200C, Winter 2009, First Homework Set
1. Polarization drift (time-varying electric field). Consider an electric field that varies in time,
starting from zero and increasing linearly, but very slowly compared to the gyroperiod.
First sketch the ion and electron orbits and show qualitatively why the particles are slowly
drifting in opposite directions due to the varying field, while they are moving in the same
direction due to the instantaneous electric field. Show that the polarization drift due to the timevarying electric field is perpendicular to the ExB drift.
Derive the polarization drift in a simple manner by conservation of energy: As the particle drifts
slowly along the electric field, it gains kinetic energy ½ mvE2. The rate of energy gain is equal to
the force applied times the particle drift in the direction of the force. Use this equation to
compute the polarization drift.
2. Bounce-average motion in constant external E, B fields. Consider a particle drifting in a
modified (distorted) but planetary field B, which is constant in time, subject to an externally
imposed planetary electric field E (e.g., corotation, convection). The particle motion is tracked
by following the equatorial crossing of the gyrocenter, r, drifting at a “bounce-averaged” motion
of its equatorial guiding center, vgc. The particle equatorial kinetic energy is a function of
position and the two adiabatic invariants: W(r, , J) and the potential energy is q, E=-.
Show that for the bounce-averaged sum of the electric and gradient/curvature motion, the total
guiding center motion is vgc = vE + vGC =(B( q+W))/qB2, i.e., the particles follow equatorial
contours of constant total energy Wtot (kinetic + potential) as follows:
First show that in an arbitrary, static E, B configuration the gradient/curvature bounce-averaged
drifts are given by: vGC = (zW)/qB, i.e. the particle moves in an “effective” potential W/q. If
no electric field is present the bounce-averaged guiding center motion is along contours of
constant W, preserving the particle kinetic energy. To do that consider conservation of the
particle energy in the equatorial plane: (vgc )Wtot= (vE + vGC) (q+W)=0, and use the fact that
since this must be satisfied for arbitrary potentials, including =0, it must be: vGC =c (zW),
where c is a constant – then determine the constant. Next show that in the electrostatic potential
and W/q add up as in vgc above.
3. Quasi-neutrality is an important property of space plasmas, where the difference in density
between the ions and electrons responsible for any space charge is much less than the number
density. As an example top show this, consider the electric field associated with a corotating
plasma. For simplicity assume the magnetic field is represented by a dipole and the dipole is
oriented along the spin axis.
First sketch the dipole field lines, and show that for the Earth, the corotation electric field
requires a negative space charge near the equatorial plane, and a positive space charge above the
magnetic pole.
Second, estimate the electric field in the equatorial plane, and show that for the Earth the electric
field is ~ 14(RE/R)2 mV/m, assuming an equatorial magnetic field of 30,000 nT at the surface of
the Earth [Here RE is the radius of the earth, and R is the radial distance[.
Third, assuming that the scale size for variation of the electric field is of order the radial distance
show that the corotation electric field requires a density of ~ 2 x10–7 electrons/cm3. Compare this
to typical magnetospheric plasma densities.
Bonus question: Determine the electron density exactly for an axisymmetric dipole, and compare
this to the estimate given in the third part of the question.
4. Derive a general form for the MHD equations, based on integrals over velocity space of
moments of the Vlasov equation. Discuss the open-ended nature of the resultant equations, and
specifically discuss how the energy equation can be closed [Hint, only the heat flux cannot be
expressed in terms of lower order moments]. What assumption is required for the energy
equation to reduce to the adiabatic form P– = constant?
5. Consider a plasma that contains a uniform magnetic field in which the plasma is flowing
radially outward (i.e., cylindrically symmetric flow). Show that in this simple geometry B dA/dt
+ A dB/dt = 0, where B is the magnetic field strength and A is the area enclosed by circle
expanding with the flow. Discuss why this is equivalent to the field lines being frozen to the
fluid.