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**SPHEROMAK CONCEPT:**

The existence of stable quantum charged atomic and nuclear particles is enabled by an electro-magnetic structure known as a spheromak, in which electric and magnetic forces cancel each other. Net charge circulating around a complex closed quasi-toroidal filament path can form the electric and magnetic field energy density distributions required for spheromak existence. The electric and magnetic fields of a spheromak maintain its stable geometry and contain a stable amount of energy.

**SPHEROMAK GEOMETRY:**

Conceptually a spheromak wall is a quasi-toroidal shaped surface formed by the closed filament winding of a spheromak. This filament path conforms to the spheromak wall surface curvature. In plan view, looking along the Z axis, a spheromak wall is round. In cross section, looking along the spheromak minor axis, a spheromak wall is pseudo elliptical.

A spheromak is cylindrically symmetric about the spheromak Z axis (major axis of symmetry) and is mirror symmetric about the spheromak's equatorial plane where Z = 0.

The linear size of a spheromak is characterized by its nominal radius about the Z axis Ro. The spheromak minimum radius about the Z axis is Rc. The spheromak maximum radius about the Z axis is Rs. The height of the spheromak, parallel to the Z axis, is 2 Hm. Hence the relative geometry of a spheromak is specified by the parameters:

(Rc / Ro), (Rs / Ro) and (Hm / Ro). These parameters are the same for all isolated spheromaks. These spheromak relative geometry parameters are the same for all isolated spheromaks and are independent of the spheromak linear size and contained energy. The total energy content Ett of a spheromak is inversely proportional to its nominal radius Ro.

**SPHEROMAK FIELDS:**

For an isolated spheromak in a vacuum, at the center of the spheromak the net electric field is zero. In the region inside the spheromak wall the field is purely toroidal magnetic and the electric field is zero. In the region outside the spheromak wall the magnetic field is purely poloidal and the electric field is locally normal to the spheromak wall. In the far field the electric field is spherically radial. The current circulates within the filament winding that forms the spheromak wall that separates the inside region from the outside region.

The spheromak's mathematical field structure allows semi-stable plasma spheromaks and discrete stable atomic charged particles to exist and act as stores of energy.

**SPHEROMAK WALL:**

Outside the spheromak wall the field energy density U has electric and poloidal magnetic field components.

Thus outside the spheromak wall:

U = Up + Ue

= ([Bp(R, Z)]^2 / 2 Muo) + ([Epsilono / 2] [E(R, Z)]^2)

Inside the spheromak wall the field is purely toroidal magnetic:

U = Ut(R)

= (Muo / 2) [Nt I / 2 Pi Ro]^2 [Ro / R]^2

At the spheromak wall:

Ut(R) = Up(R, Z) + Ue(R, Z)

or

([Bp(R, Z)]^2 / 2 Muo) + ([Epsilono / 2] [E(R, Z)]^2) = (Muo / 2) [Nt I / 2 Pi Ro]^2 [Ro / R]^2

This equation defines the spheromak wall. The functions [Bp(R, Z)]^2 and [Ep(R, Z)]^2 are developed in terms of Ro on the web page titled Theoretical spheromak.

At R = Rc, Z = 0:

Utc = Btc^2 / 2 Muo

= (Bpc^2 / 2 Muo) + (Epsilono / 2) Eec^2]

and at R = Rs, Z = 0

Uts = Bts^2 / 2 Muo

= (Bps^2 / 2 Muo) + (Epsilono / 2) Ees^2]

Bto = Muo Nt I / 2 Pi Ro

or

Uto = Muo Nt^2 I^2 / 8 Pi^2 Ro^2

and

Utc = [Muo Nt^2 I^2 / 8 Pi^2 Ro^2] [Ro^2 / Rc^2]

and

Uts = [Muo Nt^2 I^2 / 8 Pi^2 Ro^2] [Ro^2 / Rs^2]

**SPHEROMAK WINDING CONCEPT:**

An approximate plan view of the current path (filament) of a theoretical elementary spheromak with Np = 3 and Nt = 4 is shown below. The blue lines show the current path on the upper face of the spheromak. The red lines show current path on the lower face of the spheromak. **Note that the current path never intersects itself except at the point where the current starts to retrace its previous path.**

In the diagram yellow shows the region of toroidal magnetic field. Outside the yellow region is a poloidal magnetic field and a spherically radial electric field.

