__A
Stable Model for the Positron
__

All positrons consist of
electric charge q=1.602176565e-19 coul=e, mass m=9.10938356e-31 kg, and a
magnetic dipole moment m_{q}=9284.764e-27
m^{2}coul/sec.

In this article we model a
positron’s electric charge as a spinning disk with radius R, uniform charge
density s_{q}=e/pR^{2}, and spinning with an angular speed of w_{q}>0.
The formula for the magnetic dipole moment of this disk is

m_{q}=w_{q}R^{2}e/4.
(1)

Thus w_{q}R^{2}=4m_{q}/e
is a constant. We can solve for w_{q}
if we have a value for R. Let us estimate that R, the positron radius, is
R=1e-18. Then

w_{q}=4m_{q}/qR^{2}=2.318037570664338e+32.
(2)

We shall assume that a disk
lies in the xy plane and that w_{q}
points in the positive z-direction. It is readily shown that, given such a disk,
the magnetic field in the disk is *uniform*
and points in the same direction as w_{q}:

B_{z}=ew_{q}/2pe_{o}c^{2}R.
(3)

Hence an *increment* of charge, d(e), momentarily coincident with the positive
x-axis and at a distance R from the disk’s center, is subject to an
outward-pointing magnetic force:

dF_{mag x}=d(e)(w_{q}R)(ew_{q}/2pe_{o}Rc^{2})
(4)

=d(e)(w_{q}^{2}e/2pe_{o}c^{2}).

In the article __On the
Fields of a Spinning Disk of Charge __it was found that the electric field, ** E**,
points radially outward. On the disk’s periphery:

E_{x}=w_{q}^{2}R-ew_{q}^{2}/2pe_{o}c^{2}.
(5)

Thus in addition to the
magnetic force, d(e) also experiences an outward-pointing *electric* force:

dF_{elec x}=d(e)E_{x
}(6)

= d(e)(w_{q}^{2}R-ew_{q}^{2}/2pe_{o}c^{2}).

The *total* electromagnetic (or Lorentz) force on dq is

dF_{elecmag x}=dF_{elec
x}+dF_{mag x, }(7)

_{
}=d(e)(w_{q}^{2}R-ew_{q}^{2}/2pe_{o}c^{2}+ew_{q}^{2}/2pe_{o}c^{2}).

=d(e)(w_{q}^{2}R).

Now dF_{elecmag x}
points *outward *(in the positive-x
direction). If the positron is to be stable, some other force must be equal and
oppositely directed. An answer is suggested by gravitomagnetic theory.

The positron also has *mass.
*Let us model the mass portion of the particle as a disk of mass m, radius R,
uniform mass density s_{m}=m/pR^{2}, and also lying in the xy-plane and spinning with an
angular speed of w_{m,
}where the value of w_{m}
at stability is to be determined. *We shall
assume that **w*_{m}*
is parallel to w _{q}.*
This mass theoretically engenders an imaginary

O_{z}=2i|m_{gravitational}|
w_{m}G/Rc^{2}.
( 8)

Note that O_{z} is
Imaginary. An increment of mass d(i|m_{gravitational}|) on the disk’s
periphery, therefore experiences a Real *gravitomagnetic*
force*:*

dF_{gravmag x} =d(i|m_{gravitational}|)(v_{y})(O_{z})
(9)

=d(i|m_{gravitational}|)(w_{m}R)
(2i|m_{gravitational}| w_{m}G/Rc^{2})

=d(i|m_{gravitational}|)(2i|m_{gravitational}|
w_{m}^{2}G/c^{2}).

Note in this equation that dF_{gravmag x}
is *inward*.

Of course the disk of mass
also has a *gravitational* field. In the
cited recent article it was determined that the rotating disk has a
gravitational field, ** g**, which
on the periphery has a value of

g_{x}=
(iw_{m}^{2}R)(-
2G |m_{galaxy}| w_{m}^{2} / c^{2}).
(10)

Hence d(i|m_{gravitational}|)
also experiences a
Real *gravitational* force which, on the
periphery, has the value:

dF_{grav x} = d(i|m_{gravitational}|)(g_{x})
(11)

=d(i|m_{gravitational}|)
(iw_{m}^{2}R)(-
2G |m_{galaxy}| w_{m}^{2} / c^{2}).

The total grav-gravitomagnetic
force acting on dm points inward:

dF_{grav-gravmag x}=dF_{grav
x}+dF_{gravmag x}
(12)

= d(i|m_{gravitational}|)(iw_{m}^{2}R)(-
2G |m_{galaxy}| w_{m}^{2} / c^{2})+ d(i|m_{gravitational}|)(2i|m_{gravitational}|
w_{m}^{2}G/c^{2}).

=(iw_{m}^{2}R)+d(i|m_{gravitational}|)(-2G|m_{gravitational}|w_{m}^{2}/c^{2})+(2G|m_{gravitational}|
w_{m}^{2}/c^{2})

=(dm)(-iw_{m}^{2}R).

*The
positron will be stable if
*

dF_{elecmag x} + dF_{grav-gravmag
x} = 0*, *(13)

or if

d(e)(w_{q}^{2}R)-(dm)(w_{m}^{2}R)=0,
(14)

or if

d(e)/dm(w_{q}^{2})-(w_{m}^{2})=0.
(15)

Now

d(e)/dm=e/m.
(16)

Thus we have stability if

(e/m)(w_{q}^{2})-(w_{m}^{2})=0,
(17)

or if

w_{m}^{2}=(e/m)w_{q}^{2}.
(18)

This solves to

w_{m}^{2}=9.450664425311212e+75
. (19)

Thus

w_{m}=9.721452785109443e+37
(20)

At stability the disk of
mass spins faster than the disk of charge:

w_{m}/w_{q}=419382.882664983.
(21)

In summary, a positron model
of (a) a disk of charge, q=e, with radius R=1e-18, and spinning with an angular
speed of w_{q},
superposed on (b) a disk of mass, m, with radius R but spinning at a rate of w_{m}>w_{q},
is stable if

w_{m}=419382.882664983
w_{q}.