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

The purpose of this web page is to introduce the reader to various spheromak related concepts which are developed in detail on subsequent web pages.

By definition the total energy Ett of a particle is given by:
Ett - Epv = Em + Ep - Epv + Ek
Ett - Epv = total paticle energy measured with respect to a reference vacuum potential energy Epv
Ep - Epv = potential energy of particle with respect to a vacuum reference energy Epv
Ek = kinetic energy of particle with respect to a particle at rest in the observers frame of reference.
Em = energy contribution of Ett to the potential barrier Eb.

Consider a potential barrier of height (Eb - Epv) that results from ordered motion of a large number of particles.

The charged particles of a spheromak reside within a low potential valley between two high potential regions. If:
(Eb - Epv) > (Ek + Ep - Epv)
the charged particles can circulate but are trapped within a mutual potential energy well. In a plasma spheromak Eb arises from conversion of part of Ett into a common potential barrier. However, for plasma spheromaks (Ett - Epv) is positive, so the spheromaks are only semi-stable. Eventually the trapped particles find ways of escaping from the trap into the surrounding vacuum.

Due to the presence of fields from other particles the potential barrier height changes. Absent energy exchange interactions between particles conservation of energy keeps total particle energy Ett constant. However, in a plasma there are interparticle interacions that limit the spheromak lifetime.

Ek + Ep - Epv > (Eb - Epv)
everywhere then a particle is considered completely free because it can move anywhere.

Consider a particle trapping region where:
(Ek + Ep - Epv) < (Eb - Epv)
and where:
Ek > 0

In that region particles will be trapped but are free to circulate within the trap. Individual charged particles that randomly acquire sufficient additional energy to overcome the potential barrier can escape from the trap. In so doing they reduce the trapping potential height:
(Eb - Epv)
for the remaining trapped particles.

Thus a spheromak can only exist if at the position of its circulating charges there is a surrounding potential barrier of height:
(Eb - Epv) > (Ek + Ep - Epv).

In a spheromak charged particles are trapped at the charge sheet surface between the toroidal and poloidal field regions. The trapping region contains a potential well. At the bottom of the potential well Ek is positive, but Ek drops to zero as a particle attempts to leave the potential well. Absent quantum mechanical tunneling, as long as the potential barrier is maintained and the particle does not acquire additional kinetic energy from another particle, the particle is trapped.

A spheromak is formed when electrons acquire high kinetic energy from an electric field. This energy is acquired from a DC power supply. If the particle motion is suitably ordered much of the initial kinetic energy converts to magnetic field energy leaving the electrons with insufficient kinetic energy to escape from the trap. Consequently the particles are trapped and the plasma forms a spheromak. This spheromak has poloidal and toroidal magnetic field regions that contain much of the spheromaks energy and has a narrow low potential region between the poloidal and toroidal regions that contains the circulating charged particles. In the narrow low potential region:
(Ek + Ep - Epv) < (Eb - Epv)
so that these particles are trapped.

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, H)
R = radius from the main axis of cylindrical symmetry;
H = height above (or below) the spheromak equatorial plane.

The following diagram shows the approximate cross sectional shape of a real spheromak within a cylindrical enclosure.

Note that an ideal spheromak in free space has a round toroidal cross section whereas a real spheromak is somewhat radially distorted by the presence of the enclosure metal walls.

Important geometrical features of a spheromak include:
Rc = the spheromak axial core radius on the equatorial plane;
Rs = the spheromak outside radius on the equatorial plane;
Rf = the valus of R at the spheromak end;
Rx = the spheromak corner radius;
(2 |Hf|) = the overall spheromak length;
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 and their ratios to the inside radius Rc on the spheromaks equatorial plane.

It is mathematically convenient to characterize spheromaks by their parameters Ro and So which are defined by:
Ro = (Rs Rc)^0.5
So = (Rs / Rc)^0.5

When characterized in this way the spheromak total energy Ett is proportional to (1 / Ro) and the spheromak shape factor So lies in the range:
1.5077 < So < 4.3662

The mathematical structure of a spheromak can be understood by remembering that the charge sheet forming the surface of the spheromak is composed of Charge Hose. The charge hose (and hence the charged particle path) spirals around the toroid within the charge sheet providing both the poloidal and toroidal magnetic fields. The charge is uniformly distributed along the charge hose.

The total number of ions Ni and the total number of free electrons Ne are almost equal, which keeps the spheromak net positive charge:
Qs = Q (Ni - Ne)
relatively low.

Trapped electrons move along the charge hose within the plasma sheet following a constant energy path.

A spheromak has four fields, an internal toroidal magnetic field, an external poloidal magnetic field, an internal outward pointing cylindrically radial electric field and an external outward pointing spherically radial electric field.

A spheromak has a central axis of cylindrical symmetry from which all radii are measured unless otherwise indicated. The toroidal magnetic field runs around the main axis of symmetry inside the plasma sheet. The poloidal magnetic field runs through the hole in the middle of the torus with radius:
R = Rc.
and wraps around the outside of the spheromak. Within this hole at H = 0 the poloidal magnetic field is parallel to the axis of cylindrical symmetry.

