Jerk, Curvature and Torsion in Motion of Charged Particle

A. Tan and M. Dokhanian
magnetic field B
is given by the Lorentz equation. In
Gaussian system of units, we have:
dv
m qE qv B . (3)
dt
II. MOTION OF CHARGED PARTICLE IN
UNIFORM ELECTRIC FIELD
We first investigate the motion of a charged particle in a
uniform electric field. In this and the following ezamples,
we consider a positive charge (q > 0). Choose the electric
field in the positive x-direction: E Exˆ
. Without loss of
generality, choose the initial velocity of the particle as
v v xˆ
v yˆ
. Further, let the initial position of the
0
particle be at the origin. The equations of motion are the
following:
and
dv x
dt
, (4)
dv y
0 , (5)
dt
dv z 0
dt
. (6)
where qE / m . Integrating Eqs. (4-6) twice with respect
to time, we get:
and
v x
t
, (7)
C 1
1 2
x t
C1t
C2
, (8)
2
v y C 3 , (9)
y C t , (10)
3 C 4
v z C 5 , (11)
z C t . (12)
5 C 6
where C 1 , C 2 , etc., are the constants of integration. The
initial conditions give: C1 v
; C 2 0 ; C 3 v
;
C 4 0 ; C 5 C6
0 . Thus the motion is entirely in the x -
y plane with
Eliminating t between Eqs. (13) and (14), we get
v 2
x y y . (15)
v 2v
2
Thus the path of the charged particle is a parabola (Fig. 1).
FIGURE 1. Charged particle in a uniform electric field.
The acceleration and jerk vectors are obtained by
successive differentiation of the velocity vector, giving:
and
v
v
txˆ
v yˆ
, (16)
a xˆ
, (17)
0
j . (18)
The curvature and torsion can readily be calculated using
Eqs. (1) and (2), giving:
and
z
v ┴
y
װv
v
, (19)
2 2 3/ 2
v
t
v
0 . (20)
Hence, an electric field can create curvature in the motion
of a charged particle but not torsion. From Eq. (19), it
follows that as t , 0 , i.e., the curvature
diminishes in time. Also, if v 0 , then 0 . Thus, it is
the perpendicular component of the initial velocity which is
responsible for producing the curvature.
III. MOTION OF CHARGED PARTICLE IN
UNIFORM MAGNETIC FIELD
E
x
1 2
We next investigate the motion of a charged particle in a
x v t t
, (13)
2
uniform magnetic field. Choose the magnetic field in the
and
positive x-direction: B Bxˆ
. Without loss of generality,
y vt
. (14)
choose the initial velocity of the particle as
Lat. Am. J. Phys. Educ. Vol. 5, No. 4, Dec. 2011 668 http://www.lajpe.org