Questions and Answers


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.


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:

Important Dates:

Midterm (in class): Thursday April 19th


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
Final Paper: Tuesday May 22
Class presentations: May 15, 24 20

Class schedule (Rough outline - subject to change)

Week Date Topic Book reference (Chen) 
1. Jan 25
Introduction, basic definitions  Ch 1
2.  Jan 30-Feb 1
Introduction to particle motions  Ch 2
3.  Feb 6-8
Introduction to fluids, Distributions Ch 3,Ch 7 Sections 1-3
Feb 13-15 Introduction to waves, linearization
Longitudinal and transverse waves
Ch 4
5.  Feb 20-22 More on waves Ch 4
6..  Feb 27-Mar 1
More on waves Ch 4
7. Mar 6-8 Diffusion Ch 5
8. Mar 13-15 Magnetohydrodynamics Ch 5
9.  Mar 20-22 MHD waves Ch 5
10 Mar 27-29
Equilibrium and stability Ch 5
11 Apr 3-5  Instabilities Ch 6

12 Apr 17-19 More on instabilities Ch 6

Apr 19 Mid-term

13 Apr 24-26 Some effects of non-linearity  Ch 8
14 May 1-3
More non-linearity
Vlasov theory
Ch 8
Ch 7 + notes
15 May 8-10 Vlasov theory with magnetic fields
Ch 7 + notes
16 May 15
May 24 @ 1:30
Student presentations

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: Some suggestions:
The international thermonuclear reactor.
Wave-wave interactions and their effect on inertial fusion.
Conditions in the coronal plasma as determined by x-ray spectroscopy.
An analysis of the Kruskal-Schwarzschild 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, double-spaced, 10-20 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:

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 power-point 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, 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:

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 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 log-log scale for the plots.
  1. Chen 1.3 (pg 12)

  2. Include the following additional astrophysical plasmas, extending the axes as appropriate:
  3. Chen 1-5
  4. Chen 1-8,
  5. Chen 1-11 (page 17)
  6. 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

  1. Chen 2-1 (pg 25) Also compute omegac, and include

  2. e) An electron near the surface of an accreting neutron star: E = 10 keV, B = 108 T
  3. Chen 2-7
  4. Chen 2-8. (pg 35)
  5. Chen 2-12 (pg 35)
  6. Chen 2-13 (pg 49)

Problem Set 3 Due: Thursday February 15th

  1. Chen 2-16
  2. Chen 2-17
  3. Refer to problem Chen 2-8. 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

  1. 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 3-44 for a steady state. Assume isotropic pressure, E = B = 0, but non-zero gravitational field. All variations are 1-dimensional.
  2. Show that the distribution function on page 9 of Chen, f(u) = Aexp(-[(1/2)mu2 + q(phi)]/kT) satisfies the collisionless Vlasov equation.
  3. Chen 3-1 (page 58)
  4. Chen 3-7 (page 74)

Problem Set 5 Due: Thursday March 1st

  1. Calculate the plasma frequency for the plasmas in problem 1-3 of Chen (see Problem set # 1) plus, a laser fusion pellet with n = 5x1022 cm-3.
  2. Chen 4-1 (page 81)
  3. Chen 4-2 (page 86)
  4. Chen 4-3

Problem set 6 Due: Thursday March 8th

  1. Chen 4-6 (page 94)
  2. Chen 4-8 (pg 108)
  3. Chen 4-9 (pg 120)
  4. Chen problem 4-10 (pg 120)
  5. Chen 4-13

Problem Set 7 Due: Thursday March 15th.

  1. Chen 4-16 (pg 135)
  2. Chen 4-19
  3. Chen 4-21 (pg 135)
  4. Chen 4-24 (Be careful. Chen's answer is not correct.)
  5. Chen 4-25

Problem Set 8 Due: Thursday March 22nd

  1. Chen 4-26 (page 148)
  2. Chen 4-30
  3. Chen 4-32
  4. Chen 4-34
  5. Chen 4-45

Problem set 9 Due: Thursday March 29th

  1. 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).  
  2. Draw a graph of refractive index squared versus omega. Choose a particular ordering of the plasma frequency and the cyclotron frequency (ie omegap less than or greater than omegac) Mark all relevant frequencies, (eg omegaR and omegaL). 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.)
  3. Chen 5-1 (page 175-176)
  4. Chen 5-2

Problem Set 10 Due Thursday April 5th

  1. 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 = 3x1019 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 cm2. (This result is relevant to the problem of star formation.)
  2. 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.
  3. Chen 5-7 (pg 195/6)
  4. Chen 5-8
  5. Chen 5-9

Problem Set 11 Due: Thursday April 19th

  1. Chen 5-14
  2. When we derived the fluid equations we used the definition of pressure:
  3. Pij = mn<(vi-ui)(vj-uj)>
    where u is the mean fluid velocity <v> and <> denotes the average over the distribution function:
    <x> = (1/n) (integral) x f(v) d3v.
    For the single fluid MHD case, we define the pressure to be:
    PMHD,ij = (sum over e and i) mn<(vi-Vi)(vj-Vj)>
    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 Pe + Pi and PMHD is exactly balanced by the difference between the convective derivative terms niMu+ · gradu+ + ne mu- · gradu- and
    (M+m)nV · gradV, and thus that our form of the MHD equations is exact.
  4. 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 = 1024 cm-3, T = 107 K, R = 1010 cm).
  5. Chen 5-17 (pg 197)

Problem set 12 Due: Thursday April 26th

  1. Using the MHD equations, derive the dispersion relation for MHD waves propagating at an arbitrary angle theta to the B field.

Problem Set 13 Due Thusday May 3rd

  1. Chen 6-6 (pg 214)
  2. Chen 6-9 (pg 215)

Problem Set 14 Due Thursday May 17th

  1. Chen 7-1 (pg 263),
  2. Chen 7-3
  3.   Using equation (10) from the Vlasov notes, find the susceptibility for a two component plasma with protons at temperature Ti and electrons at temperature Te.  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≫vth,i=√(kBTi/M)  and  ω/k≪vth,e. To do the integral for the electrons you may want to "undo" the integration by parts, (use eqn 9).