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**INTRODUCTION:**

The material contained on this web page involves application of spheromak theory to atomic particles.

**HISTORY:**

During the period February 2016 to to August 2016 Charles Rhodes showed that elementary charged particles such as electrons and protons with quantum charges are spheromaks. A spheromak can be described in terms of a closed quantum line charge moving at the speed of light along a closed path with Np poloidal turns and Nt toroidal turns. An electromagnetic spheromak has a relative energy minimum at Np = 222 and Nt = 305, and these integers together with the speed of light C, the permiability of free space Mu, the electron charge Q and the geometric constant Pi precisely define the Planck constant h. The integer Np and Nt values also set the magnetic properties of elementary charged particles. Thus much of quantum mechanics rests on the structure and behavior of electromagnetic spheromaks.

**CHARGED PARTICLE EXISTENCE:**

A charged atomic particle, such as an electron or proton, is a localized packet of energy with a spheromak structure. A spheromak is contains energy in electric and magnetic fields as a result of circulating quantized electric charge that continuously moves along a closed spiral path at the speed of light. The charge motion causes local toroidal and poloidal magnetic fields. The charge causes an external radial electric field. The closed charge motion path traces out a toroidal shaped surface referred to as the spheromak wall. The resulting electric and magnetic fields have an energy density and are stable in time and relative position. Integrating the field energy density over all space gives the particle's rest potential energy. In its normal minimum energy state (ground state) the particle does not radiate. Hence the particle's energy is stable.

**SPHEROMAK WALL POSITION:**

The moving charge forms a charge and current sheet known as the spheromak wall. The position of this wall in space is stable if at every point on this wall the total field energy density on one side of the wall is equal to the total field energy density on the other side of the wall and if at every point on the wall the position of the wall corresponds to a total spheromak energy minimum . These conditions define the physical shape of the closed spiral charge path.

**PHOTON ABSORPTION AND EMISSION:**

If a stable particle changes energy by absorption or emission of a photon it shifts from its initial stable state to a new stable state. In order for the particle's field energy to increase the circulation frequency of the quantized charges must increase and hence the linear dimensions of the particle must decrease. The ratio:

(change in energy) / (change in frequency) = h

is known as the Planck constant

The Planck constant is a function of other physical constants.

**NUCLEAR MAGNETIC RESONANCE:**

A charged particle located in an external magnetic field can increase its energy content by orienting itself so that the particle's poloidal core magnetic field and the external magnetic field vectorially add. The same charged particle can reduce its energy content by orienting itself so that the particle's core magnetic field is opposite to the external magnetic field.

As the particle changes in orientation it either absorbs or emits a photon. The frequency of that photon is proportional to the external magnetic field strength.

With protons thes phenomena is known as nuclear magnetic resonance (NMR) or proton magnetic resonance. This phenomena has immense importance in biomedical magnetic imaging. However, NMR has limited general application because in many elements the magnetic moments of different nucleons tend to cancel each other.

**ELECTRON MAGNETIC RESONANCE:**

The electrons in a material exist in a wide range of local magnetic fields due to both nuclei and other electrons. Hence in many materials electrons absorb and emit photons over wide frequency bands. The lack of sharp frequency dependent resonances makes electron magnetic resonance a little used technique. However, broad band emission of photons by thermally energized electrons is an important source of thermal radiation.

**PARTICLE SIZE AND PARTICLE REST MASS:**

The derivation of the Planck constant is independent of the particle nominal radius (K Ro) and hence is independent of the total particle rest mass energy Ett.

If:

Rs = outside radius of spheromak wall with respect to the spheromak major axis;

Rc = inside radius of spheromak wall with respect to the spheromak major axis;

then the Planck constant effectively sets the parameter So where:

So^2 = (Rs / Rc)

but does not establish:

(K Ro) = Rc So = Rs / So

Thus the Planck constant is the same for different energy particles such as electrons and protons that have the same quantum charge magnitude.

At this time we speculate that there exists a magnetic flux quantum and that the size of stable particles such as electrons and protons is established by a requirement that the spheromak toroidal magnetic flux and the spheromak poloidal magnetic flux must both be integer multiples of the magnetic flux quantum. Work with Josephson junctions in superconducting materials indicates the existence of an apparent magnetic flux quantum of:

Phio = h / (2 Q)

= 2.0678337 X 10^-15 Weber.

