Ring electrons
5 June 2012

The electron and proton have mass, charge, angular momentum and magnetic moment which can be tied together with the dynamic structure of a ring. They are tiny machines which have energy in orbit like Bohr's planetary atoms, helical electromagnetic waves and the Cosmos.
Ring electron: Drawn with the awesome and free k3dsurf. This is a hollow torus made out of a flux tube. The tube is very much smaller than the ring so the pretty drawing on the left is not to scale. The line drawing of the ellipse on the right is much closer to scale. The radius of the ring is rring and the radius of the tube is rtube. The ratio of rring/rtube = pi/alpha = 430.511. This ratio is like a one inch diameter garden hose with a length of 225 feet making a ring with a diameter of 72 feet. When you can see all of the ring it looks like a line drawing of an ellipse not a hollow torus. The ring proton also has this same ring/tube ratio but it is much smaller. A ring proton at this scale would be a dot smaller than a forth the width of the line. A charge may travel around the ring like water in a hose or be confined to a cross sectional ring or surface or both. Does charge have volume or does it only have surface? In the ring electron, there is a flux of charge, a current, flowing around a very skinny circular flux tube at the speed of light. Since this current must flow without loss, the electron is a superconductor.

Ratio of electrostatic to gravitational forces
FCE = ce2/(4*pi*e0*r2), this is the electrostatic repulsive force between two electrons with a charge of ce at a separation of r = 1 meter but rc = ce2/(4*pi*e0*me*c2) where rc is the classical radius of the electron or me*c2*rc = ce2/(4*pi*e0).
FCE = me*c2*rc/r2, substituted me*c2*rc = ce2/(4*pi*e0) where the mass of the electron is me and c is the speed of light.

FGE = G*me2/r2, this is the gravitational force between two electrons at a separation of r = 1 meter. G is the gravitational constant.

FCE/FGE = c2*rc/(G*me) = 4.16E42, this is the huge ratio of electrostatic repulsive forces to gravitational attractive forces between two electrons.

FCE = FGE? We can write, the electrostatic repulsion of the charge of the electrons equals the gravitational attraction of the electrons, and see what happens.
me*c2*rc/r2 = G*me2/r2.
c2*rc = G*me, this is the blackhole formula with radius = rc and mass = me.
r = G*me/c2 = 6.76E-58_m, this is the radius using the mass of the electron and the black hole formula which is based on gravity holding energy in orbit. This radius is far too small for the electron to have angular momentum or spin but this radius times the ratio of the electrostatic to gravitational forces equals rc. This is the classical radius of the electron.
G*me/c2 * c2*rc/(G*me) = rc
The ring electron can be described as having energy in orbit like a black hole but based on electromagnetic not gravitational forces holding the energy equivalent of mass in orbit.
For an alternate approach see Don J. Stevens here or here.