This elementary spheromak winding pattern was generated using a polar graph and formulae of the form:

R = Rc + K [t - t(2N)] where t(2N) = (2 N) (3 Pi / 4) and t(2N) < t < t(2N+1)

and

R = Rs - K [t - t(2N + 1)]

where:

t(2N + 1) = (2N + 1)(3 Pi / 4)

and

t(2N + 1) < t < t(2N + 2)

where:

N = 0, 1, 2, 3.
Use:

Rc = 1000,

Rs = 4105,

K = (4140 / Pi)

Top to bottom connection points were depicted by adjusting the torus Rs to 4045 and Rc to 1060.

**ATOMIC PARTICLE SPHEROMAKS:**

Atomic particle spheromaks have a quantized charge that superficially appears to be at rest with respect to an inertial observer. Isolated stable atomic particles such as electrons and protons hold specific amounts of energy (rest mass). When these particles aggregate with opposite charged particles the assembly emits photons. This photon emission decreases the total amount of energy in the assembly, creating a mutual potential energy well.

In an atomic particle spheromak net charge moves at the speed of light C around a closed spiral filament path of length Lh. The spheromak net charge Qs is uniformly distributed along this current path. The uniform charge distribution along the current path and the uniform current cause constant electric and magnetic fields. The time until an element of net charge retraces its previous path is (1 / F) where:

F = C / Lh

is the characteristic frequency of the spheromak. Note that frequency F increases as filament length Lh decreases.

An isolated spheromak in free space has a distorted ellipse cross section. However, the fields of an atomic particle spheromak may be further distorted by externally imposed electric and magnetic fields.

**LOCATION IN A SPHEROMAK:**

A spheromak has both cylindrical symmetry about its main axis of symmetry and has mirror symmetry about its equatorial plane. Any position in a spheromak can be defined by:

**(R, Z)**

where:

**R** = radius from the main axis (Z axis) of cylindrical symmetry;

and

**Z** = height above (or below) the spheromak equatorial plane.

**Phi** = angle about th Z axis.

**SPHEROMAK CROSS SECTIONAL DIAGRAM:**

The following diagram shows the approximate cross sectional shape of a real spheromak.

Note that the cross section of a real spheromak is pseudo elliptical, not round.

In this diagram on the axis of symmetry is R = 0

At R = 0, Z = 0 the field energy density is maximum and is entirely due to the poloidal magnetic field.

**GEOMETRICAL FEATURES OF A SPHEROMAK:**

Important geometrical features of a spheromak include:

**Rc** = the spheromak wall inside radius on the equatorial plane;

**Rs** = the spheromak wall outside radius on the equatorial plane;

**Ro** = the radius of an imaginary ring radius on the equatorial plane which indicates the nominal linear size of a spheromak;

**Ho** = the spheromak wall height above the equatorial plane at R = Ro;

**Hm** = the maximum spheromak wall height above the equatorial plane;

**Np** = number of poloidal filament turns about the major axis of symmetry;

**Nt** = number of toroidal filament turns about the minor axis in the spheromak;

The subscript **c** refers to spheromak wall inside radius (core) on the equatorial plane;

s refers to the spheromak wall outside radius on the equatorial plane.

When the spheromak filament has passed through the spheromak central hole **Nt** times it has also circled around the main axis of spheromak symmetry **Np** times, after which it reaches the point in its closed path where it originally started.

Define:

**Lt** = 2 Pi K Ro = one purely toroidal filament turn length;

**Lp** = 2 Pi Ro = one purely poloidal filament turn length;

**SPHEROMAK CURRENT PATH LENGTH Lh:**

Electromagnetic spheromaks arise from the electric current formed by distributed net charge Qs circulating at the speed of light C around the closed spiral path of length Lh which defines the spheromak wall. On the equatorial plane the spheromak inner wall minimum radius is Rc and the spheromak's outer wall maximum radius is Rs.