However, it is possible that this apparent magnetic flux quantum is an observed change in magnetic flux due to a single electron and hence is itself an integer multiple of a smaller underlying magnetic flux quantum. To pursue this theory we must develop accurate expressions for both the poloidal magnetic flux and the toroidal magnetic flux and apply the constraint of the Planck constant.

This issue needs more work. The proof of this hypothesis will lie in the ability of the theory to accurately predict the experimentally observed rest masses and magnetic moments of an electron and a proton.

At this time, we have no idea as to the underlying mechanism of charge quantization or the underlying mechanism of magnetic flux quantization.

A possible further development of this theory is to show that the total energy versus linear velocity behavior of charged particle spheromaks is as predicted by special relativity.

**ELECTRONS, PROTONS, NEUTRONS, DEUTERONS, POSITRONS AND ANTI-PROTONS:**

In our local universe energy is primarily concentrated in the rest mass of stable particles known as electrons and protons. There are also neutrons which may be stable in an atomic nucleus but which in free space decompose into electrons, protons and neutrinos.

The net charge on a proton is 1.60217657 X 10^-19 coulombs.

The net charge on an electron is - 1.60217657 X 10^-19 coulombs.

The net charge on a neutron is zero.

The rest mass of a proton in free space is 1.67262178 X 10^-27 kg.

The rest mass of an electron in free space is 9.10938291 X 10^-31 kg.

The rest mass of a neutron in free space is slightly more than the combined rest masses of a proton plus an electron.

The rest mass of a deuteron in free space is slightly less than the combined rest masses of a neutron plus a proton./P>

There are corresponding charged particles known as positrons and anti-protons.

A positron has the same mass as an electron but opposite charge.

An anti-proton has the same mass as a proton but opposite charge.

These anti-particles are rare in our local universe. Positrons usually occur as a result of conversion of a > 1 MeV gamma photon into an energetic electron-positron pair. In some nuclear decays the energy release is sufficient to produce an energetic electron-positron pair. The positron is emitted while the net charge on the nucleus decreases by addition of one electron charge. That electron is absorbed by a proton and converts that proton into a neutron.

Note that neutrons rely on the energy fields from adjacent protons for long term stability. Absent such external energy fields free neutrons spontaneously decay with a half life of about 10 minutes.

Spin is a charged particle parameter which is related to the direction of the spheromak toroidal magnetic field with respect to the the direction of the spheromak poloidal magnetic field. For a specific poloidal magnetic field direction there are two possible toroidal magnetic field directions. In an atomic nucleus the nucleons tend to pair up so that their net external vector energy fields cancel. Similarly, in an atom the electrons tend to pair up so that their net external vector energy fields cancel. Field cancellation reduces the aggregate total energy and creates a mutual potential energy well. The combined electric and magnetic interactions between the particle spheromaks creates numerous closely spaced energy states.

A free electron is generally indistinguishable from every other similarly oriented free electron except via its spin.

A free proton is generally indistinguishable from every other similarly oriented free proton except via its spin.

A free positron is generally indistinguishable from every other similarly oriented free positron except via its spin.

An free anti-proton is generally indistinguishable from every other similarly oriented free anti-proton except via its spin.

Under circumstances of a high gamma photon concentration and a low particle concentration a photon can convert into an electron-positron pair. In circumstances of a high charged particle concentration and a low gamma photon concentration an electron and a positron can annihilate each other to form a photon. Under circumstances of a very high energy gamma photon concentration and a low particle concentration a very high energy gamma photon can convert into a proton-anti-proton pair. Under circumstances of a low gamma photon concentration and a high particle concentration a proton and an anti-proton will annhilate each other to form a new gamma photon. There is an equilibrium between radiation energy density and particle concentration at which the rate of pair production equals the rate of pair annihilation.

Our local universe has evolved to the point that, except in the cores of stars, the concentration of high energy gamma photons that can produce particle-anti-particle pairs is very low and the concentration of anti-particles is very low, so that particle generation and annihilation are no longer significant processes. However, cosmologists believe that the universe evolved from an initial high gamma photon concentration known as the "big bang".

**CHARGE HOSE:**

A Charge hose is a mathematical construct that quantitatively explains experimentally observed atomic particle phenomena. Charge hose theory may be controversial but it is a practical tool for mathematical modelling of the behavior of both charged particles and plasmas.

A charged particle can be mathematically modelled as two long thin filaments of circulating charge flowing in opposite directions within a hose. The hose coils to form a toroidal shaped charge and current sheet. The two hose ends are connected together to form a closed spiral path with length **Lh**. The charge and current sheet forms the geometric divider wall of a stable electromagnetic configuration known as a spheromak.

