Exercises with Magnetic Monopoles - Kurt Nalty

The magnitude of torsion in three dimensions is
τ =
⃗j · (⃗a × ⃗v)
(⃗a × ⃗v) · (⃗a × ⃗v)
⃗j = d⃗a
dt
F
⃗a = ⃗ m = µq eq m
4π
⃗j = µq eq m
4πm
= µq eq m
4πm
= µq eq m
4πm
= µq eq m
4πm
1
[ m
⃗a × ⃗r
r 3
⃗v × ⃗r
r 3
− 3 r 4 dr
dt (⃗v × ⃗r) ]
[ ⃗a × ⃗r
− 3 ]
⃗r · ⃗v
r 3 r 4 r (⃗v × ⃗r) [ ⃗a × ⃗r
− 3 ]
× ⃗r)
(⃗r · ⃗v)(⃗v
r 3 r2 r
[ ]
3
⃗a × ⃗r 3(⃗r · ⃗v)
− ⃗a
r 3 r 2
Before proceding, I want to comment on these jerk terms. The left portion,
⃗a × ⃗r, is a turning term, bring ⃗ J perpendicular to ⃗a and resulting in
constant acceleration for the electron. The right hand term, −3⃗a(⃗r · ⃗v)/r 2 ,
is a strong damping term for radial motion. I’ll do some simulations shortly,
but at first glance, the jerk should quickly circularize the electron in a flat
orbit.
Simplifying a bit, since ⃗a ⊥ ⃗v,
τ =
⃗j · (⃗a × ⃗v)
(⃗a × ⃗v) · (⃗a × ⃗v)
= ⃗ j · (⃗a × ⃗v)
a 2 v 2
Continuing with our development for the torsion term, ⃗a×⃗v is perpendicular
to ⃗a, so the second term in the jerk formula will drop out.
22