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**INTRODUCTION:**

The spheromak shape parameter:

So^2 = (Rs / Rc)

where:

Rs = outside radius

and

Rc = inside radius

is one of the most important spheromak characterization parameters. Experimental data shows that for plasma spheromaks:

So^2 ~ 4.2

Precise theoretical analysis elsewhere on this web site indicates that for quantum charged particles:

So^2 ~ 3.99596

On the web page titled: PLANCK CONSTANT AND FINE STRUCTURE CONSTANT it is stated that So ~ 2.0. The material on this web page supports that assertion.

The total static field energy contained in an electromagnetic spheromak can be expressed as:

Ett = [(Muo C^2 Qs^2) / 32] [1 / Ro] [4 So (So^2 - So + 1) / (So^2 + 1)^2]

where:

[1 / Ro]

= [(Pi Fh / C So)[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

Hence:

Ett = [(Muo C^2 Qs^2) / 32] [4 So (So^2 - So + 1) / (So^2 + 1)^2]

[(Pi Fh / C So)[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

= [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)^2]

[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

= [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)^2]

[Np^2 + [Nt^2 (So^2 - 1)^2 / (So^2 + 1)^2]^0.5 (So^2 + 1)

Define:

Nr = Np / Nt

and

R = (So^2 - 1) / (So^2 + 1)

Then:

Ett = [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)]

[Np^2 + Nt^2 R^2]^0.5

When a spheromak is at its stable state:

Np^2 (So^2 + 1)^2 ~ Nt^2 (So^2 - 1)^2

Hence:

Nr = (Np / Nt)

~ (So^2 - 1) / (So^2 + 1)

Hence:

Nr^2 ~ [(So^2 - 1) / (So^2 + 1)]^2

= R^2

The spheromak static field energy is given by:

Ett= [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)]

Nt [Nr^2 + R^2]^0.5

= [(Muo C Qs^2 Pi Fh Nt) / 8] [(So^2 - So + 1) / (So^2 + 1)][Nr^2 + R^2]^0.5

However, the web page titled ELECTROMAGNETIC SPHEROMAK shows that the spheromak boundary condition is:

[Nr^2 + R^2] = {1 / [(Pi / 2)^2 - (F / R)^2]}

where F indicates the electric field ratio at the spheromak wall at R = Rc, H = 0.

Generally 0.5 < F < 1.0. Note that F = 1.0 for a metal sphere and 0.5 for a flat plate.

Hence:

Ett = [(Muo C Qs^2 Pi Fh Nt) / 8] [(So^2 - So + 1) / (So^2 + 1)]

{1 / [(Pi / 2)^2 - (F / R)^2]^0.5}

In order for the Planck Constant h and Fine Structure Constant Alpha to be really constant the product:

[(So^2 - So + 1) / (So^2 + 1)]{1 / [(Pi / 2)^2 - (F / R)^2]^0.5}

must be constant at the spheromak operating point.

At So = 2.0 the equation for Ett simplifies to:

Ett = [(Muo C Qs^2 Pi Fh Nt) / 8] [(So^2 - 1) / (So^2 + 1)]

{1 / [(Pi / 2)^2 - (F / R)^2]^0.5}

= [(Muo C Qs^2 Fh Nt) / 8] [R]

{Pi^2 / [(Pi / 2)^2 - (F / R)^2]^0.5}

= [(Muo C Qs^2 Fh Nt) / 4]

{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5

Make substitution:

Muo C Qs^2 = 2 h Alpha

to get:

Ett = [(2 h Alpha Fh Nt) / 4]

{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5

Ett = h Fh

(1 / Alpha) = [ Nt / 2] {(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5

Note that Nt must be a positive integer. Comparison with experimental data shows that this equation is consistent with:

Nt = 274

and

{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5 = 1

or

{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]} = 1

Rearranging this equation gives:

(Pi R / 2)^2 = [(Pi / 2)^2 - (F / R)^2]

or

(F / R)^2 = [(Pi / 2)^2 - (Pi R / 2)^2]

or

F^2 = R^2 [(Pi / 2)^2 - (Pi R / 2)^2]

= (R Pi / 2)^2 [ 1 - R^2]

Hence:

F = (R Pi / 2) [1 - R^2]^0.5

At So = 2:

R = (So^2 - 1) / (So^2 + 1)

= (3 / 5)

Thus:

F = (R Pi / 2) [1 - R^2]^0.5

= (3 Pi / 10) [1 - (9 / 25)]^0.5

= (3 Pi / 10)(4 / 5)

= 0.24 Pi

= **0.75398**

Thus if an electric field analysis confirms that F ~ 0.75398 we can reasonably conclude that the spheromak shape factor So = 2.0

Hence, in spite of the complexity of the electric field integral calculation we are motivated to use it to solve for F in order to accurately determine So.

