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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. 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 quantized charge 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 various quantum mechanical properties. Hence the spheromak properties derived from the assumed energy distributions are the properties of real atomic particles and real spheromak plasmas.
 

SPHEROMAK WALL CONCEPT:
Assume that three dimensional space is divided into two regions by a flexible wall known as the "Spheromak Wall" which totally encloses region "t" within region "p". Assume that region "p" extends to infinity in all directions.
 

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, H, Theta)
where:
R = radius from the main axis of spheromak cylindrical symmetry;
and
H = height above (or below) the spheromak equatorial plane;
and
Theta = angle around the main axis of symetry;
 

SPHEROMAK CROSS SECTIONAL DIAGRAM:
The following diagram shows the approximate cross sectional shape of a real plasma spheromak located within a cylindrical enclosure.

Note that an ideal spheromak in free space has a nearly round toroidal cross section whereas a real electromagnetic spheromak in a laboratory may be radially distorted by the proximity of the vacuum enclosure metal walls.
 

GEOMETRICAL FEATURES OF A SPHEROMAK:
Important geometrical features of a spheromak include:
Rc = the spheromak core radius on the equatorial plane;
Rs = the spheromak outside radius on the equatorial plane;
Rf = the value of R at the spheromak end;
(2 |Hf|) = 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.
 

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

Assume that outside the spheromak wall the energy density Up as a function of position is given by:
Up = Uo [(Ro^2 / (K^2 Ro^2 + R^2 + H^2)]^2
or

Up = (Uo / K^4) Uo [((K Ro)^2 / ((K Ro)^2 + R^2 + H^2)]^2

Thus for R^2 + H^2 >> (K Ro)^2:
Up ~ Uo [(Ro^2 / (R^2 + H^2)]^2
and for R^2 + H^2 << (K Ro)^2:
Up ~ Uo / K^4

Typically:
K^4 ~ 2.0

The assumed energy distribution function gives perfect matching with the real energy density at both the center of the spheromak and at large distances from the spheromak center. However, there may be some error between the assumption and reality at positions between these two extremes. 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 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 wall:
Up = Ut.

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

This is a quadratic equation which provided that:
4 (Uto / Upo) K^2 < 1
has solutions:
R = {Ro +/- [Ro^2 - 4 (Uto / Uo) K^2 Ro^2]^0.5} / [2 (Uto / Uo)^0.5]

These solutions are:
Rc = {Ro - [Ro^2 - 4 (Uto / Upo) K^2 Ro^2]^0.5} / [2 (Uto / Upo)^0.5]
and
Rs = {Ro + [Ro^2 - 4 (Uto / Upo) K^2 Ro^2]^0.5} / [2 (Uto / Upo)^0.5]

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

Thus we have derived the important identity:
Rs Rc = K^2 Ro^2
where:
Rc < K Ro < Rs

The spheromak wall intersects the equatorial plane at both radii R = Rc and R = Rs. It is shown herein that this wall shape defines the surface of a toroid with a round 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.

This energy configuration is known as a spheromak.
 

SUMMARY:
Uo / K^4 = maximum spheromak energy density at the center of the spheromak;
Rs > K Ro > Rc;
K^2 Ro^2 = Rs Rc;
and
4 (Uto / Upo) K^2 < 1

Note that a particular value of Ro^2 defines the product (Rs Rc) / K^2 but does not define the individual Rs and Rc values. These values depend on other physical parameters.
 

SPHEROMAK SIZE:
The value of Ro is dependent on the total amount of energy present. As shown on the web page titled SPHEROMAK ENERGY integration over all space shows that the total energy present Ett is a few percent less than:
Efs = Upo (K Ro)^3 Pi^2

Thus a spheromak can potentially model a real particle or a real plasma with a total energy Ett and a nominal radius (K Ro).

In order for a spheromak to represent a real physical entity it is necessary to show that a spheromak is an energy stable configuration and that the energy density functions which cause spheromak formation arise from the electric and magnetic field energy densities due to net electric charge and electric charge flow around a closed path that is everywhere tangent to the spheromak wall.

