the Fields of a Spinning Disk of Charge
Let us begin by modeling a
spiral galaxy as a disk of single-valued mass density s=mgalaxy/pR2, lying in the xy-plane and
rotating with an angular velocity w that points in the +z
direction. In the article Galaxies and the Gravitomagnetic Force it was
found that such a galaxy theoretically has an Imaginary uniform gravitomagnetic
field O that points in the z direction:
Oz = 2pGswR/c2.
=2G i|mgalaxy| w / Rc2.
Given a star of mass mstar,
momentarily on the x-axis and at a fixed distance r from the galaxy’s center,
the gravitomagnetic force on the star is then (according to the GTM version of
the Lorentz force law)
Fgravmag x = i|mstar|
= i|mstar| wrx
2G i|mgalaxy| w/Rc2
= - (2G|mstar| |mgalaxy| w2rx) / Rc2.
And the total force on the star
is theoretically the sum of the gravitomagnetic force plus the gravitational
force (which also points toward the galactic center):
Ftotal x = Fgravmag
x + Fgrav x.
Now Newton 2 indicates that Ftotal x
Ftotal x = |mstar|
= - |mstar| w2rx.
Fgrav x = i|mstar|
is the gravitational field of the galaxy at the star’s position, we find that
gx = Fgrav x / i|mstar| (6)
= -i (Ftotal x - Fgravmag x) / |mstar|
=-i (- |mstar| w2rx - (2G |mstar| |mgalaxy| w2rx /Rc2)) / |mstar|
= -i (- w2rx)(- 2G |mgalaxy| w2rx / Rc2))
= (i w2rx)( -2G |mgalaxy| w2rx / Rc2).
Using the GMT rules that
(1) g transforms to E, (2) G transforms to 1/4peo, and (3) O transforms to
B, we find that, given a spinning disk of positive charge
with uniform charge density s, the electric field in the
disk’s plane points away from the center and has the magnitude
w2rx(-1/2peo)(qw2rx / Rc2). (7)
And, applying the same
transformations to Eq. 1, we find that
Eq. 8 agrees with the value for B specified on the Internet.