Note that an upper limit on F at So = 2 is imposed by:

(Pi / 2) > (F / R)

and

R = (3 / 5)
or

F < (R Pi / 2) = 0.30 Pi = 0.942

Since the maximum value of F = 0.942 the inequality:

together with:

F < 0.942 imposes a lower limit on R of

R = 0.942 / (Pi / 2)

= 0.5996

Recall that:

R = (So^2 - 1) / (So^2 + 1)

Thus the corresponding lower limit on So^2 is given by:

(So^2 + 1) R = So^2 - 1

or

So^2 (1 - R) = 1 + R

or

So^2 = (1 + R) / (1 - R)

= (1 + 0.59969) / (1 - 0.59969)

= 1.59969 / 0.40031

= 3.99614

which is close to where a spheromak actually operates.

**DEPENDENCE OF So ON F:**

The operation of spheromaks as contemplated on this web site relies on the spheromak shape parameter So being at or close to:

So = 2.0

This parameter value is set by the boundary condition: Nr^2 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]

The boundary condition makes Nr = (Np / Nt) a function of R which is in turn a function of So. Thus via the boundary condition F determines R and Nr which in turn determine So.

In a real spheromak typical parameter values are:

Np = 240, Nt = 274, So ~ 2.0, R ~ (3 / 5) and F ~ 0.24 Pi = .75398

When the electromagnetic field environment of a spheromak changes F will slightly change causing a corresponding small change in So.

To understand the stability of So we have to understand how the spheromak boundary condition sets So.

The boundary condition is:

Nr^2 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]

where:

R = (So^2 - 1) / (So^2 + 1)
and

Nr = (Np / Nt)

Recall that from the web page titled: ELECTROMAGNETIC SPHEROMAK in a stable spheromak:

Nr^2 ~ 2 R^2

Substitution into the boundary condition gives:
Nr^2 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]

or

3 R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]

Rearranging the boundary condition formula gives:

[(Pi / 2)^2 - (F / R)^2] [(3 R^2] = 1

or

[3 (Pi R / 2)^2 - 3 (F)^2] = 1

or

R^2 = (1 + 3 F^2) (1 / 3) (2 / Pi)^2

The parameter F lies in the range:

0.5 < F < 1.0

At F = 0.5:

R^2 = (1 + 3 F^2)(1 / 3) (2 / Pi)^2

= (1 + (3 / 4))(1 / 3) (2 / Pi)^2

= (7 / 4)(1 / 3)(2 / Pi)^2

= **0.2364**

At F = 1.0:

R^2 = (1 + 3 F^2)(1 / 3) (2 / Pi)^2
= (4)(1 / 3) (2 / Pi)^2
= **0.5403**

Thus:

0.2364 < R^2 < 0.5403

or

0.48621 < R < 0.7351

Recall that:

R = (So^2 - 1) / (So^2 + 1)

or
(So^2 - 1) = R (So^2 + 1)

or
So^2 {1 - R} = {1 + R}

or

So^2 = (1 + R) / (1 - R)

At R = 0.48621:

So^2 = (1 + R) / (1 - R)

= (1 + .48621) / (1 - .48621)

= 1.48621 / 0.51379

= **2.8926**

At R = 0.7351:

So^2 = (1 + R) / (1 - R)

= (1 + 0.7351) / (1 - 0.7351)

= (1.7351 / 0.2649)

= **6.5500**

Thus So^2 lies in the range:

2.8926 < So^2 < 6.5500

Hence:

**1.700 < So < 2.5592**

This conclusion supports the approxmate and precise mathematical analyses elsewhere on this web site which indicate that a spheromak operates at So ~ 2 with:

R = (3 / 5)

= 0.60

and

R^2 = (9 / 25)

= 0.36

This web page last updated October 10, 2018.

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