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CHARGE HOSE PROPERTIES

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

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

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

The assumed net uniform static charge per unit length (Qs / Lh) on the charge hose gives the charge hose linear charge density:
Qs / Lh = Rhoh
= [Qp Nph + Qn Nnh) / Lh]

where:
Qp = a positive charge quantum;
Qn = a negative chage quantum;
Nph = a positive integer;
Nnh = a positive integer;
and further implies that in a spheromak with constant net charge:
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 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. Thus charge hose current Ih is uniform and is given by:
Ih = (Qp Nph Vp + Qn Nnh Vn) / Lh

Note that Qpand 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. 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 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. For toroidal windings the value of Dh increases with increasing radius R. Hence when the windings have a toroidal component the average surface charge per unit area decreases with increasing spheromak radius Ro 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 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 tight closed spiral appears to be a toroidal shaped closed surface that is a geometrically stable configuration known as a spheromak wall.

The charge hose exists at the boundary between two mutually orthogonal magnetic fields. In general 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 hose position is stable because at the spheromak wall the energy density immediately inside the wall equals the energy density immediately outside the wall. In a plasma the charge hose current, which is a stream of electrons, persists as long as the free electron momentum is undisturbed.
 

SYMBOL DEFINITIONS:
Define:
Nnh = number of negative charge quanta in the charge hose;
Nph = number of positive charge quanta in the charge hose;
Lh = length of charge hose;
Vn = negative charge quanta axial velocity through the charge hose;
Vp = positive charge quanta 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;
Qs = net charge on a spheromak = Nph Qp + Nnh Qn
Dh = distance between adjacent charge hoses
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 hose turns
Nt = number of toroidal charge hose 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 HOSE IN AN ATOMIC PARTICLE:
A charge hose 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 hose 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 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 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 hose the net momentum flux along the charge hose 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 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)] [Qp Nph Vp + Qn Nnh Vn] / Lh

 

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:
Epsilono = 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 Epsilono
or
Eh = Rhoh / (2 Pi Rh Epsilono)
 

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 Epsilono)

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
= Ih Muo Ih / (2 Pi Dh)
= Muo 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:
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 hoses are in balance if:
Ih^2 / C^2 = Rhoh^2
or
Ih = C Qs / Lh
= Qs Fh

This equation is of great importance because it applies to all spheromaks. This equation relates the current through a spheromak 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 hoses if:
Ih^2 / C^2 = Rhoh^2

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.
 

This formula indicates that an electromagnetic structure can potentially be geometrically stable if it 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 hose current Ih is effectively motion of the net charge:
Qs = (Qp Nph + Qn Nnh)
along the charge hose at the speed of light. Note that due to the presence of opposite charge quanta an individual charge quantum moves at less than the speed of light.
 

AXIAL FIELD ENERGY DENSITY:
Consider a closed ring with radius Ro, net charge Qs, number of poloidal turns Np and ring current Ip where:
Ip = No Ih.

Let Z be the distance along the ring axis with Z = 0 at the center of the ring.

The electric field Ez along the ring axis is given by:
Ez = Qs Z / [4 Pi Epsilno (Ro^2 + Z^2)^1.5]

The electric field energy density Ue along the ring axis is:
Ue = (Epsilono / 2){Qs Z / [4 Pi Epsilno (Ro^2 + Z^2)^1.5]}^2
= Qs^2 Z^2 / [32 Epsilono Pi^2 (Ro^2 + Z^2)^3]

The magnetic field B along the Z axis is given by:
B = (Muo Ip / 2)[Ro^2 / (Ro^2 + Z^2)^1.5]

The magnetic field energy density Um along the Z axis is given by:
Um = B^2 / 2 Muo
= (1 / 2 Muo){(Muo Ip / 2)[Ro^2 / (Ro^2 + Z^2)^1.5]}^2
= (Muo / 8) (Ip)^2 [Ro^4 / (Ro^2 + Z^2)^3]

U = Ue + Um
= {Qs^2 Z^2 / [32 Epsilono Pi^2] + (Muo / 8) (Ip)^2 Ro^4}{1 /(Ro^2 + Z^2)^3}
= {Qs^2 Z^2 Muo C^2/ [32 Pi^2] + (Muo / 8) (Ip)^2 Ro^4}{1 /(Ro^2 + Z^2)^3}
 
= {[(Qs^2 Muo C^2) / (32 Pi^2)][Z^2 + Ro^2 (2 Pi)^2 (Ip)^2 Ro^2 / (Qs^2 C^2)]}{1 /(Ro^2 + Z^2)^3}

If:
(2 Pi)^2 (Ip)^2 Ro^2 / (Qs^2 C^2) = 1
or if:
Ip = Qs C / 2 Pi Ro
= Qs Fp

then:
U = {[(Qs^2 Muo C^2) / (32 Pi^2)][Z^2 + Ro^2]}{1 /(Ro^2 + Z^2)^3}
 
= [Qs^2 Muo C^2 / 32 Pi^2] {1 /(Ro^2 + Z^2)^2}
 

= [Qs^2 Muo C^2 / (32 Pi^2 Ro^4)] [Ro^2 /(Ro^2 + Z^2)]^2
= Uo [Ro^2 /(Ro^2 + Z^2)]^2

This equation suggests the likely form of the energy distribution function outside a spheromak wall.

Try an energy distribution function outside a spheromak wall of the form:
U = Uo [Ro^2 /(Ro^2 + (A R)^2 + Z^2)]^2
where A is a constant of the order of unity. Photographs of plasma spheromaks seem to indicate that A is slightly greater than unity.

Note that a key condition for the required outside energy density function and hence for spheromak existence is:
Ip = [Qs C / 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

Hence:
[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
[Ve / C]^2 ~ [(Ni - Ne) / Ne]^2
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, a charge hose is similar to a nearly neutral plasma in which the two charge types 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 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.
 

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 sums of the electric and magnetic field energy densities on both sides of the charge sheet are equal so that any random change in spheromak wall position normal to the charge sheet 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 hose. A charge sheet forms when there are a large number of locally parallel uniformly spaced charge hose 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 hose due to another nearby parallel charge hose 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 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 Qs that is uniformly distributed over the charge hose length Lh.

In the centwe 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 cylindrically radial. Within the sphereomak the electric fields almost cancel. In the region outside the spheromak wall the magnetic field is purely poloidal and the electric field is spherically radial. The moving charges comprising the spheromak circulate 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 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

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 hose path where it originally started.

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 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 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 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 a toroidal winding component causes Dh to vary over the spheromak surface.

The spheromak wall charge per unit area Sa is inversely proportional to Dh and plays a role in determination of the shape 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 = (Rhoh / Dh)
= [Ih / (Dh C)]

In this formula Ih is constant for a particular charge hose and hence for a particular spheromak wall formed from that charge hose. Hence:
Sa is proportional to (1 / Dh)

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

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 Qs on the charge sheet is:
Integral from X = 0 to X = Lh of:
Sa(X) Dh(X) dX
.
The product:
[Sa(X) Dh(X)]
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
 

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 hose 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 April 14, 2019.

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