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THEORETICAL SPHEROMAK

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

THEORETICAL SPHEROMAK:
Nature uses spheromaks to store energy in rest mass.

This web page makes a few simple assumptions about energy density distributions and based on these simple assumptions shows the existence of spheromaks and derives various spheromak properties. The energy content of a spheromak is derived on the web page titled: SPHEROMAK ENERGY.

Another web page titled: ELECTROMAGNETIC SPHEROMAK shows that the assumed energy density distributions correspond to the combined electric and magnetic field energy distributions in the proximity of a charge quantum that circulates around a stable closed path. The electric and magnetic field energy density distributions indicate the existence of stable charged particles such as electrons and protons, the Planck constant, and particle magnetic resonance. Hence the spheromak properties derived from the assumed energy distributions are the properties of real particles.

This concept can be extended to explain the behavior of electrons around an atomic nucleus. The nucleus provides the central electric field necessary to stabilize the walls of multi-electron spheromaks. As the positive nuclear charge increases the electron spheromaks must change to meet the spheromak wall boundary conditions on the equatorial plane. It is believed that the electron spheromaks of inert gases are exceptionally stable.

As the atomic number increases and electrons are added to the spheromaks the toroidal magnetic fields increase but the poloidal magnetic field cancels. Note that the conditions for existence of a multi-electron spheromak around an atomic nucleus are very similar to the conditions for existence of a discrete electron or proton, so it is not surprising the both systems lead to the Planck constant.

Spheromaks can also form in plasmas, but large plasma spheromaks typically only exist for short times (< 1 ms) due to complicating factors such as neutral particle interactions with spheromak plasma particles.
 

SPHEROMAK DESCRIPTION:
A spheromak consists of an external region and an internal region. The two regions are separated by a closed surface known as the spheromak wall. Each region has its own static energy field distribution. However, at the spheromak wall the field energy densities in the two regions are identical. Otherwise the position of the spheromak wall would be unstable.

Thus in addition to storing energy the static fields of a spheromak act in combination to position the spheromak wall.
 

ENERGY DENSITY BALANCE:
For a spheromak wall position to be stable the total field energy density must be the same on both sides of the spheromak wall. This requirement leads to boundary condition equations that determine the shape of spheromaks.
 

SPHEROMAK WALL CONCEPT:
Assume that three dimensional space is divided into two regions by a flexible and moveable wall known as the "Spheromak Wall" which totally toroidal shaped region "t" within region "p". Assume that region "p" extends to infinity in all directions. The spacial static field energy density functions in the "p" and "t" regions are different but are dependent on the position of the spheromak wall.

The spheromak wall is free to move. It spontaneously seeks a stable position that results in a system total field energy minimum. At this stable position the field energy densities on both sides of the spheromak wall are equal so that there is no force (change in total energy with position tending to move the spheromak wall. Since the spheromak wall is at a position corresponding to a stable total static field energy Ett minimum:
dEtt / dX = 0,
where X is the wall position in space.

Spheromaks tend to form because there is a certain degree of random energy motion. If random energy motion results in a spheromak wall configuration with a lower total potential energy and a higher total kinetic energy sometimes part of that kinetic energy becomes a photon that is radiated away. The remaining system is left trapped in a low potential energy (spheromak) state until it absorbs energy (a photon) from an external source. As long as the density of random photons in the system environment is small spheromaks are stable accumulations of field energy.
 

SPHEROMAK WALL POSITION:
A spheromak is a semi-stable energy state. The spheromak wall positions itself to achieve a total energy relative minimum. At every point on the spheromak wall the sum of the electric and magnetic field energy densities on one side of the wall equals the sum of the electric and magnetic field energy densities on the other side of the wall. This general statement resolves into different boundary conditions for different points on the wall. These boundary conditions establish the spheromak core radius Rc on the equatorial plane, the spheromak outside radius Rs on the equatorial plane and the spheromak length 2 Zf.

SPHEROMAK INTERNAL ENERGY DENSITY:
A spheromak arises from charge moving around a closed spiral that causes an internal toroidal magnetic field with an energy density Ut of the form:
Ut = Uto (Ro / R)^2
for:
Rc < R < Rs
and
-Zs < Z < Zs
 

EXTERNAL ENERGY DENSITY:
The sum of the electric field energy density Ueo and the magnetic field energy density Umo outside the spheromak wall is the total energy density function:
Up = Umo + Ueo
which decreases in proportion to [1 / (distance)^4] in the far field. The external energy density function is chosen so that it matches the total energy density on the spheromak axis of symmetry and matches the field energy density at the spheromak walls on the spheromak equatorial plane. Then the spheromak walls occur at the locus of points where:
Up = Ut

The external energy density function Up is of the form:
Up = Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2
where the value of Ro satisfies the far field boundary condition where:
((A R)^2 + (B Z)^2) >> Ro^2
and also satisfies the near field boundary condition where:
Ro^2 < ((A R)^2 + (B Z)^2).

