for ,

for , and

(820) |

In order to solve this problem, we shall employ a slightly modified form of
the well-known method of images. Since
everywhere,
the electric field can be written in terms of a scalar potential.
So,
. Consider the region .
Let us assume that the scalar potential in this region is the same as
that obtained when the whole of space is filled with the dielectric
, and, in addition to the real charge at position ,
there is a second charge at the image position (see Fig. 47).
If this is the case, then the potential at some point
in the region is given by

Consider the region . Let us assume that the scalar potential in this
region is the same as that obtained when the whole of space is filled
with the dielectric , and a charge is located at the point
. If this is the case, then the potential in this region is
given by

(825) |

It now remains to choose and in such a manner that the boundary
conditions (821)-(823) are satisfied. The boundary conditions (822) and
(823) are obviously satisfied if the scalar potential is continuous
at the interface between the two dielectric media:

(826) |

(827) |

whereas the second gives

Here, use has been made of

(830) |

(831) | |||

(832) |

The bound charge density is given by
, However, inside either dielectric,
, so
, except at the point charge .
Thus, there is zero bound charge density in either dielectric
medium. However,
there is a bound charge sheet on the interface
between the two dielectric media.
In fact, the density of this sheet is given by

(833) |

(834) |

As a second example, consider a dielectric sphere of radius , and
uniform dielectric constant , placed in a uniform
-directed electric field of magnitude . Suppose that the
sphere is centered on the origin. Now, for an electrostatic problem, we
can always write
. In the present problem,
both inside and outside the sphere, since there are no free charges, and the bound volume charge density is zero in a uniform
dielectric medium (or a vacuum). Hence, the scalar potential satisfies Laplace's equation,
, throughout space. Adopting spherical polar coordinates,
, aligned along the -axis, the boundary conditions are that
at
, and that is well-behaved at
. At the surface of the sphere, , the continuity of
implies that is continuous. Furthermore, the
continuity of
leads to the matching condition

Let us try separable solutions of the form
. It is
easily demonstrated that such solutions satisfy Laplace's equation
provided that or . Hence, the most general solution to Laplace's equation outside
the sphere, which satisfies the boundary condition at
, is

(836) |

(837) |

(838) |

(839) |

(840) | |||

(841) |

Note that the electric field inside the sphere is

(842) |

As a final example, consider a spherical cavity, of radius , in a
uniform dielectric medium, of dielectric constant , in the presence of a
-directed electric field of magnitude . This problem is analogous
to the previous problem, except that the matching condition
(835) becomes

(843) |

(844) | |||

(845) |

Note that the field inside the cavity is

(846) |