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**INTRODUCTION:**

**INTRODUCTION:**

This web page introduces charge hose 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 hose and spheromaks. The first step in understanding spheromaks is understanding the concept of charge hose and how charge hose behavior leads to the formation of spheromaks.

**UNIFORM LINEAR CHARGE DENSITY HYPOTHESIS:**

Charge hose is a mathematical construct consisting of a line current formed by a string of axially moving charge. A fundamental hypothesis about charge hose is that **the net charge per unit length (Qa / Lh) on the charge hose is uniform along the hose.**.

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

The assumed uniform static charge per unit length **(Qa / Lh)** on the charge hose gives the charge hose linear charge density:

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

and further implies that in a spheromak the surface charge per unit area decreases with increasing radius **R** from the spheromak's main axis of symmetry. This issue is crucial in the spheromak mathematical model.

It is further assumed that positive charge **+ (Np Qp)** is uniformly distributed along the charge hose length Lh and moves along the charge hose with uniform axial velocity **Vp**. It is further assumed that negative charge **(Nnh Qn)** is uniformly distributed along the charge hose length Lh and moves in the opposite direction with uniform axial velocity **Vn**. Note that **Qp**and **Qn** have opposite signs and **Vp** and **Vn** have opposite signs so that the resulting currents add. The two charge hose ends are joined together to form a closed path. The positive and negative charge streams do not collide.

Hence charge hose current **Ih** given by:

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

is uniform along the charge hose.

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

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

is uniformly distributed along the hose length **Lh** and moves at speed of light C.

Think of the charge hose as being uniformly wound in a single layer on the outside of a torus shaped form having an inner hole radius **Rc** and an outer rim radius **Rs**. The charge hose 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 **R**.

For a stable charge and energy assembly to exist there must be a stable closed spiral current path which is referred to herein as the charge hose. A charge hose 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 closed spiral appears to be a toroidal shaped surface that is a geometrically stable electromagnetic configuration known as a spheromak. It may be helpful to think of a spheromak as being somewhat like a "slinky toy" with the two ends connected together.

The charge hose exists at the boundary between two mutually orthogonal magnetic fields. On average the energy density inside the spheromak wall is less than the energy density outside the spheromak wall and hence forms a potential energy well. At the boundary of this potential energy well the charge hose position is stable. In a plasma the charge hose current is maintained as long as the free electron momentum is undisturbed.

**SYMBOL DEFINITIONS:**

Define:

**Nnh** = number of negative charge quanta forming the charge hose;

**Nph** = number of positive charge quanta forming the charge hose;

**Lh** = length of charge hose;

**Vn** = free negative charge axial velocity through the charge hose;

**Vp** = free positive charge axial velocity through charge hose;

**C** = speed of light;

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

**Qp** = positive charge quantum

**Qn** = negative charge quantum;

**Qa** = net charge on a spheromak = Nph Qp + Nnh Qn

**Qe** = free electron charge

**Qi** = ion charge

**Dh** = distance between adjacent hoses

**R** = cylindrical radius from the torus major axis

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

**Rc** = minimum value of R at H = 0

**Rs** = maximum value of R at H = 0

**Np** = number of poloidal hose turns about the torus major axis

**Nt** = number of toroidal hose turns about the torus minor axis

**REAL SPHEROMAKS:**

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

Nph = 2;

Nnh = 1;

Qp = (2 Q / 3);

Qn = (- Q / 3);

Qs = (5 Q / 3);

Qa = Q

For the case of a plasma:

Qp = Q;

Qn = -Q;

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

**CHARGE HOSE IN AN ATOMIC PARTICLE:**

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

Qa = (Qp Nph + Qn Nnh)

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

Hence the charge hose current Ih is given by:

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

However, as shown herein:

Ih = Qa C / Lh

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

**CHARGE HOSE IN A PLASMA:**

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

Consider a plasma with electrons and ions at the same location 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 hose the net momentum flux along the hose axis is zero.

**MAGNETIC FIELD AROUND A CHARGE HOSE:**

The net current **Ih** through a charge hose 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 hose;

and

**Mu** = permiability of free space.

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

**Bh 2 Pi Rh = Mu Ih**

or

**Bh = Mu Ih / (2 Pi Rh)
= [Mu / (2 Pi Rh)](1 / Lh) [Qp Nph Vp - Qn Nnh Vn]**

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

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

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

Define:

**Epsilon** = permittivity of free space.

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

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

or

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

**PARALLEL CHARGE HOSES:**

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

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

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

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

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

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

**Mu Ih^ 2 / (2 Pi Dh) = Rhoh^2 / (2 Pi Dh Epsilon)**

or

**Mu Ih^ 2 = Rhoh^2 / (Epsilon)**

or

**Epsilon Mu Ih^2 = Rhoh^2**

However, from Maxwells equations:

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

where:

**C** = speed of light.

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

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

Recall that:

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

and

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

The forces are in balance in the plane of the two adjacent charge hoses if:

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

or

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

or

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

Qa = (Qp Nph + Qn Nnh)

on the spheromak to move along the charge hose at the speed of light.

In plasmas this equation establishes a relationship between the number of ions **Ni**, the number of plasma electrons **Ne**, the ion velocity **Vi** along the charge hose and the free electron velocity **Ve** along the charge hose. This equation is of great importance because it leads to an existence condition that applies to all spheromaks.

