Ring electron

Ring electrons Modified 3-9-10
Bohr's planetary atom is the prototype of tiny machines which illuminate matter. We use these machines by analogy to explain the behavior of nature. We are very far from understanding nature. Here the electron is shown to have a dynamic structure. It is another tiny machine. It has energy in orbit like Bohr's planetary atom, 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 ellipse on the right is much closer to scale. The radius of the ring is r_ring and the radius of the tube is r_tube. The ratio of r_ring /r_tube = 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 charge of the electron 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.

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 one meter. r = 1 meter.
FCE = me*c²*r_c/r², substituted me*c²*r_c = ce²/(4*pi*e0). See the appendix.

FGE = G*me²/r², This is the gravitational force between two electrons at a separation of one meter.

FCE/FGE = c²*r_c/(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 electron equals the gravitational attraction of the electrons, and see what happens.
me*c²*r_c/r² = G*me²/r²,
c²*r_c = G*me, This is the blackhole formula with radius = r_c and mass = me.
r = G*me/c² = 6.76E-58_m, the radius of this blackhole using the black hole formula which is based on gravity holding energy in orbit and the mass of the electron. 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 r_c. This is the classical radius of the electron.
G*me/c² * c²*r_c/(G*me) = r_c
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.

Ring electrons: Electrons have a charge and a magnetic field. Moving charge produces a magnetic field so we may say the charge orbits in the electron producing its magnetic field. The electron has an angular momentum and magnetic moment which require 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 neutrons are at this site worth exploring.

Plank's law:
h_p*frequency = energy
h_p*frequency = m*c², the energy of the electron.
h_p*c/wavelength = m*c², substituted for frequency.
wavelength = h_p*c/(m*c²) = h_p/(m*c) = Compton's wavelength, isolate wavelength.
r = wavelength/(2*pi) = h_p/(2*pi*m*c) = r_c/alpha
Plank's law implies the radius of the electron is r_c/alpha.

Electron magnetic moment:
The current flowing around a loop times the area enclosed by the loop is the magnetic moment.
current *area =
charge *frequency *area =
charge *velocity/circumference *area =
ce *v/(2*pi*r) *pi*r² =
1/2 *ce *c *r, collect terms, v=c. 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.
1/2 *ce *c *r = h_p/(4*pi) *ce/me, equate with h_p/(4*pi) *ce/me, the magnetic moment of the electron.
1/2*me *c *r = h_p/(4*pi), collect terms. The ce terms cancel and me is transposed. This is the same equation we will see in the angular momentum. The incongruity between the magnetic moment and the angular momentum has been resolved in a simplistic even humorus way. Who planned this 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 pattern in the appendix, with e0*u0=1/c² and z0 = 1/(e0*c).
Spin is constant at h_p/(4*pi). The magnetic moment is spin times ce/me, h_p/(4*pi) *ce/me. The magnetic moment decreases with increasing mass of the electron.
r = h_p /(2*pi*me*c) = r_c/alpha, isolate r.
The magnetic moment implies that the radius of the electron is r_c/alpha.
The Bohr magneton is the electron magnetic moment, 1/2 *ce *v *r = 1/2 *ce *c *r_c/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 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.

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, so is the electron a disk or a ring or what?
Angular momentum = I * w, I is the moment of inertia. w is the angular velocity.
h_p/(4*pi) = I * w, substitute for the angular momentum of the electron.
w = v/r = c/r, the velocity is v=c in the electron.
h_p/(4*pi) = I * c/r, substitute for w.
r = I * 4*pi*c/h_p, isolate r.
r = m*r² * 4*pi*c/h_p, substitute for the moment of inertia of a ring. I = m*r².
r = h_p/(4*pi*m*c), isolate r. This is the radius for a ring with the correct angular momentum.
r = 1/2*m*r² * 4*pi*c/h_p, substitute for the moment of inertia of a disk. I = 1/2*m*r².
r = h_p/(2*pi*m*c), isolate r. This is the radius for a disk with the correct angular momentum. It is twice the radius of the ring.
I speculate, the radius calculated from the magnetic moment and from the angular momentum might be equal and the ambiguity in the shape of the electron, in its moment of inertia, 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 confined to the ring. The mass might simplistically be said to be confined to the disk which includes the ring.

Electron angular momentum or spin:
The mass flowing around a loop times the area enclosed by the loop is the angular momentum or spin.
mass current * area =
mass *frequency *area =
mass *velocity/circumference *area =
m *v/(2*pi*r) *pi*r² =
1/2*m *v*r, collect terms, this implies that the electron is a disk as noted above.
1/2 *me*c *r, me=m, c=v.
1/2 *me *c *r = h_p/(4*pi), equate with h_p/(4*pi) the spin of the electron. This is the same equation we found in the magnetic moment.
r = h_p /(2*pi*me*c) = r_c/alpha = r_c*137.036 = 3.86E-13_m, The radius of the ring is the radius from the spin or from the magnetic moment.
me*c² = h_p *c/(2*pi*r_c/alpha), this is h_p*frequency.
me*c² = h_p*c/(2*pi*r_ring) = h_p *1.23559E20_1/s, energy = h_p *frequency, 2*pi*r_ring = wavelength.
energy = h_p *frequency. This is Plank’s law. The spin and magnet moment imply that the radius of the electron is r_c/alpha and that the circumference is
Compton's wavelength = 2*pi*r_c/alpha = h_p/(me*c). If me increases while r decreases then the angular momentum may stay constant. If the mass and therefore the rest energy increases then the radius of the electron r_ring 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, to restrain the infinite electrostatic pressure of repulsion, 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.

