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**INTRODUCTION:**

This web page introduces charge filament theory and its role in enabling formation of spheromaks. The behavior of both charged particles and semi-stable plasmas is governed by the mathematics of charge filaments embodied in spheromaks. The first step in understanding spheromaks is understanding the concept of charge filament and how charge filament behavior enables the formation of spheromaks.

**UNIFORM LINEAR CHARGE DENSITY HYPOTHESIS:**

A charge filament is a mathematical construct consisting of a net charge Qs formed from opposite charge components that move axially along a smooth closed path of length Lh. A fundamental hypothesis about a charge filament is that **the net charge per unit length (Qs / Lh) is uniform along the length of the charge filament.**.

Charge filament current continuity implies that the net charge filament current **Ih** is also uniform along the filament.

The assumed uniform net charge per unit length **(Qs / Lh)** on the charge filament gives the charge filament the linear charge density:

**Qs / Lh = Rhoh
= [Qp Nph + Qn Nnh) / Lh]**

where:

Qp = a positive charge quantum;

Qn = a negative charge quantum;

Nph = a positive integer indicating the number of positive charge quanta;

Nnh = a positive integer indicating the number of negative charge quanta;

and further implies that in a spheromak the net charge is:

Qs = Qp Nph + Qn Nnh

.

It is further assumed that positive charge **(Nph Qp)** is uniformly distributed along the charge hose length Lh and moves along the charge filament with uniform axial velocity **Vp**. It is further assumed that negative charge **(Nnh Qn)** is uniformly distributed along the charge filament length Lh and moves in the opposite direction with uniform axial velocity **Vn**.
Thus charge hose current Ih is uniform and is given by:

Ih = (Qp Nph Vp + Qn Nnh Vn) / Lh

Note that **Qp**and **Qn** have opposite signs and **Vp** and **Vn** have opposite signs so that the resulting currents add. The two charge filament ends are joined together to form a closed path. It is assumed that the positive and negative charge streams do not collide.

The total amount of positive charge is not equal to the total amount of negative charge. The net charge:

**Qs = [(Qp Nph) + (Qn Nnh)]**

is uniformly distributed along the hose length **Lh**. Note that Qn is negative.

Hence the net charge / unit length Rhoh is given by:

Rhoh = Qs / Lh

= [(Qp Nph) + (Qn Nnh)] / Lh

Think of the charge filament as being uniformly wound in a single layer on the outside of a distorted torus shaped form having an inner hole radius **Rc** and an outer rim radius **Rs**. The charge filament winding has both toroidal and poloidal components. At any point the center-to-center spacing between adjacent windings is **Dh**. The value of **Dh** increases with increasing radius **R**. Hence the average surface charge per unit area decreases with increasing radius **R** from the spheromak's main axis of symmetry. This issue is important in the spheromak mathematical model.

For a stable charge and energy assembly to exist there must be a stable closed spiral current path which on this web page is referred to as the charge filament. A charge filament is analogous to a coil of garden hose with the two hose ends connected together to form a closed path of length **Lh**. Viewed from a distance this tight closed spiral appears to be a quasi-toroidal shaped closed surface that forms a geometrically stable configuration known as a spheromak wall.

The charge filament exists at the boundary (sspheromak wall) between two mutually orthogonal magnetic fields. The energy density inside the spheromak wall is less than or equal to the energy density outside the spheromak wall and hence forms a potential energy well. At the boundary of this potential energy well the charge filament position is stable because at the spheromak wall the field energy density immediately inside the wall equals the field energy density immediately outside the wall. In a plasma the charge filament current, which is primarily a stream of electrons, persists as long as the free electron momentum is undisturbed.

**SYMBOL DEFINITIONS:**

Define:

**Nnh** = number of negative charge quanta on the charge filament;

**Nph** = number of positive charge quanta on the charge filament;

**Lh** = overall length of charge filament;

**Vn** = negative charge quanta axial velocity through the charge filament;

**Vp** = positive charge quanta axial velocity through charge filament;

**C** = speed of light;

**Q** = net charge on a proton = 1.602 X 10^-19 coulombs;

**Qp** = positive charge quantum

**Qn** = negative charge quantum;

**Qs** = net charge on a spheromak = [Nph Qp + Nnh Qn]

**Dh** = distance between adjacent charge filament turns

**R** = cylindrical radius from the spheromak axis of symmetry

**Z** = height above the torus equatorial plane

**Rc** = minimum value of R on spheromak wall at Z = 0

**Rs** = maximum value of R on spheromak wall at Z = 0

**Np** = number of poloidal charge filament turns

**Nt** = number of toroidal charge filament turns

For the case of a plasma:

Qp = Q;

Qn = - Q;

Ih = [(Q Nph Vp) + (- Q Nnh Vn)] / Lh;

= Q [Nph Vp - Nnh Vn] / Lh

where Vn is negative.

