Name___________________
(1) [25 points] A spherical shell, radius R, has a surface charge arranged so that there is +! on a
spherical cap covering angles from ! = 0 to !/4, and !" on the larger remainder of the
spherical surface. Assume ! is positive.
(a) Find the dipole moment of this charge distribution, measured from the center of the sphere.
(b) Find the electric field, with direction, measured a long distance z directly above the center of
the sphere, in the direction ! = 0 (e.g. find the leading term at this point).
2
Name___________________
!
(2) In some region of space, a steady magnetic field is given by, B = Cx 2 ŷ , where C is a
constant.
(a) Find the current density in this region.
(b) Show that this current density satisfies the continuity equation with no charge accumulation,
as expected for a steady current distribution.
(c) Find the vector potential associated with this current. [I suggest using the gauge choice
discussed in class, though as always for such problems there are other solutions that may work as
well.]
3
Name___________________
(3) A grounded metal sphere, radius R, is placed in the center of a linear dielectric sphere, of
radius 2R. The relative permittivity of the dielectric is ! r . By an arrangement of external fields, a
potential V = ! cos" is generated at the outer surface of the dielectric (at 2R), where ! is a
constant.
(a) Write the general form for the electric potential in the region of the dielectric, R < r < 2R .
(b) Using appropriate surface conditions, solve for the potential in this region in terms of ! and
R.
(c) Find the electric field and the displacement field in this region.
(d) Find the free charge distribution on the grounded sphere at the center, and find the
distribution of bound charges in the dielectric and on the inner and outer surfaces.
4
Name___________________
(4) A hollow sphere has a single layer of windings around the z axis ( ! direction), with the turn
density (turns/length) equal to no sin ! , where no is a constant.
(a) If the wires are very thin and no is large, the resulting current density can be approximated a
surface current. Find K as a function of ! , if the current in each wire is I.
(b) In this situation, we can show that the vector potential is given by, ( C1r sin ! )"ˆ inside the
sphere, and C sin ! / r 2 "ˆ outside the sphere, where C and C are constants. For these vector
(
2
)
1
2
potentials, solve for the corresponding magnetic fields.
(c) Using appropriate boundary conditions, solve for the constants C1 and C2 .
5
Name___________________
(5) Consider a long wire, shown in cross-section. The shaded region (radius R) carries uniform
current density J into the page, while the smaller circle is a hollow region (radius R/2) running
the length of the wire.
(a) First show how to calculate the magnetic field inside and outside of a solid wire carrying
uniform current, such as the shaded wire if there were no hollow region.
(b) Now find the magnetic field of the current distribution shown, at points along the horizontal
diameter, both in the solid region of the wire and in the hollow region. [Hint: think
superposition.]
6