120 Chapter 3. Motion of a single plasma particle
- Show that of all the standard drifts (E×B,∇B, curvature, polarization) only the
polarization drift causes a change in the particle energy. Hint: consider what happens
when the following equation is dotted withv:
m
dv
dt=F+v×B- Use the numerical Lorentz solver to calculate the motion of a charged particle in a
uniform magnetic fieldB=Bzˆand an electric field given by Eq.(3.177). Compare
the motion to the predictions of drift theory (E×B,polarization). Describe the motion
for cases whereα << 1 ,α≃ 1 ,andα >> 1 whereα=mk^2 φ/qB^2 .Describe what
happens whenαbecomes of order unity. - A “magnetic mirror” field in cylindrical coordinatesr,θ,zcan be expressed asB=
(2π)−^1 ∇ψ×∇θwhereψ=B 0 πr^2 (1 + (z/L)^2 )whereLis a characteristic length.
Sketch by hand the field line pattern in ther,zplane and write out the components of
B. What are appropriate characteristic lengths, times, and velocities for an electron
in this configuration? User= (x^2 +y^2 )^1 /^2 and numerically integrate the orbit of an
electron starting atx= 0,y=L,z= 0with initial velocityvx= 0and initialvy,vz
of the order of the characteristic velocity (try different values).Simultaneously plot
the motion in thez,yplane and in thex,yplane. What interesting phenomena can be
observed (e.g., reflection)? Does the electron stay on a constantψcontour? - Consider the motion of a charged particle in the magnetic field
B=
1
2 π∇ψ(r,z,t)×∇θwhere
ψ(r,z,t) =Bminπr^2[
1 + 2λ
ζ^2
ζ^4 + 1]
and
ζ=z
L(t).
Show by explicit evaluation of theflux derivatives and also by plotting contours of
constantflux that this is an example of a magnetic mirror field with minimum axial
fieldBminwhenz= 0and maximum axial fieldλBminatz=L(t).By making
L(t)a slowly decreasing function of time show that the magnetic mirrors slowly move
together. Using numerical techniques to integrate the equation of motion, demonstrate
Fermi acceleration of a particle when the mirrors move slowly together. Do not forget
the electric field associated with the time-changing magnetic field (this electric field
is closely related to the time derivative ofψ(r,z,t);use Faraday’s law). Plot the
velocity space angle atz= 0for each bounce between mirrors and show that the
particle becomes detrapped when this angle decreases belowθtrap= sin−^1 (λ−^1 ).- Consider a point particle bouncing with nominal velocityvbetween a stationary wall
and a second wall which is approaching the first wall with speedu.Calculate the
change in speed of the particle after it bounces from the moving wall (hint: do this first
in the frame of the moving wall, and then translate back to the lab frame). Calculateτb