Electrons have a charge and a magnetic field. Moving charge is a current which produces a magnetic field so we may say the charge orbits in the electron producing its magnetic field. Does the charge loop because of the magnetic field or does the looping charge cause a magnetic field? We do know a charge and magnetic moment co-exist in the electron. The electron has an angular momentum and magnetic moment which require the energy equivalent of mass to orbit at a certain radius, which we use, so it is unlikely to be a point particle. The geometry of a ring electron can accommodate the electrodynamics necessary, for charge moving at the speed of light, to be restrained by its magnetic field.
Bergman's and Wesley's 1990 ring electron paper, ring protons and dual ring electron-proton neutrons are at this site worth exploring.
Planck's law
energy = hp*frequency, Planck's constant is hp = 6.6260755E-34_kg*m2/s.
me*c2 = hp*frequency = 8.1871044E-14_kg*m2/s2, the energy of the electron.
me*c2 = hp*c/wavelength, substituted for frequency.
wavelength = hp*c/(me*c2) = hp/(me*c) = Compton's wavelength, isolated wavelength.
r = wavelength/(2*pi) = hp/(2*pi*me*c) = rc/alpha, since hp = 2*pi*me*c*rc/alpha.
me*c2 = hp *c/(2*pi*rc/alpha), this is hp*frequency.
me*c2 = hp*c/(2*pi*rring) = hp *1.23559E20_1/s, frequency = c/(2*pi*rring).
Planck's law implies the radius of the electron is rring = rc/alpha.
Electron magnetic moment
current *area, the current flowing around a loop times the area enclosed by the loop is the magnetic moment. The electron must have a shape. Current*area suggests the electron is a disk. We seek current as a ring and magnetic field as something like a disk.
charge *frequency *area, substituted for current.
charge *velocity/circumference *area, substituted for frequency.
charge *velocity/(2*pi*r) *pi*r2, collected terms.
.5*charge*velocity*radius = .5*q*v*radius = magnetic moment.
.5*ce*c*rring = me*c2/(2*B) = magnetic moment where rring is the radius of the ring electron.
.5*ce*c*rring = hp/(4*pi) *ce/me, equate with hp/(4*pi) *ce/me, the magnetic moment of the electron.
.5*me*c*rring = hp/(4*pi), collected terms. The ce terms cancel and me is transposed. This is the same equation we see in the angular momentum. The incongruity between the magnetic moment and the angular momentum has been resolved in a simplistic even humorus way. Is this a too easy result? Are these simplistic solutions hiding a more complex underlying reality? The incongruity between two things being resolved in an odd way is one definition of a joke. We see a similar almost too simple pattern in the appendix, with e0*u0 = 1/c2 and z0 = 1/(e0*c) = c*u0_ohms.
rring = hp /(2*pi*me*c) = rc/alpha, isolated rring where hp = 2*pi *me *c *rc/alpha.
.5*q*v*rring = .5*ce*c*rc/alpha = mm = magnetic moment
The magnetic moment implies that the radius of the electron is rring = rc/alpha.

The huge magnetic field of the electron from the electron magnetic moment is
BrE = me*c2/(ce*c*rring) where the magnetic field increases as rring decreases or
BrE = me*c2/(ce*c*rc/alpha) = 4.41399827402E9_kg/(A*s2), 4 billion Tesla? A magnetic field this big would disrupt orbits in the atom were it not concealed in a loop. See the Beaty video.

Proton magnetic moment
.5 *charge *velocity *radius = magnetic moment.
.5 *ce *c *rRP = hp/(4*pi) *ce/mp, equate with hp/(4*pi) *ce/mp, the magnetic moment of the proton. Spin is constant in both the electron and proton at hp/(4*pi). The magnetic moment is spin times the charge divided by the mass, spin *ce/mp, hp/(4*pi) *ce/mp. The magnetic moment decreases with increasing mass of the proton.
.5*mp *c *rRP = hp/(4*pi), collected terms.
rRP = hp /(2*pi*mp*c), isolated the radius of the ring proton.
rRP = 2*pi*me*c*rc /(2*pi*mp*c*alpha), substituted for hp = 2*pi*me*c*rc/alpha.
rRP = rRE * me/mp = rc/alpha *me/mp =
The radius of the ring proton = 2.10309114269E-16_m which is mp/me, 1836.15267145 times smaller than the ring electron.
The radius of the tube of the ring proton is 4.88510138648E-19_m.
Proton magnetic field
B = mp*c2/(ce*c*rp), the huge magnetic field of the proton from the proton magnetic moment.
BrP = BrE *(mp/me)2 = mp*c2/(ce*c *rc/alpha *me/mp) = 1.48816037584E16_kg/(A*s2), a magnetic field this big would disrupt orbits in the atom were it not concealed in a loop.

Bohr magneton or electron magnetic moment
mm = .5*charge*velocity*radius.
.5 *ce *v *rring = .5 *ce *c *rc/alpha = 9.27E-24_A*m2. The accepted value of the magnetic moment, mm = 9.284770E-24_A*m2, is about (1+1/862) or (1+alpha/(2*pi)) times bigger than the Bohr magneton. Where do all those decimal places come from? We use a lot of decimal places only as a means of finding errors. We paint with a broad brush omitting small corrections. Small corrections imply perfect knowledge. We will see that alpha/pi is the ratio of the radii of the ring electron.

