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SPHEROMAK Lt CALCULATION:

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

WEB PAGE DESCRIPTION:
This web page is devoted to evaluation of the line integral for accurate evaluation of toroidal turn length Lt which is identified on the web page titled:
Spheromak Approximation

This winding turn along its length is everywhere tangent to the spheromak wall.
 

SPHEROMAK ANALYSIS:
Accurate spheromak analysis requires precise knowledge of the function Zw(Rw), which function specifies the position in space of the spheromak wall. This function is required in order to accurately calculate energy density integrals over volume both inside and outside the spheromak wall. On this web page we develop an accurate equation for the spheromak wall position of the form:
1 / (Zw^2 + Rw^2)^2 = (A / Rw^2) - B^2
= B^2 [(A / B^2 Rw^2) -1].

On this web page it is further shown that the fundamental radius Ro of the spheromak is:
Ro^2 = [A / 2 B^2]
so that the equation for the spheromak wall position in the range:
Rc < Rw < Rs
becomes:
1 / (Zw^2 + Rw^2)^2 = B^2 [(2 Ro^2 / Rw^2) - 1]

The object of this web page is to accurately evaluate the constant B^2.

Calculating B^2 involves finding the winding ratio:
(Np / Nt)
which first involves finding the winding turn length ratio:
(Lp / Lt).
The poloidal turn length Lp is accurately known as:
Lp = 2 Pi Ro
but calculating the toroidal turn length Lt involves a line integral of such complexity that its numerical evaluation needs this dedicated web page.

To obtain a rough idea as to the size of B^2, we can use the approximation that:
[(Np Lp) / (Nt Lt)] ~ [1 / 2].
For accurate calculation of B^2 it is necessary to use the relationship:
[(Np Lp) / (Nt Lt)] =[Mp / Mt]
where, as shown on the web page titled (A HREF="GF Spheromak Winding Constraints.htm"> Spheromak Winding Constraints
[Mp / Mt] involves multiple possible pairs of prime numbers. Then:
[Np / Nt] =[Mp / Mt][Lt / Lp]

In summary, finding B^2 requirs finding [Np / Nt] which in turn requires finding [Lt / Lp], which means that Lt must be accurately calculated. That calculation is the subject of this web page.
 

SOLUTION APPROXIMATION:
Note that a neutral spheromak cannot exist because a spheromak relies on the distributed charge on the current filament to balance the attractive magnetic foces between adjacent current filaments. An electriclly neutral particle such as a neutron must be composed of at least two spheromaks.

For Family A spheromaks, as defined on the web page titled: Spheromak Winding Constraints to a good approximation:
[(Np Lp) / (Nt Lt)] ~ [1 / 2]

This approximation is accurate to about 1% but is not good enough for precision calculations. However, this approximation does indicate general spheromak behaviour not dependent on the winding ratio.

Note that Np and Nt are both integers which ultimately enable a unique solution.

A real charged spheromak in a vacuum has an external radial electric field which contributes to the field energy distribution.
A real spheromak case might be an electron spheromak around a positive nucleus. At large distances the electric fields cancel. Inside the spheromak walls the electric field is zero. Everywhere on the spheromak walls the poloidal magnetic energy density just outside the wall plus the external electric field energy density normal to the wall equals the toroidal magnetic field energy density just inside the wall.

In a plasma or a crystal the external electric fields almost cancel so spheromak energy is largely magnetic. Hence it is informative to calculate the theoretical Planck constant for the purely magnetic case.
 