**CHARGE FILAMENT IN AN ATOMIC PARTICLE:**

A charge filament in an atomic particle is simply a net charge Qs uniformly distributed along a closed path of length Lh where:

Qs = (Qp Nph + Qn Nnh)

The charge quanta move along the current path with velocities Vp and Vn.

Hence the charge filament current Ih is given by:

Ih = (Qp Nph Vp / Lh) + (Qn Nnh Vn / Lh)

Note that in an atomic particle the charge quanta have no mass so there is no inertial force acting on the charge.

**CHARGE FILAMENT IN A PLASMA:**

In a plasma free electrons and ions must both follow the charge filament path.

Consider a plasma with electrons and ions at the same location but moving in exact opposite directions in magnetic field B. The electron velocity is Ve. The ion velocity is Vi.

The magnetic force on the electrons is given by:

F = - Q (Ve X B)

The inertial force on the electrons is:

F = Me Ve^2 / Re

where Re is the electron cyclotron radius.

Thus for the electrons:

- Q Ve B = Me Ve^2 / Re

or

Re = Me Ve^2 / (- Q Ve B)

The magnetic force on the ions is given by:

F = + Q (Vi X B)

The inertial force on the ions is:

F = Me Vi^2 / Ri

where R is the ion cyclotron radius.

Thus for the ions:

Q Vi B = Mi Vi^2 / Ri

or

Ri = Mi Vi^2 / (Q Vi) B

In order for electrons and ions to follow the same current path:

Re = Ri

or

Me Ve^2 / (- Q Ve B) = Mi Vi^2 / (Q Vi) B

or

- Me Ve = Mi Vi

or

**Me Ve + Mi Vi = 0**

Recall that Ve and Vi have opposite signs. Thus in a plasma charge filament the net momentum flux along the charge filament axis is zero. If the circulating charged particles in a plasma impact neutral atoms there is unequal exchange of momentum which leads to randomization of a plasma spheromak.

**MAGNETIC FIELD AROUND A CHARGE FILAMENT:**

The net current **Ih** through a charge filament is given by:

**Ih = [(Qp Nph Vp + Qn Nnh Vn) / Lh]**

In this equation **Qp and Vp** are both positive and **Qn and Vn** are both negative.

Define:

**Rh** = radial distance from the axis of the plasma charge filament;

and

**Mu** = permiability of free space.

The magnetic field **Bh** around a charge filament at distance **Rh** is given by:

**Bh 2 Pi Rh = Mu Ih**

or

**Bh = Mu Ih / (2 Pi Rh)
= [Mu / (2 Pi Rh)] [Qp Nph Vp + Qn Nnh Vn] / Lh**

**RADIAL ELECTRIC FIELD AROUND A CHARGE FILAMENT:**

The net charge per unit length **Rhoh** on the charge filament is given by:

**Rhoh = [(Qp Nph + Qn Nnh) / Lh]**

Define:

Epsilono = permittivity of free space.

The radial electric field **Eh** at radial distance **Rh** from the axis of the charge filament is given by;

**Rhoh Lh = 2 Pi Rh Lh Eh Epsilono**

or

**Eh = Rhoh / (2 Pi Rh Epsilono)**

**PARALLEL CHARGE FILAMENTS:**

Now consider two identical parallel charge filaments, each with current **Ih** in the same direction as the current in the other charge filament. The two charge filaments are separated by center to center distance **Dh**.

