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**INTRODUCTION:**

This web page introduces charge and current filament theory and its role in enabling formation of stable charge and vector field energy structures known as spheromaks. The behavior of both charged particles and semi-stable plasmas is governed by the mathematics of charged current filaments embodied in spheromaks. The first step in understanding spheromaks is understanding how charged current filament behavior enables the formation of a closed spheromak wall.

**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 closed filament path of length Lh. A fundamental hypothesis about a charge and current filament is that **the net charge per unit length is uniform along the length of the filament.**.

Filament current continuity implies that the 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 filament length Lh and moves along the filament with uniform axial velocity **Vp**. It is further assumed that negative charge **(Nnh Qn)** is uniformly distributed along the filament length Lh and moves in the opposite direction with uniform axial velocity **Vn**.
Thus filament 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.

For a charged particle 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 filment 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 filament as being wound in a single layer on the outside of a distorted torus shaped form having an inner hole radius **Rc** at the centre of the torus and an outer rim radius **Rs**. The 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** slowly changes over the torus surface. 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 filament. The two charge filament ends are 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 filament exists at the boundary (spheromak wall) between two mutually orthogonal magnetic fields. The field energy density inside the spheromak wall is less than or equal to the field energy density outside the spheromak wall and hence forms a potential energy well. The spheromak wall 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 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 filament;

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

**FILAMENT IN AN ATOMIC PARTICLE:**

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

Qs = (Qp Nph + Qn Nnh)

The charge quanta move along the filament 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.

**MAGNETIC FIELD AROUND A FILAMENT:**

The net current **Ih** through a 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

**Muo** = permiability of free space.

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

**Bh 2 Pi Rh = Muo Ih**

or

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

**RADIAL ELECTRIC FIELD AROUND A FILAMENT:**

The net charge per unit length **Rhoh** on the 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 FILAMENTS:**

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

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

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

The magnetic force per unit length causing the two 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 filaments are in a common plane. Within that plane the 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 if:

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

or

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

However, from Maxwells equations:

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

where:

**C** = speed of light.

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

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

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 hence the natural frequency:

**Fh = C / Lh**.

**FORCE BALANCE:**

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 enables 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. It is possible that it is not necessary to have distinct charged filaments as a charged conducting sheath that simultaneously rotates about the spheromak major and minor axes would likely have the same effect.

**GEOMETRIC STABILITY:
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 per unit length 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 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.

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 filament theory is fundamental to modern physics.

**THEORETICAL COMPLEXITY:**

There is theoretical complexity with the concept of a charged filament. In classical electrodynamics the moving negative and positive charges comprising parallel charged filaments will electrically repel and magnetically attract each other.

**NATURAL COILING TO FORM 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 I 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**

A current carrying filament with a net charge will tend to naturally curl upon itself until the magnetic, electric and inertial forces are in balance. In essence a charged 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 an outside 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 everywhere 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 CONCEPT:**

Conceptually a spheromak is a wound filament sheet in the shape of a quasi-toroid which provides a closed path for the current in the charged filament. The direction of the charged filament axis 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 quasi-toroid.

In a spheromak filament 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 filament 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 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 at R = 0, Z = 0 the electric fields 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 filament current **I** 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.

**SUMMARY:**

A charged filament forming a spheromak is characterized by a net charge **Qs**, a net filament current **I**, 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 an external radial electric field. The charge motion gives the spheromak an external poloidal magnetic field and an internal toroidal magnetic field. Note that the internal toroidal magnetic field has two possible directions with respect to the external poloidal magnetic field.

**SPECIAL CASE:**

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;

This web page last updated July 5, 2024.

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