SPHEROMAK GEOMETRY:
Assume that the spheromak shape is a symmetrical distorted toroid.
Rs = maximum spheromak wall radius in the equatorial plane;
Rc = minimum spheromak wall radius in the equatorial plane:
Ro = Spheromak characteristic radius where:
Rs^2 + Rc^2 = 2 Ro^2;
H = Z value of a point on the spheromak wall;
Ho = Z value at R = Ro;
Outside zone nearly radial electric fieldis set by an imaginary charged ring of radius Ro with charge Qs located at (R = Ro, Z = 0);
Outside zone poloidal magnetic field set by an imaginary ring of radius Ro carring current Np I located at (R = Ro, Z = 0);
Inside zone toroidal magnetic field Bt = Muo Nt I / 2 Pi R

DETERMINE THE SPHEROMAK WALL FUNCTION:
Recall that the spheromak wall function is:
1 / (Zw^2 + Rw^2)^2 = (A / Rw^2) - B^2
= [4 (Nt / Lh)^2 (1 / Rw^2)] - [16 Np^2 / Lh^2(Rs - Rc)^2]

Lh^2 / (Zw^2 + Rw^2)^2 = [4 (Nt / Rw)^2] - [16 Np^2 / (Rs - Rc)^2]
2020 = 4 [(Nt / Rw)^2 - 4 (Np^2 / (Rs - Rc)^2)]

[Lh^2 / 4] / (Zw^2 + Rw^2)^2 = [(Nt / Rw)^2 - 4 (Np^2 / (Rs - Rc)^2)]
 

FIND THE Rw^2 = Rc^2 AND Rw^2 = Rs^2 VALUES WHERE Zw^2 = 0:
Recall that:
1 / (Zw^2 + Rw^2)^2 = (A / Rw^2) - B^2

At Zw = 0:
1 / Rw^4 = A / Rw^2 - B^2
or
1 = A Rw^2 - B^2 Rw^4
or
B^2 Rw^4 - A Rw^2 + 1 = 0
or
Rw^2 = {A +/- [A^2 - 4 B^2]^0.5} / 2 B^2
= [A / 2 B^2]{1+/- [1 - (4 B^2 / A^2)]^0.5}

Note that Rc and Rs are the radii at which the spheromak wall intersects the equatorial plane.

Rc^2 = [A / 2 B^2]{1 - [1 - (4 B^2 / A^2)]^0.5}
= Ro^2{1 - [1 - (4 B^2 / A^2)]^0.5

Rs^2 = [A / 2 B^2]{1 + [1 - (4 B^2 / A^2)]^0.5}
= Ro^2 [1 + [1 - (4 B^2 / A^2)]^0.5}

Note that:
Rs^2 / Ro^2 = [1 + [1 - (4 B^2 / A^2)]^0.5}
and
Rc^2 / Ro^2 = [1 - [1 - (4 B^2 / A^2)]^0.5}

Hence:
(Rs^2 / Ro^2) - 1 = - [(Rc^2 / Ro^2) - 1]
or
Rs^2 + Rc^2 = 2 Ro^2
 

FIND (Rs - Rc) in terms of Ro:
Recall that:
Rs^2 = [A / 2 B^2]{1 + [1 - (4 B^2 / A^2)]^0.5}
= Ro^2 {1 +[1 + [1 - (1 / (Ro^2 B^2)]^0.5

Rc^2 = [A / 2 B^2]{1 - [1 - (4 B^2 / A^2)]^0.5}
= Ro^2 {1 - [1 - (1 /(Ro^2 B^2)]^0.5

Rs = [A / 2 B^2]^0.5 {1 + [1 - (4 B^2 / A^2)]^0.5}^0.5
and
Rc = [A / 2 B^2]^0.5 {1 - [1 - (4 B^2 / A^2)]^0.5}^0.5

(Rs - Rc) = Ro ({1 + [1 - (1 / Ro^2 B^2)]^0.5}^0.5 - {1 - [1 - (1 / Ro^2 B^2)]^0.5}^0.5)
which is critically dependent on the B value.

Clearly a spheromak is characterized by two constants. They are:
[1 / B] = [Lh (Rs - Rc) / 4 Np]
and
[A / 2 B] = [4 Nt / Lh]^2 [1 / 2] = B Ro
where:
[2 B / A]^2 < 1

Note that as an alternative the spheromak wall can be specified via B^2 and Ro.