**CHARACTERISTIC FREQUENCY:**

If C = speed of light, then the time required for cyclic net charge propagation around the closed charge hose path of length **Lh** gives the charged particle at rest a characteristic frequency **Fh** where:

**Fh = C / Lh**

For a given charge the smaller a spheromak is the more energy that it contains and the higher is its characteristic frequency **Fh**. If the charged particle's energy changes due to photon capture or photon emission while the particle's net charge remains constant there is a corresponding change in spheromak size and hence there is a corresponding change in the charged particle's characteristic frequency **Fh**. The emitted or absorbed photon must reflect both the change in energy and the change in the charged particle characteristic frequency. The proportionality constant between the change in energy and the emitted or absorbed photon frequency is known as the Planck constant **h**. Note that the emitted or absorbed photon frequency is the beat frequency difference between the initial particle characteristic frequency **Fha** and the final particle characteristic frequency **Fhb**.

**ENERGY STABILITY:**

As long as charge is uniformly distributed along the length of the charge hose and moves at a uniform axial velocity along the charge hose, and as long as the charge hose coil is dimensionally stable, there is no change in the spacial distribution of charge with time and hence there is no change in field geometru or spheromak energy. Hence there is no emitted or absorbed electromagnetic radiation and the particle is stable.

**DISCRETE SOLUTIONS:**

There is a further aspect of charge hose that is important in quantum mechanics. In order for the particle to be stable over time each turn of the charge hose of length **Lp** around the charge hose coil's main axis of symmetry must be the same length and geometry as every other similar such turn. Likewise each turn of length **Lt** around the toroidal axis must be the same length and geometry as every other similar such turn. The number of poloidal turns **Np** and the number of toroidal turns **Nt** must both be exact integers. The poloidal and toroidal directions are orthogonal. Hence:

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

**NEUTRON SPHEROMAK:**

Consider two charged particle spheromaks, one inside the other, sharing common major and minor axes, with opposite net static charges. If these opposite charges are identical the result is a particle with no net charge but with an external magnetic moment. This is the assumed structure of neutrons and anti-neutrons.

The electric field between the inner spheromak and the outer spheromak contains part of the neutron rest mass energy.

**NEUTRAL ASSEMBLY:**

Consider two spheromaks, one inside the other, sharing common major axes but with different minor axis radii, and with opposite net static charges. The positive spheromak has much more energy than the negative spheromak and hence fits inside the core of the electron spheromak. If these opposite charges are identical the result is a an assembly with no net charge known as a hydrogen atom.

The electric field between the inner spheromak and the outer spheromak contains part of assembly rest mass energy.

**NUCLEAR PARTICLE BINDING:**

A proton spheromak can readily magnetically bind to 0, 1 or 2 neutron spheromaks to form hydrogen, deuterium and tritium nuclei. A tritium nucleus is unstable and absent intense gamma ray radiation decays into a He-3 nucleus. A particularly stable combination is two protons plus two neutrons forming a He-4 nucleus.

**CHARGED PARTICLE MODEL:**

The following structure summarizes my understanding of the physical properties of simple charged particles.

A free charged particle can be viewed as 3 quanta of charge (3 quarks) that are uniformly distributed along closed spiral paths. The net charge motion along the spiral path is at speed of light C. The period required for the charge quantum to cycle around the closed path length Lh has a corresponding frequency Fh which gives the charged particle a characteristic frequency. The resulting electric and magnetic field energies together form the particle's rest mass energy. The closed spiral charge path forms a toroidal shaped charge and current sheet known as the spheromak wall. Inside the spheromak wall the magnetic field is purely toroidal. Outside the spheromak wall the magnetic field is poloidal. Due to the closed spiral current path geometry the charge per unit area on the spheromak wall varies with the radius from the toroid's main axis of symmetry. This configuration forms a stable spheromak. Outside the spheromak wall the total energy density **U** takes the form:

**U = Uo [Ro^2 / ((K Ro)^2 + R^2 + H^2)]^2**

where:

**Uo** = a theoretical peak energy density;

**(K Ro)** = a characteristic radius from the toroid main axis of symetry which determines the particle rest mass;

**R** = radius of a point from the toroid's main axis of symmetry;

**H** = distance of a point from the toroid's equitorial plane.

Define:

**Rc** = inner radius of spheromak wall on the equatorial plane

**Rs** = outside radius of spheromak wall on the equatorial plane.

Inside the spheromak wall, where the toroidal magnetic field is proportional to (1 / R), for **Rc < R < Rs** the field energy density is given by:

**Ut = Uto [Ro / R]^2**

Note that **U** is the sum of the electric field energy density plus the magnetic field energy density and that for spheromak stability the expressions for **U** inside nd outside the wall must be equal everywhere on the spheromak wall.

