1.13 Assignments 27
(ii) Define a Cartesian coordinate system such that thezaxis is parallel to the incident
relative velocity vectorv∞and goes through the scattering center. Let the impact
parameter be in theydirection so that the incident particle is traveling in they−z
plane. Make the graphics display span− 50 ≤z/bπ/ 2 ≤ 50 and− 50 ≤y/bπ/ 2 ≤ 50.
(iii) Set the magnetic field to be zero, and let the electric field be
E=−∇φ
whereφ=?soEx=?etc.
(iv) By usingr^2 =x^2 +y^2 +z^2 calculate the electric field at each particle position,
and so determine the particle trajectory.
(v) Demonstrate that the scattering is indeed at 90 degrees whenb=bπ/ 2. What
happens whenbis much larger or much smaller thanbπ/ 2? What happens whenq 1 ,q 2
have the same or opposite signs?
(vi) Have your code draw the relevant theoretical scattering angleθsand show that the
numerical result is in agreement.
- Collision relations- Show thatσntlmfp= 1whereσis the cross-section for a colli-
sion,ntis the density of target particles andlmfpis the mean free path. Show also
that the collision frequency is given byυ=σntvwherevis the velocity of the in-
cident particle. Calculate the electron-electron collision frequency for the following
plasmas: fusion (n∼ 1020 m−^3 , T ∼ 10 keV), partially ionized discharge plasma
(n∼ 1016 m−^3 , T∼ 10 eV). At what temperature does the conductivity of plasma
equal that of copper, and of steel? Assume thatZ= 1.
- Cyclotron motion- Suppose that a particle is immersed in a uniform magneticfield
B=Bzˆand there is no electric field. Suppose that att= 0the particle’s initial
position is atx= 0and its initial velocity isv=v 0 x.ˆUsing the Lorentz equation,
calculate the particle position and velocity as a function of time (be sure to take initial
conditions into account). What is the direction of rotation for ions and for electrons
(right handed or left handed with respect to the magnetic field)? If you had tomake
up a mnemonic for the sense of ion rotation, would it be Lions or Rions? Now, repeat
the analysis but this time with an electric fieldE= ˆxE 0 cos(ωt).What happens in the
limit whereω→ΩwhereΩ =qB/mis the cyclotron frequency? Assume that the
particle is a proton and thatB= 1Tesla,v 0 = 10^5 m/s, and compare your results with
direct numerical solution of the Lorentz equation. UseE 0 = 10^4 V/m for the electric
field.
- Space charge limited current- When a metal or metal oxide is heatedto high tem-
peratures it emits electrons from its surface. This process calledthermionic emission
is the basis of vacuum tube technology and is also essential when high currentsare
drawn from electrodes in a plasma. The electron emitting electrode is called acath-
odewhile the electrode to which the electronsflow is called ananode.An idealized
configuration is shown in Fig.1.5.