Note that the ratio:
(Nt / Np) is tied to the ratio (Lp / Lt) through the stability requirement that:
(Np Lp / Nt Lt)= [Mp / Mt] ~ (1 / 2).
The ratio:
(Lp / Lt)
comes from the spheromak geometry.
 

SPHEROMAK SHAPE AND CONSTANT B^2:
Recall that:
1 / (Zw^2 + Rw^2)^2 = (A / Rw^2) - B^2

Since via [Mp / Mt] there are multiple possible solutions for B^2 and the real solution is selected by minimization of the total system energy. The other issue is that [Np / Nt] is a ratio of integers which will be tied to the integer ratio in [Mp / Mt].

Note that in the region Rc < Rw < Rs
2 Ro^2 > Rw^2

Note that the (Np / Nt) ratio can be obtained via the Lt / Lp ratio obtained from the surface line integral that determines Lt and the ratio:
(Np Lp / Nt Lt) =[Mp / Mt] ~ [1 / 2],
 

DETERMINE THE SPHEROMAK'S Lt VALUE:
Lt / 2 = Integral from R = Rc to R = Rs of:
dLt
= Integral from R = Rc to R = Rs of:
[dR^2 + dZ^2)^0.5
= Integral from R = Rc to R = Rs of:
[(dZw / dRw)^2 + 1]^0.5 dRw

Recall that:
Zw^2 + Rw^2 = Rw /(A - B^2 Rw^2)^0.5
or
Zw^2 = [Rw / (A - B^2 Rw^2)^0.5] - [Rw^2]

2 Zw dZw = [{(A - B^2 Rw^2)^0.5 dRw - Rw 0.5 (A - B^2 Rw^2)^-0.5 (- 2 B^2 Rw)dRw} / (A - B^2 Rw^2)] - [2 Rw dRw]

dZw / dRw = (1 / 2 Zw)[{(A - B^2 Rw^2)^0.5 - Rw 0.5 (A - B^2 Rw^2)^-0.5 (- 2 B^2 Rw)} / (A - B^2 Rw^2)] - [2 Rw /2 Zw]
= (1 / 2 Zw)[{(Rw / (Zw^2 + Rw^2)) - (Rw (1 / 2)(Zw^2 + Rw^2) / Rw)(- 2 B^2 Rw)} / (Rw^2 / (Zw^2 + Rw^2)^2] - [ Rw / Zw]
= (1 / 2 Zw)[{(1 / (Zw^2 + Rw^2)) + ( (Zw^2 + Rw^2))(+ B^2)} / (Rw / (Zw^2 + Rw^2)^2] - [Rw / Zw]
= (1 / 2 Zw)[{((Zw^2 + Rw^2)) + ( (Zw^2 + Rw^2)^3)(+ B^2)} / Rw ] - [Rw / Zw]

CONTINUE - FIND Lt from line integral, relate to (Rs - Rc), Zo^2.
 

FIND Zw = Ho AT Rw^2 = Ro^2

At Rw^2 = Ro^2:
Rw^2 = Ro^2 = [1 / 8] [Nt (Rs - Rc) / Np]^2

Recall that the spheromak wall function is:

[Lh^2 / 4] / (Zw^2 + Rw^2)^2 = [(Nt / Rw)^2 - 4 (Np^2 / (Rs - Rc)^2)]
or
(Zw^2 + Rw^2)^2 = [Lh^2 / 4] / [(Nt / Rw)^2 - 4 Np^2 / (Rs - Rc)^2]

At:
Rw^2 = Ro^2 = [1 / 8] [Nt (Rs - Rc) / Np]^2

[Ho^2 + Ro^2]^2 = [Lh^2 / 4] / {[((Nt / Ro )^2] - [4 Np^2 / (Rs - Rc)^2]}

Ho^4 + 2 Ro^2 Zo^2 + (Ro^2)^2 = [Lh^2 / 4] / {[((Nt / Ro)^2] - [4 Np^2 / (Rs - Rc)^2]}
or
Ho^4 + 2 Ro^2 Zo^2 + Ro^4 (1-F) = 0
where:
F = [Lh^2 / {4 Ro^4 [((Nt / Ro )^2] - [4 Np^2 / (Rs - Rc)^2]}
= [Lh^2 / {(4 Nt^2 Ro^2) - 16 Ro^4 Np^2 / (Rs - Rc)^2}
= [Lh / 2 Ro]^2 {1 / [(Nt^2) - (4 Ro^2 Np^2 / (Rs - Rc)^2)]}