Right hand rule: 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.

Faraday's and Ampere's laws:
The red E electric field has units of volts/meter, kg*m/(A*s³) = kg*m/(s²) *1/(A*s),
E = force per (amp*second) or force per charge.
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.

The above two forces are equal in an electromagnetic wave or ring electron where the velocity = c.
q*E = q*c*B, or canceling q
E = c*B, in an electromagnetic wave.

Poincaré stress and energy density: “Poincaré 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 or kg*m/(A*s³)
E² = B²*c², squared.
E² = B² *1/(u0*e0), c² = 1/(u0*e0)
E²*e0 = B²/u0 = 1.5504E25_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.

Converting Ampere's law to Faraday's law/(c*u0):
d(E*pi*r²)/dt *e0 = 2*pi*r*B/u0 = amps. Start with Ampere's law.
d(B*c*pi*r²)/dt *1/(c²*u0) = 2*pi*r*E/(c*u0) = amps. Substitute E=B*c, e0=1/(c²*u0), B=E/c.
d(B*pi*r²)/dt *1/(c*u0) = 2*pi*r*E/(c*u0) = amps, collect terms, Faraday's law/(c*u0).

First on the left, in the last figure, Faraday's law/(c*u0): The red ring is the moving electron charge which is a toroidal electron current in amps.
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 poroidal and the former green poroidal 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.

Click images to enlarge! Another view of the ring electron. 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 without the red torus. 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 of a red helix becomes a pink torroidal current. As the distance between the red loops becomes infinitesmal, the field between the neighbor loops cancel as they are in opposite directions. The loops at the center of the tube, by the pink current, reinforce each other as they are in the same direction. The residual field is the original electron current which is here shown as pink.
Forth: A red helix on on a green helix on a pink torus of current. This is the view when all the layers are stacked up. The red electron current is here shown as pink for clarity.

Cyclotron formula:
B*q*v = mass*v²/r, cyclotron formula. mass*v²/r = centrifugal force, v is velocity. r is radius.
force = B*q*v = B*A*s*m/s = B*A*m. This B*q*v force is what keeps charge in orbit in the electromagnetic wave, cyclotron, tokamak and ring electron.
B*q*v 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 B*q*v. 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.
B = mass*v/(q*r), collected terms from the cyclotron formula.
B = me*c/(ce*r_ring) = 2*pi*me²*c²/(ce*h_p) = me*c*alpha/(ce*r_c), me = mass of the electron. v = c.
B = me*c²/(mm*2) = 4.414E9_kg/(A*s²), Teslas. This is B at r_ring which holds the charge in orbit. This same equal B is next used at r_tube. It is used with the current and Ampere's law to define the radius of the tube.
c*ce*r_c/(alpha*2) = c*ce*r_ring/2 = ce*h_p/(4*pi*me) = mm, magnetic moment from Bohr.
2*mm*B = me*c²2*mm*B = h_p *frequency, This is the electron spin resonance formula.

Toroidal electron current:
amps = current = charge *frequency = charge * velocity/circumference
amps = ce *c/(2*pi*r_ring) = ce *c *me*c/h_p = ce *m*c²/h_p = 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*r_tube*B/u0 = amps, The loop around r_tube times B/u0 equals the toroidal current.
r_tube = amps/B *u0/(2*pi), radius of the tube
r_tube = ce*c/(2*pi*r_ring) *ce*r_ring/(me*c) *u0/(2*pi) = ce²/(2*pi*me) *u0/(2*pi) =
r_tube = ce²/(4*pi*e0*me*c² *pi) = r_c/pi = 8.97E-16_m
r_ring = h_p/(2*pi*me*c) = r_c/alpha = 3.86E-13_m
r_tube = alpha/pi *r_ring = alpha/pi *r_c/alpha = r_c/pi = r_ring /430.511
The ratio r_ring/r_tube = 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.

Volume and density:
2*pi*r_ring *2*pi*r_tube = 1.367E-26_m², ring electron area.
Ce /(2*pi*r_ring *2*pi*r_tube) = 11.716E6_A*s/m², surface charge density.
2*pi*r_ring *pi*r_tube² = 2*pi*r_c/alpha *pi*r_c²/pi² = 2*r_c³/alpha = 6.1328E-42_m³, ring electron volume.
Ce *alpha/(2*r_c³) = 2.61246_A*s/m³, charge volume density.
me *alpha/(2*r_c³) = 1.4853E11_kg/m³. Ring electron density. Nuclear density is much larger at 10E21_kg/m³.
me*c² *alpha/(2*r_c³) = 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*r_c³) / {B^2/u0} = 861.0224
me*c² *alpha/(2*r_c³) / {(me*c*alpha/(ce*r_c))^2/u0} =
me*c² *alpha/(2*r_c³) / {me²*c²*alpha²/(ce²*r_c²*u0)} =
1/(2*r_c) / {me*alpha/(ce²*u0)} =
ce²*u0/(2*r_c* me*alpha) = 861.0224 = 2*pi/alpha
ce²/(2*r_c* me*c²*e0*alpha) = 861.0224 = 2*pi/alpha or
r_c = ce²/(4*pi*e0 *me*c²), This is true.

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 may illuminate some of these questions but we are very far from understanding nature.

*Index* *Next, Bohr Atom*

Appendix 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) = 376.73_kg*m²/(A²*s³) = ohms, impedance of space
4*pi/u0 = e0*4*pi*c² = ce²/(me*r_c) = 1E7_A²*s²/(kg*m)

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

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

me*c²*r_c = 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 ?

*Index* *Next, Bohr Atom*