XYLENE POWER LTD.

SPHEROMAK WINDING CONSTRAINTS

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

CURRENT FILAMENTS:
Our universe is composed of a large number of circulating closed charge filaments. Each closed charge filament contains one quantum of net electric charge, approximately 1.602 X 10^-19 coulombs which flows along the filament at the speed of light C, approximately 3 X 10^8 m / s. The net charge is uniformly distributed along the filament. In a stable charged particle at every point along the filament the electric and magnetic forces are in balance. For an isolated charged particle in a vacuum that geometry is a spheromak. Hence isolated electrons and protons exhibit spheromak geometry.
 

SPHEROMAK FREQUENCY Fh:
Each spheromak has a current path length Lh and a nominal radius Ro where the ratio:
(Lh / 2 Pi Ro)
is a constant common to all spheromaks. Thus:
Lh = (Lh / 2 Pi Ro) (2 Pi Ro)
and:
Lh Fh = C
or
Fh = C / Lh
= [C /(Lh / 2 Pi Ro)][1 / 2 Pi Ro]

Hence:
Fh = [C / (Lh / 2 Pi Ro)][1 / 2 Pi] [1 / Ro]
 

RADIATION AND MATTER:
A spheromak is a mathematical representation of an isolated charged particle. Electro-magnetic spheromaks are stable energy states. These stable states are reached by emission or absorption of radiation. During radiant energy emission and absorption total system energy and total system momentum are conserved. Charged particles and radiation, both have characteristic frequencies. During photon emission the emitting spheromak's frequency Fh decreases and the amount of propagating radiant energy increases. During photon absorption the absorbing spheromak's natural frequency Fh increases and the amount of propagating radiant energy decreases.

A spheromak has an electric current which follows a closed spiral path that traces out the shape of the wall of a quasi-toroid. The closed current path has both has both toroidal and poloidal circulation components. The quasi-toroidal surface is referred to as the spheromak wall.

The current circulates at the speed of light. At the spheromak geometry the total field energy density just inside the spheromak wall equals the total field energy density just outside the spheromak wall. Hence the electric and magnetic forces are in balance everywhere on the spheromak wall making the spheromak geometrically stable.

The existence, mass and other properties of each spheromak and hence each real atomic particle is governed in part by the prime number P that simultaneously satisfies all of the spheromak constraint equations. Since P, and the spheromak winding parameters Np and Nt must be integers and Np and Nt cannot share common factors the number such prime numbers and hence the number of real atomic particle possibilities is distinctly limited.

If a spheromak's static electromagnetic field energy Ett changes from Ea to Eb and the spheromak frequency Fh changes from Fa to Fb then:
dEtt = (Ea - Eb)
= h (Fa - Fb)
= h dFh

Over time electromagnetic spheromaks in free space will absorb or emit energy until they reach a stable state.
At this stable state the value of (dEtt / dFh) for an electromagnetic spheromak is given by:
(dEtt / dFh) = h, where:
Fh = the natural frequency of the circulating quantum net charge that forms an electromagnetic spheromak and dFh is the frequency of a radiation emitted or absorbed.

This formula is the basis of quantum mechanics. Spheromaks form the static field structure of charged particles with rest mass. Since spheromaks are the main sources and sinks of radiant energy, spheromak properties in large measure determine the radiant energy absorption and emission properties of matter.

SUMMARY OF SPHEROMAK WINDING CONSTRAINTS:
The spheromak winding is governed by multiple spheromak existence constraints including:
1) Np and Nt must both be whole positive integers;
2) Np and Nt cannot be equal;
3) Np and Nt cannot share any integer factors other than unity;
4) If is a change in spheromak state such that Np and/or Nt change the new values of Np and Nt cannot share any integer factors other than unity;
5) Let dNp be a small change in Np. Let dNt be a small change in Nt. Both dNp and dNt must be whole integers;
6) It can be shown that in order to comply with the aforementioned constraints either:
dNp = - 2 dNt
or
2 dNp = - dNt
7) Each prime number has associated with it two families of potential Np and Nt values;
8) Spheromaks stabilize at:
Np = Nt +/- 1;
9) The requirement for spheromaks existence by avoiding common factors in Np and Nt demands that:
dNp = -2 dNt,
which in effect forces the spheromak to operate at its relative energy maximum at:
Np = Nt + 1
 

SPHEROMAK FORMATION:
During formation a spheromak initially contains a higher than normal energy at a non-optimal prime number. The spheromak then proceeds through a series of prime number and dNp and dNt steps such that the spheromak adopts a prime number corresponding to the spheromak energy.
 

