- Starting from Equation (201), derive the result
.
- Consider an electron of charge
moving in a circular orbit of radius
about a charge
in a field directed at right angles to the
plane of the orbit. Show that the polarizability
is approximately
.
- A point charge
is located in free space a distance
from the center of a dielectric sphere of radius
(
) and
dielectric constant
. Find the potential at all points in space as an expansion in spherical harmonics. Calculate
the rectangular components of the electric field in the vicinity of the center of the sphere.
- A dielectric sphere of radius
and dielectric constant
is imbedded in an infinite dielectric block of
dielectric constant
. The block is placed in a uniform electric field
. In other words, if
then the electric field would be
. Find the potential both inside
and outside the sphere (assuming that
), and the distribution of bound charges on the surface of the sphere.
- An electric dipole of moment
lies at the center of a spherical cavity of radius
in a uniform dielectric material of relative dielectric constant
. Find the electrostatic potential
throughout space. Find the bound charge sheet density on the surface of the
cavity.
- A cylindrical coaxial cable consists of a thin inner conductor of radius
,
surrounded by a dielectric sheath of dielectric constant
and outer radius
, surrounded by a second dielectric sheath of
dielectric constant
and outer radius
, surrounded
by a thin outer conductor. All components of the cable are touching. What
is the capacitance per unit length of the cable?
- A very long, right circular, cylindrical shell of dielectric constant
and inner and outer radii
and
, respectively,
is placed in a previously uniform electric field
with its axis perpendicular to the field. The medium inside and outside the
cylinder has a dielectric constant of unity. Determine the potential in the three regions, neglecting end effects.
Discuss the limiting forms of your solutions for a solid dielectric cylinder in a uniform field, and a cylindrical cavity in a
uniform dielectric.
- Suppose that
- Show that for an
*electret*(i.e., a material of fixed ) the integral over all space vanishes. - Two long, coaxial, cylindrical conducting surfaces of radii
and
(
) are lowered vertically into a
liquid dielectric. If the liquid rises a mean height
between the electrodes when a potential difference
is
established between them, show that the susceptibility of the liquid is