PHYSICS 712
PHYSICS OF PLASMAS
SPRING 2007
Questions
and Answers
SUSAN LEA
In this course we shall study the basics of plasma physics, with
emphasis
on waves and instabilities.
TEXT: Required: Chen, Francis: Introduction
to Plasma Physics and Nuclear
fusion,
Volume I, (Plenum)
Supplementary: Lea, Susan: Mathematics
for Physicists (Brooks/Cole)
We shall cover almost the entire text during the semester, with a
small
amount of supplementary material. Please give me feedback as we go
along:
I want to know when you are having difficulties so that we can clear
them
up.
Part of the class assignment is a term paper in
which you can explore some topic in Plasma Physics in more depth than
we
can get into in class. The paper may be a review paper based on a
literature
search, or it may involve original calculations or a computer project.
Please begin to think about your paper right away.
Grading:
Problem sets 
Midterm examination 
Term Paper 
30% 
30% 
40% 
In the problems I am looking for an honest
effort,
and a steady growth in capability. Some of the problems have solutions
in the back of the book. Simply copying these solutions will not help
you
learn. Try to use the solutions only as a guide if you really get
stuck.
(It takes will power!)
Supplementary texts: There are many interesting and helpful
books on Plasma Physics. I especially recommend:
 Electrodynamics of particles and plasmas: Clemmow and
Dougherty,
Addison Wesley.
 Principles of Plasma Physics: Krall and Trivelpiece (Out
of
Print,
but is in the library.)
 Introduction to plasma physics: Nicholson. (Wiley).
 The framework of plasma physics: Hazeltine and
Waelbroeck,
Cambridge.
 Introduction to Plasma Physics: Goldston and
Rutherford,
IOP 1995
 The Physics of Plasmas: Boyd
and Sanderson, Cambridge, 2003
 Fundamentals of Plasma Physics:
Bellan, CUP 2006
 Physics of Space plasmas.
George Parkes, Wetsview, 2004
Important Dates:
Midterm (in class): Thursday April 19th
Paper

