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SPHEROMAK STRUCTURE:

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

SPHEROMAK STRUCTURE:
This web page uses a simplified analysis to show that net charge circulating around a closed path forms the external field energy density distribution required for the existence of a spheromak. The more elaborate mathematical proof of spheromak wall position stability is developed on the web page titled:THEORETICAL SPHEROMAK. The electric and magnetic fields of a spheromak maintain its quasi-toroidal geometry and to hold a stable amount of energy.

A spheromak quasi-toroid has a major axis and a minor axis. The current follows a closed spiral path which is characterized by Np turns around the quasi-toroid's major axis and by Nt turns around the quasi-toroid's minor axis. This current path has the quasi-toroidal shape of the glaze on a doughnut and is referred to as the spheromak wall. Inside the spheromak wall the magnetic field is toroidal. Outside the spheromak wall the magnetic field is poloidal. Inside the spheromak wall the electric field for an isolated spheromak is zero. Outside the spheromak wall the electric field is normal to the spheromak surface. Within the core of the spheromak the electric field components parallel to the equitorial plane cancel. In the far field the electric field is nearly spherically radial. The spacial distribution of the circulating charge is constant over time. Hence the electric and magnetic field geometry is constant over time.

The spheromak field structure allows semi-stable plasma spheromaks and discrete stable atomic charged particles to exist and act as stores of energy.
 

ELEMENTARY SPHEROMAK:
A plan view of the current path of an elementary spheromak with Np = 3 and Nt = 4 is shown below. The blue lines show the current path on the upper face of the torus. The red lines show current path on the lower face of the torus. Note that the current path never intersects itself except at the point where the current path starts to retrace its previous path.

In the diagram yellow shows the region of toroidal magnetic field. Outside the yellow region is a poloidal magnetic field and a spherical electric field.

In a real stable charged particle typically Np = 222 ____and Nt = 305 _____so the numbers of current path poloidal and toroidal turns is much greater in a real spheromak than is shown on this simple diagram.

This elementary spheromak winding pattern was generated using a polar graph and formulae of the form:
R = Rc + K [t - t(2N)] where t(2N) = (2 N) (3 Pi / 4) and t(2N) < t < t(2N+1)
and
R = Rs - K [t - t(2N + 1)]
where:
t(2N + 1) = (2N + 1)(3 Pi / 4)
and
t(2N + 1) < t < t(2N + 2)
where:
N = 0, 1, 2, 3. Use:
Rc = 1000,
Rs = 4105,
K = (4140 / Pi)
Top to bottom connection points were depicted by adjusting the torus Rs to 4045 and Rc to 1060.
 

ATOMIC PARTICLE SPHEROMAKS:
Atomic particle spheromaks have a quantized charge that superficially appears to be at rest with respect to an inertial observer. Isolated stable atomic particles such as electrons and protons hold specific amounts of energy (rest mass). When these particles aggregate with opposite charged particles the assembly emits photons. This photon emission decreases the total amount of energy in the assembly creating a mutual potential energy well.

In an atomic particle spheromak current moves uniformly around a closed spiral path. The apparent net charge motion is at the speed of light. The spheromak net charge is uniformly distributed along the current path. The uniform charge distribution along the current path and uniform current cause constant electric and magnetic fields. The time until the current retraces its previous path is (1 / Fh) where Fh is the characteristic frequency of the spheromak.

A spheromak in free space has a slightly elliptical cross section. However, the fields of an atomic particle spheromak may be distorted by external electric and magnetic fields.
 

PLASMA SPHEROMAKS:
Plasma spheromaks are composed of free electrons and ions that move along a stable closed path. The individual particles move at less than the speed of light but the net charge circulates at the speed of light.

Plasma spheromaks can be generated and photographed in a laboratory. A plasma spheromak in free space has a nearly round cross section. However, in a laboratory the field configuration of a plasma spheromak may be affected by the proximity of a metal enclosure containment wall.

The image below shows a plasma spheromak photograph made by General Fusion Inc.