NO COMMON FACTORS:
A spheromak will collapse if the number of poloidal turns Np and the number of toroidal turns Nt share a common factor other than one. The spheromak turns must not overlap or intersect. For the same reason Np can never be an integer multiple of Nt or vice versa. Hence Np cannot equal Nt except at:
Np = Nt = 1
 

PRIME NUIMBER THEORY:
Prime number theory together with the requirement for no common factors in Np and Nt gives two formulae for sets of number pairs Np and Nt that share no common factors other than one. Those formula are:
Family A:
P = Np + 2 Nt
or
Np = P - 2 Nt;
where:
P = controlling prime number.
Note that Np is always odd and that:
dNp = - 2 dNt.

The spheromak solution lies at:
Np = Nt + 1
Hence:
P = 3 Nt + 1
or
Nt = (P - 1) / 3
and
Np = (P + 2) / 3

Family B:
P = 2 Np + Nt
where:
P = prime number.
Note that Nt is always odd and that:
dNt = - 2 dNp

Spheromak solution lies at:
Nt = Np + 1
Hence:
P = 3 Np + 1
or
Np = (P - 1) / 3
and
Nt = (P + 2) / 3

If Family A is involved [dNp / dNt] would be - 2 .
If family B is involved:
dNp / dNt = - (1 / 2).

The consequence of using Family B instead of Family A would be that Np would be much smaller with respect to Nt causing the poloidal magnetic field at R = 0, Z = 0 to be insufficient to meet the spheromak's:
Upo = [Uo / K^4]
energy density requirement at R = 0, Z = 0. Hence Family B is excluded from electromagnetic spheromaks by this magnetic field strength requirement. Only Family A corresponds to real electromagnetic spheromaks.

For Family A:
P = Np + 2 Nt
or
Np = (P - 2 Nt)
which gives:
dNp = - 2 dNt
which restricts real spheromaks to Family A.

P = Np + 2 Nt

Note that as Nt increases (Np / Nt) decreases.

At P = 7 the available states are:
Np = 1, Nt = 3
Np = 3, Nt = 2

At P = 11 the available states are:
Np = 1, Nt = 5;
Np = 3, Nt = 4;
Np = 5, Nt = 3;
Np = 7, Nt = 2;
Np = 9, Nt = 1;

At P = 13 the available states are:
Np = 1, Nt = 6;
Np = 3, Nt = 5;
Np = 5, Nt = 4;
Np = 7, Nt = 3;
Np = 9, Nt = 2;
Np = 11, Nt = 1;
Note that between adjacent states:
dNp / dNt = - 2.

When P is large the number of possible states is about half the P value.
 

SPHEROMAK FORMATION:
At the commencement of spheromak formation there is an initial packet of energy. That packet of energy is consistent with a prime number P value under the Np, Nt conditions yhat initially prevail. That choice of P value enables a spheromak existence along a line of constant P of the form:
P = Np + 2 Nt.
Then over time Np and Nt can step along this constant P line in accordance with:
dNp = - 2 dNt
Until the spheromak reaches its energy min‌imum.

At the energy minimum:
dLh / dNt = 0
wh‎ile
Np = (P - 2 Nt)

Recall that when:
d{[Lh / (2 Pi Ro)]^2} = 0
then:
dNp / dNt = - [Nt / Np] {Lt / 2 Pi Ro]^2 / [Lp / 2 Pi Ro]^2
and
dNp / dNt = -2
giving:
2 = [Nt / Np] {Lt / 2 Pi Ro]^2 / [Lp / 2 Pi Ro]^2}

On the spheromak existence line:
Np = (P - 2 Nt)

Hence at the minimum energy state:
2 = [Nt / (P - 2 Nt)] {Lt / (2 Pi Ro)]^2 / [Lp / (2 Pi Ro)]^2}
or
[2 (P - 2 Nt) / Nt] = {Lt / (2 Pi Ro)]^2 / [Lp / (2 Pi Ro)]^2}

The web page titled: Theoretical Spheromak gives:
[Lp / (2 Pi Ro)]^2 = 1
and
[Lt / (2 Pi Ro)]^2 = [Ho / Ro]^2 [(Ho / Ro) (1 / 4 K^2) + 1]
~ 2

Hence:
[2 (P - 2 Nt) / Nt] ~ 2
or
(P - 2 Nt) / Nt ~ 1
or
(Np / Nt) ~ 1

Thus at the spheromak energy minimum:
Np = Nt + 1
or
Np = Nt + 2

This is a very important result in terms of solving for the Fine Structure Constant.

In summary for real spheromaks to exist, Np and Nt conform to the equation:
P = Np + 2 Nt
In each case there is a minimum energy spheromak solution at:
Np = Nt +/- 2

Then Np and Nt have no common factors, which is a condition for spheromak existence. To some extent spheromak sizes are constrained by the availability of suitable prime numbers.
 