Let Np be the integer number of poloidal currrent path turns in filament length Lh and let Nt be the integer number of toroidal current path turns in filament length Lh.

The spheromak wall contains Nt quasi-toroidal turns equally spaced around 2 Pi radians about the main spheromak axis of symmetry.

Each purely toroidal winding turn has length:

Lt = 2 Pi K Ro

so the purely toroidal spheromak winding length is:

(Nt Lt)

The spheromak wall contains Np poloidal turns which are spaced around the spheromak wall perimeter. The spheromak field calculations are based on the poloidal turns being concentrated at R = Ro. The purely poloidal turn length is:

Lp = 2 Pi Ro

and the purely poloidal winding length is:

(Np Lp)

If the spheromak wall was a straight round solenoid the total spheromak winding length Lh would be given by:

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

However, in reality, due to toroid curvature the formula for Lh is more complicated.

In one spheromak cycle period the poloidal angle advances Np (2 Pi) radians.
In the same spheromak cycle period the toroidal angle advances Nt (2 Pi) radians.

Hence on average:

(poloidal angle advance) / (toroidal angle advance) = Np / Nt

Note that to prevent spheromak collapse, Np and Nt cannot be equal.

**SPHEROMAK SHAPE PARAMETERS:**

The relative shape of a spheromak is defined by the ratios:

(Rc / Ro), (Rs / Ro) and (Hm / Ro)

The winding of a spheromak is defined by the number of poloidal filament turns **Np** and the number of toroidal filament turns **Nt**.

The energy and frequency of a spheromak are a function of Lh and hence Ro.

**SPHEROMAK PARAMETER DEFINITIONS:**

Define:

**R** = radial distance of a point from the major axis of symmetry of spheromak;

**Z** = axial distance of a point above the spheromak equatorial plane (Z is negative for points below the equatorial plane);

**H** = distance of a point on the spheromak wall above the spheromak equatorial plane;

**Ro** = characteristic spheromak radius

**Ho** = H|(R = Ro)

**Hm** = H|(R = Rm)

**Ue** = electric field energy density as a function of position outside the spheromak wall;

**Up** = magnetic field energy density as a function of position outside the spheromak wall;

**U = Ue + Up + Ut** = total field energy density at any position;

**Upor = (Bpo^2 / 2 Muo)** = magnetic field energy density at R = 0, Z = 0;

**Lp = 2 Pi Ro**

**Lt** = 2 Pi K Ro = single turn toroidal winding length

**Utc** = toroidal field energy density at **R = Rc, Z = 0**

**Ut = Uto (Ro / R)^2**

= toroidal magnetic energy density function inside the spheromak wall

**Uts = Uto (Ro / Rs)^2** = toroidal magnetic energy density at R = Rs

**Utc = Uto (Ro / Rc)^2** = toroidal magnetic energy density at R = Rc

**Upor** = _________

= poloidal magnetic field energy density at the origin on the spheromak axis.

**SPHEROMAK FILAMENT PARAMETERS**

Define:

**Ih** = filament current;

**Lh** = overall length of closed filament loop;

**Dh** = center to center distance between adjacent filament paths

**As** = outside surface area of spheromak wall

**Q** = proton net charge

**Qs** = net charge on spheromak

**Nnh** = integer number of negative charge quanta

**Nph** = integer number of positive charge quanta

**Vn** = velocity of negative charge quanta along charge hose

**Vp** = velocity of positive charge quanta along charge hose

**C** = speed of light

**Nr** = Np / Nt

= ratio of two integers which have no common factors. This ratio must be inherently stable.

**SPHEROMAK CHARGE DISTRIBUTION ASSUMPTION:**

Assume that the spheromak charge is uniformly distributed over the filament length.

**EQUATORIAL PLANE:**

On the spheromak's equatorial plane:

**Z = 0**

For points on the spheromak's equatorial plane the following statements can be made:

For **R = 0 ** the net electric field is zero;

For **R < Rc** the toroidal magnetic field is zero;

For **R < Rc** the magnetic field **Bp** is purely poloidal;

For **R = 0** the magnetic field is parallel to the axis of cylindrical symmetry;

For **Rc < R < Rs** the electric field is zero;

For **Rc < R < Rs** the poloidal magnetic field is zero;

For **Rc < R < Rs** the toroidal magnetic field **Bt** is proportional to **(1 / R)**.

