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- Two concentric spheres have radii
,
(
) and are each divided into two hemispheres
by the same horizontal plane. The upper hemisphere of the inner sphere and the lower hemisphere
of the outer sphere are maintained at potential
. The other hemispheres are at zero potential.
Demonstrate that the potential in the region
can be written

where

Here,
,
,
are conventional spherical coordinates whose origin coincides with the common center of the spheres, and are such that the dividing plane corresponds to
.

- A spherical surface of radius
has charge uniformly distributed over its surface with
density
, except for a spherical cap at the north pole, defined by the cone
. Here,
,
,
are conventional spherical
coordinates whose origin coincides with the center of the surface.
- Show that the potential inside the spherical surface can be expressed as

where
.
- Show that the electric field at the origin is

- Show that in the limit
,

- Show that in the limit
,

- The Dirichlet Green's function for the unbounded space between planes at
and
allows a
discussion of a point charge, or a distribution of charge, between parallel conducting planes held at zero potential.
- Using cylindrical coordinates, show that one form of the Green's function is

- Show that an alternative form of the Green's function is

- From the results of the previous exercise, show that the potential due to a point charge
placed between two infinite parallel conducting
planes held at zero potential can be written as

where the planes are at
and
, and the charge is on the
-axis at
. Show that induced surface charge densities on the
lower and upper planes are

respectively.

- Show that the potential due to a conducting disk of radius
carrying a charge
is

in cylindrical coordinates (whose origin coincides with the center of the disk, and whose symmetry axis coincides with that of the disk.)

- A conducting spherical shell of radius
is placed in a uniform electric field
. Show that the force tending to
separate two halves of the sphere across a diametral plane perpendicular to
is given by

** Next:** Electrostatics in Dielectric Media
** Up:** Potential Theory
** Previous:** Poisson's Equation in Cylindrical
Richard Fitzpatrick
2014-06-27