On the equatorial plane (H = 0), for:
R < Rc
the electric field is zero.

On the equatorial plane the net positive charge on the plasma sheet at the surface of the central hole at radius:
R = Rc
causes a cylindrically radial electric field for:
Rc < R < Rs,
where Rs is the spheromak charge sheet outside radius on the equatorial plane.

Let Rw = inside radius of a metal enclosure in which the spheromak is situated.

On the equatorial plane, at:
R = Rs,
the internal cylindrical electric field caused by the surface charge at;
R = Rc
adds to the electric field caused by the surface charge at:
R = Rs.
Hence if the inequality:
Rw >> Rs
is not true then Rs slightly reduces to keep the net external electric field for:
Rs < R < Rw
cylindrically radial on the equatorial plane while being spherically radial off the equatorial plane.

At the charge sheet the spiraling charge is trapped in a narrow mutual potential well.

General Fusion Inc. is a leader in the laboratory production of spheromaks. General Fusion has reported on issues such as compressing a spheromak, which involves increasing the spheromak's energy while decreasing the spheromak's linear size via use of a conical plasma injector. General Fusion has also reported on on the effect of spheromak compression on the spheromak free electron kinetic energy and on the spheromak lifetime.

When a spheromak is compressed it absorbs energy and shrinks. During spheromak compression the net positive charge on the spheromak:
Qs = Q (Ni - Ne)
remains constant. As shown in the web page titled PLASMA SHEET PROPERTIES the spheromak compression causes electron-ion recombination within the plasma sheet. The resulting change in the ratio:
(Ni - Ne) / Ne = Ve / C
increases the free electron kinetic energy Eke and increases both the spheromak's electric field energies Eep and Eet and the spheromak's magnetic field energies Emp and Emt.

General Fusion has reported spheromak free electron kinetic energies ranging from 20 eV - 25 eV for low energy density spheromaks at the spheromak generator to 400 ev - 500 eV for higher energy density spheromaks at the downstream end of the conical plasma injector. General Fusion reports a spheromak linear size reduction between these two positions of between 4X and 5X. The corresponding observed apparent electron densities rise from 2 X 10^14 cm^-3 to 2 X 10^16 cm^-3. The corresponding observed axial magnetic field increases from .12 T to 2.4 T to 3 T.

At this time this author does not know for certain: where on the spheromak the electron kinetic energy was measured, where on the spheromak the apparent electron density was measured or the absolute dimensions of the measured spheromaks and their enclosure.

As a charged circulates within the spheromak charge sheet, it follows a spiral constant energy path. Along that path for a particular particle there may be energy exchange back and forth between electric field energy and magnetic field energy. However, as long as the charged particles remain equally spaced along the plasma hose, implying that Eke and Eki are both constant, the spheromaks total energy remains constant.

Plasma sheet theory indicates that the free electron kinetic energy Eke is almost uniform throughout a spheromak.

The spheromak shown in the photograph on the General Fusion website shows the following apparent dimension ratios:
Hfa = 2.7 Rca
Rxa = 0.3 Rca
Rfa = 3.0 Rca
Rsa = 4.2 Rca
Note that the trailing subscript a indicates a value measured before spheromak compression. The corresponding trailing subscript b indicates a value measured after spheromak compression.

This author believes that the experimentally measured magnetic field strength is the poloidal magnetic field strength in the spheromak core. One of the complications is that General Fusion did not indicate the (Rs / Rc) value corresponding to the spheromak's magnetic field measurements.

Eke = average kinetic energy of a single free electron;
Eki = average kinetic energy of a single ion;
Eep = electric field energy of a spheromak outside the charge sheet
Eet = electric field energy of a spheromak inside the charge sheet
Emp = magnetic field energy of a spheromak outside the charge sheet
Emt = magnetic field energy of a spheromak inside the charge sheet
Ett = total available energy contained in a spheromak
Ett = Eep + Eet + Emp + Emt + Ne Eke + Ni Eki

To appreciate the relative size relationship between magnetic field energy Em and total electron kinetic energy (Ne Eke) it is helpful to consider a solenoid of axial length Lp realized by Ne electrons evenly distributed along the length Lp revolving around the solenoid axis with tangential velocity Ve along toroidal circumference length Lt. The kinetic energy Eke of each electron is given by:
Eke = (Me / 2) Ve^2
Me = electron mass.