Electron angular momentum or spin
mass current * area, the mass flowing around a loop times the area enclosed by the loop is the angular momentum or spin. This implies that the electron is a disk as noted above
mass *frequency *area, substituted for mass current.
mass *velocity/circumference *area, substituted for frequency.
me *c/(2*pi*r) *pi*r2, collected terms.
.5 *me *c *rring = hp/(4*pi), equate with hp/(4*pi) the spin of the electron. This is the same equation we found in the magnetic moment.
rring = hp /(2*pi*me*c) = rc/alpha = rc*137.036 = 3.86E-13_m, isolated rring.
The angular momentum implies that the radius of the electron is rring = rc/alpha.

Electron spin
Spin is constant at hp/(4*pi). We can write hp = 2*pi*me*c*rring. If the relativistic mass of me increases as the radius of rring decreases then the spin at hp/(4*pi) stays constant. The magnetic moment however hp/(4*pi) *ce/me decreases with increasing mass.

Planck's law, the spin and the magnet moment of the electron imply that the radius of the electron is rring = rc/alpha and that the circumference is,
Compton's wavelength = 2*pi*rc/alpha = hp/(me*c). If me increases while rring decreases then the angular momentum may stay constant. If the mass me and therefore the rest energy increases then the radius of the electron rring must decrease. The electron ring must show a very small radius at high energy in a particle accelerator. The electron tube is pi/alpha = 430.511 times smaller. The radii are ever smaller at higher energy. This is not evidence that the electron is a point particle. A point particle must incorporate infinite magnetic pinch pressure, force/area, to restrain the infinite electrostatic pressure of repulsion, energy/volume, due to the charge of the electron being confined to the infinitely small volume of a point. This makes a point particle electron infinitely improbable. While we abhor infinities, the gigantic ring currents and magnetic fields we calculate also strain credibility however, so much energy in such a small volume has to come from somewhere.

The g-factor
This is a quote from Wiki, "The spin of a charged particle is associated with a magnetic dipole moment with a g-factor differing from 1. This is incompatible with classical physics, assuming that the charge and mass of the particle are distributed evenly in spheres of equal radius." Of course, from the name of this paper, one might astutely infer we are not taking the electron as a sphere.
The electron is not a sphere
Is the electron a disk or a ring or what?
Angular momentum = I *w, where I is the moment of inertia and w is the angular velocity.
hp/(4*pi) = I *w, substitute for the angular momentum of the electron,
where w = v/r = c/r and the velocity is v = c in the electron.
hp/(4*pi) = I *c/r, substitute for w.
r = I * 4*pi*c/hp, isolate r.
r = m*r2 * 4*pi*c/hp, substitute for the moment of inertia of a ring: I = m*r2.
r = hp/(4*pi*me*c), isolate r. This is the radius for a ring with the correct angular momentum.

r = I * 4*pi*c/hp, from above.
r = 1/2*m*r2 * 4*pi*c/hp, substitute for the moment of inertia of a disk: I = 1/2*m*r2.
r = hp/(2*pi*me*c) = rc/alpha = 3.86E-13_m, isolate r. This is the radius for a disk with the correct angular momentum.

I speculate, the ambiguity in the shape of the electron, in its moment of inertia, might be resolved in favor of the disk shaped electron. We have toroidal currents along the circumference of the ring of the electron and poloidal magnetic flux through the area of the ring of the electron. Both of which have energy and therefore mass. It would be hard to say the mass is only confined to the ring. The mass might be said to be confined to something like a disk which includes the ring.