An important feature of this photograph is the ratio of the spheromak outside radius Rs to its inside radius Rc. On this photograph the ratio:
(Rs / Rc) ~ 4.0.
However, this ratio may be affected by the proximity of the enclosure wall. Theoretical analysis on this web site indicates that in free space this ratio is close to:
(Rs / Rc) = 4.1047.

For plasma spheromak stability there must be a slight separation between the positive ion and negative electron paths to prevent inter-particle scattering. Further, to prevent scattering causing loss of energy from circulating plasma free electrons the concentration of neutral ionizable atoms in the vacuum chamber containing the spheromak must be very small.

The amount of field energy trapped by a plasma spheromak can be significant. In the Plasma Impact Fusion (PIF) process the total field energy Ett of each injected plasma spheromak, above and beyond the kinetic energy of the plasma spheromak's constituant particles, must be at least: 3000 J.
 

SPHEROMAK CONCEPT:
Conceptually a spheromak wall is a quasi-toroidal surface formed from the closed current path of a spheromak. The current path within the spheromak wall conforms to the quasi-toroid surface curvature.

The magnetic field of a spheromak has both toroidal and poloidal components. The current path axis gradually changes direction over the surface of the quasi-toroid following a closed spiral.

In a spheromak quantized positive and/or negative charge moves along a closed spiral path tangent to the spheromak wall. A spheromak is cylindrically symmetric about the spheromak major axis and is mirror symmetric about the spheromak's equatorial plane. The net charge Qs is uniformly distributed over the current path length Lh.

For an isolated spheromak in a vacuum, at the center of the spheromak the net electric field is zero. Inside the spheromak wall the magnetic field is purely toroidal and the electric field is zero. In the region outside the spheromak wall the magnetic field is purely poloidal. Outside the spheromak wall the electric field is normal to the spheromak wall and is radial in the far field. The current circulates within the spheromak wall which forms the interface between the toroidal and poloidal magnetic fields.
 

LOCATION IN A SPHEROMAK:
A spheromak has both cylindrical symmetry about its main axis of symmetry and has mirror symmetry about its equatorial plane. A position in a spheromak can be defined by:
(R, Z)
where:
R = radius from the main axis of cylindrical symmetry;
and
Z = height above (or below) the spheromak equatorial plane.
 

SPHEROMAK CROSS SECTIONAL DIAGRAM:
The following diagram shows the approximate cross sectional shape of a real spheromak.

In this diagram:
Rc = 1.0
Rs = 4.1
Rf = 2.55
Zf = 2.0
The axis of symmetry is R = 0.0

At R = 0, Z = 0 the energy density is maximum and is entirely due to the poloidal magnetic field.

Note that a theoretical spheromak in free space is quasi-toroidal with an apparent elliptical cross section.
 

GEOMETRICAL FEATURES OF A SPHEROMAK:
Important geometrical features of a spheromak include:
Rc = the spheromak wall inside radius on the equatorial plane;
Rs = the spheromak wall outside radius on the equatorial plane;
Rf = the value of R at the spheromak end where: Z = Zf;
(2 |Zf|) = the overall spheromak length;
Np = number of poloidal charge hose turns about the major axis of symmetry;
Nt = number of toroidal charge hose turns about the minor axis in the spheromak;

The subscript c refers to spheromak "core" surface on the equatorial plane;
The subscript f refers to the "funnel edge" at the spheromak end;
The subscript s refers to the spheromak outer "surface" on the equatorial plane.

In order to understand the material on this web page it is essential for the reader to study the spheromak cross sectional diagram and to identify the above mentioned parameters.

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

Define:
Lt = one purely toroidal turn length;
Lp = one average purely poloidal turn length.
 

SPHEROMAK CURRENT PATH LENGTH Lh:
Electromagnetic spheromaks arise from the electric current formed by distributed net charge Qs circulating at the speed of light C around the closed spiral path of length Lh which defines the spheromak wall. On the equatorial plane measured from the main axis of symmetry the spheromak inside radius is Rc and the spheromak outside radius is Rs.

