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**SPHEROMAK STRUCTURE:**

This web page uses a simplified analysis to show that circulating static charge forms the external field energy density distribution required for the existence of a spheromak. The mathematical proof of spheromak wall position stability is developed on the web page titled:THEORETICAL SPHEROMAK. The external electric and magnetic fields of a spheromak are required to maintain its toroidal geometry and to hold a stable amount of energy.

A spheromak toroid has a major axis and a minor axis. The charge motion closed spiral path is characterized by **Np** turns around the toroid's major axis and by **Nt** turns around the toroid's minor axis. This charge motion path has the toroidal shape of the glaze on a doughnut and is referred to as the spheromak wall. Inside the spheromak wall the magnetic field is toroidal. Outside the spheromak wall the magnetic field is poloidal. Inside the spheromak wall the electric field is cylindrically radial. Outside the spheromak wall the electric field is spherically radial. The spacial distribution of the circulating charge is constant over time. Hence the electric and magnetic field geometry is constant over time.

The spheromak field structure allows semi-stable plasma spheromaks and discrete stable atomic charged particles to exist and act as stores of energy.

**ATOMIC PARTICLE SPHEROMAKS:**

Atomic particle spheromaks consist of quantized charge that superficially appears to be at rest with respect to an inertial observer. Isolated stable atomic particles such as electrons and protons hold specific amounts of energy (rest mass). When these particles aggregate the assembly emits photons. This photon emission decreases the total amount of energy in the assembly creating a mutual potential energy well.

In an atomic particle spheromak current moves uniformly around a closed spiral path. The net charge motion is at the speed of light. The spheromak net charge is uniformly distributed along the charge path. The uniform charge distribution along the charge motion path and uniform charge motion cause constant electric and magnetic fields.

A spheromak in free space has a nearly round cross section. However, the magnetic field of an atomic particle spheromak may be distorted by an external magnetic field.

Einstein's formula:

**Ett = Mp C^2**

gives the total energy of an isolated proton at rest as:

Ett = 1.67262 X 10^-27 kg X (2.99792458 X 10^8 m / s)^2

= **15.03275887 X 10^-11 J**.

The magnetic moment of a proton is:

1.4106067 X 10^-26 J / T

The Larmor precession frequency for a proton is:

42.57748 MHz / T

Einstein's formula:

**Ett = Me C^2**

gives the total energy of an isolated electron at rest as:

Ett = 9.109383 X 10^-31 kg X (2.99792458 X 10^8 m / s)^2

= **81.87105146 X 10^-15 J**.

The magnetic moment of an electron is:

-928.4763 X 10^-26 J / T

**PLASMA SPHEROMAKS:**

Plasma spheromaks are composed of free electrons and ions that move along a stable closed path at less than the speed of light.
Plasma spheromaks can be generated and photographed in a laboratory. A plasma spheromak in free space has a nearly round cross section. However, in a laboratory the field configuration of a plasma spheromak is affected by the proximity of a metal enclosure containment wall. If the equatorial outside radius **Rs** of a spheromak approaches the inside radius **Rw** of an enclosure's cylindrical wall the spheromak loses stability because the external field energy distribution becomes cylindrical rather than spherical.

The image below shows a plasma spheromak photograph made by General Fusion Inc.

An important feature of this photograph is the ratio of the spheromak outside radius Rs to its inside radius Rc. On this photograph the ratio:

**(Rs / Rc) ~ 4.2**.

However, this ratio may be affected by the proximity of the enclosure wall. Theoretical analysis on this web site indicates that in free space this ratio is close to:

**(Rs / Rc) = 4.8**.

For plasma spheromak stability there must be a slight separation between the positive ion and negative electron paths to prevent inter-particle scattering. Further, to prevent further scattering and loss of energy from energetic plasma free electrons the concentration of neutral ionizable atoms in the vacuum chamber containing the spheromak must be very small.

The amount of field energy trapped by a plasma spheromak can be significant. In the Plasma Impact Fusion (PIF) process the total field energy **Ett** of each injected plasma spheromak, above and beyond the total field energy of the plasma spheromaks constituant particles, must be at least: **3000 J**.

**SPHEROMAK CONCEPT:**

Conceptually a spheromak is a closed charge sheet in the shape of a toroid that provides a closed path for the current in the charge hose. The direction of the charge hose axis within the charge sheet conforms to the toroid surface curvature.