For a spheromak in free space:
A = B = 1.0
For experimental plasma spheromaks:
A > 1
due to proximity of cylindrical enclosure side walls. If the length of the enclosure along the spheromak axis of symmetry is not much greater than its radius then B would also be slightly greater than unity.

This external energy density function, while it has not been proven to be exact, has the benefit that it yields a practical closed form mathematical model for a spheromak which seems to be in general agreement with experimental observations.
 

SPHEROMAK GENERAL SHAPE:
A spheromak has a toroidal shape. The surface of the torus is the spheromak wall. It is shown on this web page that the spheromak geometry is a toroid which may have an elliptical cross section and a ratio of internal radius Rc to external radius Rs of:
So^2 =(Rs / Rc)
 

SPHEROMAK GEOMETRY:
The geometry of a spheromak in free space can be characterized by the following parameters:
R = radius of a general point from the spheromak's axis of cylindrical symmetry;
Rc = spheromak's minimum core radius;
Rs = spheromak's maximum equatorial radius;
Rf = spheromak's funnel radius;
Z = distance of a general point from the spheromak's equatorial plane;
Zs = height of a point on the spheromak wall above the spheromak's equatorial plane;
(2 |Zf|) = spheromak's overall length measured at R = Rf;
So = [Rs / Rc]^0.5 = spheromak shape parameter;
 

SPHEROMAK CROSS SECTIONAL DIAGRAM:
The following diagram shows the cross sectional shape of a spheromak. On this diagram:
Rc = 1.00
Rs = 4.10
Zf = 2.00

Note that an ideal spheromak in free space is toroidal with a slightly elliptical cross section. A real plasma spheromak in a laboratory may be radially distorted by the proximity of the vacuum chamber metal walls.
 

POSITION IN A SPHEROMAK:
A spheromak wall has cylindrical symmetry about its main axis of symmetry and has mirror symmetry about its equatorial plane. A position in a spheromak can be referenced by:
(R, Z, Theta)
where:
R = radius from the main axis of spheromak cylindrical symmetry;
and
Z = height above (or below) the spheromak equatorial plane;
and
Theta = angle around the main axis of symetry;
 

GEOMETRICAL FEATURES OF A SPHEROMAK:
Important geometrical features of a spheromak include:
Rc = the spheromak wall core radius on the equatorial plane;
Rs = the spheromak wall outside radius on the equatorial plane;
Rf = the value of R at the spheromak top and bottom;
(2 |Zf|) = the overall spheromak length;
The subscript c refers to spheromak wall "core" surface at the equatorial plane;
The subscript f refers to the "funnel edge" at the spheromak top and bottom;
The subscript s refers to the spheromak outer "surface" at 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.
 

SPHEROMAK STRUCTURAL ASSUMPTIONS:
1) A spheromak wall is composed of a closed spiral of charge hose or plasma hose of overall length Lh which has both poloidal and toroidal turns;
2) Spheromak net charge Qs is uniformly distributed over charge hose or plasma hose length Lh.
3) At a particular spheromak energy the charge hose current:
Ih = Q C / Lh
is constant;
4) The charge hose current causes a toroidal magnetic field inside the spheromak wall and a poloidal magnetic field outside the spheromak wall;
5) There is a net zero electric field inside the spheromask wall and a spherically radial electric field at large distances outside the spheromak wall;
6) At the center of the spheromak at R = 0, Z = 0) the electric field is zero;
7) Inside the spheromak wall where:
Rc < R < Rs and |Z| < |Zs|
the total field energy density U takes the form:
Ut = Uto (Ro / R)^2
8) Outside the spheromak wall the total field energy density takes the form:
Up = Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2;
9) Everywhere on the spheromak wall surface:
Up = Ut
 

ENERGY DENSITY FUNCTIONS:
Assume that within the region enclosed by the spheromak wall the field energy density Ut as a function of position is:
Ut = Uto (Ro / R)^2
where Uto and Ro are constants. Note that this energy density function is cylindrically radial and:
Uto = Ut|R = Ro.