SPHEROMAK SOLUTION:
The method of finding the mathematical model for an isolted spheromak is:
1) Find equations for the field energy density both inside and outside the spheromak wall;

2) Solve the field energy density equations simultaniouly to find the equation for the spheromak wall;

3) Find equations for the length Lp of a poloidal turn and for the length Lt of a toroidal turn;

4) Find the equation for the length Lh of the current filament closed loop.

5) Since Lh is inversely proportional to the spheromak energy, at the spheromak energy minimum:
dLh = 0

6) At the spheromak energy minimum apply the condition that:
dNp = - 2 dNt

7) The resulting equation describes spheromak behavior.

8) the solution lies at:
Np = Nt +/- 1
or
Np = Nt +/- 2

9) The current filament length Lh in a spheromak is given by:
[Lh / 2 Pi Ro]^2 = Np^2 [Lp / 2 Pi Ro]^2 + Nt^2 [Lt / 2 Pi Ro]^2

The contained energy of an isolated spheromak is constant. Hence, for an isolated spheromak:
d{[Lh / 2 Pi Ro]^2} = 0.

Hence:
2 Np dNp [Lp / 2 Pi Ro]^2 + 2 Nt dNt [Lt / 2 Pi Ro]^2 = 0
or
dNp / dNt = - [Nt / Np] {Lt / 2 Pi Ro]^2 / [Lp / 2 Pi Ro]^2
where Np and Nt are positive integers and dNp and dNt are small integers. This equation allows a spheromak to make small tradeoffs between Np and Nt with minimum energy consequences.

During these tradeoffs the spheromak must survive, meaning that at all times Np and Nt must have no shared integer factors other than one, which forces:
dNp / dNt = -2.

Typically:
Lp = 2 Pi Ro
and Lt is calculated from the spheromak wall equation.
 

Each prime number P yields a family of (Np / Nt) integer pairs. The spheromak steps along its existence line to find its low energy state which has a specific Nt value from this family.

A stable spheromak needs to be tolerant of outside disturbances that cause fluctuations in Np and Nt. In normal spheromak operation:
Np = Npo and Nt = Nto dNp = - 2 dNt However, for good spheromak stability:
If:
dNt = 1,
then:
dNp = -2
and if:
dNt = - 1
then:
dNp = + 2
to prevent a spheromak collapse.

The issue is that the prime number P will not change but an external disturbance can potentially cause Np and Nt to temporarily deviate from their normal values Npo and Nto. A stable spheromak should not collapse under such a condition.

This tolerance is improved if P is located within a tight cluster of prime numbers such as 1087, 1091, 1093 and 1097.

P = 1087 implies:
Nt = 360, Np = 367;
Nt = 361, Np = 365;
Nto = 362, Npo = 363;
Nt = 363, Np = 361;
Nt = 364,Np = 359;

P = 1091 implies:
Nt = 361, Np = 369;
Nt = 362, Np = 367;
Nto = 363, Npo = 365;
Nt = 364, Np = 363;
Nt = 365, Np = 361;

P = 1093 implies:
Nt = 362, Np = 369;
Nt = 363, Np = 367;
Nto = 364, Npo = 365;
Nt = 365, Np = 363;
Nt = 366, Np = 361;

P = 1097 implies:
Nt = 363, Np = 371;
Nt = 364, Np = 369;
Nto = 365, Npo = 367;
Nt = 366, Np = 365;
Nt = 367, Np = 363;
 

Note that if Np = 363, Nt can be 362, 364, 365, 367
Note that if Np = 365, Nt can be either 361, 363, 364, 366
Note that if Np = 367, Nt can be 360, 362, 363 or 365

Note that if Nt = 365, Np can be 367, 363, 361
Note that if Nt = 364, Np can be 369, 365, 363, 359
Note that if Nt = 363, Np can be 371, 367, 365, 361

For this group of primes the central point appears to be Npo = 365, Nto = 364, 363
The exact Nto value and hence which P value is dominant is not obvious. [Lh / 2 Pi Ro] = [Np P]^0.5

Real spheromaks seem to have a P value of:
P = 1087.

Thus in summary the Fine Structure constant is possibly in part a result of the tight group of the prime numbers 1087, 1091, 1093 and 1097.

However, there is comparable prime number tight groups at:
223, 227, 229, 233;
and at
821, 823, 827, 829;
and at
877, 881, 883, 887;
and at
1297,1301, 1303, 1307;
and at:
1483, 1487, 1489, 1493;
and at 1867, 1871, 1873 and 1877
and at
1993, 1997, 1999, 2003

Thus a prime number group alone is not responsible for the prime number that sets the Fine Structure constant. This prime number selection appears to be primarily the result of electromagnetic theory setting K.
 

PRIME NUMBERS
Prime numbers less than 2003 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193 , 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003
 

This web page last updated October 24, 2022.