For **Rs < R** in free space the electric field **Ero** is spherically radial;

For **Rs < R** in the far field the electric field **Ero** is proportional to **(1 / R^2)**;

For **Rs < R** in free space the toroidal magnetic field is zero;

For **Rs < R** in the far field the poloidal magnetic field **Bp** is proportional to **(1 / R^3)**;

**FILAMENT CURRENT:**

**Ih = [Qp Np Vp + (- Q Ne Ve)] / Lh**

**SPHEROMAK FILAMENT WINDING GEOMETRY:**

A very important issue in understanding natural spheromaks is grasping that:

**Np** cannot equal **Nt** and that Np and Nt can have no common factors other than one. Otherwise the windings would fall on top of one another or the spheromak would collapse.

**SPHEROMAK FILAMENT PATH:**

Consider a quasi-toroidal winding with axial length along the minor axis of:

2 Pi Ro

This winding has Np round poloidal turns with radius Ro at Z = 0 and has Nt distorted elliptical turns about the minor axis with radius [(Rs - Rc) / 2] and height parallel to the Z axis of 2 Ho.

The minor axis of this winding lies along R = Ro, Z = 0 .

The filament winding path is described by:

Z = Ho sin(Theta)
R = Ro + [(Rs - Rc) / 2] cos(Theta)

where Ro > (Rs - Rc)

and

Phi / Theta = Np / Nt

where:

Phi is an angle about the Z axis.

Differentiation of Z and R gives:

dZ = Ho cos(Theta) d(Theta)

dR = - [(Rs - Rc) / 2] sin(Theta) d(Theta)

dY = R d(Phi)

d(Phi) / d(Theta) = (Np / Nt)

= Ho^2 cos^2(Theta)[d(Theta)]^2 + [(Rs - Rc) / 2]^2 sin^2(Theta)[d(Theta)]^2 + R^2 [d(Theta) d(Phi) / d(Theta)]^2

= Ho^2 cos^2(Theta)[d(Theta)]^2 + [(Rs - Rc) / 2]^2 sin^2(Theta)[d(Theta)]^2 + [R^2 Np^2 / Nt^2][d(Theta)]^2

= {Ho^2 cos^2(Theta) + [(Rs - Rc) / 2]^2 sin^2(Theta) + [R^2 Np^2 / Nt^2]}[d(Theta)]^2

dL = {Ho^2 cos^2(Theta) + [(Rs - Rc) / 2]^2 sin^2(Theta) + [R^2 Np^2 / Nt^2]}^0.5 [d(Theta)]

**For the general case of
R = Ro + [(Rs - Rc) / 2] cos(Theta)**

or

R^2 = Ro^2 + 2 Ro [(Rs - Rc) / 2] cos(Theta) + [(Rs - Rc) / 2]^2 cos^2(Theta)

due to zero averaging of terms propotional to cos(Theta) in calculation of L

Hence:

R^2 Np^2 / Nt^2 = Ro^2 Np^2 / Nt^2 + [Np^2 / Nt^2][(Rs - Rc) / 2]^2 cos^2(Theta)

Hence:

dL = {Ho^2 cos^2(Theta) + [(Rs - Rc) / 2]^2 sin^2(Theta) + [R^2 Np^2 / Nt^2]}^0.5 [d(Theta)]

= {Ho^2 cos^2(Theta) + [(Rs - Rc) / 2]^2 sin^2(Theta) + (Ro^2 Np^2 / Nt^2) + [Np^2 / Nt^2][(Rs - Rc) / 2]^2 cos^2(Theta)}^0.5 [d(Theta)]

= {[(Rs - Rc) / 2]^2 + [Ho^2 - [(Rs - Rc) / 2]^2]cos^2(Theta) + (Ro^2 Np^2 / Nt^2) + [Np^2 / Nt^2][(Rs - Rc) / 2]^2 cos^2(Theta)}^0.5 [d(Theta)]