Thus spheromak mathematics explains the existence and behavior of both stable charged atomic particles and semi-stable toroidal plasmas.

Typical plasma spheromaks produced in a laboratory have shape factors of about:
(Rs / Rc) = 4.2
 

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 = (Rs + Rc) / 2 = spheromak's top and bottom radius;
H = distance of a general point from the spheromak's equatorial plane;
Hs = height of a point on the spheromak wall above the spheromak's equatorial plane;
(2 |Hf|) = spheromak's overall length measured at R = Rf;
So = [Rs / Rc]^0.5 = spheromak shape ratio;
 

SPHEROMAK STRUCTURAL ASSUMPTIONS:
1) A spheromak wall is composed of a closed spiral of charge hose or plasma hose of overall length Lh;
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 is constant;
4) The charge hose current causes a toroidal magnetic field inside the hose spiral and a poloidal magnetic field outside the hose spiral;
5) The net charge causes a cylindrically radial electric field inside the charge hose spiral and a spherically radial electric field outside the charge hose spiral;
6) At the center of the spheromak at (R= 0, H = 0) the electric field is zero;
7) Inside the spheromak wall where:
Rc < R < Rs and |H| < |Hs|
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 / (K^2 Ro^2 + R^2 + H^2)]^2;
9) Everywhere at the spheromak surface:
Up = Ut
 

ROLES OF SPHEROMAK FIELDS:
The fields of a spheromak serve two important purposes storing energy and acting in combination to position and stabilize 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 POSITION 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 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 Hf.
 

SPHEROMAK WALL POSITION:
A stable spheromak charge sheet wall will be located at a position of force balance (energy density balance) which is also a spheromak energy minimum.
Define:
Ut = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [Rc / R]^2
= total field energy density inside the spheromak charge wall

Up = Uo [Ro^2 / (K^2 Ro^2 + R^2 + H^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
d(Ett) = change in Ett due to a change in spheromak wall position dX normal to spheromak surface at X.
(Rw, Hw) = an arbitrary point on the spheromak wall;
 

SPHEROMAK INSIDE ENERGY DENSITY:
A spheromak arises from charge moving around a closed spiral that causes an internal toroidal magnetic field and an internal cylindrically radial electric field with a combined energy density Ut of the form:
Ut = Uc (Rc / R)^2
for:
Rc < R < Rs
and
-Hs < H < = Hs
Note that Ut has both electric and magnetic field components.
 

TOTAL EXTERNAL ENERGY FIELD APPROXIMATION:
The sum of the electric field energy density Ue and the magnetic field energy density Um outside the spiral charge motion path walls is the total energy density function:
Up = Um + Ue
which decreases in proportion to [1 / (distance)^4] in the far field. This energy density function is chosen so that it matches the magnetic field energy density in the near field and matches the electric field energy density in the far field. Then the spheromak walls occur at the locus of points where:
Us = Ut

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

This external energy density function, while it may not be exact, has the benefit that it yields a practical closed form mathematical model for a spheromak.

At the point R = Rc, H = 0, where the inner spheromak wall crosses the spheromak equatorial plane:
Up = Ut
giving:
Uc = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2

Hence:
Ut = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 (Rc / R)^2

At R = Ro, Ut = Uto. Hence:
Uto = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 (Rc / Ro)^2
= Uo [Ro Rc / (K^2 Ro^2 + Rc^2)]^2
= Uo [Ro Rc / (Rs Rc + Rc^2)]^2
= Uo [Ro / (Rs + Rc)]^2
 

SPHEROMAK WALL POSITION:
The 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 spheromak 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 Hf.
 

