3.10 Assignments 121
the time for the particle to make one complete bounce between the walls if thenomi-
nal distance between walls isL.Calculate∆L,the change inLduring one complete
bounce and show that ifu << v,thenLvis a conserved quantity. By considering
collisionless particles bouncing in a cube which is slowly shrinking self-similarly in
three dimensions show thatPV^5 /^3 is constant whereP =nκT,nis the density of
the particles andTis the average kinetic energy of the particles. What happens if the
shrinking is not self-similar (hint: consider the effect of collisions and see discussion
in Bellan (2004a)).
- Using numerical techniques to integrate the equation of motion illustratehow a charged
particle changes from being non-axis-encircling to axis-encircling when a magnetic
fieldB=(2π)−^1 ∇ψ(r,z,t)×∇θreverses polarity att= 0.For simplicity useψ=
B(t)πr^2 , i.e., a uniform magnetic field. To make the solution as general as possi-
ble, normalize time to the cyclotron frequency by definingτ=ωct,and setB(τ) =
tanhτto represent a polarity reversing field. Normalize lengths to some reference
lengthLand normalize velocities toωcL.Show that the canonical angular momen-
tum is conserved. Hint - do not forget about the inductive electric field associated with
a time-dependent magnetic field.
- Consider a cusp magnetic field given byB=(2π)−^1 ∇ψ(r,z)×∇θwhere theflux
function
ψ(r,z) =Bπr^2
z
√
1 +z^2 /a^2
.
is antisymmetric inz.Plot the surfaces of constantflux. Using numerical techniques to
integrate the equation of motion demonstrate that a particle incident atz <<−a and
r=r 0 with incident velocityv=vz 0 ˆzwill reflect from the cusp ifvz 0 <r 0 ωcwhere
ωc=qB/m.
- Consider the motion of a charged particle starting from rest in a simpleone dimen-
sional electrostatic wave field:
m
d^2 x
dt^2
=−q∇φ(x,t)
whereφ(x,t) =φ ̄cos(kx−ωt).How large does ̄φhave to be to give trapping of
particles that start from rest. Demonstrate this trapping threshold numerically.
- Prove Equation (3.218).
- Prove that
δ(z) = lim
N→∞
sin(Nz)
πz
is a valid representation for the delta function.
- As sketched in Fig.3.18, a current loop (radiusr,currentI) is located in thex−y
plane;the loop’s axis defines thezaxis of the coordinate system, so that the center
of the loop is at the origin. The loop is immersed in a non-uniform magnetic field
Bproduced by external coils and oriented so that the magnetic field lines converge
symmetrically about thez-axis. The currentIis small and does not significantly
modifyB.Consider the following three circles: the current loop, a circle of radius
bcoaxial with the loop but with center atz=−L/ 2 and a circle of radiusawith