Let Np be the integer number of poloidal currrent path turns in Lh and let Nt be the integer number of toroidal current path turns in Lh.

The spheromak wall contains Nt quasi-toroidal turns equally spaced around 2 Pi radians in angle Theta about the main spheromak axis of symmetry.
Each purely toroidal winding turn has length:
2 Pi (Rs - Rc) Kc / 2 = Pi (Rs - Rc) Kc
so the purely toroidal spheromak winding length is:
Nt Pi (Rs - Rc) Kc

Note that for a round spheromak cross section toroid Kc = 1. If for an elliptical cross section spheromak:
A > 1
then:
Kc > 1

The spheromak wall contains Np poloidal turns which are equally spaced around the ellipse perimeter. The average purely poloidal turn length is:
2 Pi (Rs + Rc) / 2 = Pi (Rs + Rc)
and the purely poloidal winding length is:
Np Pi (Rs + Rc)

In one spheromak cycle period the poloidal angle advances Np (2 Pi) radians. In the same spheromak cycle period the toroidal angle advances Nt (2 Pi) radians.
Hence:
(poloidal angle advance) / (toroidal angle advance) = Np / Nt

While a current point moves radially outward from Rc to Rs the toroidal angle advance is Pi radians and the toroidal travel is Lt / 2. The corresponding distance along the equatorial outer circumference is:
Pi (Np / Nt) Rs.
Thus Pythagoras theorm gives the current point travel distance along the winding for the toroidal half turn as:
[(Lt / 2)^2 + ( Pi Np Rs / Nt)^2]^0.5

While the current point moves radially inward from Rs to Rc the toroidal angle advance is Pi radians and the toroidal point travel is Lt / 2. The corresponding distance along the equatorial inner circumference is:
Pi (Np / Nt) Rc.
Thus Pythagoras theorm gives the current point travel distance along this winding toroidal half turn as:
[(Lt / 2)^2 + ( Pi Np Rc / Nt)^2]^0.5

Thus the total winding length Lh is:
Lh = Nt [(Lt / 2)^2 + (Pi Np Rs / Nt)^2]^0.5
+ Nt [(Lt / 2)^2 + ( Pi Np Rc / Nt)^2]^0.5
 
= [(Nt Lt / 2)^2 + (Pi Np Rs)^2]^0.5
+ [(Nt Lt / 2)^2 + (Pi Np Rc)^2]^0.5

Recall that:
Lt = [Kc Pi (Rs - Rc)]
and from spheromak geometry:
Rs = Ro So / A
and
Rc = Ro / A So

Thus:
Lh = {(Nt Lt / 2)^2 + (Pi Np Rs)^2}^0.5
+ {(Nt Lt / 2)^2 + (Pi Np Rc)^2}^0.5
 
= {(Nt [Kc Pi (Rs - Rc)] / 2)^2 + (Pi Np Rs)^2}^0.5
+ {(Nt [Kc Pi (Rs - Rc)] / 2)^2 + (Pi Np Rc)^2}^0.5
 
= {[Nt Kc Pi (Rs - Rc) / 2]^2 + (Pi Np Rs)^2}^0.5
+ {[Nt Kc Pi (Rs - Rc) / 2]^2 + (Pi Np Rc)^2}^0.5
 
= {[Nt Kc Pi ((Ro So / 2 A) - (Ro / 2 A So))]^2 + (Pi Np (Ro So / A))^2}^0.5
+ {[Nt Kc Pi ((Ro So / 2 A) - (Ro / 2 A So))]^2 + (Pi Np (Ro / A So))^2}^0.5
 
= [Pi Ro / A]{[Nt Kc ((So / 2) - (1 / 2 So))]^2 + [Np So]^2}^0.5
+[Pi Ro / A] {[Nt Kc ((So / 2) - (1 / 2 So))]^2 + [(Np / So)]^2}^0.5
 