Thus at R = Rc, H = 0:

U = Uo [Ro^2 / ((K Ro)^2 + Rc^2)]^2 = Uto [Ro / Rc]^2

and at R = Rs, H = 0:

U = Uo [Ro^2 / ((K Ro)^2 + Rs^2)]^2 = Uto[Ro / Rs]^2

Thus equating the expressions for Uto gives:

[Ro Rc / ((K Ro)^2 + Rc^2)]^2 = [Ro Rs / ((K Ro)^2 + Rs^2)]^2

or

[Ro Rc / ((K Ro)^2 + Rc^2)] = [Ro Rs / ((K Ro)^2 + Rs^2)]

or

((K Ro)^2 + Rs^2) Rc = ((K Ro)^2 + Rc^2) Rs

or

Rs Rc (Rs - Rc) = (K Ro)^2 (Rs -Rc)

or

**Rs Rc = (K Ro)^2**

The spheromak shape factor So is defined as:

So = (Rs / Rc)^0.5

Note that for **R^2 + H^2 >> Ro^2** the dependence of **U** on radial distance is the same as for a theoretical point charge.

Spheromaks tend to emit photons until they reach their stable low energy state which occurs at:

**So^2 ~ 4.1047**

As shown on other web pages on this web site this spheromak mathematical model accurately predicts the Planck constant and appears to explain the experimentally observed parameters of particle charge, particle rest mass, particle magnetic moment and particle spin.

**ATOMIC PARTICLE SPHEROMAK:**

The charge on an atomic particle spheromak is quantized. The mechanism of this charge quantization is not known. The net charge circulates at the speed of light. The characteristic frequency Fh of an atomic particle spheromak is:

Fh = C / Lh

where:

C = speed of light

and

Lh = charge hose length

= [(Np Lp)^2 + (Nt Lt)^2]^0.5

where:

Lt = toroidal turn length

Lp = toroidal magnetic path length at R = Rf

Nt = number of toroidal turns in length Lh

The characteristic frequency F of an atomic particle is given by:

**F = C / Lh**

= C / [(Np Lp)^2 + (Nt Lt)^2]^0.5

Recall that:
Lp = Pi (Rs + Rc)

and

Lt = Pi (Rs - Rc)

and

K Ro / Rc = So = Rs / K Ro

Thus:

**Fh** = C / [(Np Lp)^2 + (Nt Lt)^2]^0.5

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

or

** Fh = C / {Pi K Ro [[(Np (Rs + Rc))^2 / (K Ro)^2] + [(Nt (Rs - Rc))^2 / (K Ro)^2]]^0.5}**

Hence:

**(1 / K Ro)**

= {Pi Fh / C) [[(Np (Rs + Rc))^2 / (K Ro)^2] + [(Nt (Rs - Rc))^2 / (K Ro)^2]]^0.5} (Fh / C)

= {Pi Fh / C) [[(Np Rc (So^2 + 1))^2 / (K Ro)^2] + [(Nt Rc (So^2 - 1))^2 / (K Ro)^2]]^0.5}

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

= Qs C / [(Np Lp)^2 + (Nt Lt)^2]^0.5

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

**TOTAL ISOLATED PARTICLE ENERGY:**

As shown on the web page titled SPHEROMAK ENERGY the total energy of an isolated elementary particle is given by:

**Ett = (Ett / Efs) Uo Pi^2 (K Ro)^3**

where:

Efs = Uo (K Ro)^3 Pi^2

and

(Ett / Efs) ~ 0.95 at So^2 = 4.0

Let:

Uoe = Uo for a free electron

Uop = Uo for a free proton

Roe = Ro for a free electron

Rop = Ro for a free proton

We have excellent experimental data which gives:

[(proton rest energy / electron rest energy] = [(Ett / Efs)|proton (Uop Roe^3)] / [(Ett / Efs)|electron (Uoe Rop^3)]

= [(Uop Roe^3)] / [(Uoe Rop^3)]

= (Mp / Me)

= (1.67262178 X 10^-27 kg) / (9.10938291 X 10^-31 kg)

= 1836.152675

If Uoe = Uop

then:

(Roe / Rop) = (1836.152675)^0.3333

= **12.145**

**OTHER ATOMIC PARTICLE MODELS:**

The Standard Model a proton consists of two quarks each with (2 / 3) of a proton charge and one quark with (1 / 3) of an electron charge. Quarks seem to exist in various combinations but do not exist in isolation.

