / Rc) - 1]
/ {4 [(Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2}^0.5}

Hence:
(Lh / Pi Rc)^2 = P^2 {[(Rs / Rc) + 1]}^2 [(Rs / Rc) - 1]^2
/ {4 [(Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2}}

Note that since (Lh / Pi Rc)^2 is an integer then:
N = {[(Rs / Rc) + 1]}^2 [(Rs / Rc) - 1]^2
/ {4 [(Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2}}
is an integer. Hence:
{[(Rs / Rc) + 1]}^2 [(Rs / Rc) - 1]^2
= N {4 [(Rs / Rc) + 1]^2 + [(Rs / Rc) - 1]^2}}
or
[(Rs / Rc)^2 - 1]^2 = N [4 (Rs / Rc)^2 + 8 (Rs / Rc) + 4 + (Rs / Rc)^2 - 2 (Rs / Rc) + 1]
or
[(Rs / Rc)^4 - 2 (Rs / Rc)^2 + 1] = N [5 (Rs / Rc)^2 + 6 (Rs / Rc) + 5]
or
N = [(Rs / Rc)^4 - 2 (Rs / Rc)^2 + 1] / [5 (Rs / Rc)^2 + 6 (Rs / Rc) + 5]

Try (Rs / Rc) = 2, then:
N = [9 / 37]

Try (Rs / Rc) = 3, then:
N = [81 - 18 + 1] / [45 + 18 + 5] = [64 / 68]
~ 1

Try (Rs / Rc) = 4, then:
N = [256 -32 + 1] / [80 + 24 + 5]
= 225 / 109
~ 2

Try (Rs / Rc) = 5. Then:
N = [625 -50 + 1] / [125 + 30 + 5]
=  / 
~ 4

Thus the viable choices for N are 1, 2, 3 or 4 corrresponding to (Rs / Rc) values in the range 3 to 5. These equations can be used to find exact values of (Rs / Rc ) for N = 2, 3, 4.

ACCURATE DETERMINATION OF (Rs / Rc):
Recall that:
[(Rs / Rc)^4 - 2 (Rs / Rc)^2 + 1] = N [5 (Rs / Rc)^2 + 6 (Rs / Rc) + 5]
or
[(Rs / Rc)^4 + (- 2 - 5 N)(Rs / Rc)^2 + [- 6 N (Rs / Rc) - 5 N + 1] = 0
or
(Rs / Rc)^2 = {(3 + 5 N) +/- [(2 + 5 N)^2 - 4 (1)(- 6 N (Rs / Rc) - 5 N + 1]^0.5} / 2
= {(3 + 5 N) +/- [(2 + 5 N)^2 + 4 (6 N (Rs / Rc) + 5 N - 1]^0.5} / 2
which can be itterated to find the exact value of (Rs / Rc)^2 and hence (Rs / Rc).

Recall that:
[P / Nto] = 2 + {[(Rs / Rc) - 1]^2 / (2 [((Rs / Rc) + 1]^2)}

Knowing the exact value of (Rs / Rc) and with the aid of a table of prime numbers we can find P and Nto.

With knowledge of both N and P we can calculate (Lh / Pi Rc)^2 using the equation:
(Lh / Pi Rc)^2 = N P^2.

Then we can find Npo using the equation:
Npo = P - 2 Nto

SPHEROMAK WALL:
The spheromak wall separates the toroidal magnetic field region enclosed by the spheromak wall and the poloidal magnetic and radial electric field region outside the spheromak wall.

Hence a stable isolated spheromak has radial electric, poloidal magnetic and toroidal magnetic field energy components.

At every point on the spheromak wall there is perfect balance between the local internal toroidal magnetic field energy density and the local sum of the external field energy densities. This energy density balance sets the position of the spheromak wall.

Due to a combination of electromagnetic field theory, wall geometry and prime number theory only certain values of Nto and Npo can exist in a stable spheromak.

Since the inner spheromak wall and the outer spheromak wall have different radii with respect to the main axis of symmetry, at the inner wall the separation between adjacent current filaments is less than in the outer wall.