= [Pi Ro / 2 So A] {[Nt Kc ((So^2) - (1))]^2 + [Np (2 So^2)]^2}^0.5
+[Pi Ro / 2 So A] {[Nt Kc ((So^2) - (1))]^2 + [2 Np]^2}^0.5
 
= [Pi Ro / 2 So A] {[Nt Kc (So^2 - 1)]^2 + [Np (2 So^2)]^2}^0.5
+[Pi Ro / 2 So A] {[Nt Kc (So^2 - 1)]^2 + [2 Np]^2}^0.5
 
= [Pi Ro Nt / 2 So A] {[Kc (So^2 - 1)]^2 + [(Np / Nt) (2 So^2)]^2}^0.5
+[Pi Ro Nt / 2 So A] {[Kc (So^2 - 1)]^2 + ([Np / Nt][2])^2}^0.5

Thus:
[Lh A / 2 Pi Ro]
= [Nt / 4 So] {[Kc (So^2 - 1)]^2 + (Np / Nt)^2 [2 So^2]^2}^0.5
+ [Nt / 4 So]{[Kc (So^2 - 1)]^2 + (Np / Nt)^2 [2]^2}^0.5

 
= Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [2 So^2 / 4 So]^2}^0.5
+ Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [(2 / 4 So)]^2}^0.5
 
= + Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
+ Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

= [Lh A / 2 Pi Ro]
=La + Lb
where:
La = Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
and
Lb = Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

This equation is the result of spheromak geometric analysis.
 

SPHEROMAK SHAPE:
Lp = 2 Pi [(Rs + Rc) / 2]
= Pi (Rs + Rc)

Lt = 2 Pi [(Rs - Rc) Kc / 2]
= Pi (Rs - Rc) Kc

Note that for a spheromak with a round cross section:
Kc = 1.0000

In general:
(Lp / Lt) = [Pi (Rs + Rc)] / [Pi (Rs - Rc) Kc]
= (Rs + Rc) / (Rs - Rc) Kc
or
Kc (Rs - Rc)(Lp / Lt) = (Rs + Rc)
or
Rs [(Kc Lp / Lt) - 1] = Rc [(Kc Lp / Lt) + 1]
or
(Rs / Rc) = [(Kc Lp / Lt) + 1] / [(Kc Lp / Lt) - 1]
= [Kc Lp + Lt] / [Kc Lp - Lt]

Thus the shape of the spheromak is defined by the ratio:
So^2 = (Rs / Rc)
= [Kc Lp + Lt] / [Kc Lp - Lt].

The energy and frequency of a spheromak also involve Np and Nt.

There are Nt nearly parallel charge hoses that go through the equatorial plane in the central core of the spheromak and hence form the spheromak inner wall.
 

SPHEROMAK PARAMETER DEFINITIONS:
The remainder of this web page is devoted to mathematical analysis of undistorted ideal spheromaks.

Define:
Ue = electric field energy density as a function of position, Ue = 0 at R = 0, Z = 0;
Um = magnetic field energy density as a function of position;
U = Ue + Um = total field energy density as a function of position;
Uo = maximum total field energy density at R = 0, Z = 0;
Umo = (Bpo^2 / 2 Mu) = Uo = maximum magnetic field energy density at R = 0, Z = 0;
Upo = Uo = maximum poloidal field energy density at R = 0, H = 0;
R = radial distance of a point from the major axis of symmetry of spheromak;
Z = distance of a point above the spheromak equatorial plane (Z is negative for points below the equatorial plane);
Ro = nominal spheromak radius defined by: Ro^2 = Rs Rc
(Rx, Zx) = a point X located at R = Rx, Z = Zx
Zf = maximum value of |Z| on the spheromak wall
2 Zf = spheromak overall length
Rf = radius of spheromak charge sheet at Z = Zf and at Z = - Zf
Lpo = 2 Pi Ro = 2 Pi (Rs Rc)^0.5
Lp = Pi (Rs + Rc)
Lt = Pi (Rs - Rc) Kc
Uc = total field energy density at R = Rc, Z = 0
Zs = spheromak wall height at radius R defined by:
Zs^2 = (A / B)^2 (Rs - R)(R - Rc)
Ut(R) = Uc (Rc / R)^2
= energy density function inside the spheromak wall
Up(R, Z) = Upo [Ro^2 / (Ro^2 + (A R)^2 + (Z)^2)]^2
= poloidal magnetic field plus radial electric field energy density function outside the spheromak wall
(Rx, Zx) = a point located at R = Rx, Z = Zx
 