The charge hose winding forming a spheromak has both toroidal and poloidal magnetic field components. The charge hose axis gradually changes direction over the surface of the toroid following a spiral. the charge sheet is known as the spheromak wall.

In a spheromak quantized positive and negative charge moves along a spiral path tangent to the spheromak wall. A spheromak is cylindrically symmetric about the spheromak main axis and is mirror symmetric about the spheromak's equatorial plane. The charge hose has a net charge **Qs** that is uniformly distributed over the charge hose length **Lh**.

In the central core of the spheromak the electric field is zero. In the region enclosed by the spheromak wall the magnetic field is purely toroidal and the electric field is cylindrically radial. In the region outside the spheromak wall the magnetic field is purely poloidal. Outside the spheromak wall the electric field is spherically radial. The charged particles circulate in the narrow low magnetic field region at the interface between the toroidal and poloidal magnetic fields within the spheromak wall.

**LOCATION IN A SPHEROMAK:**

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)**

where:

**R** = radius from the main axis of cylindrical symmetry;

and

**H** = height above (or below) the spheromak equatorial plane.

**SPHEROMAK CROSS SECTIONAL DIAGRAM:**

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.

**GEOMETRICAL FEATURES OF A SPHEROMAK:**

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 value of R at the spheromak end where: H = Hf;

**(2 |Hf|)** = the overall spheromak length;

**Np** = number of poloidal charge hose turns about the major axis in the spheromak;

**Nt** = number of toroidal charge hose turns about the minor axis in the spheromak;

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.

When a unit of charge has passed through the spheromak core hole **Nt** times it has also circled around the main axis of spheromak symmetry **Np** times, after which it reaches the point in its closed path where it originally started.

Define:

**Lt** = one purely toroidal turn length;

**Lp** = one average purely poloidal turn length;

Then due to the orthogonality of toroidal and poloidal directions the charge hose or plasma hose length **Lh** is given by:

**Lh^2 = (Nt Lt)^2 + (Np Lp)^2**

**SPHEROMAK SHAPE:**

Lp = 2 Pi [(Rs + Rc) / 2]

= Pi (Rs + Rc)

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

= Pi (Rs - Rc)

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

= (Rs + Rc) / (Rs - Rc)

or

(Rs - Rc)(Lp / Lt) = (Rs + Rc)

or

Rs [(Lp / Lt) - 1] = Rc [(Lp / Lt) + 1]

or

**(Rs / Rc)** = [(Lp / Lt) + 1] / [(Lp / Lt) - 1]

= **[Lp + Lt] / [Lp - Lt]**

Thus the shape of the spheromak is defined by the ratio:

**(Rs / Rc)**.

The energy and frequency of a spheromak also involve **Np** and **Nt**.

There are **Nt** parallel charge hoses that go through the equatorial plane in the central core of the spheromak and hence form the spheromak inner wall.

**SPHEROMAK PARAMETER DEFINITIONS:**

The remainder of this web page is devoted to mathematical analysis of undistorted ideal spheromaks.

Define:

**Ue** = electric field energy density;

**Um** = magnetic field energy density

**U** = Ue + Um = total field energy density as a function of position;

**Umo** = Upo = maximum magnetic field energy density at R = 0, H = 0;

**R** = radial distance of a point from the major axis of symmetry of spheromak;

**H** = distance of a point above the spheromak equatorial plane (H is negative for points below the equatorial plane);

**J** = slowly varying function related to the field energy density;

**Jn** = near field value of J;

**Jf** = far field value of J;

**(Jf / Jn)^4 ~ 0.5**;

**Ro** = nominal spheromak radius defined by: **Ro^2 = J^2 Rs Rc**

**(Rx, Hx)** = a point **X** located at **R = Rx, H = Hx**

**Hf** = maximum value of **|H|** on the spheromak charge sheet

**2 Hf** = spheromak overall length

**Rf** = radius of spheromak charge sheet at **H = Hf** and **H = -Hf**

**Lpo = 2 Pi Ro = 2 Pi (Rs Rc)^0.5**

**Lp = Pi (Rs + Rc)**

**Lt = Pi (Rs - Rc)**

**Uc** = field energy density at **R = Rc, H = 0**

**Upo** = maximum field energy density at **R = 0, H = 0** = **(Bpo^2 / 2 Mu)**

**Uto** = field energy density at **R = Ro, H = 0**;

**Hs** = spheromak wall height at radius R defined by **Hs^2 = (Rs - R)(R - Rc)**

**Ut(R) = Uc (Rc / R)^2** = cylindrical energy density function inside the spheromak wall

**Up(R, H) = Uo [Ro^2 / (Ro^2 + J^2 R^2 + J^2 H^2)]^2** = spherical energy density function outside the spheromak wall

**(Rx, Hx)** = a point located at **R = Rx, H = Hx**

**SPHEROMAK CHARGE HOSE PARAMETERS**

Define:

**Ih** = charge hose current;

**Lh** = length of closed loop of charge hose;

**Dh** = center to center distance between adjacent charge hoses

**As** = outside surface area of spheromak wall

**Q** = proton net charge

**Qs** = net charge on spheromak

**Nnh** = integer number of negative charge quanta

**Nph** = integer number of positive charge quanta

**Vn** = velocity of negative charge quanta along charge hose

**Vp** = velocity of positive charge quanta along charge hose

**C** = speed of light

**CHARGE HOSE LENGTH:**

Due to orthogonality of toroidal and poloidal charge hose turns:

**Lh^2 = (Np Lp)^2 + (Nt Lt)^2**

**SPHEROMAK CHARGE DISTRIBUTION ASSUMPTION:**

Assume that the spheromak charge is uniformly distributed along the charge hose length.

**EQUATORIAL PLANE:**

On the spheromak's equatorial plane:

**H = 0**

For points on the spheromak's equatorial plane the following statements can be made:

For **R < Rc** the radial electric field is zero;

For **R < Rc** the toroidal magnetic field **Btoc = 0**

For **R < Rc** the magnetic field **Bpoc** is purely poloidal;

For **R = 0** the magnetic field is parallel to the axis of cylindrical symmetry;

For **Rc < R < Rs** the electric field **Er** is cylindrically radial;

For **Rc < R < Rs** the electric field **Er** is proportional to **(1 / R)**;

For **Rc < R < Rs** the poloidal magnetic field **Bpit = 0**;

For **Rc < R < Rs** the toroidal magnetic field **Btit** is proportional to **(1 / R)**.

For **Rs < R** in free space the electric field **Ero** is spherically radial;

For **Rs < R** in free space the electric field **Ero** is proportional to **(1 / R^2)**;

For **Rs < R** in free space the toroidal magnetic field **Bto = 0**;

For **Rs < R** in free space the poloidal magnetic field **Bpo** is proportional to **(1 / R^3)**;

**CHARGE HOSE CURRENT:**

**Ih = [Qp Np Vp + (- Q Ne Ve)] / Lh**

**CHARGE HOSE LENGTH Lh:**

Lh = {[2 Pi Np (Rs + Rc) / 2]^2 + [2 Pi Nt (Rs - Rc) / 2]^2}^0.5

= Pi {[Np (Rs + Rc)]^2 + [Nt (Rs - Rc)]^2}^0.5

= Pi Nt {[Nr (Rs + Rc)]^2 + [(Rs - Rc)]^2}^0.5

= Pi Nt Rc {[Nr (So^2 + 1)]^2 + [(So^2 - 1)]^2}^0.5

**CHARACTERISTIC FREQUENCY Fh:**

A spheromak's characteristic frequency Fh is given by:

Fh = C / Lh

where:

C = speed of light.

**AXIAL ELECTRIC FIELD ENERGY DENSITY DUE TO A RING OF CHARGE:**

Assume that a thin ring of radius "Ra" has net charge Qa. Then the linear charge density along the thin ring is:

Qa / (2 Pi Ra)

and an element of charge is:

dQa = [Qa / (2 Pi Ra)] dL

where dL is an element of the ring's circumferential length.

Consider a point at distance "Zb" along the ring axis, where Zb = 0 on the ring plane.