SPHEROMAK STABILITY:
Apart from the constraints on Np and Nt detailed analysis shows that spheromaks are stable due to formation of an energy well inside the spheromak wall.

Due to a combination of electromagnetic field theory, ellipsoid geometry and prime number theory only certain values of Nt and Np can exist in a stable spheromak.

The optimum spheromak geometry is stable over a wide range of spheromak energy.

The stable spheromak structure enables the existence of isolated charged particles and semi-stable plasmas. 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.

INNER WALL BOUNDARY CONDITION:
1) In the toroidal magnetic field region:
Bt = Muo Nt I / 2 Pi R
Note that inside the spheromak wall this field is everywhere proportional to (1 / R). Inside the sphedromak wall at R = Ro:
Bto = Muo Nt I / 2 Pi Ro

2) Inside the spheromak wall at R = Rc, Z = 0:
Btc = Muo Nt I / 2 Pi Rc.

At R = Rc, Z = 0:
Bpc = Btc
where:
Bpc = Muo Npo I / Lt
or
Npo / Lt = Nto / 2 Pi Rc
or
Npo / Nto = Lt / 2 Pi Rc

For a spheromak with a round cross section:
Lt = 2 Pi (Rs - Rc) / 2
= Pi (Rs - Rc)

Hence for a spheromak with a round cross section:
Npo / Nto = Pi (Rs - Rc) / 2 Pi Rc
= (Rs - Rc) / 2 Rc
= (1 / 2)[(Rs / Rc) - 1]

Rearranging this equation gives:
2 Rc (Npo / Nto) = (Rs - Rc)
or
Rs = Rc [1 + 2 (Npo / Nto)]

APPLICATION OF INNER WALL BOUNDARY CONDITION TO DETERMINATION OF (Lh / Pi Rc)

Recall that:
(Pi / Lh){Npo^2 (Rs + Rc)^2 + Nto^2 (Rs - Rc)^2}^0.5 = 1
or
(Pi / Lh){Npo^2 Rc^2 [2 + 2 (Npo / Nto)]^2 + Nto^2 (2 Rc Npo / Nto)^2}^0.5 = 1
or
(Pi / Lh){Npo^2 Rc^2 4 [1 + (Npo / Nto)]^2 + (Rc^2 Npo^2)}^0.5 = 1
or
(Pi Npo Rc / Lh){4 [1 + (Npo / Nto)]^2 + 1}^0.5 = 1
or
(Pi Npo Rc / Lh){[5 + (Npo / Nto)^2 + 2 (Npo / Nto)}^0.5 = 1
or
[5 + (Npo / Nto)^2 + 2 (Npo / Nto)} = [Lh / Pi Npo Rc]^2
or
(Npo / Nto)^2 + 2 (Npo / Nto) + [5 - (Lh / Pi Npo Rc)^2] = 0
(Npo / Nto) = {-2 + / - [4 - 4 (1)[5 - (Lh / Pi Npo Rc)^2]^0.5} / 2
or
Npo = Nto {- 1 +/- [(Lh / Pi Npo Rc)^2 - 4]^0.5}
or
Npo = Nto {- 1 + [(Lh / Pi Npo Rc)^2 - 4]^0.5}
or
(Npo + Nto)^2 = [(Lh / Pi Npo Rc)^2 - 4]
or
(Npo + Nto)^2 + 4 = (Lh / Pi Npo Rc)^2
or
Npo^2 [(Npo + Nto)^2 + 4] = (Lh / Pi Rc)^2

Since Npo and Nto are both integers then:
(Lh / Pi Rc)^2 is an integer. However, that fact is not helpful in determination of Npo and Nto because all integer values of Npo and Nto result in integer values of (Lh / Pi Rc)^2. However, knowledge of (Lh / Pi Rc)^2 imposes upper limits on Npo and Nto.