Due date

% of paper grade 
Topic statement: 
Thursday, February 22 
5

Abstract and outline: 
Tuesday March 20 
5

First draft: 
Tuesday April 24 
10

Second draft: 
Tuesday May 8

10

Final Paper: 
Tuesday May 22

50

Class presentations: 
May 15, 24 
20

Class schedule (Rough outline  subject to change)
More about the paper.
The paper will take the form of a literature review, or a summary
of your own work, if you did an original project.
If you choose to do a literature review, I expect to see sources
from
the refereed journals that are of a technical nature. Avoid the
temptation
to stick with Scientific American and the Web! This is a graduate
physics
course, and I expect to see a paper at the appropriate level, with
technical
detail included.
If you do an original calculation, I expect to see more detail than
would be typical in a journal article. Show me what you
did!
If you use a computer program, include the source code as an appendix.
Topic statement.
Your topic statement should include:
 A title.
 A one or two paraagraph discussion of the topic.
 At least one major reference.
Some suggestions:
The international thermonuclear reactor.
Wavewave interactions and their effect on inertial fusion.
Conditions in the coronal plasma as determined by xray spectroscopy.
An analysis of the KruskalSchwarzschild instability using an energy
method.
Propagation of whistlers in the Earth's upper atmosphere.
Determination of the intergalactic magnetic field.
Magnetic field reconnection.
(Don't know what all those things are? Look them up!)
Abstract and outline.
Abstract: A one or two paragraph summary of your paper.
Outline: A list of sections and subsections, with a one or two
sentence description of the contents of each.
A list of references. (You should have a dozen or so by now.)
First draft.
An almost complete paper, typed, doublespaced, 1020 pages. (A paper
that
contains original work may be shorter than a review paper.) A few
things
may be left to discover. Your reference list should be essentially
complete.
Art work may need a final polish. Paper must include:
 Title
 Author name (you!)
 Abstract
 Introduction
 Body (divided into sections and subsections as necessary)
 Conclusion
 List of references
 Figures and figure captions where appropriate.
Second draft.
Research should now be complete. You should have responded to comments
made on the first draft. Reference list must be complete, art work
should
be in final form.
Final paper:
Your best effort, including response to comments on the second draft.
Paper
should be in a form that you would be ready to send to a journal or
submit
to your supervisor.
Class presentation.
Have clear, legible viewgraphs or powerpoint slides (if you use them)
that show at most one or two big ideas. One clear diagram is worth 6
messy,
cluttered ones. Have an outline of your talk written out (some people
like
to use index cards). Time your presentation beforehand to make sure the
important stuff is included. Remember: your audience knows less about
your
topic than you do  go slowly and explain everything!
Citing references
There are several styles for citing references. Within the body of your
paper you may use a number to label your sources in the order they
appear.
Then you list the references by number at the end of the paper. The
style
used by the Astrophysical Journal (and most astrophysics journals) is
to
cite the reference by Author and year (Lea, 2007). At the end of the
paper
you list the references alphabetically by author. Convention dictates
that
more than three authors become et. al. (Lea et.al.,
2007).
In the reference list, the entry should be:
Lea, S.M., 2007, Journal of irreproducible results, 15,
75.
Here the 15 is the volume number and the 75 is the page number.
To cite references on the web, give the author's name, title of the
work in quotes, title of complete work (if appropriate) in italics, the
complete URL, and the date of the document or of your visit to the
site.
See the following site for more info:
http://ia.juniata.edu/citation/cse/cse05.htm
A word of warning: The web is not refereed! Look carefully at
the credibility of your site. A NASA site probably has reliable
information,
but Joe Bloe's home page may not. Use some common sense.
A paper with only (or even predominantly) web site references
is not acceptable.
I expect to see some papers from the professional literature as well.
Here is a useful reference
on how to write a good paper.
PROBLEM SETS
Problem Set #1 Due: Thursday February 1
These problems will get you used to the orders of magnitude encountered
in laboratory and space plasmas. Use a loglog scale for the
plots.
 Chen 1.3 (pg 12)
Include the following additional astrophysical plasmas, extending the
axes as appropriate:
 8) cluster of galaxies: n_{e }= 10^{3} m^{3},
T = 10 keV.
 9) Supernova remnant: n_{e} = 10^{5} m^{3},
T
= 1 keV
 10) Solar interior: n_{e} = 10^{32} m^{3},
T =
1 keV
 11) Solar corona: n_{e} = 10^{14} m^{3},
T =
100
eV.
 Chen 15
 Chen 18,
 Chen 111 (page 17)
 Compute the magnitude of the shielded point source potential
(equation
(5) in Debye notes) and numerically compare the result with the
unshielded potential at distances 0.1, 1 and 5 times the Debye length
from
the charge. Comment.
Problem set #2 Due: Thursday February 8th
 Chen 21 (pg 25) Also compute omega_{c}, and include
e) An electron near the surface of an accreting neutron star: E = 10
keV, B = 10^{8} T  Chen 27
 Chen 28. (pg 35)
 Chen 212 (pg 35)
 Chen 213 (pg 49)
Problem Set 3 Due: Thursday February 15th
 Chen 216
 Chen 217
 Refer to problem Chen 28. Calculate the bounce frequency for the
1 eV
protons and 30 keV electrons. (You'll need to use the formula for field
strength in a dipole field.) The last part involves some numerical
calculations.
Problem Set 4 Due: Thursday February 22nd
 Read carefully sections 3.3.1 to 3.3.5. In a steady state there
is no
explicit
time dependence (partial time derivatives identically zero). Consider
equation
