Fundamentals of Plasma Physics

3.10 Assignments 119

If the distribution function has the Maxwellian formf∼exp(−v^20 / 2 v^2 T)wherevTis the
thermal velocity, and ifω/k >>vTthen
[
d
dv 0

(f(v 0 )v 0 )

]

v 0 =ω/k

=

[

v 0

d

dv 0

(f(v 0 )) +f(v 0 )

]

v 0 =ω/k

=

[

− 2

v^20

vT^2

f(v 0 ) +f(v0)

]

v 0 =ω/k

(3.253)

showing that the derivative offis the dominant term. Hence, Eq.(3.252) becomes
〈
dWtotal
dt

〉

=−

πmω^4 bω

2 k^4

[

d

dv 0

(f(v 0 ))

]

v 0 =ω/k

. (3.254)

Substituting for the bounce frequency using Eq.(3.223) this becomes

〈

dWtotal

dt

〉

=−

πmω

2 k^2

(

qkφ 0

m

) 2 [

d

dv 0

(f(v 0 ))

]

v 0 =ω/k

. (3.255)

Thus particlesgainkinetic energy at the expense of the wave if the distribution function
has negative slope in the rangev∼ω/k. This process is called Landau damping and will
be examined in the Chapter 5 from the wave viewpoint.

3.10 Assignments

1. A charged particle starts fromrestin combined static fieldsE=EyˆandB=Bˆzwhere
   E/B <<candcis the speed of light. Calculate and plot its exact trajectory (do this
   both analytically and numerically).


2. Calculate (qualitatively and numerically) the trajectory of a particle starting from rest
   atx= 0, y= 5ain combinedEandBfields whereE=E 0 ˆxandB=ˆzB 0 y/a.What
   happens toμconservation on the liney= 0? Sketch the motion showing both the
   Larmor motion and the guiding center motion.


3. Calculate the motion of a particle in the steady state electric field produced by a line
   chargeλalong thezaxis and a steady state magnetic fieldB=B 0 z.ˆObtain an approx-
   imate solution using drift theory and also obtain a solution using Hamilton-Lagrange
   theory. Hint -for the drift theory show that the electric field has the formE=ˆrλ/ 2 πr.
   Assume thatλis small for approximate solutions.


4. Consider the magnetic field produced by a toroidal coil system;this coil consists of
   a single wire threading the hole of a torus (donut)Ntimes with theN turns evenly
   arranged around the circumference of the torus. Use Ampere’s law to show that the
   magnetic field is in the toroidal direction and has the formB=μNI/ 2 πrwhereN
   is the total number of turns in the coil andI is the current through the turn. What are
   the drifts for a particle having finite initial velocities both parallel and perpendicular
   to this toroidal field.