Npo^4 < (Lh / Pi Rc)^2
and
Nto^2 < (Lh / Pi Rc)^2

Recall that: Npo and Nto are prime numbers that satisfy:
2 Npo + Nto = P
or
2 Nto + Npo = P
where P = prime

Try Nto = 1, Npo = 3, both of which are prime, then:
2 Nto + Npo = 5 which is also prime

This test will identify all sequential odd number pairs which are prime.

Try Nto = 2, Npo = 3, both of which are prime, then:
2 Nto + Npo = 7 which is also prime

Try Nto = 2, Npo = 7, both of which are prime, then: 2 Nto + Npo = 11, which is also prime.

Try Nto = 3, Npo = 7, both of which are prime. Then: 2 Nto + Npo = 13, which is also prime.

Thus the prime number test gives many potential Np, Nt pairs. A further test is required to select the correct Npo, Nto pair.

FINDING Npo AND Nto:
Recall that:
Npo^2 [(Npo + Nto)^2 + 4] = (Lh / Pi Rc)^2
where:
P = Npo + 2 Nto or
Npo^2 [(Npo + ((P - Npo) / 2))^2 + 4] = (Lh / Pi Rc)^2
or
Npo^2 [(Npo^2 + Npo (P - Npo) + (P - Npo)^2 / 4 + 4] = (Lh / Pi Rc)^2
or
Npo^2 [Npo P + [(P^2 - 2 P Npo + Npo^2) / 4] + 4] = (Lh / Pi Rc)^2
or
Npo^2 [(Npo P / 2) + ((P^2 + Npo^2) / 4) + 4] = (Lh / Pi Rc)^2
or
Npo^2 [[(Npo + P) / 2]^2 + 4] = (Lh / Pi Rc)^2

Thus it is helpful if we can quantify:
(Lh / Pi Rc)^2
which should be a large integer.

SPHEROMAK STRUCTURE:
The geometry of a spheromak is symmetric about its central point:
R = 0, Z = 0
and can be characterized by:
its outer radius R = Rs at Z = 0 ,
its inner radius R = Rc at Z = 0
and its height Z = +/- H at R = [(Rs + Rc) / 2].

For the stable spheromak geometry the ratios (H / Rs) and (Rc / Rs) are constants irrespective of the value of Rs.

Note that a spheromak with a minor axis at:
R = (Rs + Rc) / 2, Z = 0
has a surface defined by:
{Z^2 / H^2} + {[R - ((Rs + Rc) / 2)]^2 / [(Rs - Rc) / 2]^2} = 1.

It is shown that the total electromagnetic field energy Ett trapped by a spheromak is proportional to (1 / Rs).

Current moves along the spheromak closed path length Lh at speed of light C. Hence the natural frequency F of the path length Lh is:
F = C / Lh.

The electromagnetic field energy Ett is proportional to (1 / Rs) which in turn is shown to be proportional to (1 / Lh).

Hence:
Ett is proportional to frequency F. The proportionality constant is known as the Planck constant h where:
Ett = h F

Thus the spheromak structure is the cause of photon energy quantization.

SPHEROMAK FIELD ENERGY DENSITY FUNCTIONS:
A spheromak has three cylindrically symmetric field energy density functions:
Ut(R,Z) = toroidal magnetic field energy density function
Up(R,Z) = poloidal magnetic field enegry density function
Ue(R,Z) = radial electric field energy density function

The spheromak has two regions which are separated by the thin closed spheromak wall.

Everywhere in the region inside the spheromak wall:
Ut(R, Z) < Up(R, Z) + Ue(R,Z)
and
U = Ut(R, Z).

Everywhere in the region outside the spheromak wall:
Ut(R, Z) > Up(R, Z) + Ue(R, Z)
and U = Up(R,Z) + Ue(R, Z)

Hence the spheromak forms an energy well.

Everywhere on the wall:
Ut(R, Z) = Up(R, Z) + Ue(R,Z).
Ths equation defines the position of the spheromak wall.