344 for a steady state. Assume isotropic pressure, E = B = 0, but
nonzero
gravitational field. All variations are 1dimensional.
 (i) Write down a modified version of 344 appropriate to these
assumptions.
(Hint: gravitational force replaces Lorentz force.
 (ii) Derive Bernoulli's equation.
 Show that the distribution function on page 9 of Chen,
f(u) = Aexp([(1/2)mu^{2} + q(phi)]/kT)
satisfies the collisionless Vlasov equation.
 Chen 31 (page 58)
 Chen 37 (page 74)
Problem Set 5 Due: Thursday March 1st
 Calculate the plasma frequency for the plasmas in problem 13 of
Chen
(see
Problem set # 1) plus, a laser fusion pellet with n = 5x10^{22}
cm^{3}.
 Chen 41 (page 81)
 Chen 42 (page 86)
 Chen 43
Problem set 6 Due: Thursday March 8th
 Chen 46 (page 94)
 Chen 48 (pg 108)
 Chen 49 (pg 120)
 Chen problem 410 (pg 120)
 Chen 413
Problem Set 7 Due: Thursday March 15th.
 Chen 416 (pg 135)
 Chen 419
 Chen 421 (pg 135)
 Chen 424 (Be careful. Chen's answer is not correct.)
 Chen 425
Problem Set 8 Due: Thursday March 22nd
 Chen 426 (page 148)
 Chen 430
 Chen 432
 Chen 434
 Chen 445
Problem set 9 Due: Thursday March 29th
 On a plot of frequency omega versus wave number k, draw
the
dispersion
relations for all the waves we have studied. Label each wave and
indicate
its nature ( EM or electrostatic).
 Draw a graph of refractive index squared versus omega. Choose a
particular
ordering of the plasma frequency and the cyclotron frequency (ie omega_{p}
less than or greater than omega_{c}) Mark all relevant
frequencies,
(eg omega_{R} and omega_{L}). Show the dispersion
relations
for all the waves we have studied. Use different colors or otherwise
distinguish
the waves that propagate across B from the waves that propagate
along B. Label all the waves. Estimate where the relations for
waves
propagating at an intermediate angle to B would lie, and sketch them
in.
(You should have a total of 4 waves, including plasma oscillations.
Assume
T = 0 throughout.)
 Chen 51 (page 175176)
 Chen 52
Problem Set 10 Due Thursday April 5th
 In the interstellar medium the electron density is about 10^{2}
cm^{3}, B is about 3x10^{6} Gauss and the density of
neutral hydrogen atoms is about 1 cm^{3}. The temperature is
about
100 K and is the same for electrons and ions. If a typical interstellar
cloud has dimension r approximately 10 pc = 3x10^{19}
cm,
how long does it take for the magnetic field to escape from a cloud if
the only means of escape is electron and ion diffusion across B? Assume
cylindrical symmetry, and neglect velocities along B. The collision
cross
section for collisions with neutrals is about 10^{16} cm^{2}.
(This result is relevant to the problem of star formation.)
 Here is another exercise in ambipolar diffusion across B.
Consider a
weakly
ionized plasma in a conducting box which has finite dimensions in the x
and z directions, but which is effectively infinite in the y
direction. The magnetic field is in the z direction.
 a) Write the 2d continuity equation appropriate to this
problem.
 b) Neglecting mobility across field lines, show that the
continuity
equation
may be written:
partial dn/dt = D_{parallel} partial d^{2}n/dz^{2}
+ D_{perp} partial d^{2}n/dx^{2}
and find expressions for D_{parallel} and D_{perp.}  c)
Compare your results with field free diffusion and the
results we found
in class for a different case of ambipolar diffusion. Comment.
 Chen 57 (pg 195/6)
 Chen 58
 Chen 59
Problem Set 11 Due: Thursday April 19th
 Chen 514
 When we derived the fluid equations we used the definition of
pressure:
P_{ij} = mn<(v_{i}u_{i})(v_{j}u_{j})>
where u is the mean fluid velocity <v> and
<> denotes
the average over the distribution function:
<x> = (1/n) (integral) x f(v)
d^{3}v.
For the single fluid MHD case, we define the pressure to be:
P_{MHD,ij }= (sum over e and i) mn<(v_{i}V_{i})(v_{j}V_{j})>
where V = (Mu_{+} + mu_{})/(M+m),
u_{+}
= <v> for the ions, and u_{} = <v>
for the electrons.
Show that the difference between the two expressions P_{e}
+ P_{i} and P_{MHD} is exactly balanced
by
the difference between the convective derivative terms n_{i}Mu_{+}
· gradu_{+} + n_{e} mu_{}
· gradu_{ }and
(M+m)nV · gradV, and thus that our
form of the MHD equations is exact.  Referring to problem 1 of the
previous set above, calculate the
resistivity
for this plasma (the interstellar medium) (ignoring the neutrals) and
calculate the time for the
field to decay. Repeat the calculation for a stellar interior ( n = 10^{24}
cm^{3}, T = 10^{7} K, R = 10^{10} cm).
 Chen 517 (pg 197)
Problem set 12 Due: Thursday April 26th
 Using the MHD equations, derive the dispersion relation for MHD
waves
propagating
at an arbitrary angle theta to the B field.
 a) Show that there are three waves, and sketch the phase speed
of the
waves
as a function of theta.
 b) Find the velocity eigenvectors for the three waves. Show
that one
wave
(the Alfven wave) is always transverse, while the other two are neither
longitudinal nor transverse.
 c) Evaluate the phase velocities and eigenvectors of the mixed
waves
when
v_{ A} > v_{ S}, and show that one wave has v along
B (or
mostly along B) and one has v almost perpendicular to B (Alfven type).
What happens when v_{S} > v_{A}?
 d) Show how these waves fit on the dispersion diagram (n^{2}
versus
omega) that you drew in problem set 9.
Problem Set 13 Due Thusday May 3rd
 Chen 66 (pg 214)
 Chen 69 (pg 215)
Problem Set 14 Due Thursday May 17th
 Chen 71 (pg 263),
 Chen 73
 Using equation (10) from the Vlasov notes, find the
susceptibility for a two component plasma with protons at temperature T_{i}
and electrons at temperature T_{e}. Hence find the
dielectric constant, and the normal mode frequencies. Show that
you obtain ion sound waves, and find their damping rate in terms of the
two temperatures. What condition is needed for the damping to be small?
In doing the integrals, make the following
assumptions: ω/k≫v_{th,i}=√(k_{B}T_{i}/M)
and ω/k≪v_{th,e}. To do the integral for the electrons
you may want to "undo" the integration by parts, (use eqn 9).