Solving the quadratic equation for Ho^2 gives:
Ho^2 = { - 2 Ro^2 +/- [4 Ro^4 - 4 (1) Ro^4 (1 - F)]^0.5 } / 2
= Ro^2 {-1 +/- [1 - (1 - F)]^0.5}
= Ro^2 {-1 + [F^0.5]}

For a real solution F > 1 and typically is in the range:
4 < F < 16

Note that:
Nt^2 ~> 4 Ro^2 Np^2 / (Rs - Rc)^2
or
[Nt^2 / Np^2] ~> [4 Ro^2 / (Rs - Rc)^2]
 

In the Family A winding stability analysis it is shown that:
(Np Lp)/ (Nt Lt) ~ (1 / 2)
giving:
[(Nt Lt) / (Np Lp)]^2 ~ 4

Hence:
[Nt / Np]^2 ~ 4 {Lp / Lt}^2

Recall that:
[Nt^2 / Np^2] ~> [4 Ro^2 / (Rs - Rc)^2]

Recall that:
Lp = 2 Pi Ro

Thus:
4 [2 Pi Ro / Lt]^2 ~> [4 Ro^2 / (Rs - Rc)^2]
or
[4 Pi^2 / Lt ^2] ~> [1 / (Rs- Rc)^2]
or
[1 / Lt^2] ~> [1 / 4 Pi^2 (Rs - Rc)^2]
or
[1 / Lt] ~> [1 / 2 Pi (Rs - Rc)]

This inequality describes a real Lt path that is a squished circle. This inequality also limits the range of the constant B^2.

1 / (Zw^2 + Rw^2)^2 = [4 (Nt / Lh)^2 (1 / Rw^2)] - [16 Np^2 / Lh^2(Rs - Rc)^2]
where:
A = 4 (Nt / Lh)^2
and
B^2 = 16 Np^2 / Lh^2 (Rs - Rc)^2
or
B = [4 Np / Lh (Rs - Rc)]

The spheromak characterizing constants are:
[A / 2 B] = [4 Nt^2 / Lh^2] / [8 Np / Lh (Rs - Rc)]
= [Nt^2 / 2 Lh Np] = B Ro
and
B = [4 Np / Lh (Rs - Rc)]

An alternative way of expressing the spheromak wall position formula is:

[Lh^2 / 4] / (Zw^2 + Rw^2)^2 = [(Nt / Rw)^2 - 4 (Np^2 / (Rs - Rc)^2)]
where:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2

For real (Family A) spheromaks:
[(Np Lp) / (Nt Lt)] ~ (1 / 2)

Lp = 2 Pi Ro

Ro = [1 / 8]^0.5 [Nt (Rs - Rc) / Np]
Rc^2 = Ro^2 - [1 / 8][Lh (Rs - Rc) / Np] {(Nt^4 (Rs - Rc)^2)/ (Lh^2 Np^2))- 4}^0.5
Rs^2 = Ro^2 + [1 / 8][Lh (Rs - Rc) / Np] {(Nt^4 (Rs - Rc)^2)/ (Lh^2 Np^2))- 4}^0.5

A spheromak has a geometric height parameter:
[Ho^2 / Ro^2] = [F^0.5 -1]
where:
F = [Lh / 2 Ro]^2 {1 / [(Nt^2) - (4 Ro^2 Np^2 / (Rs - Rc)^2)]}

The parameter F must be greater than one. Typically F is in the range:
4 < F < 9

The spheromak will likely adopt a F value that minimizes the spheromak total energy content.