The electric force per unit length causing the two charge filaments to repel each other is:

**Rhoh Eh = Rhoh^2 / (2 Pi Dh Epsilono)**

The magnetic force per unit length causing the two charge filaments to attract each other is:

**[(Qp Nph Vp Bh) + (Qn Nnh Vn Bh)] / Lh
= Ih Bh
= Ih Muo Ih / (2 Pi Dh)
= Muo Ih^2 / (2 Pi Dh)**

The two parallel charge filaments are in a common plane. Within that plane the charge filaments locally exert no net force on each other if the electric and magnetic forces are in balance. That force balance will exist if:

**Muo Ih^2 / (2 Pi Dh) = Rhoh^2 / (2 Pi Dh Epsilono)**

or

**Muo Ih^2 = Rhoh^2 / (Epsilono)**

or

**Epsilono Mu Ih^2 = Rhoh^2**

However, from Maxwells equations:

**Epsilon Muo = 1 / C^2**

where:

**C** = speed of light.

Hence the forces between two parallel charge filaments are in balance if:

**Ih^2 / C^2 = Rhoh^2**

or

**Ih = C Qs / Lh
= Qs Fh**

where:

Qs = Rhoh Lh

is the net charge on the charge filament;

and

Qs / Lh = net charge per unit length along the charge filament;

and

Ih = C (Qs / Lh)

= charge filament current

The charge filament current can be expressed as:

Ih = Qs Fh

where:

Fh = C / Lh

which is the spheromak natural frequency.

Thus:

Ih = Qs C / Lh

= charge filament current.

Recall that:

Ih = [(Qp Nph Vp + Qn Nnh Vn) / Lh]

Equate the two expressions for Ih to get:

Qs C = (Qp Nph Vp + Qn Nnh Vn)

or

Qs = (Qp Nph Vp + Qn Nnh Vn) / C

In a plasma Nph and Nnh are large so that Vp and Vn are small compared to speed of light C. In a metal Vp may be zero.

The equation:

Ih = Qs C / Lh

is of great importance because it applies to all spheromaks. This equation relates the current through a spheromak's current path filament to the speed of light C, the net charge on the filament Qs, the total length of the filament Lh and the natural frequency:

**Fh = C / Lh**.

Recall that:

**Ih = [(Qp Nph Vp + Qn Nnh Vn) / Lh]**

and

**Rhoh = [(Qp Nph + Qn Nnh) / Lh]**

Hence the forces are in balance in the plane of the two adjacent charge filaments if:

**Ih^2 / C^2 = Rhoh^2**

which enagles spheromak existence.

Notice that this force balance condition is independent of the actual center to center distance **Dh** between adjacent charge filaments. Hence if **Dh** varies slowly over the charge filament length force balance between adjacent charge filament turns is maintained.

**This formula indicates that the electromagnetic structure of a spheromak can potentially be geometrically stable if the spheromak is formed from a long filament in which the current is equal to the net charge multiplied by the speed of light.**

**Ih^2 = Rhoh^2 C^2
= [(Qp Nph Vp + Qn Nnh Vn) / Lh]^2
= [(Qp Nph + Qn Nnh) / Lh]^2 C^2
= [Qs C / Lh]^2
= [Qs Fh]^2**

which implies that in a spheromak the charge filament current Ih is effectively motion of the net charge:

Qs = (Qp Nph + Qn Nnh)

along the charge filament at the speed of light. Note that in a plasma due to the presence of nearly equal numbers of positive and negative charges the net charge is small so that individual charged particles move at much less than the speed of light.

Note that a key condition for the required spheromak outside energy density function and hence for spheromak existence is:

**Ip = [Qs C / 2 Pi Ro]**

Note that Ro is the characteristic radius of the spheromak and Qs is its net charge. However, the spheromak frequency is much reduced because:

Fh = C / Lh

where:

Lh > Np 2 Pi Ro

In a plasma spheromak:

Qp = Q = proton charge

Qn = - Q = electron charge

Nph = Ni = number of ions

Nn = Ne = number of electrons

Vp = ion velocity

Vn = electron velocity

In atomic particles the equation:

**Ih^2 = [Qs C / Lh]^2**

eventually leads to the Planck constant, which is fundamental to quantum mechanics. Hence charge hose theory is fundamental to modern physics.

**FOR THE SPECIAL CASE OF A PLASMA:**

Nph = Ni = number of positive ions

Nnh = Ne = number of free electrons

Ni ~ Ne

Qp = Q

Qn = - Q

Vp = Vi

Vn = Ve

Vi << Ve

[Qp Nph Vp + Qn Nnh Vn]^2 / (Qp Nph + Qn Nnh)^2 = C^2

becomes:

[Q Ni Vi - Q Ne Ve]^2 / (Q Ni - Q Ne)^2 = C^2

or

[Ni Vi - Ne Ve]^2 / (Ni - Ne)^2 = C^2

or

[Ne Ve]^2 / (Ni - Ne)^2 ~ C^2

or

This equation is fundamental to analysis of plasma spheromaks.