SPHEROMAK CHARGE HOSE PARAMETERS
Define:
Ih = charge hose current;
Lh = length of closed loop of charge hose;
Dh = center to center distance between adjacent charge hoses
As = outside surface area of spheromak wall
Q = proton net charge
Qs = net charge on spheromak
Nnh = integer number of negative charge quanta
Nph = integer number of positive charge quanta
Vn = velocity of negative charge quanta along charge hose
Vp = velocity of positive charge quanta along charge hose
C = speed of light
Nr = Np / Nt
= ratio of two integers which have no common factors. This ratio must be inherently stable.
 

SPHEROMAK CHARGE DISTRIBUTION ASSUMPTION:
Assume that the spheromak charge is uniformly distributed along the charge hose length.
 

EQUATORIAL PLANE:
On the spheromak's equatorial plane:
Z = 0
For points on the spheromak's equatorial plane the following statements can be made:

For R < Rc the net electric field is zero;
For R < Rc the toroidal magnetic field Btoc = 0
For R < Rc the magnetic field Bpoc is purely poloidal;
For R = 0 the magnetic field is parallel to the axis of cylindrical symmetry;

For Rc < R < Rs the electric field is zero;
For Rc < R < Rs the poloidal magnetic field Bp = 0;
For Rc < R < Rs the toroidal magnetic field Bt is proportional to (1 / R).

For Rs < R in free space the electric field Ero is spherically radial;
For Rs < R in the far field the electric field Ero is proportional to (1 / R^2);
For Rs < R in free space the toroidal magnetic field Bt = 0;
For Rs < R in the far field the poloidal magnetic field Bp is proportional to (1 / R^3);

CHARGE HOSE CURRENT:
Ih = [Qp Np Vp + (- Q Ne Ve)] / Lh
 

SPHEROMAK WINDING GEOMETRY:
A very important issue in understanding natural spheromaks is grasping that:
Np cannot equal Nt and that they can have no common factors other than one. Otherwise the windings would fall on top of one another.
 

SPHEROMAK CHARACTERISTIC FREQUENCY Fh:
A spheromak's characteristic frequency Fh is given by:
Fh = C / Lh
where:
C = speed of light.

This equation relates the change in spheromak frequency to a change in Lh assuming that the spheromak geometry is stable so that the parameters Np, Nt, So and Kc all remain constant. To find the Planck constant we need to find the change in spheromak energy with respect to a change in Ro.

The spheromak energy will have to be expressed in terms of the toroidal magnetic energy density:
Bt = Muo Nt Ih / (2 Pi R)
= Muo Nt Qs C / (2 Pi R Lh)
or
Ut = Bt^2 / 2 Muo
= [Muo Nt Qs C / (2 Pi R Lh)]^2 / 2 Muo
= [Nt Qs C / Pi R Lh]^2 / 8 Muo
which is the energy density inside the spheromak wall.

To proceed we need to quantify the total spheromak energy and find how it changes with Rc.

FIND AN EXPRESSION FOR THE ENERGY DENSITY AS A FUNCTION OF POSITION OUTSIDE THE SPHEROMAK WALL:

AXIAL ELECTRIC FIELD ENERGY DENSITY DUE TO A RING OF CHARGE:
Assume that a thin ring of radius "Ro" has net charge Qs. Then the linear charge density along the thin ring is:
Qs / (2 Pi Ro)
and an element of charge is:
dQ = [Qs / (2 Pi Ro)] dL
where dL is an element of the ring's circumferential length.

