__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 *gravitomagnetic*
field

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

Note that O_{z} is
Imaginary and positive, since i|m_{grav}| is Imaginary and w_{m} is
positive. An increment of mass i(dm) on the disk’s periphery, therefore
experiences a Real *gravitomagnetic*
force*:*

dF_{gravmag x} =i(dm)(v_{y})(O_{z})
(9)

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

=-(dm)(2|m_{grav}|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}=i(w_{m}^{2}R-2mw_{m}^{2}G/c^{2}).
(10)

Hence dm also experiences a
Real *gravitational* force inward which
has the value:

dF_{grav x} = (dm)(g_{x})
(11)

=-(dm)(w_{m}^{2}R-2Gmw_{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)

=(dm)(-w_{m}^{2}R+2Gm/c^{2})+(dm)(-2mw_{m}^{2}G/c^{2})

=(dm)(-w_{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)

Now

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

Thus we have stability if

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

or if

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

This solves to

w_{m}^{2}=9.450665030982446e+75.
(19)

Thus

w_{m}=9.721453096622153e+37.
(20)

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

w_{m}/w_{q}=419397.
(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}=419397
w_{q}. (22(