As shown on the web page titled:Charge Hose Properties outside a spheromak wall the energy density Up as a function of position is given by:
Up = Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2

Thus for:
[(A R)^2 + (B Z)^2] >> Ro^2:
Up ~ Uo [Ro^2 / ((A R)^2 + (B Z)^2)]^2
and for:
[(A R)^2 + (B Z)^2] << Ro^2:
Up = Uo

The assumed external energy density function allows perfect matching with the real energy density at both the center of the spheromak and at large distances from the spheromak center. However, there could be some error between the assumption and reality at positions between these two extremes.

Experimental photographs of plasma spheromaks indicate that the cross section of the spheromak seems elliptical. Hence in general A is not equal to B.

The choice of this particular energy density function, although it seems simple, required a lot of work.
 

SPHEROMAK WALL
Differential forces on the flexible spheromak wall due to changes in spheromak energy with respect to wall position will cause the spheromak wall to spontaneously position itself so that the energy densities on both sides of the wall are equal and so that the wall position is at a total energy minimum. At this wall position at every point on the spheromak wall:
Up = Ut
or
Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2 = Uto [Ro / R]^2

At the intersections of the spheromk wall with the equatorial plane:
Z = 0
and equal energy densities on both sides of the spheromak wall give:
Uto (Ro / R)^2 = Uo [(Ro^2 / (Ro^2 + (A R)^2)]^2
or
(Uto / Uo) = [(Ro^2 / (Ro^2 + (A R)^2)]^2 (R / Ro)^2
= [(Ro R / (Ro^2 + (A R)^2)]^2
or
(Uto / Uo)^0.5 = [(Ro R / (Ro^2 + (A R)^2)]
or
[Ro^2 + (A R)^2](Uto / Upo)^0.5 = (Ro R)
or
(A R)^2 (Uto / Upo)^0.5 - Ro R + Ro^2 (Uto / Uo)^0.5 = 0

This is a quadratic equation in R which has solutions:
R = {Ro +/- [Ro^2 - 4 A^2 (Uto / Upo) Ro^2]^0.5} / [2 A^2 (Uto / Upo)^0.5]

These solutions which define Rc and Rs are:
Rc = {Ro - [Ro^2 - 4 A^2 (Uto / Uo) Ro^2]^0.5} / [2 A^2 (Uto / Uo)^0.5]
and
Rs = {Ro + [Ro^2 - 4 A^2 (Uto / Uo) Ro^2]^0.5} / [2 A^2 (Uto / Uo)^0.5]

These solutions are real provided that:
4 A^2 (Uto / Uo) < 1

Thus:
(Uto / Uo) < [1 / (4 A^2)]
is a requirement for spheromak existence.

Note that:
Rs > Ro > Rc

The product of these two solutions is:
Rs Rc = {Ro^2 - [Ro^2 - 4 A^2 (Uto / Uo) Ro^2]} / [4 A^4 (Uto / Uo)]
= {4 A^2 (Uto / Upo) Ro^2} / [4 A^4 (Uto / Upo)]
= Ro^2 / A^2

Thus we have derived the important identity:
A^2 Rs Rc = Ro^2

Define So by:
So = (A Rs) / Ro

Then the identity:
A^2 Rs Rc = Ro^2
can be expressed as:
(A Rs / Ro) = (Ro / A Rc)
= So

which gives:
So^2 = (Rs / Rc)
 

The spheromak wall intersects the equatorial plane at both radii R = Rc and R = Rs. Hence:
So^2 = Rs / Rc
is a good indicator of the spheromak shape on the Z = 0 plane.
 

DETERMINATION OF Uto:
Let Zs be the value of Z on the spheromak wall.

Let R = Rc , Z = 0 be the inner intersection of the spheromak wall with the equatorial plane. At R = Rc, Z = 0: Upc = Utc

Upc = Uo [Ro^2 / (Ro^2 + A^2 Rc^2)]^2
and
Utc = Uto [Ro / Rc]^2

Hence:
Uo [Ro^2 / (Ro^2 + A^2 Rc^2)]^2 = Uto [Ro / Rc]^2
or
Uto = Uo [Ro^2 / (Ro^2 + A^2 Rc^2)]^2 [Rc / Ro]^2
= Uo [(Ro Rc) /(Ro^2 + A^2 Rc^2)]^2
= Uo [(Ro^2 Rc^2) /(Ro^2 + A^2 Rc^2)^2]
= Uo [(A^2 Rs Rc Rc^2) /(A^2 RsRc + A^2 Rc^2)^2]
= [Uo / A^2] [(Rs Rc) /(Rs + Rc)^2]
= [Uo / A^2] [(Rs / Rc) /((Rs/ Rc) + 1)^2]
= [Uo / A^2] [So^2 /(So^2 + 1)^2]
This equation is of great practical importance in characterization of electromagnetic spheromaks.