The axial magnetic field Bt within the solenoid is given by:
Bt = (Mu Q Ne Ve) / (Lt Lp)
Q = proton charge
Mu = permiability of free space
Pi = 3.14159

The magnetic energy Emt within a round cross section toroidal solenoid is given by:
Emt = (Bt^2 / 2 Mu) (Pi (Lt / 2 Pi)^2 Lp)
= [(Mu Q Ne Vet) / (Lt Lp)]^2 (Lt^2 Lp / 4 Pi) / (2 Mu)

= (Mu^2 Q^2 Ne^2 Vet^2) / (Lp 8 Pi Mu)
= (Mu Q^2 Ne^2 Vet^2) / (8 Pi Lp)
= [(Mu Q^2 Ne^2) / (4 Pi Lp Me)] Eket

(Emt / Ne Eket) = [(Mu Q^2 Ne) / (4 Pi Lp Me)]

Numerical substitution into this expression with practical values for Ne and Lu gives:
Mu = 4 Pi X 10^-7 T^2 m^3 / J
Q = 1.602 X 10^-19 coul
Lp = 2.0 m
Me = 9.11 x 10^-31 kg
Ne = 3 X 10^16 electrons
(Em / Ne Eket) = [(Mu Q^2 Ne) / (4 Pi Lp Me)]
= (10^-7 X (1.602)^2 X 10^-38 X 3 X 10^16) / (4.0 m X 9.11 X 10^-31 kg)

= [((1.602)^2 X 3) /(4.0 X 9.11)] X 10^2
~ 21

Hence, for the practical spheromaks under consideration on this web site:
(Emt / (Ne Eket)) >> 1

To appreciate the relative size relationship between electric field energy Eep and total electron kinetic energy:
(Ne Eke)
it is helpful to consider a sphere of surface area:
As = (Lt Lp) realized by Ne electrons evenly distributed over its surface area
all moving around the sphere with tangential velocity Ve.
The sphere radius Rs is given by:
4 Pi Rs^2 = Lp Lt
Rs = [(Lp Lt) / (4 Pi)]^0.5

The kinetic energy Eke of each electron is given by:
Eke = (Me / 2) Ve^2
Me = electron mass.

Recall that for a charge sheet:
[Ih / (Dh C)]^2 = Sa^2

For this spherical approximation:
(Ih / Dh) = (Q Ne Ve) / (As)
[(Q Ne Ve) / (As C)]^2 = Sa^2

The surface electric field Es for a sphere is given by:
Es = (Sa / Epsilon)
Epsilon = permittivity of free space

The electric field energy around a sphere is given by:
Eep = Integral from R = Rs to R = infinity of:
[Epsilon / 2][Qs / Epsilon 4 Pi R^2]^2[4 Pi R^2]dR
= [Qs^2 / Epsilon 8 Pi Rs]
= [(Sa 4 Pi Rs^2)^2 / Epsilon 8 Pi Rs]
= [(Sa^2 2 Pi Rs^3) / Epsilon]
= [([(Q Ne Ve) / (As C)]^2 2 Pi Rs^3) / Epsilon]
= [([(Q Ne Ve) / (4 Pi Rs^2 C)]^2 2 Pi Rs^3)/ Epsilon]
= [(Q Ne Ve)^2 / (8 Pi Rs C^2 Epsilon)]
= [(Q Ne)^2 (Eke / Me)/ (4 Pi Rs C^2 Epsilon)]
= [(Q Ne)^2 (Eke / Me)/ (4 Pi [(Lp Lt) / (4 Pi)]^0.5 C^2 Epsilon)]

(Eep / Ne Eke) = [(Q^2 Ne) / (Me [4 Pi Lp Lt]^0.5 C^2 Epsilon)]
= [1 / Mu Epsilon C^2][4 Pi Lp / Lt]^0.5 [(Mu Q^2 Ne) / (4 Pi Lp Me)]
= [4 Pi Lp / Lt]^0.5 [(Mu Q^2 Ne) / (4 Pi Lp Me)]
= [4 Pi Lp / Lt]^0.5 (Emt / Ne Eket)

Lp ~ Lt
and if:
Eket ~ (Eke / 2)
( Eep / Ne 2 Eket) = [4 Pi Lp / Lt]^0.5 (Emt / Ne Eket)
~ [4 Pi]^0.5 [Emt / Ne Eket]

(Eep / Emt) ~ 2 [4 Pi]^0.5
= 7.09

Hence for a practical spheromak poloidal region electric field energy Eep is of the order of 7 X the toroidal region magnetic field energy Emt.

The magnitude relationship between the spheromak energy components is:
Eep > Emt >> Ne Eke >> Ni Eki
Hence the ion kinetic energy Ni Eki is negligibly small.

The above solutions for (Eep / Ne Eke) and (Emt / Ne Eket) are extremely crude due to assumptions about the spheromak geometry and current flow that are only very approximate. However, the calculation is sufficiently accurate to show that the total available spheromak energy Ett is dominated by its electric and magnetic field components. The particle kinetic energy is by comparison relatively small.

However, some free electron kinetic energy is required to trap the spheromak's field energy. The spheromak releases its stored field energy if the spheromak's circulating free electrons and ions become disordered. Hence the spheromak lifetime is principally governed by the rate of loss of order of the spheromak free electrons and ions.

As is shown later herein, a practical value for Eke is about:
Eke = 500 eV

Due to the limited total energy in a compressed spheromak the above derived expression:
Emt = [(Mu Q^2 Ne^2) / (4 Pi Lp Me)] Eke
effectively sets a practical upper limit on Ne which is at least three orders of magnitude below the number of D-T ions required per fusion pulse for practical fusion energy production.

Hence subsequent D-T ion injection is required and production of spheromaks is only the first step in the multi-step process required for energy production via thermonuclear fusion.

This web page last updated April 5, 2015.

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