SPHEROMAK WALL DIAGRAM:
The following diagram shows the approximate cross sectional shape of a spheromak wall in free space. The spheromak outside radius on its equatorial plane is:
R = Rs,
the spheromak inside radius on its equatorial plane is:
R = Rc,
and the maximum height of the spheromak is:
Z = Hm.

SPHEROMAK TOTAL FIELD ENERGY:
The position of the spheromak wall corresponds to the lowest available spheromak energy state. At this state 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) + Ue(R,Z)
and at the spheromak wall:
Ut(R, Z) = Up(R, Z) + Ue(R,Z)

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

Spheromaks are used by nature to form particles that are local concentrations of electromagnetic field energy. We commonly refer to this locally concentrated electromagnetic field energy as rest mass. Spheromaks are instrumental in nuclear and atomic particle interactions. Spheromaks also have important roles in semi-stable plasmas, chemical binding and thermal radiation.

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

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

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

Hence:
Ett = Integral from Z = -infinity to Z = infinity of:
Integral from R = 0 to R = infinity of:
U(R, Z) 2 Pi R dR dZ
= Integral from Z = 0 to Z = infinity of:
U(R, Z) 4 Pi R dR dZ

SPHEROMAK STABILITY:
Recall that a spheromak will collapse if Np and Nt have a common integer factor. This issue confines the possible Np and Nt values in a stable spheromak.

ODD Npj VALUES:
Equations that give pairs of Npj, Ntj values with no common integer factor where Npj is odd are:
P = Npo + 2 Nto
P = Npj + 2 Ntj
Npj = Npo + 2j
Ntj = Nto - j
where P is a prime number and j is a positive or negative integer.

Note that in this case at a stable energy minimum Npo and Nto are such that:
dNp = - 2 dNt

POSSIBLE Nt, 2 Nt, Np, P VALUES:
Nt, 2 Nt, Np, P = 2 Nt + Np

1, 2, 3, 5;
1, 2, 5, 7;
1, 2, 11, 13;
1, 2, 17, 19;
1, 2, 29, 31;
1, 2, 41, 43;

2, 4, 3, 7;
2, 4, 7, 11;
2, 4, 13, 17;
2, 4, 19, 23;
2, 4, 37, 41;
2, 4, 43, 47;
2, 4, 47, 51;

3, 6, 1, 7
3, 6, 5, 11;
3, 6, 7, 13;
3, 6, 11, 17;
3, 6, 13, 19;
3, 6, 17, 23;
3, 6, 23, 29;
3, 6, 31, 37;
3, 6, 37, 43;

4, 8, 3, 11;
4, 8, 5, 13;
4, 8, 11, 19;
4, 8, 23, 31;
4, 8, 29, 37;
4, 8, 43, 51;
4, 8, 51, 59;
4, 8, 53, 61;
4, 8, 59, 67;

5, 10, 1, 11;
5, 10, 3, 13;
5, 10, 7, 17;
5, 10, 13, 23;
5, 10, 19, 29;
5, 10, 31, 41;
5, 10, 37, 47;
5, 10, 41, 51;
5, 10, 43, 53;
5, 10, 47, 57;
5, 10, 51, 61;

7, 14, 3, 17;
7, 14, 5, 19;
7, 14, 17, 31;
7, 14, 23, 37;
7, 14, 29, 43;
7, 14, 37, 51;
7, 14, 43, 57;
7, 14, 47, 61;
7, 14, 53, 67;
7, 14, 57, 71;

11, 22, 1, 23;
11, 22, 7, 29;
11, 22, 19, 41;
11, 22, 31, 53;
11, 22, 37, 59;
11, 22, 51, 73;

In order to properly choose the appropriate stable Np, Nt values we need to accurately quantify (Np / Nt).

EVEN Npj VALUES:
Equations that give pairs of Npj, Ntj values with no common factor where Ntj is odd are:
P = 2 Npo + Nto
P = 2 Npj + Ntj
Npj = Npo + j
Ntj = Nto - 2j
where P is a prime number and j is a positive or negative integer.

