Fundamentals of Plasma Physics

2.8 Assignments 59

this ratio constant or not (explain your answer)? Let this ratio be denoted byλ

(this is called a Lagrange multiplier).

(f) Show that the conclusion reached in ‘e’ above lead to thefME(v)having to be

a Maxwellian.

3. Suppose that a group ofNparticles with chargeqσand massmσare located in an
   electrostatic potentialφ(x).What is the maximum entropy distribution function for
   this situation (give a derivation)?


4. Prove that ∫
   ∞

−∞

dxe−ax

2

=

√

π

a

(2.125)

Hint: Consider the integral

∫∞

−∞

dxe−x

2

∫∞

−∞

dye−y

2

=

∫∞

−∞

∫∞

−∞

dxdye−(x

(^2) +y (^2) )
,
and note thatdxdyis an element of area. Then instead of using Cartesian coordinates
for the integral over area, use cylindrical coordinates and express allquantities in the
double integral in cylindrical coordinates.

5. Evaluate the integrals

∫∞

−∞

dxx^2 e−ax

2

,

∫∞

−∞

dxx^4 e−ax

2

.

Hint: Take the derivative of both sides of Eq.(2.125) with respect toa.

6. Suppose that a particle starts at timet=t 0 with velocityv 0 at locationx 0 and is
   located in a uniform, constant magnetic fieldB=Bz.̂There is no electric field. Cal-
   culate its position and velocity as a function of time. Make sure that your solution
   satisfies the initial conditions on both velocity and position. Be careful totreat mo-
   tion parallel to the magnetic field as well as perpendicular. Express your answer in
   vector form as much as possible;use the subscripts‖,⊥to denote directions parallel
   and perpendicular to the magnetic field, and useωc=qB/mto denote the cyclotron
   frequency. Show thatf(x 0 )is a solution of the Vlasov equation.


7. Thermal force (Braginskii 1965)- If there is a temperature gradient then because of
   the temperature dependence of collisions, there turns out to be an additional subtle
   drag force proportional to∇T. To find this force, suppose a temperature gradient
   exists in thexdirection, and consider the frictional drag on electrons passing a point
   x=x 0 .The electrons moving to the right (positive velocity) atx 0 have travelled
   without collision from the pointx 0 −lmfp, where the temperature wasT(x 0 −lmfp),
   while those moving to the left (negative velocity) will have come collisionlessly from
   the pointx 0 +lmfpwhere the temperature isT(x 0 +lmfp). Suppose that both electrons
   and ions have no mean velocity atx 0 ;i.e.,ue=ui= 0.Show that the total drag force
   on all the electrons atx 0 is

Rthermal=− 2 menelmfp

∂

∂x

(νeivTe).