The electric field along distance (Ra^2 + Zb^2)^0.5 due to charge dQa is:

dE =(1 / 4 Pi Epsilon) [dQa / (Ra^2 + Zb^2)]

where:

Epsilon = permittivity of free space

The component of this electric field along the ring axis is:

dE = (1 / 4 Pi Epsilon) [dQa / (Ra^2 + Zb^2)] cos(Theta)

where:

cos(Theta) = Zb / (Ra^2 + Zb^2)^0.5

The net electric field E at distance Zb along the ring axis is:

E = (1 / 4 Pi Epsilon) [Qa / (Ra^2 + Zb^2)] cos(Theta)

= (1 / 4 Pi Epsilon) [Qa / (Ra^2 + Zb^2)] [Zb / (Ra^2 + Zb^2)^0.5]

= (1 / 4 Pi Epsilon) [Qa Zb / (Ra^2 + Zb^2)^1.5]

The electric field energy density Ue at Z = Zb, R = 0 is given by:

Ue = (Epsilon / 2) E^2

= (Epsilon / 2){(1 / 4 Pi Epsilon) [Qs Zb / (Ra^2 + Zb^2)^1.5]}^2

= [Qa^2 / (32 Pi^2 Epsilon)] [Zb^2 / (Ra^2 + Zb^2)^3]

= [Mu C^2 Qa^2 / (32 Pi^2)] [Zb^2 / (Ra^2 + Zb^2)^3]

Note that at Zb = 0 the net electric field is zero and the electric field energy density is zero..

In the far field where:

Zb >> Ra

then along the ring axis:

Ue = [Qs^2 / (32 Pi^2 Epsilon)] [1 / Zb^4]

Thus for Zb >> Ra the electric field energy density Ue is proportional to (1 / Zb)^4.

Note that if the charge is evenly distributed over the surface of a sphere instead of a ring:

At a distance (R^2 + H^2)^0.5 from the center of the sphere the electric field E is given by:

E = Qa / 4 Pi Epsilon (R^2 + H^2)

and

Ue = (Epsilon / 2) E^2
= (Epsilon / 2) [Qa / 4 Pi Epsilon (R^2 + H^2)]^2

= [Qa^2 / (32 Pi^2 Epsilon (R^2 + H^2)^2)]

From Maxwells equations:

[1 / (Mu Epsilon)] = C^2

or

[1 / (Epsilon)] = Mu C^2

giving:

**Ue = [(Mu C^2 Qa^2) / (32 Pi^2 (R^2 + H^2)^2)]**

**AXIAL EXTERNAL MAGNETIC FIELD ENERGY DENSITY DUE TO A RING OF CURRENT:**

The law of Biot and Savart gives an element of magnetic field **dB** at a measurement point on the axis of a ring due to an electric current I is:

**dB** = (Mu / 4 Pi) Ip **dL X R** / |R|^3

where:

Ip = poloidal current around the ring

|R| = distance from current element Ip **dL** to the measurement point

Mu = magnetic permeability of free space

**dL** = an element of length along the direction of electric current flow around the ring

[**R / |R|]** = unit vector along the direction of R

Consider a ring of radius "Ra" and an axial measurement point at distance "Zb" from the current ring along the ring axis.

Then the net magnetic field **B** at the measurement point is axial and is given by:

**B** = [(Mu / 4 Pi) Ip 2 Pi Ra / (Ra^2 + Zb^2)] sin(Theta)

where:

(Theta) = angle between unit vector (**R** / |R|) and the ring axis.

However:

sin(Theta) = Ra / (Ra^2 + Zb^2)^0.5

Thus the net magnetic field **B** along the current ring axis due to ring current I is given by::

**B** = [(Mu / 4 Pi) Ip 2 Pi Ra / (Ra^2 + Zb^2)] sin(Theta)

= [(Mu / 4 Pi) Ip 2 Pi Ra / (Ra^2 + Zb^2)][ Ra / (Ra^2 + Zb^2)^0.5]

= [(Mu Ip) / 2] [Ra^2 / (Ra^2 + Zb^2)^1.5]

The magnetic field energy density along the ring axis at Z = Zb is:

Um = B^2 / 2 Mu

= [(Mu Ip) / 2]^2 [Ra^2 / (Ra^2 + Zb^2)^1.5]^2 / 2 Mu

= [Mu Ip^2 / 8] [Ra^4 / (Ra^2 + Zb^2)^3]

Thus for Zb >> Ra the magnetic field energy density Um is proportional to (1 / Zb)^6. Hence at large distances the magnetic field energy density becomes negligibly small as compared to the electric field energy density.