Note that via:
[(Np Lp) / (Nt Lt)] ~ (1 / 2)
the (Np / Nt) ratio is set by the (Lp / Lt) ratio, which is set by the Lt line integral.

The Lt line integral must be accurately calculated.

Recall that:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2
and (Np Lp / Nt Lt) = (Mp / Mt) ~ (1 / 2)
[(Np Lp) / (Nt Lt)]^2 = (Mp / Mt)^2 ~ (1 / 4)
Lh^2 = (Np Lp)^2 + (Mt / Mp)^2 (Np Lp)^2
= [1 + (Mt / Mp)^2](Np Lp)^2
~ 5 (Np Lp)^2
This calculation is only approximate. For exact results it is necessary to replace the 5 with the quantity:
[1 + (Mt / Mp)^2].

Thus in exact calculations:
Lh^2 = [1 + (Mt / Mp)^2][Np Lp]^2
 

SPHEROMAK PROFILE FUNCTION FOR F = 4:
Recall that the general spheromak profile function is:
Zw^2 + Rw^2 = Rw [Lh (Rs - Rc)] /{4 Nt^2 (Rs - Rc)^2 - 16 Np^2 Rw^2}^0.5

Recall that if F = 4 then:
[Ho^2 / (Rs - Rc)^2] = Lh / [4 Np (Rs - Rc)]
or
Lh (Rs - Rc) = {Ho^2 /(Rs - Rc)^2} [4 Np (Rs - Rc)] (Rs - Rc)
= Ho^2 4 Np

Thus the spheromak profile function for F = 4 becomes:
Zw^2 + Rw^2 = Rw Ho^2 4 Np /{4 Nt^2 (Rs - Rc)^2 - 16 Np^2 Rw^2}^0.5
= Rw Ho^2 2 Np /{Nt^2 (Rs - Rc)^2 - 4 Np^2 Rw^2}^0.5
= 2 Rw Ho^2 /{(Nt / Np)^2 (Rs - Rc)^2 - 4 Rw^2}^0.5
 

MAGNETIC FIELD ENERGY CONTAINED INSIDE THE SPHEROMAK WALL:
Toroidal Field Energy = Integral from Rw = Rc to Rw = Rs of:
2 Zw 2 Pi Rw dRw [Bto^2 / 2 Muo](Ro / Rw)^2

Recall that:
(Zw^2 + Rw^2 = Rw /(A - B^2 Rw^2)^0.5
or
Zw^2 = [Rw /(A - B^2 Rw^2)^0.5] - Rw^2
or
Zw = {[Rw /(A - B^2 Rw^2)^0.5] - Rw^2}^0.5
= Rw {1 / Rw (A - B^2 Rw^2)^0.5 - 1}^0.5

Hence:
Toroidal Field Energy = Integral from Rw = Rc to Rw = Rs of:
{[1 / (Rw (A - B^2 Rw^2)^0.5)] - 1}^0.5 dRw] [Bto^2 / 2 Muo] 4 Pi Ro^2

Recall that:
A = 4 [Nt / Lh]^2
and
B = 16 Np^2 / [Lh^2 (Rs - Rc)^2]

Note that the integrand has a sharp peak at:
Nt^2 = [4 Np^2 Rw^2 / (Rs-Rc)^2
or
Nt = 2 Np Rw / (Rs - Rc)
or
Rw = [(Rs - Rc) / 2] [Nt / Np]

Note that the small difference between Nt^2 and [4 Np^2 Rw^2 / (Rs - Rc)^2 causes Nt to have many turns that increase the value of Bto and hence the spheromak energy and the Planck constant.