In atomic particles the equation:

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

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

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

**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 charge hose. In classical electrodynamics the negative and positive charges comprising a charge hose will electrically and magnetically attract each other. However, at dimensions of the order of a charge hose diameter classical electrodynamics may be no longer valid. The two charge strings move in opposite directions without collision.

**COLLISION AVOIDANCE:**

Imagine a red ant and a black ant crawling in opposite axial directions along the surface of a closed coil of garden hose. Due to twists that arise during hose coiling, the ants' paths actually spiral around the hose axis once per coil turn. If the two ants follow paths that are suitably staggered around the garden hose axis the two ant's paths are identical in overall length but never collide. The red ant's path represents positive charges moving along the plasma hose with axial velocity **Vp**. The black ant's path represents negative charges moving along the plasma hose with axial velocity **Vn**. Note that velocities **Vp** and **Vn** have opposite signs and that the net charge is non-zero.

**NATURAL COILING:**

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

Note that a charge sheet's position will not be physically stable until the sums of the electric and magnetic field energy densities on both sides of the charge sheet are equal and until any random change in charge sheet position normal to the charge sheet increases the total system energy. Generally there must be curvature in the charge sheet to meet this wall position stability requirement.

The curling of the charge sheet causes it to form a closed surface known as the 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 surface charge on the wall causes a radial electric field. This charge and charge motion configuration is known as a spheromak.

**SPHEROMAK WALL:**

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

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

Since:

|Qp Nph Vp + Qn Nnh Vn| = |net charge| X (speed of light)

quarks do not affect the spheromak's electromagnetic structure.

Note that a spheromak wall's position will not be physically stable unless the sums of the electric and magnetic field energy densities on both sides of the wall are equal and unless any random change in wall position in either direction normal to the wall increases the total spheromak energy. Generally there must be some curvature in the spheromak wall to meet this stability requirement.

**SPHEROMAK CONCEPT:**

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

Hence the charge hose winding forming a spheromak has both toroidal and poloidal magnetic field components. The spiral charge hose axis gradually changes direction over the surface of the 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 **Qa** that is uniformly distributed over the charge hose length **Lh**.

In the central core of the spheromak the electric field is zero. In the region inside the spheromak wall the magnetic field is purely toroidal and the electric field is cylindrically radial. In the region outside the spheromak wall the magnetic field is purely poloidal and the electric field is spherically radial. The plasma electrons and ions circulate in the narrow region at the boundary between the toroidal and poloidal magnetic fields.

**STABLE SPHEROMAK:**

In a spheromak the motion of the positive and/or negative charges along the charge hose causes current **Ih** and hence poloidal and toroidal magnetic fields. The net charge on the charge hose produces electric fields. The magnetic force between adjacent charge hoses balances the electric force between adjacent charge hoses, allowing the charge hose to form a stable closed spiral that forms the toroidal shaped plasma sheet that is the wall of a spheromak.

**ENERGY STABILITY:**

As long as both negative and positive charges are uniformly distributed along the length of the charge hose and move at uniform velocities along the charge hose, and as long as the charge hose 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 HOSE CURRENT:**

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

**CHARGE HOSE LENGTH:**

Define:

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

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

**Np** = number of poloidal turns

**Nt** = number of toroidal turns

Then due to the orthogonality of toroidal and poloidal directions the charge hose length **Lh** is given by:

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

There are **Nt** parallel plasma hoses that go through the equatorial plane in the central core of the spheromak and hence form the spheromak core walls.

**DISCRETE INTEGER SOLUTIONS:**

There is a further aspect of charge hose and plasma hose that is important. In order for a spheromak to be stable over time each circuit of the charge hose or plasma hose 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 hose path of length **Lh** gives the spheromak a characteristic frequency **Fh** where:

**Fh = C / Lh**

For a given net electric charge Qa 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 Qa 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 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 **Qa** on the spheromak is given by:

**Qa = Qp Nph + Qn Nnh**

**SPHEROMAK SURFACE CHARGE DENSITY:**

Plasma hose current continuity means that **Ih** is everywhere constant for a particular spheromak. Force balance between adjacent charge hoses causes the charge hose 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 Dh varies over the spheromak surface.

The spheromak wall charge per unit area **Sa** is inversely proportional to **Dh** and plays an important role in determination of the shape and stability of the spheromak.

**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^2 = (Rhoh / Dh)^2
= [Ih / (Dh C)]^2**

In this formula

Recall that:

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

Hence at any particular point on a charge sheet:

**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 surface.

The total charge **Qa** on the charge sheet 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 hose current:

Ih = Qs C / Lh

**SUMMARY:**

A charge hose forming a spheromak is characterized by a net charge **Qa**, a net axial current **Ih**, a stored electromagnetic energy Ett, and a length Lh. Spheromaks have a characteristic frequency **Fh = C / Lh**. The net charge **Qa** gives the spheromak an external electric field. The energy and charge motion gives the spheromak an external poloidal magnetic field and an internal toroidal magnetic field. The poloidal magnetic field may be interpreted as angular momentum and the toroidal magnetic field may be interpreted as spin. Note that the toroidal magnetic field has two possible directions with respect to the poloidal magnetic field.

This web page last updated August 22, 2016.

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