plasma lect.3

Motion of Charged Particle in Fields
Plasma are complicated because motions of electrons and ions are determined by the
electric and magnetic fields but also change the fields by the currents they carry.
For now we shall ignore the second part of the problem and assume that fields are
prescribed.
Even so, calculating the motions of a charged particle can be quite hard.
Equation of motion:
m dv
dt
q
E  v  B
v: velocity, q: charge, E: Electric Field, B: magnetic field
q
E  v  B: Lorentz Force
Have to solve this differential equation, to get position (r) and velocity (v
r
) given E(r,t),
B(r,t).
Approach: start simple, gradually generalize.
1. Uniform B field, E
0
m v
qv  B
Qualitatively in the plane perpendicular to B: Acceleration is perpendicular to v
so particle moves in a circle whose radius rL is such as to satisfy
mrL2
m v

2
rL

|q|v
B
: angular (velocity) frequency
1st equality shows 2
v

2
rL 2 (rL
v


)
Hence second gives m v



2
|q|v
B
1
i.e.

|q|B
m
Particle moves in a circular orbit with angular velocity  (“Cyclotron frequency”) and radius
rL (“Larmor Radius”, “Gyro Radius”).
Some Vector Algebra:

Particle Energy is constant. Proof : take v
Eq of motion then
mv

v
d
dt 12
mv2
qv
v  B
0

Parallel and perpendicular motions separate. v//
const. because accelerate is
perpendicular to B.
Perpendicular Dynamics:
Take B in z direction and write components
mv x
qvyB, mv y

qvxB
Hence v x
qB
m v y


qB
m 2vx

2vx
Solution: vx
v
cost (choose zero of time)
Substitute back: vy
m
qB v x

|q|
q v
sint
Integrate:
x
x0  v


sint, y
y0  q
|q|
v


cost
This is the equation of a circle with center r0
x0, y0 and radius rL
v


: Gyro Radius
Direction of motion is as indicated opposite for opposite sign of charge:
Ions rotate anticlockwise
Electrons clockwise about the magnetic field.
2
The