SPHEROMAK WALL LOCATION:
The spheromak wall exists on the locus of points where:
Up = Ut
or
Uo [Ro^2 / (K^2 Ro^2 + R^2 + Hs^2)]^2 = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 (Rc / R)^2
or
[(K^2 Ro^2 + Rc^2)]^2 (R / Rc)^2 = [(K^2 Ro^2 + R^2 + Hs^2)]^2
or
(K^2 Ro^2 + Rc^2)(R / Rc) = (K^2 Ro^2 + R^2 + Hs^2)
or
Hs^2 = (K^2 Ro^2 + Rc^2)(R / Rc) - K^2 Ro^2 - R^2
= R (K^2 Ro^2 / Rc) + R Rc - K^2 Ro^2 - R^2
= - R (R - Rc) + (K^2 Ro^2 / Rc) (R - Rc)
= [(K^2 Ro^2 / Rc) - R] [R - Rc]
= [(Rs - R)(R - Rc)]
where:
Rs = (K^2 Ro^2 / Rc

Hence the general spheromak wall position equation is:
Hs^2 = [(Rs - R) (R - Rc)]
or
Hs = +/- [(Rs - R) (R - Rc)]^0.5
This equation describes the cross sectional profile of a spheromak in free space.
Note that Hs = 0 at R = Rc and at R = Rs and that in the range:
Rc < R < Rs
Hs has real values that are both positive and negative.
 

SPHEROMAK CROSS SECTION:
Recall that at the spheromak wall:
Hs^2 = [Rs - R][R - Rc]

Make substitution:
R = X + [(Rs + Rc) / 2]
Then:
Hs^2 = [Rs - R][R - Rc]
= [Rs - X - [(Rs + Rc) / 2]] [ X + [(Rs + Rc) / 2] - Rc]
= [[(Rs - Rc) / 2] - X] [X + [(Rs - Rc) / 2]]
= [[(Rs - Rc) / 2]^2 - X^2
or
Hs^2 + X^2 = [[(Rs - Rc) / 2]^2
which is the well known equation of a circle.

Thus the spheromak wall is circular in cross section with a minor radius of [[(Rs - Rc) / 2] and a cross section center at:
R = (Rs + Rc) / 2
 

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

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

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

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

These identities are extensively used in spheromak characterization.
 

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

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

Hence the energy density Ut inside the wall is given by:
Ut = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [Rc / R]^2
or
(Ut / Uo)
= [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [Rc / Ro]^2 [Ro / R]^2
= [Ro Rc / (K^2 Ro^2 + Rc^2)]^2 [Ro / R]^2
= [Ro Rc / (Rs Rc + Rc^2)]^2 [Ro / R]^2
= [Ro / (Rs + Rc)]^2 [Ro / R]^2
= [(Ro / Rc) / ((Rs / Rc) + 1)]^2 [Ro / R]^2
= [(So / K) / (So^2 + 1)]^2 [Ro / R]^2
= {(So^2) / [K (So^2 + 1)]^2} [Ro / R]^2

Thus a stable spheromak results from a central region with a peak energy density:
[Uo / K^4]
and a spherical radial decrease in energy density Up(R,H) containing within it a region with a cylindrical radial decrease in energy density Ut(R). On the spheromak equatorial plane these two functions are as shown on the below graph where the abcissa is:
X = [R / Ro]
and the ordinate
[Up (R / Ro), 0) / (Uo)] 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:
Rs Rc = K^2 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, H) = Uo [Ro^2 / (K^2 Ro^2 + R^2 + H^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:
So^2 = 3.765
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, (K Ro), So
where:
Uo / K^4 is the highest field energy density at the center of the spheromak;
and
(K Ro)^2 = Rs Rc
and
So^2 = (Rs / Rc)
 

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

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

The difference between these two spheromaks lies in their contained energy. When a spheromak initially forms its energy is too high causing its So^2 value to be either too small or too large. The spheromak must spontaneiously emit photons until it reaches its stable state where:
So^2 = 3.765.
Thereafter the spheromak can absorb or emit photons by changing its value of (K 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) which also corresponds to a spheromak energy minimum.
Define:
Ut = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [Rc / R]^2
= total field energy density inside the spheromak charge wall
Up = Uo [Ro^2 / (K^2 Ro^2 + R^2 + H^2)]^2
= total field energy density outside the spheromak charge 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, Hw) = an arbitrary point on the spheromak wall;
 

EVALUATE DERIVATIVES:
Ut = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [Rc / R]^2
dUt / dR = Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 2 [Rc / R][- Rc / R^2]
= Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [-Rc^2 / R^3]

