Problem Set 1

(Due: April 10, 1998)

1. Using the field line equation, write the magnitude of the dipole field in terms of (a) the L parameter and the radial distance r; and (b) L and the latitude l (eliminating r).

2. (Problem 3.3 of Parks) Show that the arc length of a dipole field line is given by:


Note that the arc length can be defined (in Cartesian coordinates) by:


3. Suppose that the IMF is southward, with a magnitude of 10 nT. Calculate the invariant latitude of the separatrix for the Earth's dipole field using vacuum superposition.

4. (Problem 3.14 of Parks) Show that the vector potential for a dipole field can be written in terms of its azimuthal (j) component as:


Show that lines of constant A_j are always perpendicular to the magnetic field line.

5. Consider a Harris field geometry of the form
   .

   Determine the current density (magnitude and direction) and the plasma pressure distribution for this equilibrium.