- Prove the
*mean value theorem*: for charge-free space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any sphere centered on that point. - Prove
*Green's reciprocation theorem*: if is the potential due to a volume charge density within a volume and a surface charge density on the conducting surface bounding the volume , while is the potential due to another charge distribution and (non-simultaneously occupying the same volume and the same surface, respectively), then - Two infinite grounded parallel conducting planes are separated by a distance
. A point charge
is
placed between the planes. Use Green's reciprocation theorem to prove that the total charge induced on
one of the planes is equal to
times the fractional perpendicular distance of the point charge from the
other plane. [Hint: Choose as your comparison electrostatic problem with the same surfaces one whose charge
densities and potential are known and simple.]
- Consider two insulated conductors, labeled 1 and 2. Let
be the
potential of the first conductor when it is uncharged and the second conductor holds a
charge
. Likewise, let
be the potential of the second conductor
when it is uncharged and the first conductor holds a charge
. Use Green's
reciprocation theorem to demonstrate that
- Consider two insulated spherical conductors. Let the
first have radius
. Let the second be sufficiently small that it
can effectively be treated as a point charge, and let it also be located a distance
from
the center of the first. Suppose that the first conductor is uncharged, and that the second carries a charge
. What is the potential of the first conductor?
[Hint: Consider the result proved in Exercise 1.]
- Consider a set of
conductors distributed in a vacuum. Suppose that
the
th conductor carries the charge
and is at the scalar potential
. It follows from the linearity of Maxwell's equations and Ohm's law
that a linear relationship exists between the potentials and the charges:
that is,
*coefficients of potential*. Demonstrate that for all . [Hint: Consider the result proved in Exercise 1.] Show that the total electrostatic potential energy of the charged conductors is - Demonstrate that the Green's function for Poisson's equation in two
dimensions (i.e.,
) is
- A electric dipole of fixed moment
is situated at position
in a non-uniform external electric field
. Demonstrate
that the net force on the dipole can be written
, where
- Demonstrate that the electric field generated by an electric dipole of dipole moment
is
- Show that the torque on an electric dipole of moment
in a uniform
external electric field
is
- Consider two coplanar electric dipoles with their centers a fixed distance apart. Show that if the angles the dipoles
make with the line joining their centers are
and
, respectively, and
is held fixed, then