Note that:
dUt / dH = 0
and
dJ / dR ~ 0

Up = Uo [Ro^2 / (K^2 Ro^2 + R^2 + H^2)]^2
gives:
dUp / dR = 2 Uo [Ro^2 / (K^2 Ro^2 + R^2 + H^2)][- Ro^2 2 R / (K^2 Ro^2 + R^2 + H^2)^2]
= - 2 Uo Ro^4 R / (K^2 Ro^2 + R^2 + H^2)^3
and

dUp / dH = 2 Uo [Ro^2 / (K^2 Ro^2 + R^2 + H^2)][- Ro^2 2 H / (K^2 Ro^2 + R^2 + H^2)^2]
= - 2 Uo Ro^4 H / (K^2 Ro^2 + R^2 + H^2)^3
 

SPHEROMAK WALL POSITION STABILITY:
The criteria for spheromak wall position stability is that at every point on the spheromak wall:
d2(Ett / dXw^2) > 0
 

SPHEROMAK WALL POSITION STABILITY AT (Rs, 0):
At R = Rs:
d(Ett) / dXw = d(Ett) / dRs
= (Ut - Up) dAs
= (Ut - Up) 2 Pi Rs dH

Hence:
d^2(Ett) / dRs^2 = [d(Ut - Up) / dRs] 2 Pi Rs dH + [Ut - Up] 2 Pi dH
However:
Ut - Up = 0
giving:
d^2(Ett) / dRs^2 = [d(Ut - Up) / dRs] 2 Pi Rs dH

Thus wall stability requirement at (Rs, 0) is:
[d(Ut - Up) / dR]|R = Rs , H = 0 > 0
or
[dUt / dR] |R = Rs, H = 0 > [dUp / dR]|R = Rs, H = 0
or
Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [-2 Rc^2 / Rs^3] > - 4 Uo Ro^4 Rs / (K^2 Ro^2 + Rs^2)^3 or
2 Uo Ro^4 Rs / [K^2 Ro^2 + Rs^2]^3 > Uo [Ro^2 / ((K^2 Ro^2 + Rc^2)]^2 [Rc^2 / Rs^3] or
2 Rs / (K^2 Ro^2 + Rs^2)^3 > [1 / (K^2 Ro^2 + Rc^2)]^2 [Rc^2 / Rs^3] or
2 [ Rs^4 / Rc^2][(K^2 Ro^2 + Rc^2)^2 / (K^2 Ro^2 + Rs^2)^3 > 1
or
2 [ Rs^4 / Rc^2][(Rs Rc + Rc^2)^2 / (Rs Rc + Rs^2)^3 > 1
or
2 [ Rs^4 / Rc^2][Rc (Rs + Rc)]^2 / [Rs (Rc + Rs)]^3 > 1
or
2 [ Rs] / [(Rc + Rs)] > 1

Therefore, since Rs > Rc, spheromak wall stability is proven at (Rs, 0)
Any disturbance of the wall position at (Rs, 0) causes an increase in total spheromak energy. Hence the outside wall position on the spheromak equatorial plane is at a spheromak energy minimum.
 

SPHEROMAK WALL POSITION STABILITY AT (Rc, 0):
At R = Rc:
d(Ett) / dXw = d(Ett) / dRc
= (Up - Ut) dAs
= (Up - Ut) 2 Pi Rc dH

Hence:
d^2(Ett) / dRc^2 = [d(Up - Ut) / dRc] 2 Pi Rc dH + [Up - Ut] 2 Pi dH
= [d(Ut - Up) / dRc] 2 Pi Rc dH
since:
[Up - Ut] = 0