At this time I do not understand the structure of neutrinos. They appear to be analogous to a photons but have different properties. They have no charge and little or no rest mass but they appear to convey energy and spin.

**CHARGE HOSE SUMMARY:**

A charge hose is characterized by a net charge **Qa**, an axial current **Ih** and a frequency **Fh**. The net charge gives the particle an external electric field. The charge hose current:

**Ih = (Qa C / Lh)**

gives the charged particle an external poloidal magnetic field and an internal toroidal magnetic field. Physical motion of the spheromak with respect to an external observer gives the particle linear momentum. Interaction of neighboring spheromaks converts potential energy into kinetic energy.

**DEFINITIONS:**

Define:

**Nnh** = number of negative charge quanta forming the charge hose;

**Nph** = number of positive charge quanta forming the charge hose;

**Lh** = length of charge hose;

**Vn** = negative charge axial velocity through the charge hose;

**Vp** = positive charge axial velocity through charge hose;

**Qp** = one quantum of positive charge

**Qn** = one quantum of negative charge

**Q** = net charge on a proton

**QUARK BEHAVIOR:**

Simple atomic particles are composed of quarks that move around the closed path at less than the speed of light. Hence if the net apparent static charge Qa is:

Qa = (Qp Nph + Qn Nnh)

the effective charge hose current Ih is:

Ih = [(Qp Nph Vp + Qn Nnh Vn)] / Lh

= Qa C / Lh

Under quark theory:

Qa = (Qp Nph + Qn Nnh)

= Q

For a proton:

Qp = (2 / 3) Q

Nph = 2

Qn = - (1 / 3) Q

Nnh = 1

giving:

(Qp Nph + Qn Nnh) = [(2 Q / 3) (2)] + [(- Q / 3) (1)]

= Q = Qa

For a proton:

Ih = [(Qp Nph Vp + Qn Nnh Vn] / Lh

= [(2 Q / 3) (2) Vp + (- Q / 3)(1) Vn] / Lh

= Q C / Lh

**ATOMIC PARTICLES:**

Spheromaks account for the existence of stable elementary particles. In these particles net charge moves along a closed spiral path at the speed of light C and adopt the shape of a spheromak and hence form stable atomic particles. The rest mass of each atomic particle comes from the energy contained within its electric and magnetic fields. The externally observed quantum charge units are themselves believed to be composed of opposite but unequal sub-quantum charges known as quarks.

**NUCLEAR BINDING:**

The elementary stable charged particles are electrons and protons. A proton can give stability to up to two adjacent coupled neutrons. Stable atomic nuclei appear to be magnetically coupled collections of H-1, H-2, H-3, He-3 and He-4 nuclei. This structure is experimentally observed via the net nuclear magnetic moment.

At very short ranges the magnetic field energy density of a proton spheromak exceeds the electric field energy density. A neutron's magnetic moment is similar to a proton's magnetic moment. Hence adjacent protons and neutrons in appropriate ratios can magnetically couple together in spite of the apparent high electric field repulsive forces.

These binding mechanisms seem to account for the experimentally observed changes in nuclear magnetic moment and nuclear structure stability depending on the numbers of protons and neutrons in the observed atomic nuclei.

**DEFINITIONS:**

Nne = number of equivalent electron charges in an atomic nucleus;

Nnp = number of equivalent positron charges in an atomic nucleus;

Nip = number of equivalent proton charges in an atomic nucleus;

Nie = number of equivalent anti-proton charges in an atomic nucleus

Thus:

Number of protons in a nucleus = (Nip - Nne)

Number of neutrons in a nucleus = Nne

**NATURAL RADIO ISOTOPE DECAY:**

An atomic nucleus consists of an assembly of charged particles. If this nucleus can change into a similar assembly with a lower energy by emission of an electron, positron or alpha particle (He-4 nucleus) and/or a photon this change may spontaneously occur. Such a change is known as radio active decay. The time required for half of the nuclei of a particular isotope to spontaneously change is known as that isotope's half life.

The range of possible changes is constrained by the requirement for simultaneous compliance with the laws of conservation of energy, conservation of momentum and conservation of charge.

A decrease in nuclear energy occurs making the nucleus more stable when emission of an electron decrements the number of neutrons Nne and increments the number of protons (Nip - Nne).