**THEORETICAL COMPLEXITY:**

There is theoretical complexity with the concept of a charge filament. In classical electrodynamics the negative and positive charges comprising a charge filament will electrically and magnetically attract each other. However, a charge filament is similar to a nearly neutral plasma in which the two charge types move in opposite directions without collision.

**NATURAL COILING:**

A current carrying plasma filament with a net charge will tend to naturally curl upon itself until the magnetic, electric and inertial forces are in balance. In essence charge filament spontaneously coils until it reaches a dimensionally stable low energy state in the form of a spheromak wall.

In the region inside this wall the magnetic field is purely toroidal. In the region outside this wall the magnetic field is purely poloidal. The net surface charge on the wall causes a radial electric field. This net charge and charge motion configuration is known as a spheromak.

Note that a spheromak wall position will not be physically stable until the field energy densities on both sides of the spheromak wall are equal so that any random deviation in spheromak wall position increases the total system energy. Generally there must be continuous curvature in the spheromak wall to meet this stability requirement.

**SPHEROMAK WALL:**

Imagine that there is a single layer coil of charge filament. A charge sheet forms when there are a large number of locally parallel uniformly spaced charge filament coil turns each with the same current in the same direction. As shown above in the local plane of the spheromak wall there is no net force on any charge filament due to another nearby parallel charge filament provided that:

**[Qp Nph Vp + Qn Nnh Vn]^2 / (Qp Nph + Qn Nnh)^2 = C^2**

**SPHEROMAK CONCEPT:**

Conceptually a spheromak is a closed charge sheet in the shape of a quasi-toroid which provides a closed path for the current in the charge filament. The direction of the charge filament axis within the charge sheet conforms to the spheromak wall surface curvature.

Hence the current path through the charge filament forming a spheromak has both toroidal and poloidal magnetic field components. The spiral charge filament axis gradually changes direction over the surface of the quawi-toroid.

In a spheromak positive and negative charge quanta move along a spiral path that is continuously tangent to the spheromak wall. A spheromak is cylindrically symmetric about the spheromak main axis and is mirror symmetric about the spheromak's equatorial plane. The spheromak wall has a net charge **Qs** that is uniformly distributed over the charge hose length **Lh**.

In the center of the spheromak at R = 0 and Z = 0 the electric field is zero. In the region inside the spheromak wall the magnetic field is purely toroidal and for a simple spheromak the electric field is zero. Outside the spheromak wall the magnetic field is purely poloidal and the electric field is spherically radial. Within the spheromak core the electric fields partially cancel. The net charge circulates at the speed of light along the current path of length Lh within the thin spheromak wall at the boundary between the toroidal and poloidal magnetic fields.

**STABLE SPHEROMAK:**

In a spheromak the motion of the positive and/or negative quantum charges along the charge filament causes current **Ih** and hence poloidal and toroidal magnetic fields. The net charge on the charge filament produces the external electric field. The magnetic force between adjacent charge filament turns balances the electric force between adjacent charge filament turns, allowing the charge filament to form a stable closed path that is the spheromak wall.

**ENERGY STABILITY:**

As long as both negative and positive charge is uniformly distributed along the length of the charge filament and moves a uniform velocity along the charge filament, and as long as the charge filament coil is dimensionally stable, there is no change in the spacial distribution of charge with time and hence there is no emitted or absorbed electromagnetic radiation.

**CONSTANT CHARGE FILAMENT CURRENT:**

An important issue in spheromak analysis is that the charge filament current Ih is the same everywhere on the charge filament.

**CHARGE FILAMENT GEOMETRY:**

Define:

**Lt** = one purely toroidal filament turn length;

**Lp** = one average purely poloidal filament turn length;

**Np** = number of poloidal turns

**Nt** = number of toroidal turns

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

There are **Nt** parallel filament turns that go through the equatorial plane in the central core of the spheromak.

**DISCRETE INTEGER SOLUTIONS:**

There is a further aspect of charge filaments that is important. In order for a spheromak to be stable over time each circuit of the charge filament must be identical to every other such circuit. Hence for the spheromak to be stable the number of toroidal turns **Nt** and the number of poloidal turns **Np** included in length **Lh** must both be integers.