Consider a point at distance "Z" along the ring axis, where Z = 0 on the ring plane.

The electric field along distance (Ro^2 + Z^2)^0.5 due to charge dQ is:
dE =(1 / 4 Pi Epsilon) [dQ / (Ro^2 + Z^2)]
where:
Epsilon = permittivity of free space

The component of this electric field along the ring axis is:
dE = (1 / 4 Pi Epsilon) [dQ / (Ro^2 + Z^2)] cos(Theta)
where:
cos(Theta) = Z / (Ra^2 + Z^2)^0.5

The net electric field E at distance Z along the ring axis is:
E = (1 / 4 Pi Epsilon) [Qs / (Ro^2 + Z^2)] cos(Theta)
= (1 / 4 Pi Epsilon) [Qs / (Ro^2 + Z^2)] [Z / (Ro^2 + Z^2)^0.5]
= (1 / 4 Pi Epsilon) [Qs Z / (Ro^2 + Z^2)^1.5]

The electric field energy density Ue at Z = Z, R = 0 is given by:
Ue = (Epsilon / 2) E^2
= (Epsilon / 2){(1 / 4 Pi Epsilon) [Qs Z / (Ro^2 + Z^2)^1.5]}^2
= [Qs^2 / (32 Pi^2 Epsilon)] [Z^2 / (Ro^2 + Z^2)^3]
= [Mu C^2 Qs^2 / (32 Pi^2)] [Z^2 / (Ro^2 + Z^2)^3]

Note that at Z = 0 the net electric field is zero and the electric field energy density is zero.

In the far field where:
Z >> Ro
then along the ring axis in the far field:
Ue = [Qs^2 / (32 Pi^2 Epsilon)] [1 / Z^4]

Thus for Z >> Ro along the ring axis the electric field energy density Ue is proportional to (1 / Z)^4.
 

AXIAL EXTERNAL MAGNETIC FIELD ENERGY DENSITY DUE TO A RING OF CURRENT:
The law of Biot and Savart gives an element of magnetic field dB at a measurement point on the axis of a ring due to an electric current I is:
dB = (Mu / 4 Pi) Ip dL X R / |R|^3
where:
Ip = poloidal current around the ring
|R| = distance from current element Ip dL to the measurement point
Mu = magnetic permeability of free space
dL = an element of length along the direction of electric current flow around the ring
[R / |R|] = unit vector along the direction of R

Consider a ring of radius "Ro" and an axial measurement point at distance "Z" from the current ring along the ring axis.

Then along the ring axis the net magnetic field B at the measurement point is axial and is given by:
B = [(Mu / 4 Pi) Ip 2 Pi Ro / (Ro^2 + Z^2)] sin(Theta)
where:
(Theta) = angle between unit vector (R / |R|) and the ring axis.

However:
sin(Theta) = Ro / (Ro^2 + Z^2)^0.5

Thus the net magnetic field B along the current ring axis due to ring current I is given by::
B = [(Mu / 4 Pi) Ip 2 Pi Ro / (Ro^2 + Z^2)] sin(Theta)
= [(Mu / 4 Pi) Ip 2 Pi Ro / (Ro^2 + Z^2)][ Ro / (Ro^2 + Z^2)^0.5]
= [(Mu Ip) / 2] [Ro^2 / (Ro^2 + Z^2)^1.5]

The magnetic field energy density along the ring axis at Z = Z is:
Um = B^2 / 2 Mu
= [(Mu Ip) / 2]^2 [Ro^2 / (Ro^2 + Z^2)^1.5]^2 / 2 Mu
= [Mu Ip^2 / 8] [Ro^4 / (Ro^2 + Z^2)^3]

Thus for Z >> Ro the magnetic field energy density Um is proportional to (1 / Z)^6. Hence at large distances the magnetic field energy density becomes negligibly small as compared to the electric field energy density.