At So = 2:
(Uto / Uo) = (1 / A)^2 (4 / 25) = 0.16 / A^2
%nbsp;

SPHEROMAK WALL POSITION:
The spheromak wall exists on the locus of points where:
Up = Ut
or
Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Zs)^2)]^2
= Uto [Ro / R]^2
= Uo [(Ro Rc) /(Ro^2 + A^2 Rc^2)]^2 [Ro / R]^2
= Uo [Ro^2 / (Ro^2 + (A Rc)^2)]^2 (Rc / R)^2

or
[(Ro^2 + (A Rc)^2)]^2 (R / Rc)^2 = [(Ro^2 + (A R)^2 + (B Zs)^2)]^2
or
(Ro^2 + (A Rc)^2)(R / Rc) = (Ro^2 + (A R)^2 + (B Zs)^2)
or
(B Zs)^2 = (Ro^2 + (A Rc)^2)(R / Rc) - Ro^2 - (A R)^2
= R (Ro^2 / Rc) + A^2 R Rc - Ro^2 - (A R)^2
= - R A^2 (R - Rc) + (Ro^2 / Rc) (R - Rc)
= [(Ro^2 / Rc) - A^2 R] [R - Rc]

Recall the identity that:
Ro^2 = A^2 Rs Rc

Hence:
(B Zs)^2 = [(Ro^2 / Rc) - A^2 R] [R - Rc]
= [(A^2 Rs Rc / Rc) - A^2 R] [R - Rc]

= [A^2 (Rs - R)(R - Rc)]

To better appreciate the shape of the spheromak wall use the change of variables:
R = [(Rs + Rc) / 2] + X
which gives:
(B Zs)^2 = [A^2 (Rs - [(Rs + Rc) / 2] - X)([(Rs + Rc)/2 + X - Rc)]
= A^2 [((Rs - Rc) / 2) - X][((Rs - Rc) / 2) + X]

Define a by:
a = (Rs - Rc) / 2

Then:
(B Zs / A)^2 = [(a - X)(a + X)]
= a^2 - X^2
or
(B Zs / A)^2 + X^2 = a^2
or
(B Zs / A a)^2 + (X^2 / a^2) = 1
or
(Zs^2 / b^2) + (X^2 / a^2) = 1
where:
a = (Rs - Rc) / 2
and
b = (A / B) a
= (A / B)(Rs - Rc) / 2

Typically b > a which implies that (A / B) > 1

This wall shape defines the surface of a toroid with an elliptical cross section with a minimum inside radius Rc (core radius) and a maximum outside radius Rs. This torus shaped surface is known as the spheromak wall. The wall position corresponds to a stable total energy minimum.

FIND Zf:
The spheromak wall position is described by the formula:
Zs = +/- [A / B] [(Rs - R) (R - Rc)]^0.5

The parameter Zs reaches its peak value Zs = Zf at:
dZs / dR = 0

dZs = (1/ 2) [A / B] [(Rs - R) dR + (- dR) (R - Rc)] / [(Rs - R) (R - Rc)]^0.5 = 0
Implies that at R = Rf:
[Rs + Rc - 2 Rf] = 0
giving:
Rf = (Rs + Rc) / 2
 

Recall that:
(B Zs)^2 = [A^2 (Rs - R)(R - Rc)]

At R = Rf, Zs = Zf:
(B Zf)^2 = [A^2 (Rs - (Rs + Rc) / 2)((Rs + Rc) / 2 - Rc)]
= A^2 [(Rs - Rc) / 2]^2

Thus:
Zf = (A / B)(Rs - Rc) / 2

SUMMARY:
Uo = maximum spheromak energy density at the center of the spheromak;
(A Rs) > Ro > (A Rc);
Ro^2 = A^2 Rs Rc;
So^2 = Rs / Rc;

Hence:
(Utc / Uts) = (Rs^2 / Rc^2)
= So^4

So = (A Rs / Ro) = [Ro /(A Rc)];
and
(Uto / Uo) = [1 / (A^2)][So / (So^2 + 1)]^2

Zf = (A / B)(Rs - Rc) / 2

For real solutions:
So > 1
and
(Uto / Uo) < [1 / (4 A^2)]

Note that a particular value of Ro^2 = A^2 Rs Rc defines the product (Rs Rc) but does not define the individual Rs and Rc values. These values depend on Ro and the spheromak shape parameter So.
Rs = (Ro / A) So
Rc = (Ro / A) (1 / So)
 

IMPORTANT CHANGE OF VARIABLES:
Recall that:
(B Zs)^2 = [A^2 (Rs - R)(R - Rc)]

Define X by:
X = A R / Ro

Hence the general spheromak wall position equation is:
(B Zs / Ro)^2 = [(A Rs/ Ro) - (A R / Ro)] [ (A R /Ro) - (A Rc / Ro)]
= [So - X] [X - (1 / So)] or
Zs = +/- (1 / B) [(So - X) (X - (1 / So))]^0.5
This equation describes the cross sectional outline of a spheromak wall.
Note that Zs = 0 at X = (1 / So) where R = Rc and Zs = 0 at X = So where R = Rs and that in the range:
(1 / So) < X < So
or
Rc < R < Rs
Zs has real values that are both positive and negative.
 