= {[(Rs - Rc) / 2]^2 + Ho^2 cos^2(Theta) + [(Rs - Rc) / 2]^2][(Np^2 / Nt^2)- 1]cos^2(Theta) + (Ro^2 Np^2 / Nt^2)}^0.5 [d(Theta)]

Integrating dL from Theta = 0 to Theta = (Nt 2 Pi) gives:

L ={[(Rs - Rc) / 2]^2 + (Ho^2 / 2) + [(Rs - Rc) / 2]^2][(Np^2 / Nt^2)- 1] / 2 + (Ro^2 Np^2 / Nt^2)}^0.5 [Nt 2 Pi]

or

[L / 2 Pi]^2 = {[(Rs - Rc) / 2]^2 + (Ho^2 / 2) + [(Rs - Rc) / 2]^2][(Np^2 / Nt^2)- 1] / 2 + (Ro^2 Np^2 / Nt^2)} [Nt^2]

= {[(Rs - Rc) / 2]^2 Nt^2 + (Ho^2 / 2) Nt^2 + [(Rs - Rc) / 2]^2][(Np^2 / 2) - (Nt^2 / 2)] + (Ro^2 Np^2)}

= {[(Rs - Rc) / 2]^2 Nt^2 / 2 + (Ho^2 / 2) Nt^2 + [(Rs - Rc) / 2]^2][(Np^2 / 2)] + (Ro^2 Np^2)}

= **{[(Rs - Rc) / 2]^2 / 2 + (Ho^2 / 2)} Nt^2 + {[(Rs - Rc) / 2]^2][(1 / 2)] + (Ro^2)}Np^2}**

For stable operation:

Np^2 = (P - 2 Nt)^2

which gives:

[L / 2 Pi]^2 = {[(Rs - Rc) / 2]^2 / 2 + (Ho^2 / 2)} Nt^2 + {[(Rs - Rc) / 2]^2][(1 / 2)] + (Ro^2)} [P - 2 Nt]^2}

or

{[(Rs - Rc) / 2]^2 / 2 + (Ho^2 / 2)}2 Nt + {[(Rs - Rc) / 2]^2][(1 / 2)] + (Ro^2)} 2 [P - 2 Nt][-2]} = 0

or 2 Np {[(Rs - Rc) / 2]^2][(1 / 2)] + (Ro^2)} 2 = {[(Rs - Rc) / 2]^2 [(1 / 2)]+ (Ho^2 / 2)}2 Nt

or [Np / Nt] = {[(Rs - Rc) / 2]^2 [1 / 2] + (Ho^2 / 2)} / {[(Rs - Rc) / 2]^2](1 / 2) + (Ro^2)} 2

= [1 / 2] {[(Rs - Rc) / 2]^2 + (Ho^2)} / {[(Rs - Rc) / 2]^2 + (2 Ro^2)}

This equation has a stable solution at:

**(Np / Nt) = (1 / 2)**

and

**Ho^2 = 2 Ro^2**

Thus the spheromak wall cross section is a pseudo-ellipse with:

Ho = 2^0.5 Ro

In order for this solution to exist:

(Rs - Rc) < 2 Ro

Note that;

(Np / Nt) ~ (1 / 2)

is independent of the size of (Rs - Rc)

There is a complication that the peak in H for the ellipse occurs at:

R = (Rs + Rc) / 2

whereas the field equations indicate that the peak in H occurs at R = Rm, H = Hm, where:

Rm > [(Rs + Rc)/ 2]

Note that a circular path around the spheromak minor axis is not a solution for Np ~ Nt

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX**EFFECT OF SOLENOID CURVATURE:**

The effect of spheromak curvature is to force the spheromak wall to be pseudo elliptical in cross section.

The ellipse specifying the spheromak wall position is:

(H^2 / 2 Ro^2) + (R - Ro)^2 / [(Rs - Rc) / 2]^2 = 1

Hence the spheromak wall position can be approximated by:

**H^2** = **2 Ro^2 {1 - (R - Ro)^2 / [(Rs - Rc) / 2]^2}**

or

H = +/- 2^0.5 Ro {1 - (R - Ro)^2 / [(Rs - Rc) / 2]^2}^0.5

or

This web page last updated on August 8, 2024.

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