A decrease in nuclear energy occurs making the nucleus more stable when there is electron-positron pair production immediately followed by positron emission and absorption of the remaining electron which decrements the number of protons (Nip - Nne) and increments the number of neutrons Nne.

**ELECTRON AND POSITRON:**

In each case the total rest energy is the sum of the external electric field energy, the internal electric field energy, the external poloidal magnetic field energy and the internal toroidal magnetic field energy.

The poloidal circulation of charge around the electron spheromaks main axis of symmetry gives the electron its external magnetic moment. The toroidal circulation of charge gives the electron its spin. Since the toroidal charge motion has two direction possibilities with respect to a particular poloidal magnetic field direction, an electron aligned to an external magnetic field has two possible spin states which are of equal energy. In some materials application of a strong external magnetic field can partially align the electron poloidal magnetic fields causing a phenomena known as Electron Spin Resonance (ESR). ESR has limited general application because in many materials the electron magnetic moments cancel each other.

**PROTON:**

The Standard Model indicates that a proton is composed of three quarks, two of which each have charge:

**(2 Q / 3)**

and one with charge:

**(- Q / 3)**.

Note that isolated quarks have never been experimentally observed. However, they exist as mathematical constructs that explain the existance of stable charged particles. Quarks seem to behave as separate entities within a proton.

**FREE NEUTRON:**

A special case particle is a free neutron (a neutron outside an atomic nucleus) which has zero net charge and which for thermal neutrons has an apparent half life of about 11 minutes.

A free neutron contains more energy than the sum of the energies of its main components which are a free electron and a free proton. On neutron decay the extra energy forms a neutrino which conveys spin but no charge and which only rarely interacts with charged particle matter.

A free neutron has no net external electric field, which absence prevents external electric field based spheromak wall stability. The neutron mass and neutron decay characteristics suggest that a neutron spheromak consists of a proton charge plus an electron charge plus a small amount of additional energy. Charge hose current appears to account for the neutrons external magnetic field. However, outside of an atomic nucleus a free neutron is unstable.

Free neutrons spontaneously decompose into an electron plus a proton plus a neutrino. A neutron within a stable atomic nucleus can be viewed as consisting of a proton plus an electron plus some additional energy and opposed spin which on neutron decay are carried away by the neutrino.

I suspect that a free neutron is a an unstable structure consisting of a positive spheromak and a negative spheromak sharing the same minor axis. This structure lacks its own stabilizing external electric field and hence is unstable in free space. This structure can be stable in a nucleus where other nearby protons provide the required external stabilizing electric field.

**NEUTRON MODEL:**

Assume that a neutron can be modelled as a spheromak with a net positive charge located inside another spheromak with a net negative charge. The two spheromaks have a common major axis and a common minor axis. The major radius is Ra. The minor radius of the outer spheromak wall is Rb. The minor radius of the inner spheromak wall is Rc.

Definitions:

a = subscript referring to a property at the major radius

b = subscript referring to a property at the larger minor radius

c= suscript referring to a property at the smaller minor radius

Q = quantum of charge (1.6 X 10^-19 coulombs)
Ra = major radius

Rb = larger minor radius

Rc = smaller minor radius

Nb = number of poloidal turns on outer charge sheet

Nc = number of toroidal turns on inner charge sheet

Lb = outer path length

Lc = inner path length

Bt = toroidal magnetic field for R < Rc

Mu = permiability of free space = 4 Pi X 10^-7

Epsilon = permittivity of free space

C = speed of light

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

Geometry gives:

Lc^2 = (2 Pi Rc Nc)^2 + 2 Pi Ra

Lb^2 = (2 Pi Ra Nb)^2 + 2 Pi Rb

Fc = C / Lc

Fb = C / Lb

Charge / unit length on the inner charge hose is:

(- Q / Lc)

Charge per unit length on the outer charge hose is:

(Q / Lb)

The current due to the inner charge hose circulating around the toroidal axis is:

Q Nc Fc

The toroidal magnetic field inside the inner torus due to current in the inner torus is:

[(Mu Q Nc Fc) / (2 Pi Ra)]

The toroidal magnetic field inside the inner torus due to current in the outer torus is:

[(Mu Q Fb) / (2 Pi Rb)]

The total toroidal magnetic field inside the inner torus is:

Bt = [(Mu Q Nc Fc) / (2 Pi Ra)] + [(Mu Q Fb) / (2 Pi Rb)]