q*E = q*c*B
The charge of the electron is q = ce, the electric field is E or force per charge. B is the magnetic field or force per charge per velocity. q*E is a radial force like centrifugal force. q*c*B is a center seeking force like gravity. E = c*B when canceling q in an electromagnetic wave.
force = q*v*B = A*s*m/s*B = amp*second*meters/second*B = amp*meter*B = A*m*B. This q*v*B force is what keeps charge in orbit in the electromagnetic wave, cyclotron, tokamak and ring electron and what causes the spiral of charged particles in the magnetic field of a bubble chamber as the particles loose velocity. q*v*B acts like a central force to keep the charge and its energy equivalent of mass in orbit. We are comfortable with the not-so-obvious central force of gravity holding the planets in orbit. This central force is more obscure. There is no central object to provide a central force. There is only q*v*B. The direction of the moving charge is changed by B. How does this work? We know how it acts sometimes. We seek a metaphor to describe this peculiar force.
Poincare stress and energy density
"Poincare stress" has to be present to prevent the electric charge of an electron from flying apart due to the Coulomb repulsion". See Paul Marmet.
q*E = q*c*B: The electrostatic force of repulsion of the charge equals the magnetic pinch force of attraction of the charge when the charge moves at the speed of light.
E = c*B, cancelled q, units are volts per meter, force per charge or kg*m/(A*s3)
E2 = B2*c2, squared.
E2 = B2 *1/(u0*e0), substituted c2 = 1/(u0*e0)
E2*e0 = B2/u0 = 1.55E25_kg/(m*s2), The energy density or pressure of the E and B fields are equal.
force/area = energy/volume = kg/(m*s2) = pressure = Pascals
kg*m/s2 /m2 = kg*m2/s2 /m3 = kg/(m*s2)

This is the magnetic pinch pressure equals the electrostatic pressure of repulsion. This magnetic pinch pressure restrains the charge to the thin ring of the electron like a hose restrains water.

Electron energy
force *radius = energy, in an orbiting system.
q*v*B *rring = me*c2, energy of the orbiting electron where v is velocity c and rring is radius.
Cyclotron formula and the magnetic field
q*c*B = me*c2/rring, the cyclotron formula with centrifugal force = mass*v2/rring.
The q*c*B force holds the mass energy of the electron charge in orbit.
B = me*c/(ce*rring) = me*c*alpha/(ce*rc), isolated B with q = ce.
B = me*c2/(mm*2) = 4.414E9_kg/(A*s2), Four billion Teslas. This is the huge magnetic field calculated from the mass and the magnetic moment. A magnetic field this big would disrupt orbits of atoms were the magnetic field not contained within a loop. See the Beaty video. This is the B at rring which holds the charge in orbit. This same value of B is next used at rtube. It is used with the current and Ampere's law to define the radius of the tube. The huge magnetic field of the electron is derived from the well measured mm = electron magnetic moment.
mm = .5*charge*velocity*radius = magnetic moment
mm = .5*ce*c*rring = .5*ce*c*rc/alpha = ce*hp/(4*pi*me).
2*mm*B = 2 * .5*ce*c*rring * me*c/(ce*rring) = me*c2
2*mm*B = me*c2, the magnetic potential energy is the rest mass energy.
2*mm*B = hp *frequency, this is the electron spin resonance formula. The amount of rotational work is the torque times the angle to rotate the electron from alignment with the magnetic field to alignment plus 180 degrees which is a spin flip.
me*c2/hp = 1.23559E20_1/s, the frequency of the ring electron.
me*c2*alpha2/hp = 6.57968E15_1/s, the frequency of the Bohr atom.

Toroidal electron current
amps = current = charge *frequency = charge * velocity/circumference
amps = ce *c/(2*pi*rring) = ce *c *me*c/hp = ce *me*c2/hp = ce2*c2/(4*pi*mm) = 19.8_A

Ampere’s law
The poloidal magnetic field around the tube of the electron is associated with the toroidal current along the ring of the electron.
2*pi*rtube*B/u0 = amps, The loop around rtube times B/u0 equals the toroidal current.
rtube = amps/B *u0/(2*pi), radius of the tube
rtube = ce*c/(2*pi*rring) *ce*rring/(me*c) *u0/(2*pi) = ce2/(2*pi*me) *u0/(2*pi) =
rtube = ce2/(4*pi*e0*me*c2 *pi) = rc/pi = 8.97E-16_m
rring = hp/(2*pi*me*c) = rc/alpha = 3.86E-13_m
rtube = alpha/pi *rring = alpha/pi *rc/alpha = rc/pi = rring /430.511

The ratio rring/rtube = pi/alpha = 430.511 stays constant with increasing rest energy so that the relative proportions of the electron stay the same with decreasing size of the electron at higher electron masses.

Bergman and Wesley in their version of the ring electron used charge/area = surface charge density. I see current through the tube which implies charge/volume = charge density. The ring electron model may illuminate some of these questions but we are very far from understanding nature.