Recall that for the case of F = 4:
Lh (Rs - Rc)= Zo^2 4 Np
or
Np / (Rs - Rc) = [Lh / 4 Zo^2]
giving:
Rw (A -B^2 Rw^2)^0.5 = (2 Rw / Lh){Nt^2 - 4 Np^2 Rw^2 / (Rs - Rc)^2}^0.5
= (2 Rw / Lh){Nt^2 - 4 Lh^2 Rw^2 / 16 Zo^4}^0.5
= {[4 Rw^2 Nt^2 / Lh^2] - [Rw^4 / Zo^4]}^0.5

Then:
Integral from Rw = Rc to Rw = Rs of:
{[1 / (Rw (A - B^2 Rw^2)^0.5)] - 1}^0.5 dRw]
= Integral from Rw = Rc to Rw = Rs of:
= {[1 / {[4 Rw^2 Nt^2 / Lh^2] - [Rw^4 / Zo^4]}^0.5] - 1}^0.5 dRw

This integration may need to be numerical.
 

EFFECT OF EXTERNAL MAGNETIC FIELD:
Recall that: Bpor = Muo Np I / 2 Ro
and
Bto = Muo Nt I / 2 Pi Ro

Hence:
Bpor / Bto = Np Pi / Nt
or
(Bpor / Bto)^2 = Np^2 Pi^2 / Nt^2

Hence:
Bto^2 = Bpor^2 Nt^2 / Np^2 Pi^2

Similarly if the spheromak poloidal magnetic field and the external magnetic field are not aligned: Bpor^2 changes to Bpor^2 - 2 Bpor Be

Change in toroidal field energy density is = 4 Bpor Be / 2 Muo

The change in spheromak contained toroidal magnetic field energy is:
Delta E = Integral from R = Rc to R = Rs of:
{[1 / (Rw (A - B^2 Rw^2)^0.5)] - 1}^0.5 dRw] [4 Bpor Be / 2 Muo] [4 Ro^2 Nt^2 / (Np^2 Pi)]
CONTINUE XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Q = 1.602 X 10^-19 coul
Muo = 4 Pi X 10^-7 web / amp -m
C = 2.997 X 10^8 m / s

Q^2 Muo C / 8 = [2.5664 X 10^-38 / 8][4 Pi X 10^-7][2.997 X 10^8]
= 12.082 X 10^-37

 

Muo = 4 Pi X 10^-7 ______
Qs = 1.602 X 10^-19 coul
C = 2.997 X 10^8 m / s
giving:
h = Np [(Muo Qs^2 C) / 2^1.5 Pi]
= Np X 4 Pi X 10^-7____ X (1.602)^2 X 10^-38 coul X 2.997 X 10^8 m / s X (1 / 2^1.5 Pi) x [(4 / 3) + (Pi / 4)]
= Np X 2 X 5.429 X 10^-37 joule second X [(4 / 3) + (Pi / 4)]
~ Np X 23 X 10^-37 joule second

The experimental value of h is:
h = 6.62607015 × 10−34 joule second
suggesting that:
Np ~ Nt
~ [6.62607015 × 10−34 joule second] / [23 X 10^-37 joule second]
~ 288

Thus we have developed a crude expression for the Planck constant:
h = (E / F)

However, to get the lead coeeficient right we cannot use the approximation that (Np Lp) / (Nt Lt) = (1 / 2). Instead exact expressions must be used. that are mathematically much more complex.

We need a more accurate calculation of the field energy distribution outside the spheromak wall.

The spheromak wall will tend to adopt the fixed geometric parameter values (Rc / Ro), (Rs / Ro) and (Ho / Ro) that result in a stable spheromak wall position. The ratios (Rc / Ro) and Rs / Ro) are set by boundary conditions at (Rc, 0) and (Rs, 0). The integers Np and Nt arise from application of prime number theory to the aforementioned parameters.

The boundary condition at (Rc, 0) generates a factor of about Pi in the ratio of (Np / Nt) which in large measure determines the integers Np and Nt. At any particular controlling prime number P the Np and Nt values increment and decrement together to find the best Np, Nt number pair for meeting this boundary condition. The mechanism by which a spheromak finds its controlling prime number P is not yet fully understood.
The likey source is 2^0.5 relationship which tends to produce a P = 577. 8^0.5 P = integer
8^0.5 (577) = 1632.002451
 

This web page last updated Aug.17, 2025.