ELLIPSE PERIMETER LENGTH:
On the spheromak wall:
(Zs B / A)^2 = [Rs - R] [R - Rc]
= - R^2 + R (Rs + Rc) - Rs Rc

Differentiating gives:
2 Zs dZs B^2 / A^2 = - 2 R dR + (Rs + Rc) dR
or
dZs / dR = {- R + [(Rs + Rc) / 2]} A^2 / [B^2 Zs]

Let Lt = perimeter length of ellipse measured around the toroidal axis:
Z = 0, R = (Rs + Rc) / 2.
dLt = [dZs^2 + dR^2]^0.5
= dR [(dZs / dR)^2 + 1]^0.5
= dR [{{- R + [(Rs + Rc) / 2]} A^2 / B^2 Zs}^2 + 1]^0.5
= dR [{{- R + [(Rs + Rc) / 2]}^2 A^4 B^2 / B^4 A^2([Rs - R] [R - Rc])} + 1]^0.5
 
= dR [{{- R + [(Rs + Rc) / 2]}^2 A^2 / B^2 ([Rs - R] [R - Rc])} + 1]^0.5

Make substitutions:
Y = {R - [(Rs + Rc) / 2]}
a = (Rs - Rc) / 2

Then:
dY = dR
and
[Rs - R] [R - Rc]
= [Rs - (Y + [(Rs + Rc) / 2])][(Y + [(Rs + Rc) / 2]) - Rc]
= [[(Rs - Rc) / 2] - Y][Y + [(Rs - Rc) / 2]]
= [a - Y] [Y + a]
= a^2 - Y^2

Hence:
dLt = dR [{{- R + [(Rs + Rc) / 2]}^2 A^2 / B^2 ([Rs - R] [R - Rc])} + 1]^0.5
= dY [{{- Y}^2 A^2 / B^2 (a^2 - Y^2)} + 1]^0.5
= dY [(Y^2 A^2 + B^2 (a^2 - Y^2)) / B^2 (a^2 - Y^2)]^0.5
= dY [(Y^2 (A^2 - B^2) + B^2 (a^2)) / B^2 (a^2 - Y^2)]^0.5

For the special case of a circle where A = B this integrand simplifies to:
dLt = dY [(a^2)) / (a^2 - Y^2)]^0.5

Then:
Lt = 2 Integral from R = Rc to R = Rs of:
dY [(a^2) / (a^2 - Y^2)]^0.5

Recall that:
Y = {R - [(Rs + Rc) / 2]}

At R = Rc:
Y = {Rc - [(Rs + Rc) / 2]}
= - (Rs - Rc) / 2
= - a

At R = Rs:
Y = {Rs - [(Rs + Rc) / 2]}
= (Rs - Rc) / 2
= + a

Hence:
Lt = Integral from Y = - a to Y = + a of:
2 dY [a^2 / (a^2 - Y^2)]^0.5
 
= Integral from Y = -a to Y = a of:
2 a dY [1 / ([a^2 - Y^2])]^0.5
= 2 a [arc sin(a / a) - arc sin(-a / a)]
= 2 a [(Pi / 2) - (- Pi / 2)]
= 2 a Pi
= 2 (Rs - Rc) Pi / 2
= (Rs - Rc) Pi

Hence Lt = Pi (Rs - Rc)
as expected for a spheromak with a circular cross section.