Right hand rule modified 20111224
When you grab a ring with your right hand, the thumb points in the toroidal direction along the ring while the fingers curl through the area of the ring in the perpendicular poloidal direction around the tube. Poloidal flux, which occurs through an area, and perpendicular toroidal looping around, which occurs along a circumference, are always associated. A flux or current has magnitude and direction. It has a rate of change d/dt, if its direction changes, even if its magnitude, area or circumference do not change. The flux of red E and green B, seen below, are as real as a mudslide. If they are real, what are they? We know their units. The mathematical interaction of units tells us something about the consistancy of our world model but one might mistake superficial knowledge of units for the deep knowledge of reality.
Ampere's and Faraday's laws
The red E electric field has units of volts/meter, kg*m/(A*s3) = kg*m/(s2) *1/(A*s),
E = force/q, force per charge or force per (amp*second).
force = q*E. q = A*s is charge.
The green B magnetic field has units of Weber's/m2 = Teslas = kg/(A*s2) = kg*m/(s2) *1/(A*m)
B = force/(charge*velocity), charge*velocity = amps*seconds*meters/second = amps*meters.
force = B*q*v = B*A*s*m/s = B*A*m. v is velocity. This B*q*v force is what keeps charge in orbit in the electromagnetic wave, cyclotron, tokamak and ring electron.
B*q*v = kg*v2/r, where kg*v2/r = centrifugal force. r is the orbital radius.
B*q*v acts like a central force to keep the charge and its energy equivalent of mass in orbit.
First on the left, Faraday's law/(c*u0): The red ring is the moving electron charge which is a toroidal electron current in amps. The green arrow is the magnetic flux.
d(B*pi*r2)/dt /(c*u0) = 2*pi*r*E/(c*u0) = amps, Faraday's law/(c*u0).
The poloidal green flux of B/(c*u0) times the area of the ring equals the toroidal red E/(c*u0) times the circumference of the ring. Red exerts a torque around green.
Second: The poloidal green flux which was shown as a green arrow is now shown as a green poloidal looping around the red torroidal current. The green flux is still out of the ring like the north pole of a magnet. We will use the cyclotron formula to calculate the poloidal green flux of B.
Third, Ampere's law:
e0*d(E*pi*r2)/dt = 2*pi*r*B/u0 = amps. Maxwell's changing red poloidal displacement current times the area of the tube equals the toroidal green current times the circumference of the tube.
Green exerts a torque around red. This is a cross section through the second figure. It shows a single green loop of the poloidal flux around the tube and a piece of the red ring is now shown as a red arrow. In this cross section, showing the tube of the electron, the former red toroidal is now red poloidal and the former green poloidal is now green toroidal. This perpendicular transformation, in going to a cross section, changes our viewpoint from Faraday to Ampere.
Forth: The poloidal red flux which was shown as a red arrow is now shown as a red poloidal looping around a green torroidal current. The red flux is still out of the cross section of the tube. The red flux is the toroidal electron current along the ring of the electron.
Another view of ring electrons
Click images to enlarge! They are something like a wave guide.
First on the left: A green helix on a red torus. The red electron current is surrounded by a green magnetic helix. As the distance between the green loops becomes infinitesmal, the green helix becomes a green tube, the green totally encloses the red like a hose encloses water.
Second: A red helix on the green helix. The red torus is omitted for clarity. The green magnetic helix is surrounded by a red helical current layer. The red loops separation become infinitesmal in a different way than the green loops separation become infinitesmal.
Third: The red helix on a green helix becomes a pink torroidal electron current. At the center of the tube the green helix is omitted for clarity. As the distance between the red loops becomes infinitesmal, the field between the neighbor loops cancel where they are in opposite directions. The loops at the center of the tube, the pink current inside the green layer, reinforce each other where these loops are pointing in the same direction. The loops on the outside of the tube, outside of the green layer, reinforce each other, where they are pointing in the same direction. The residual field is the original electron current which is here shown as pink in the center of the tube, a green magnetic layer which is not shown and a current on the outside of the torus flowing in a direction opposite to the current in the center of the torus.
Forth: A red helix on on a green helix on a pink torus of current. This is the view when all three layers are stacked up. The red electron current, in the center of the torus, is here shown as pink for clarity.
Helical loops cancel when oppositely directed and add when commonly directed:
These helical current loops wrap around the green magnetic field which wraps around the torus. The red side of the loops and the blue side of the loops go in different directions, so they have a tendency to cancel when they are close together. Where the loops point in a common direction they have a tendency to add like the blue line at the top, which touches the torus, and the red line at the bottom, which is a little farther out from the torus.