Thus the requirement for wall stability requirement at (Rc, 0) is:
[d(Up - Ut) / dR]|R = Rc , H = 0 > 0
or
[dUp / dR] |R = Rc, H = 0 > [dUt / dR]|R = Rc, H = 0
or
- 2 Uo Ro^4 Rc / (K^2 Ro^2 + Rc^2)^3 > - Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2[1 / Rc]
or flipping sign
Uo [Ro^2 / (K^2 Ro^2 + Rc^2)]^2 [1 / Rc] > 2 Uo Ro^4 Rc / (K^2 Ro^2 + Rc^2)^3
or
[1 / (K^2 Ro^2 + Rc^2)]^2[1 / Rc] > 2 Rc / (K^2 Ro^2 + Rc^2)^3
or
(K^2 Ro^2 + Rc^2)^3 / 2 Rc^2 (K^2 Ro^2 + Rc^2)^2 > 1 or
(K^2 Ro^2 + Rc^2) / 2 Rc^2 > 1
or
(Rs Rc + Rc^2) / 2 Rc^2 > 1
or
(Rs + Rc) / 2 Rc > 1

Hence spheromak wall stability has been proven at (Rc, 0)
 

SPHEROMAK WALL POSITION STABILITY AT (Rf, Hf):
At R = Rf, H = Hf:
d(Ett) / dXw = d(Ett) / dHf
= (Ut - Up) dAs
= (Ut - Up) 2 Pi Rf dR

Hence:
d^2(Ett) / dHf^2 = [d(Ut - Up) / dHf] 2 Pi Rf dR + [Ut - Up] 2 Pi d(Rf dR) / dHf
Note that:
d(Rf dR) / dHf = 0
giving:
d^2(Ett) / dHf^2 = [d(Ut - Up) / dHf] 2 Pi Rf dR

Thus the condition for wall stability requirement at (Rf, Hf) is:
[d(Ut - Up) / dHf]|R = Rf , H = Hf > 0
or
[dUt / dH] |R = Rf, H = Hf > [dUp / dH]|R = Rf, H = Hf
or
0 > - 2 Uo Ro^4 Hf / (K^2 Ro^2 + Rf^2 + Hf^2)^3
which is immediately true by inspection.

Hence we have proven spheromak wall position stability at (Rf, Hf).
 

SPHEROMAK GEOMETRY SUMMARY:
As shown above:
Lt = Pi (Rs - Rc)
Hence a spheromak in free space is a round toroid with an core (central hole) radius Rc and an outside radius Rs. The toroidal region minor diameter is:
(Rs - Rc)
and the toroidal region radius equals:
Hf = (Rs - Rc) / 2
and the toroidal region centerline is at R = Rf where:
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 magnetic field, which determines Upo, the effective current ring radius is NOT at R = Rf.
p  

NUMERICAL EXAMPLE:
For a typical spheromak:
So = [Rs / Rc]^0.5
= 2

giving:
So^2 = (Rs / Rc) = 4.0
and
Rc Rs = K^2 Ro^2
or
K Ro = [(Rc Rs)^0.5]
= [Rc^0.5 Rc^0.5 2]
= 2 Rc

For this spheromak:
So^2 = (Rs / Rc)
= 4.0

and
Rf = (Rc + Rs) / 2
= (Rc + 4 Rc) / 2
= 2.5 Rc

Then:
Lt / Rc = [Pi (Rs - Rc) / Rc]
= Pi (So^2 - 1)
= 3 Pi

and
Lp / Rc = Pi (Rs + Rc) / Rc
= Pi (So^2 + 1)
= 5 Pi

giving:
(Lp / Lt) = (5 / 3)
 

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

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

The exact surface area As of a spheromak in free space is given by:
As = Integral from R = Rc to R = Rs of:
2 X d(Lt) (2 Pi R)
 
= Integral from Phi = 0 to Phi = Pi of:
2 X (2 Pi R) [(Rs - Rc) / 2] d(Phi)
 
= Integral from Phi = 0 to Phi = Pi of:
(2 Pi) [(Rs - Rc)] R d(Phi)
 
= Integral from Phi = 0 to Phi = Pi of:
(2 Pi) [(Rs - Rc)][((Rs + Rc) / 2) - ((Rs - Rc) / 2)cos(Phi)] d(Phi)
 
= Integral from Phi = 0 to Phi = Pi of:
(Pi) [(Rs - Rc)][(Rs + Rc) - (Rs - Rc)cos(Phi)] d(Phi)
 
= Pi^2 [Rs^2 - Rc^2] - Pi (Rs - Rc) [sin(Pi) - sin(0)]
= Pi^2 [Rs^2 - Rc^2]
= Lp Lt

Note that the spheromak surface area As is smaller than the surface area of a sphere of radius Rs.
 