Note that in this case at a stable energy minimum Npo and Nto are such that:
dNt = - 2 dNp

ENHANCED STABILITY:
Spheromak stability is enhanced if a small energy disturbance does not lead to spheromak collapse. Ideally, in addition to P being prime, Npo and Nto are both prime numbers. Then if there ⁮is a disturbance that affects only Np or only Nt the spheromak remains stable.

NATURAL FREQUENCY:
The spheromak wall consists of a long filament of charge that forms a complex closed spiral current path with length Lh. This current path contains Nt toroidal turns and Np poloidal turns. The numbers Np and Nt have no common factors so the current path never crosses itself. The current flows along this path at the speed of light. The length of this current path and the speed of light together give the spheromak a characteristic frequency Fh.

The natural frequency Fh given by:
Fh = C / Lh.

It is shown on this web site that:
dEtt / dFh = h,
where:
h = Planck Constant

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

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

PLANCK CONSTANT AND FINE STRUCTURE CONSTANT:
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)
where:
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.

Note that during a change in energy the spheromak relative geometry remains constant. As the spheromak absorbs energy Rc, Rs and H all reduce by the same fraction so that the relative spheromak geometry parameters (Rc / Rs), and (H / Rs) remain constant.

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

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

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

APPROXIMATION FOR Lh:
Let Lt = length of one toroidal current path turn
Let Lp = length of one poloidal current path turn

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

BOUNDARY CONDITIONS:
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 is more complicated.

For an isolated spheromak the field energy density outside the wall is the sum of the energy densities caused by the electric field due to fixed charge Q and by the polodal magnetic field arising from current circulation within the spheromak wall. The field energy density inside the wall is caused only by the toroidal magnetic field that arises from current circulation in the spheromak wall.

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
where:
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 magnetic and electric fields produced by a current ring located at R = (Rc + Rs) / 2, Z = 0 carrying a current (Np I) and having net charge Q.

INNER WALL BOUNDARY CONDITION:
1) Inside the spheromak wall:
Bt = Muo Nt I / 2 Pi R
Note that inside the spheromak wall this toroidal magnetic field is everywhere proportional to (1 / R). Inside the spheromak wall at radius R:
Bt = Muo Nt I / 2 Pi R

2) Inside the spheromak wall at R = Rc, Z = 0:
Btc = Muo Nt I / 2 Pi Rc.

3) Inside the spheromak wall at R = Rs:
Bts = Muo Nt I / 2 Pi Rs

INNER WALL BOUNDARY CONDITION:
Due to spheromak symmetry, for R < Rc on the equatorial plane where Z = 0 the electric field R and Z components all cancel. Hence at R= Rc, Z = 0 the poloidal magnetic 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
= [Bt|(R = Rc)]^2 / 2 Muo
or
[Bp|(R = Rc)] = [Bt|(R = Rc)]
or
(Muo Np I / Lt) = Muo Nt I / 2 Pi Rc
or
Np / Lt = Nt / 2 Pi Rc
or
Np / Nt = Lt / 2 Pi Rc

This result is the inside wall boundary condition. Note that this inner wall boundary condition is independent of charge Q.

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

Note that Er contains electric field contributions from both the inner and outer spheromak walls.

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.

FURTHER SPHEROMAK ATTRIBUTES:
Multi-quantum charge spheromaks can exist. A free neutron is a spheromak assembly with no net charge. In a nuclear 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.

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.

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 TRANSIENT BEHAVIOR:
When a spheromak first forms 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.

NO RADIATION IN THE GROUND STATE:
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.

RELATED MATERIAL ON THIS WEB SITE:
At other web pages 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.

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.

The result is a practical mathematical model that gives relatively simple closed form solutions to problems that would otherwise likely require extensive computing power.

Other spheromak issues include: plasma spheromak circulating electron kinetic energy, the number of free electrons in a plasma spheromak, the plasma spheromak enclosure size and plasma spheromak lifetime.

This web page last updated October 18, 2022.