- Given that
- A quasi-neutral slab of cold (i.e.,
) plasma whose bounding surfaces are normal to the
-axis consists of electrons of mass
, charge
, and
mean number density
, as well as ions of mass
, charge
, and mean number density
.
The slab is fully magnetized by a uniform
-directed magnetic field of magnitude
. The slab is then subject to an externally generated, uniform,
-directed
electric field that is gradually ramped up to a final magnitude
. Show
that, as a consequence of ion polarization drift, the final magnitude of the electric field inside the plasma is
- A linear magnetic dipole consists of two infinite straight wires running parallel to the
-axis. The first wire lies at
,
and carries a steady current
.
The second lies at
,
and carries a steady current
. Let
and
.
Demonstrate that the magnetic field generated by the dipole in the region
can be written
- Consider a particle of charge
, mass
, and energy
, trapped on a field-line of the linear magnetic
dipole discussed in the previous exercise. Let
. Suppose that the field-line crosses the ``equatorial'' plane
at
, and that the magnetic field-strength at this point is
.
Suppose that the particle's mirror points lie at
. Assume that the particle's gyroradius is much
smaller than
, and that the electric field-strength is negligible.
- Demonstrate that the variation of the particle's perpendicular and parallel velocity components with the ``latitude''
is

respectively. - Demonstrate that the particle's bounce period is
- Demonstrate that the particle drifts in the
-direction with the mean velocity

- Demonstrate that the variation of the particle's perpendicular and parallel velocity components with the ``latitude''
is
- A charged particle of mass
is trapped in a static magnetic mirror field given by
- A particle of charge
, mass
, and energy
, is trapped in a one-dimensional magnetic well of the form

Here, is the perpendicular energy [i.e., ], and is the parallel energy [i.e., ], both evaluated at and . Assume that the particle's gyroradius is relatively small, and that the electric field-strength is negligible. - Consider the static magnetic field
- Consider a particle orbit that does not cross the neutral plane, but
is instead confined to the region
, where
.
Demonstrate that the mean drift velocity of the particle in the
-direction
can be written
- Consider a particle orbit that is confined to the region
, where
, and is
such that
when
. Demonstrate that the mean drift velocity in
the
-direction is

- Consider a particle orbit that does not cross the neutral plane, but
is instead confined to the region
, where
.
Demonstrate that the mean drift velocity of the particle in the
-direction
can be written