For (Nc Fc / Ra) >> (Fb / Rb):

Bt ~ [(Mu Q Nc Fc) / (2 Pi Ra)]

The surface area of the inner torus is:

(2 Pi Ra 2 Pi Rc)

The electric field Erc at the outer surface of the inner torus is:

Erc = [- Q / (Epsilon 2 Pi Ra 2 Pi Rc)]

The electric field at radius R for Rc < R < Rb is:

Er = [- Q / (Epsilon 2 Pi Ra 2 Pi R)]

The poloidal magnetic field at the outer surface of the inner torus due to current in the inner torus is:

[(Mu Q Fc) / (2 Pi Rc)]

The toroidal magnetic field at the outer surface of the inner torus due to current in the outer torus is:

[(Mu Q Fb) / (2 Pi Rb)]

Field energy density matching at the surface of the inner torus requires that:

[Bt^2 / 2 Mu] = [(Epsilon) Erc^2 / 2] + {[(Mu Q Fc) / (2 Pi Rc)]^2 / 2 Mu} + {[(Mu Q Fb) / (2 Pi Rb)]^2 / 2 Mu}

Assume that:

[(Epsilon) Erc^2 / 2] >> {[(Mu Q Fc) / (2 Pi Rc)]^2 / 2 Mu} + {[(Mu Q Fb) / (2 Pi Rb)]^2 / 2 Mu}

Then:

[Bt^2 / 2 Mu] = [(Epsilon) Erc^2 / 2]

or

[Bt^2 / 2 Mu] = [(Epsilon) [- Q / (Epsilon 2 Pi Ra 2 Pi Rc)]^2 / 2]

or

Bt^2 = [- Q / (C Epsilon 2 Pi Ra 2 Pi Rc)]^2

or

Bt = [Q / (C Epsilon 2 Pi Ra 2 Pi Rc)]

Recall that:

Bt = [(Mu Q Nc Fc) / (2 Pi Ra)]

Equating the two expressions for Bt gives:

[(Mu Q Nc Fc) / (2 Pi Ra)] = [Q / (C Epsilon 2 Pi Ra 2 Pi Rc)]

or

[(Mu Nc Fc)] = [1 / (C Epsilon 2 Pi Rc)]

or

[(Nc Fc)] = [C / (Epsilon 2 Pi Rc)]

The toroidal magnetic field energy inside the inner torus is:

[(2 Pi Ra) Pi Rc^2 Bt^2 / 2 Mu]

= [(2 Pi Ra) Pi Rc^2 [(Mu Q Nc Fc) / (2 Pi Ra)]^2 / 2 Mu]

= [Rc^2 [(Mu Q Nc Fc)^2 / (2 Ra)] / 2 Mu]

Recall that:

Fc = C / Lc

~ (C / 2 Pi Rc Nc)

Hence toroidal magnetic the field energy inside the inner torus is:

[Rc^2 [(Mu Q Nc Fc)^2 / (2 Ra)] / 2 Mu]

= [Rc^2 [(Mu Q Nc (C / 2 Pi Rc Nc))^2 / (2 Ra)] / 2 Mu]

= [Mu (Q C / Pi)^2 / (16 Ra)]

= [Mu (Q C / Pi)^2 (2 Pi Nb Fb / C)/ (16 Ra)]

= **[(Mu Q^2 C Nb Fb) / (8 Pi)]**

Fb = C / Lb ~ C / 2 Pi Ra Nb

The electric field energy between the inner torus and the outer torus is:

Integral from R = Rc to R = Rb of:

(2 Pi Ra) (2 Pi R) dR Epsilon Er^2 / 2

= Integral from R = Rc to R = Rb of:

(2 Pi Ra) (2 Pi R) dR Epsilon [- Q / (Epsilon 2 Pi Ra 2 Pi R)]^2 / 2

= Integral from R = Rc to R = Rb of:

dR Epsilon [ Q^2 / (Epsilon^2 (2 Pi Ra) (2 Pi R)] / 2

= [ Q^2 / (Epsilon (8 Pi^2 Ra)] Ln(Rb / Rc)

= [(Q^2 Mu C^2) / (8 Pi^2 Ra)] Ln(Rb / Rc)

Recall that:

Fb = C / Lb ~ C / 2 Pi Ra Nb

Hence the electric field energy between the inner torus and the outer torus is:

[(Q^2 Mu C^2) / (8 Pi^2 Ra)] Ln(Rb / Rc)
= **[(Q^2 Mu C Nb Fb) / (4 Pi)] Ln(Rb / Rc)**

Thus the total interior energy for a neutron is:

[(Mu Q^2 C Nb Fb) / (8 Pi)] + [(Q^2 Mu C Nb Fb) / (4 Pi)] Ln(Rb / Rc)

= **[(Mu Q^2 C Nb) / (4 Pi)][(1 / 2) + Ln(Rb / Rc)] Fb**

~ h Fb

which to a first approximation gives the Planck constant h. To get a better theoretical value for h we must develop an accurate expression for the external poloidal field energy.