At the center of the ring where Z = 0 the magnetic field is:
Bpo = [(Mu Ip) / (2 Ro)

The corresponding magnetic field energy density Umo at the center of the ring is:
Umo = Bpo^2 / 2 Mu
= [Mu Ip^2 / 8 Ro^2]

Note that Umo is the magnetic field energy density at R = 0, Z = 0 corresponding to current Ip circulating at R = Ro.

Along the ring axis the total electromagnetic energy density U is given by:
U = Ue + Um
= [Mu C^2 Qs^2 / (32 Pi^2)] [Z^2 / (Ro^2 + Z^2)^3] + [Mu Ip^2 / 8] [Ro^4 / (Ro^2 + Z^2)^3]
= [Mu C^2 Qs^2 / (32 Pi^2)] {[Z^2 / (Ro^2 + Z^2)^3] + {[Mu Ip^2 Ro^2 / 8] /[Mu C^2 Qa^2 / (32 Pi^2)]} [Ro^2 / (Ro^2 + Z^2)^3]

If:
{[Mu Ip^2 Ro^2 / 8] /[Mu C^2 Qs^2 / (32 Pi^2)]} = 1
or if:
{[ Ip^2 Ro^2 ] / [ C^2 Qs^2 / (4 Pi^2)]} = 1
or if
{[ Ip^2 Ro^2 4 Pi^2 ] / [ C^2 Qs^2]} = 1
or if:
{[ Ip Ro 2 Pi ] / [ C Qs]} = 1
or if
Ip = C Qs / (2 Pi Ro)
then the total energy density U is given by:
U = [Mu C^2 Qs^2 / (32 Pi^2)] [1 / (Ro^2 + Z^2)^2]

Assume that the net static charge Qs circulates around the ring at the speed of light or that Qs is a net charge consisting of both positive and negative charges circulating in opposite directions so that the requirement on Ip is met. This is the basic condition for the existence of a spheromak. In order to exist a spheromak requires both a net charge Qs and a poloidally circulating current Ip. As shown on the web page titled PLASMA HOSE the charge hose existence requires that:
Ih = Qs C / Lh

Note that the net charge Qs of an atomic particle might consist of the sum of positive and negative charge quanta known as quarks that individually circulate at less than the speed of light, but due to the constraints of charge hose behave as if the net charge is circulating at the speed of light. Hence:
Ip = [Qs C / 2 Pi Ro]
giving:
Upo = [Mu Ip^2 / 8 Ra^2]
= [Mu [Qs C / 2 Pi Ro]^2 / 8 Ro^2]
= [(Mu Qs^2 C^2) / (32 Pi^2 Ro^4)]

The magnetic field energy density Um along the ring axis is given by:
Um = [Mu Ip^2 / 8] [Ro^4 / (Ro^2 + Z^2)^3]
= [Mu [Qs C / 2 Pi Ro]^2 / 8] [Ro^4 / (Ro^2 + Z^2)^3]
= [Mu Qs^2 C^2 / 32 Pi^2] [Ro^2 / (Ro^2 + Z^2)^3]
 

Ue = [Mu C^2 Qs^2 / (32 Pi^2)] [Z^2 / (Ro^2 + Z^2)^3]
 

TOTAL FIELD ENERGY DENSITY OUTSIDE THE SPHEROMAK WALL:
The magnetic field energy density Um along the ring axis is given by:
Um = [Mu Qs^2 C^2 / 32 Pi^2] [Ro^2 / (Ro^2 + Z^2)^3]
 

Recall that the electric field energy density Ue along the ring axis was given by:
Ue = [Mu C^2 Qs^2 / (32 Pi^2)] [Z^2 / (Ro^2 + Z^2)^3]