Volume and density:
2*pi*rring *2*pi*rtube = 1.367E-26_m2, ring electron torus surface area.
Ce /(2*pi*rring *2*pi*rtube) = 11.716E6_A*s/m2, surface charge density.
2*pi*rring *pi*rtube2 = 2*pi*rc/alpha *pi*rc2/pi2 = 2*rc3/alpha = 6.1328E-42_m3, ring electron torus volume.
Ce *alpha/(2*rc3) = 2.61246E22_A*s/m3, charge volume density.
me *alpha/(2*rc3) = 1.4853E11_kg/m3. Ring electron density. Nuclear density is much larger at 10E21_kg/m3.

me*c2 *alpha/(2*rc3) = 1.3349E28_kg/(m*s2). Ring electron energy density using only the volume of the ring. This is 861.0224 times the energy density of B2/u0 or E2*e0.
me*c2 *alpha/(2*rc3) / {B2/u0} = 861.0224
me*c2 *alpha/(2*rc3) / {(me*c*alpha/(ce*rc))2/u0} =
me*c2 *alpha/(2*rc3) / {me2*c2*alpha2/(ce2*rc2*u0)} =
1/(2*rc) / {me*alpha/(ce2*u0)} =
ce2*u0/(2*rc* me*alpha) = 861.0224 = 2*pi/alpha
ce2/(2*rc* me*c2*e0*alpha) = 861.0224 = 2*pi/alpha
rc = ce2/(4*pi*e0 *me*c2), This is true.

me = 9.109E-31_kg, mass of the electron
ce = 1.602E-19_A*s, charge of the electron
e0 = 1E7/(4*pi*c2)_A2*s2/(kg*m), permittivity of free space.
u0 = 4*pi/1E7_kg*m/(A2*s2), permeability of free space
e0*u0 = 1/c2
z0 = 1/(e0*c) = c*u0 = 376.73_kg*m2/(A2*s3) = ohms, impedance of space
4*pi/u0 = e0*4*pi*c2 = ce2/(me*rc) = 1E7_A2*s2/(kg*m)

rc = ce2/(4*pi*e0*me*c2) = 2.82E-15_m, the classical radius of the electron.
me*c2 = ce2/(4*pi *e0 *rc), The rest energy me*c2 equals the energy associated with the charge of the electron of ce and a radius of rc. If the rest energy increases then rc decreases.

alpha = fine structure constant = .007297 or 1/137.036
hp = Planck's constant = 2*pi*me*c*rc/alpha = 6.626E-34 _kg*m2/s
frequency*wavelength = c
me*c2 = hp*frequency
me*c2 = hp*c/wavelength
wavelength = hp/(me*c) = Compton's wavelength = 2.426E-12_m,
the circumference of the ring electron.
hp*alpha = 2*pi*me*c*rc = ce2 /(2*c*e0)
Any symbolic definition of hp uses alpha and any symbolic definition of alpha uses hp. This gives us at least two definitions each of both hp and alpha. Combinations of the above three terms are seen. Electron spin comes from the first and second while rc comes from the first and second or the second and third.

me*c2*rc = ce2/(4*pi*e0) = 2.307E-28_kg*m3/s2, for comparison with,
1.41E-28_m3/(kg*s3) = c3/Mc = G/age, Mc is the mass of the cosmos about 1.91E53_kg. G about 6.67 E-11_m3/(kg*s2) is the gravitational constant. Age is the age of the cosmos about 15 billion years = 4.74E17_s.
Are these small and similar numbers a clue linking the force between electrons and the cosmological constants or is this merely numerology ?

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