Numerical evaluation gives:

[(Mu Q^2 C) / (4 Pi)] = 10^-7 X 2.56 X 10^-38 X 3 X 10^8

= 7.68 X 10^-37

indicating that Nb must be several hundred in order for theory to match experimental measurements of h.

**FIELD ENERGY BALANCE AT THE OUTSIDE WALL:**

For a neutron the electric field Erb at the inside surface of the outside wall is:

Erb = [Q /(Epsilon 2 Pi Ra 2 Pi Rb)]

The toroidal magnetic field at the inside surface of the outside wall is:

(Mu Q Fb) / (2 Pi Rb)

The poloidal magnetic field at the outside surface of the outside wall is:

Bp = Mu Q Nb Fb / 2 Pi Rb

Note that Nb can be determined from an experimental measurement but determining it via theoretical analysis requires a more elaborate calculation than is done herein which properly takes into account the exact toroidal geometry.

However, this analysis demonstrates the physical origin of the Planck constant.

Note that for a proton the energy that is internally contained in a neutron is contained in the external field of a proton. An electron is structually different again because with the same charge it contains much less energy. An electron may consist of only one charge type.

**NUCLEAR BINDING:**

Nuclear particle binding appears to involve magmetic fields arising from charge motion along a closed path at the speed of light. In a neutron it appears that one proton and one electron exist within the same spheromak structure.

**ATOMIC NUCLEUS:**

An atomic nucleus can be viewed as being an assembly of protons and neutrons that are trapped within a mutual potential energy well. The mutual potential energy well is deep and is stable only at certain values of the number of protons and the number of neutrons. The depth of the well is characterized by its negative binding energy. When a nucleus absorbs or emits energy it usually does so via stable or nearly stable particles such as photons, electrons, positrons, protons, neutrons or alpha particles (He-4 nuclei).

**Pb-207:**

An example of a stable nucleus is a lead-207 where:

**Nip = 2 X 207 = 414**

**Nnp = 207**
and from the atomic number:

**Nne = 207 - 82 = 125**

For Pb-207:

**(Nnp / Nne) = 207 / 125 = 1.656**

It has been experimentally observed that atomic nuclei with **(Nnp / Nne) < 1.66** are unstable and hence are radio active (eg uranium, plutonium, etc).

It has further been experimentally observed that atomic nuclei with **(Nnp / Nne) > 2.0** are likewise radioactive (eg beryllium-7, boron-8, carbon-9,carbon-10, carbon-11, etc.)

There is only a relatively small range of **(Nnp / Nne)** values that result in highly stable atomic nuclei.

**SEMI-STABLE ATOMIC PARTICLE ASSEMBLIES:**

There are about 1000 well documented semi-stable elementary particle assemblies that are known as atomic nuclei. Many of these assemblies are unstable and hence exhibit radio activity.

then the nucleus tends to decay by electron emission which decrements

If:

**(Nnp / Nne) > stable value**

and if:

**Nnp < 207**

then the nucleus tends to decay by electron-positron pair production immediately followed by positron emission which together increment **Nne** and hence decrements (Nip - Nne).

If:

**Nnp > 207**

then the nucleus tends to decay by alpha particle (helium-4 nucleus) emission which decrements **Nnp** by 4 and decrements **Nne** by 2, followed by electron emission(s).

Some nuclei are meta-stable energy states that spontaneously absorb or emit small amounts of energy by photon absorption/emission with no change in either **Nnp** or **Nne**.

**NUCLEAR FORCES:**

The above work suggests that so called nuclear forces are largely the result of overlapping magnetic fields between adjacent elementary particle spheromaks. When two spheromaks are physically adjacent to one another their mutual binding energy is much greater than if the spheromaks were rigid spheres.

There are a large number of short lived particles that can be formed via high energy collisions between nuclei. However, these particles are short lived energy states that have little impact on the real world and hence are outside the scope of this web site.

This web page last updated April 21, 2016.

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