Then along the ring axis the total field energy density U is given by:
U = Um + Ue
= [Mu Qs^2 C^2 / 32 Pi^2] [Ro^2 / (Ro^2 + Z^2)^3] + [Mu C^2 Qs^2 / (32 Pi^2)] [Z^2 / (Ro^2 + Z^2)^3]
= [Mu Qs^2 C^2 / 32 Pi^2]{[Ro^2 / (Ro^2 + Z^2)^3] + [Z^2 / (Ro^2 + Z^2)^3]}
= [Mu Qs^2 C^2 / 32 Pi^2]{[(Ro^2 + Z^2) / (Ro^2 + Z^2)^3]
= [Mu Qs^2 C^2 / 32 Pi^2] / [(Ro^2 + Z^2)^2]

This expression, which applies along the axis of a thin charge and current ring of radius Ro, suggests several likely features of the energy density function of a spheromak outside the spheromak wall. At the center of the spheromak where Z = 0, Ue = 0 the energy density must be finite at about:
Uo = Umo = [Mu Qs^2 C^2 / 32 Pi^2 Ro^4]
and elsewhere along the spheromak axis the total energy density will likely be given by:
U = [Mu Qs^2 C^2 / 32 Pi^2] / [(Ro^2 + Z^2)^2]

The spacial geometry of a spheromak is more complicated than a thin charge and current ring. However, the above simple analysis gives an indication of the general form of a spheromak's energy density function outside the spheromak wall. The advantage of the above analysis is that it is simple. It is easy to get lost in the mathematical complexity of the detailed spheromak analysis.
 

However, in general the distance from the center of the spheromak is (R^2 + Z^2) and the effective spheromak radius is Ro so we can reasonably guess that the spheromak total energy density function outside the spheromak wall will take the form:
U = [Mu Qs^2 C^2 / 32 Pi^2] / [(Ro^2 + (A R)^2 + (B Z)^2]^2
= [Mu Qs^2 C^2 / 32 Pi^2 Ro^4] [(Ro^2) / (Ro^2 + (A R)^2 + (B Z)^2)]^2
= Uo [(Ro^2) / (Ro^2 + (A R)^2 + (B Z)^2)]^2
where:
Uo = [(Mu Qs^2 C^2) / (32 Pi^2 Ro^4)]

Then for R > Rc the magnetic field energy density outside the spheromak wall will likely be:
Um = Umo [(Ro^2 Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
and the electric field energy density outside the spheromak wall will likely be:
Ue = Ueo [(((A R)^2 + (B Z)^2) Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
where:
Umo = Ueo = Uo

For R < Rc the magnetic field energy density outside the spheromak wall will likely be:
Um = Umo [(Ro^2 + (A R)^2) Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
and the electric field energy density outside the spheromak wall will likely be:
Ue = Ueo [(Z^2 Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
where:
Umo = Ueo = Uo

Note that both Ue and Um have step changes at R = Rc but the steps are equal so that U is a smoothly continuous function.

Note that for R > Rc:
U = Um + Ue
= Umo [(Ro^2 Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
+ Ueo [(((A R)^2 + (B Z)^2) Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
= Uo [((Ro^2 Ro^4) + ((A R)^2 + (B Z)^2) Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
= Uo [Ro^4 / (Ro^2 + (A R)^2 + (B Z)^2)^2]
= Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2
as expected.
 

Note that for R < Rc:
U = Um + Ue
= Umo [(Ro^2 + (A R)^2) Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
+ Ueo [(Z^2 Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
where:
Umo = Ueo = Uo

Hence:
U = Uo [(Ro^2 + (A R)^2 + (B Z)^2) Ro^4) / (Ro^2 + (A R)^2 + (B Z)^2)^3]
= Uo [Ro^4 / (Ro^2 + (A R)^2 + (B Z)^2)^2]
= Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2
as expected
 

Note that both Ue and Um have step changes at R = Rc but the steps are equal and opposite in magnitude so that the total energy density U remains a smoothly continuous function.

Note that there is a geometric relationship between A, B and Kc. The parameter Kc involves the perimeter length of an ellipse whereas the parameters A and B involve the ratio of the major and minor axis of that same ellipse.

This web page last updated April 17, 2021.

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