**CHARACTERISTIC FREQUENCY:**

In an atomic particle the time required for movement of net charge:

(Qp Nph + Qn Nnh)

at velocity **C** around the closed charge filament path of length **Lh** gives the spheromak a characteristic frequency **Fh** where:

**Fh = C / Lh**

For a given net electric charge Qs the smaller a spheromak is the more total energy Ett that it traps and the higher is its characteristic frequency **Fh**. If an atomic particle spheromak's energy changes due to photon capture or photon emission while the spheromak net charge Qs remains constant there is a corresponding change in spheromak size and hence there is a corresponding change in the spheromak characteristic frequency **Fh**. An emitted or absorbed photon must reflect both the change in total spheromak energy

**(Ettb - Etta)**

and the change in the spheromak characteristic frequency

(Fhb - Fha).

Note that the emitted or absorbed photon frequency is at the beat frequency difference between the initial spheromak frequency **Fha** and the final spheromak frequency **Fhb**.

Expressed mathematically:

**(Ettb - Etta) = h (Fhb - Fha)
= h [(C / Lhb) - (C / Lha)]**

**SPHEROMAK NET CHARGE:**

Net charge **Qs** on the spheromak is given by:

**Qs = Qp Nph + Qn Nnh**

**SPHEROMAK SURFACE CHARGE DENSITY:**

Charge filament current continuity means that **Ih** is everywhere constant for a particular spheromak. Force balance between adjacent charge filament turns causes the charge filament linear charge density:

**Rhoh = [(Qp Nph + Qn Nnh) / Lh]**

to be uniform everywhere on that spheromak.

The charge per unit area **Sa** on the spheromak surface is:

**Sa = Rhoh / Dh**

where **Dh** is the distance between adjacent plasma hoses. Note that the toroidal winding component causes Dh to vary over the spheromak surface.

The spheromak wall charge per unit area **Sa** is inversely proportional to **Dh**.

**TOTAL SURFACE CHARGE:**

The net charge per unit area **Sa** at any point on spheromak wall is:

**Sa = (Rhoh / Dh)**

where **Dh** is position dependent.

Recall that:

**(Ih / C)^2 = Rhoh^2**

Hence the local spheromak wall surface charge per unit area **Sa** is given by:

**Sa = (Rhoh / Dh)
= [Ih / (Dh C)]**

In this formula

CHECK FROM HERE ON

Recall that:

**Ih = [Qp Nph Vp + (Qn) Nnh Vn] [ 1 / Lh]**

Hence at any particular point on a spheromak wall:

**Sa^2 = [Ih / (Dh C)]^2
= [Qp Nph Vp + (Qn) Nnh Vn]^2 [ 1 / Lh]^2 / (Dh C)^2
= [Qp Nph Vp + (Qn) Nnh Vn]^2 [1 / (Lh Dh C)^2]**

or

Note that because **Sa** is proportional to **(1 / Dh)** both the left hand side and the right hand side of this equation are constant independent of position on the spheromak wall.

The total charge **Qs** on the spheromak wall is:

**Integral from X = 0 to X = Lh of:
Sa(X) Dh(X) dX**.

The product:

is constant.

Hence:

**Qs^2 = [Qp Nph Vp + (Qn) Nnh Vn]^2 / C^2**

**SPECIAL CASES:**

For a plasma with **Ve >> Vi** and **Nnh ~ Nph** the equation for **Qs** simplifies as follows:

**Qs^2 = [Q Ni Vi + (-Q) Ne Ve]^2 / C^2
~ [Q Ne Ve]^2 / C^2**

For an atomic particle:

**Qs^2 C^2 = [Qp Nph Vp + (Qn) Nnh Vn]^2**

This equation indicates that a free atomic charged particle is simply a spheromak with charge filament current:

Ih = Qs C / Lh

**NUCLEAR PARTICLE SPHEROMAKS:**

Quark theory indicates that for the special case of a proton:

Nph = 2;

Nnh = 1;

Qp = (2 Q / 3);

Qn = (- Q / 3);

Qs = 2 Qp + Qn = Q;

**SUMMARY:**

A charge filament forming a spheromak is characterized by a net charge **Qs**, a net filament current **Ih**, a stored static electromagnetic energy Ett, and a filament length Lh. Spheromaks have a characteristic frequency **Fh = C / Lh**. The net charge **Qs** gives the spheromak both internal and external electric fields. The charge motion gives the spheromak an external poloidal magnetic field and an internal toroidal magnetic field. Note that the toroidal magnetic field has two possible directions with respect to the poloidal magnetic field.

This web page last updated October 16, 2022.

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