For the general case of an ellipse described by:
(Y^2 / a^2) + (Z^2 / b^2) = 1
where:
a = ellipse minor radius
b = ellipse major radius
mathemeaticians have shown that the ellipse perimeter length Lt is given by:
Lt = Pi (a + b) Kh
= Pi (a + b) [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

where:
h = (a - b)^2 / (a + b)^2

For a spheromak with an elliptical cross section:
Zs = (A / B)[(Rs - R)(R- Rc)]^0.5
Zf = (A / B)[(Rs - (Rs + Rc) / 2)((Rs + Rc) / 2 - Rc)]^0.5
= (A / B)[(Rs - Rc) / 2)((Rs - Rc) / 2)]^0.5
= (A / B)[(Rs - Rc) / 2]

Hence:
b = (A / B) a
or
b B = a A

and
(b / a) = (A / B)
where if b > a then (A / B) > 1.0

Recall that:
a = (Rs - Rc) / 2

Hence:
(a + b) = [(Rs - Rc) / 2] [1 + (A / B)]
and
(b - a) = [(Rs - Rc) / 2] [(A / B) - 1]
and
h = (a - b)^2 / (a + b)^2
= [((A / B) - 1)]^2 / [(A / B) + 1]^2
= [A - B]^2 / [A + B]^2

Recall that:
Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

Note that Kh is always greater than unity.
Then:
Lt = Pi (a + b) Kh
= Pi [(Rs - Rc) / 2] [1 + (A / B)] Kh
= 2 Pi [(Rs - Rc) / 2] [1 + (A / B)] [Kh / 2]

Define the lumped constant Kc by:
Kc = [1 + (A / B)] [Kh / 2]
= {Perimeter length of ellipse with a = (Rs - Rc) / 2}
/{Perimeter length of circle with radius a = (Rs - Rc) / 2}

Then:
A^2 / (B^2 Kc) = A^2 / [(1 + (A / B))(B^2 Kh / 2)]
= {2 A^2 / [(B^2 + (A B))(Kh)]}

Kc = [1 + (A / B)] [Kh / 2]

dKc = d(A / B) [Kh / 2] + [(1 + (A / B)) / 2] dKh
or
dKc / d(A / B) = [Kh / 2] + [(1 + (A / B)) / 2] [dKh / d(A / B)]
or
dKc / d(A / B) = [Kh / 2] + [(1 + (A / B)) / 2] [dKh / dh][dh / d(A / B)]

*********************************************************

Recall that:
h = [(A - B)]^2 / [A + B]^2

dh = {[A + B]^2 2 (A - B) (dA - dB) - [(A - B)]^2 2 [A + B] (dA + dB)} / [A + B]^4
 
= {[A + B] 2 (A - B) (dA - dB) - [(A - B)]^2 2 [dA + dB] / [A + B]^3
 
CONTINUE FROM HERE = {(A - 1) {[A + 1] 2 dA - [(A - 1)] 2 dA} / [A + 1]^3
 
= [(A - 1) 4 dA] / [A + 1]^3

Hence:
[dh / dA] = [(A - 1) 4] / [A + 1]^3

Hence:
dKc / dA = [Kh / 2] + [(1 + A) / 2] [dKh / dh][dh / dA]
= [Kh / 2] + [(1 + A) / 2]{[(A - 1) 4] / [A + 1]^3} [dKh / dh]
 
= [Kh / 2] + {[2 (A - 1)] / [A + 1]^2} [dKh / dh]

Recall that:
Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

dKh = [(dh / 2^2) + (2 h dh / 2^6) + (3 h^2 dh / 2^8)
+ (5^2 4 h^3 dh / 2^14) + (7^2 5 h^4 dh / 2^16) + (21^2 6 h^5 dh / 2^20) +
....]

Thus:
[dKh / dh] = [(1 / 2^2) + (h / 2^5) + (3 h^2 / 2^8)
+ (5^2 h^3 / 2^12) + (7^2 5 h^4 / 2^16) + (21^2 6 h^5 / 2^20) + ....]

Note that if b > a then A > 1 and since Kh > 1 thus Kc > 1.

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

The spheromak wall position is fully defined by:
a, b and [(Rs + Rc) / 2]
or by
a = [(Rs - Rc) / 2],
b = [A / B] [(Rs - Rc) / 2],
and by:
Rf = [(Rs + Rc) / 2]
 

SPHEROMAK SHAPE PARAMETER So:
Define spheromak shape parameter So by:
So^2 = (Rs / Rc)
or
So = (Rs / Rc)^0.5

Recall that:
Ro^2 = A^2 Rs Rc

Thus:
So = (Rs / Rc)^0.5
= (Rs)^0.5 [A^2 Rs / Ro^2]^0.5
= A Rs / Ro

Similarly:
So = (Rs / Rc)^0.5
= (Ro^2 / A^2 Rc)^0.5 (1 / Rc)^0.5
= (Ro / A Rc)

These identities are extensively used in spheromak characterization.
 

A Rs = Ro^2 / A Rc = Ro So

A Rc = Ro^2 / A Rs = Ro / So

A (Rs - Rc) = Ro [So - (1 / So)]
= (Ro / So)[So^2 - 1]

A (Rs + Rc) = Ro [So + (1 / So)]
= (Ro / So) [So^2 + 1]

Hence:
Ro, So and A fully specify the position of a spheromak wall.
 

