This preview shows page 1. Sign up to view the full content.
Unformatted text preview: simple cyclotron gyration.
The equation of motion
dv
m
= qv × B
dt
Taking z to be the direction of B = Bz mvx = qBv y mv y = − qBvx mvz = 0 2 qB
⎛ྎ qB ⎞ྏ v
vx =
v y = −⎜ྎ ⎟ྏ x m
⎝ྎ m ⎠ྏ (2.1) 2 − qB
⎛ྎ qB ⎞ྏ v
vy =
vx = −⎜ྎ ⎟ྏ y m
⎝ྎ m ⎠ྏ (2.2) ± The denoting the sign of q.
The solution to this equation
vx = v⊥eiωct = x (2.4a) where v⊥ is a positive constant denoting the speed in the plane
perpendicular to B. then m vy =
vx
qB
= ±iv ⊥ e iωct =y Integrating once again, we have
v
x − x0 = −i ⊥ eiωct ωc y − y0 = ± v⊥ ωc e iωct (2.4b) We define the Larmor radius to be v⊥ mv⊥
rL ≡
=
ωc q B
Taking the real part of eq. (2.5) we have x − x0 = rL sin ωct
y − y0 = rL cos ωct x B  + electron Larmor orbits in a magnetic field ion Figure 2.2
This describes a circular orbit in a guiding center (x0, y0) which is fixed.
The direction of the gyration is always such that the magnetic field
generated by the charged particle gyration is opposite to the external...
View
Full
Document
This document was uploaded on 01/17/2014.
 Winter '14

Click to edit the document details