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))^2]^0.5
= Nt Pi [(Nr (Rs + Rc))^2 + ((Rs - Rc))^2]^0.5
= Nt Pi Rc [(Nr ((Rs / Rc) + 1))^2 + ((Rs / Rc) - 1)^2]^0.5
= Nt Pi Rc [(Nr (So^2 + 1))^2 + (So^2 - 1)^2]^0.5
= Nt Pi Ro (Rc / Ro) [(Nr (So^2 + 1))^2 + (So^2 - 1)^2]^0.5
= Nt Pi Ro (K / So) [(Nr (So^2 + 1))^2 + (So^2 - 1)^2]^0.5
 

SPHEROMAK SURFACE CHARGE PER UNIT AREA:
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. However, Dh varies in a complex way over the spheromak surface.

Consider a poloidal strip of spheromak surface of length 2 Pi R, width dLt and area 2 Pi R dLt.

The number of toroidal turns of charge hose passing through this strip is Nt.

The number of poloidal turns of charge hose passing through this strip is:
[dLt / (Pi (Rs - Rc))] Np

The toroidal length of charge hose in this strip is:
Nt dLt

The poloidal length of charge hose in this strip is:
[dLt / (Pi (Rs - Rc))] Np 2 Pi R
= [dLt / (Rs - Rc)] Np 2 R

Due to orthogonality of poloidal and toroidal turns, the total length of plasma hose in this strip is:
{(Nt dLt)^2 + [(Np 2 R dLt / (Rs - Rc)]^2}^0.5
= dLt {Nt^2 + [(Np 2 R) / (Rs - Rc)]^2}^0.5

Define:
Rhoh = Qs / Lh
= charge per unit length on the charge hose.

The total charge on the aforementioned strip is:
Rhoh dLt {Nt^2 + [(Np 2 R) / (Rs - Rc)]^2}^0.5

Hence the charge per unit area Sa on the strip is:
Sa = Rhoh dLt {Nt^2 + [(Np 2 R) / (Rs - Rc)]^2}^0.5 / dLt 2 Pi R
= Rhoh {Nt^2 + [(Np 2 R) / (Rs - Rc)]^2}^0.5 / 2 Pi R
= Rhoh {[Nt / (2 Pi R)]^2 + [Np / Pi (Rs - Rc)]^2}^0.5

Thus:
At R = Rc:
Sac = Rhoh {[Nt / (2 Pi Rc)]^2 + [Np / Pi (Rs - Rc)]^2}^0.5
= (Qs / Lh){[Nt / (2 Pi Rc)]^2 + [Np / Pi (Rs - Rc)]^2}^0.5
= Qs {[Nt / (2 Pi Rc)]^2 + [Np / Pi (Rs - Rc)]^2}^0.5
/ [(Np Pi (Rs + Rc))^2 + (Nt Pi (Rs - Rc))^2]^0.5

Hence:
Sac^2 = Qs^2 {[Nt / (2 Pi Rc)]^2 + [Np / Pi (Rs - Rc)]^2}
/ [(Np Pi (Rs + Rc))^2 + (Nt Pi (Rs - Rc))^2]

 
= (Qs^2 / Pi^2) {[Nt / (2 Pi Rc)]^2 + [Np / Pi (Rs - Rc)]^2}
/ [(Np (Rs + Rc))^2 + (Nt (Rs - Rc))^2]
 
= (Qs^2 / Pi^4) {[Nt / (2 Rc)]^2 + [Np / (Rs - Rc)]^2}
/ [(Np (Rs + Rc))^2 + (Nt (Rs - Rc))^2]
 
= (Qs^2 / Pi^4) {[1 / (2 Rc)]^2 + [Nr / (Rs - Rc)]^2}
/ [(Nr (Rs + Rc))^2 + ((Rs - Rc))^2]

This equation is of fundamental importance in determination of spheromak wall position on the equatorial plane.
 

This web page last updated August 30, 2016.

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