SPHEROMAK POTENTIAL ENERGY WELL:
Outside the spheromak wall the field energy density is given by:
Up = Uo [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2
or
(Up / Uo) = [Ro^2 / (Ro^2 + (A R)^2 + (B Z)^2)]^2
= [1 / (1 + (A R / Ro)^2 + (B Z / Ro)^2)]^2

At R = Rc and Z = 0 Up = Uc
giving:
Uc = Uo [Ro^2 / (Ro^2 + (A Rc)^2)]^2

The energy density Ut inside the spheromak wall on the spheromak equatorial plane is given by:
Ut = Uto [Ro / R]^2
or
(Ut / Uo)
= {Uto [Ro / R]^2 / Uo}
= {Uo [1 / (1 + A^2)]^2 [Ro / R]^2 / Uo}
= {[1 / (1 + A^2)]^2 [Ro / R]^2}

On the spheromak equatorial plane for A = 1 these two energy density functions are as shown on the below graph where the abcissa is:
X = [R / Ro]
and the ordinate
[Up (R / Ro), 0) / (Upo)] is the pink line
and the ordinate
[Ut (R / Ro) / Uo] for (Rc / Ro) = 0.5 is the dark blue line
and the ordinate
[Ut (R / Ro) / Uo] for (Rc / Ro) = 0.4 is the light blue line

On the spheromak's equatorial plane the graph lines intersect at:
Up(Rc, 0) = Ut(Rc)
and at
Up(Rs, 0) = Ut(Rs)
where:
A^2 Rs Rc = Ro^2
 


 

Note that in the region Rc < R < Rs the energy density inside the spheromak wall is lower than what it would be if the external energy density function:
Up(R, Z) = Upo [Ro^2 / (Ro^2 + (A R)^2 + Z^2)]^2
prevailed everywhere. Hence the spheromak forms a potential energy well. This potential energy well gives a spheromak its inherent stability.

Note that increasing the ratio:
So^2 = (Rs / Rc)
increases the depth of the potential well and hence increases spheromak energy stability. This issue indicates that for highly stable charged atomic particle spheromaks:
So is significantly greater than unity. Typically for atomic particle spheromaks the theoretical value of So^2 is given by:
So^2 = 4.1
and typically for plasma spheromaks the experimental value of So^2 is given by:
So^2 ~ 4.2

Hence from an energy stability perspective an ideal spheromak in free space is defined by its parameters:
Uo, Ro, So, A
where:
Uo is the highest field energy density at the center of the spheromak;
Ro^2 = A^2 Rs Rc
and
So^2 = (Rs / Rc)
and
A relates to Kc via a series expansion.
 

STABLE SPHEROMAK:
Note that a spheromak with:
So = [A Rs / Ro] = 2.0
and
(1 / So) = [A Rc / Ro] = 0.5
forms a stable potential energy well with a nominal radius of:
R = Ro.

Note that a spheromak with:
So = [A Rs / Ro] = 2.5
and
(1 / So) = [A Rc / Ro] = 0.4
also forms a stable potential energy well with a nominal radius of:
R = Ro.

The difference between these two spheromaks lies in their contained energies. When a spheromak initially forms its energy is too high causing its So^2 value to be too small. The spheromak must spontaneiously emit photons until it reaches its stable state where:
So^2 ~ 4.1.
Thereafter the spheromak can absorb or emit photons by changing its value of Ro while keeping So^2 constant.
 

SPHEROMAK WALL POSITION:
A stable spheromak wall will be located at a position of force balance (energy density balance).
Define:
Ut = Uo [1 / (1 + A^2)]^2 [Ro / R]^2
FIX FIX = total field energy density inside the spheromak wall
Up = Upo [Ro^2 / (Ro^2 + A^2 R^2 + (B Z)^2)]^2
= total field energy density outside the spheromak wall
X = wall position vector normal to the plasma sheet
dAs = element of plasma sheet surface area at wall position X
Ett = total spheromak energy
DeltaEtt = an element of Ett corresponding to element of surface area dAs
d(DeltaEtt) = change in DeltaEtt due to a change in spheromak wall position dX normal to spheromak surface at dAs.
(Rw, Zw) = an arbitrary point on the spheromak wall;
 

SPHEROMAK WALL POSITION STABILITY:
There are two energy density distributions. One energy density distribution is:
U = Uo [Ro^2 / (Ro^2 + (A R)^2 + Z^2)]^2

The other energy density distribution is:
U = Uo [1 / (1 + A^2)]^2 [Ro / R]^2

A spheromak naturally seeks a minimum energy state.