At the center of the ring where Zb = 0 the magnetic field is:

**Bpo = [(Mu Ip) / (2 Ra)**

The corresponding magnetic field energy density Umo at the center of the ring is:

Umo = [Mu Ip^2 / 8 Ra^2]

Note that Umo is the magnetic field energy density at R = 0, H = 0 corresponding to current circulating at R = Ra.

Assume that the net static charge circulates at the speed of light. This is the basic condition for the existence of a spheromak. In order to exist a spheromak requires both a net static charge Qs and a poloidally circulating current Ip. As shown on the web page titled PLASMA HOSE the charge hose existence requires that:

Ih = Qs C / Lh

Note that the net static charge of an atomic particle may consist of the sum of positive and negative charge quanta known as quarks that individually circulate at less than the speed of light, but due to the constraints of charge hose behave as if the net static charge is circulating at the speed of light. Hence:

Ip = [Qs C / 2 Pi Ra]

giving:

Upo = [Mu Ip^2 / 8 Ra^2]

= [Mu [Qs C / 2 Pi Ra]^2 / 8 Ra^2]

= [(Mu Qs^2 C^2) / (32 Pi^2 Ra^4)]

The magnetic field energy density Um along the ring axis is given by:

Um = [Mu Ip^2 / 8] [Ra^4 / (Ra^2 + Zb^2)^3]

= [Mu [Qs C / 2 Pi Ra]^2 / 8] [Ra^4 / (Ra^2 + Zb^2)^3]

= [Mu Qs^2 C^2 / 32 Pi^2] [Ra^2 / (Ra^2 + Zb^2)^3]

**TOTAL EXTERNAL FIELD ENERGY DENSITY:**

The magnetic field energy density Um along the ring axis is given by:

Um = [Mu Qs^2 C^2 / 32 Pi^2] [Ra^2 / (Ra^2 + Zb^2)^3]

Recall that the electric field energy density Ue along the ring axis was given by:

= [Mu C^2 Qa^2 / (32 Pi^2)] [Zb^2 / (Ra^2 + Zb^2)^3]

Then along the ring axis the total field energy density U is given by:

U = Um + Ue

= [Mu Qs^2 C^2 / 32 Pi^2] [Ra^2 / (Ra^2 + Zb^2)^3] + [Mu C^2 Qa^2 / (32 Pi^2)] [Zb^2 / (Ra^2 + Zb^2)^3]

Assume that **Qa = Qs**. Then:

U = [Mu Qa^2 C^2 / 32 Pi^2]{[Ra^2 / (Ra^2 + Zb^2)^3] + [Zb^2 / (Ra^2 + Zb^2)^3]}

= [Mu Qa^2 C^2 / 32 Pi^2]{[(Ra^2 + Zb^2) / (Ra^2 + Zb^2)^3]

= [Mu Qa^2 C^2 / 32 Pi^2] / [(Ra^2 + Zb^2)^2]

This expression which applies along the axis of a thin charge and current ring suggests several features of the outside energy density function of a spheromak. At the center of the spheromak where Zb = 0 the energy density must be finite at about:

Umo = [Mu Qa^2 C^2 / 32 Pi^2 Ra^4]

and elsewhere along the ring axis:

U = [Mu Qa^2 C^2 / 32 Pi^2] / [(Ra^2 + Zb^2)^2]

However, in general the distance from the center of the spheromak is (R^2 + H^2) so the contemplated spheromak external field energy density function will have to be of the approximate form:

**U** = [Mu Qa^2 C^2 / 32 Pi^2] / [(Ra^2 + R^2 + H^2)^2]

= [Mu Qa^2 C^2 / 32 Pi^2 Ra^4] [ Ra^2 / (Ra^2 + R^2 + H^2)]^2

= **Uo [Ra^2 / (Ra^2 + R^2 + H^2)]^2**

where:

**Uo = [Mu Qa^2 C^2 / 32 Pi^2 Ra^4]**

In reality the spacial geometry of a spheromak is much more complicated than a thin charge and current ring. However, the above simple analysis gives an indication of the general form of a spheromak's external energy density function. The advantage of the above analysis is that it is simple. It is easy to get lost in the mathematical complexity of a detailed spheromak analysis.

This web page last updated August 21, 2016.

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