Inside the spheromak wall:
[Ro^2 / (Ro^2 + (A R)^2 + Z^2)]^2 > [1 / (1 + A^2)]^2 [Ro / R]^2 so inside the spheromak wall the energy density distribution adopted by the spheromak is:
U = Uo [1 / (1 + A^2)]^2 [Ro / R]^2

FIX FIX

Outside the spheromak wall:
[Ro^2 / (Ro^2 + (A R)^2 + Z^2)]^2 < [1 / (1 + A^2)]^2 [Ro / R]^2 FIX FIX so outside the spheromak wall the energy density distribution adopted by the spheromak is:
U = Uo [Ro^2 / (Ro^2 + (A R)^2 + Z^2)]^2

The spheromak wall is located at the junction between these two energy density distributions on the locus of points where the energy density is the same for both energy density distributions.
 

SPHEROMAK GEOMETRY SUMMARY:
As shown above:
Lt = Pi (Rs - Rc) Kc
Hence a spheromak in free space is an elliptical cross section toroid which on the equatorial plane has a core (central hole) radius Rc and an outside radius Rs. The cross section diameter on the equatorial plane is:
(Rs - Rc)
and the cross section radius parallel to the axis of symmetry equals:
Zf = (A / B)(Rs - Rc) / 2
and the toroidal region centerline is at:
Z = 0, R = Rf = (Rs + Rc) / 2

Hence from a structural perspective the poloidal winding turn length Lp is given by:
Lp = 2 Pi Rf
= 2 Pi (Rs + Rc) / 2
= Pi (Rs + Rc)

However, from the perspectives of determination of the peak central poloidal magnetic field, which determines Uo, the effective current ring radius R = Ro, NOT R = Rf.
 

TOTAL SPHEROMAK WALL SURFACE AREA As:
For a spheromak in free space the approximate surface area As of the spheromak wall is given by:

As = Lp Lt [2 Pi (Rs + Rc) / 2] [2 Pi (Rs - Rc) Kc / 2]
= Pi^2 (Rs^2 - Rc^2) Kc

 

CHARGE HOSE LENGTH:
Define:
Nt = integer number of toroidal charge hose turns; Nt = 1, 2,3,....
Np = integer number of poloidal charge hose turns; Np = 1, 2 , 3, ....

The total charge hose length Lh is given by:
Lh = [(Np Lp)^2 + (Nt Lt)^2]^0.5
= [(Np Pi (Rs + Rc))^2 + (Nt Pi (Rs - Rc) Kc)^2]^0.5
= Nt Pi [(Nr (Rs + Rc))^2 + ((Rs - Rc) Kc)^2]^0.5
= Nt Pi Rc [(Nr ((Rs / Rc) + 1))^2 + ((Rs / Rc) - 1)^2 Kc^2]^0.5
= Nt Pi Rc [(Nr (So^2 + 1))^2 + (So^2 - 1)^2 Kc^2]^0.5
= Nt Pi Ro (Rc / Ro) [(Nr (So^2 + 1))^2 + (So^2 - 1)^2 Kc^2]^0.5
= Nt Pi Ro (1 / A So) [(Nr (So^2 + 1))^2 + (So^2 - 1)^2 Kc^2]^0.5
 

SPHEROMAK SURFACE CHARGE PER UNIT AREA Sa:
Assume that the spheromak net charge is uniformly distributed along the charge hose length Lh. Then the spheromak surface charge per unit area Sa is inversely proportional to the winding center to center distance Dh. The spheromak achieves a zero net internal electric field by making Dh proportional to R.

Sac = surface charge density at R = Rc.

At radius R due to increasing winding spacing the surface charge density Sa is given by:
Sa = Sac (Rc / R)

Consider a strip of spheromak surface of length 2 Pi R, width dLt and hence area 2 Pi R dLt.
The charge on this strip is:
2 Pi R dLt Sa
= 2 Pi R dLt Sac (Rc / R)
= 2 Pi dLt Sac Rc

Then the total charge Qs on the spheromak is given by:
Qs = 2 Pi Lt Sac Rc
or
Sac = Qs / 2 Pi Lt Rc

Similarly:
Qs = 2 Pi Lt Sas Rs
or
Sas = Qs / 2 Pi Lt Rs

These equations are required to determine the radial electric field on the equatorial plane.
 

Now assume that the radial electric field inside the spheromak is zero. Hence:
Sac (Rc / Rs) = Sas
or
Sac^2 (Rc / Rs)^2 = Sas^2

This web page last updated December 29, 2019.

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