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## Boundary value problems with dielectrics

Consider a point charge embedded in a semi-infinite dielectric a distance away from a plane interface which separates the first medium from another semi-infinite dielectric . The interface is assumed to coincide with the plane . We need to find solutions to the equations
 (818)

for ,
 (819)

for , and
 (820)

everywhere, subject to the boundary conditions at that
 (821) (822) (823)

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

 (824)

where and , when written in terms of cylindrical polar coordinates, , aligned along the -axis. Note that the potential (824) is clearly a solution of Eq. (818) in the region . It gives , with the appropriate singularity at the position of the point charge .

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)

The above potential is clearly a solution of Eq. (819) in the region . It gives , with no singularities.

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)

The boundary condition (821) implies a jump in the normal derivative of the scalar potential across the interface:
 (827)

The first matching condition yields
 (828)

whereas the second gives
 (829)

Here, use has been made of
 (830)

Equations (828) and (829) imply that
 (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)

Hence,
 (834)

In the limit , the dielectric behaves like a conducting medium (i.e., in the region ), and the bound surface charge density on the interface approaches that obtained in the case where the plane coincides with a conducting surface (see Sect. 5.10).

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

 (835)

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)

Likewise, the most general solution inside the sphere, which satisfies the boundary condition at , is
 (837)

The continuity of at yields
 (838)

Likewise, the matching condition (835) gives
 (839)

Hence,
 (840) (841)

Note that the electric field inside the sphere is uniform, parallel to the external electric field outside the sphere, and of magnitude . Moreover, , provided that . Finally, the density of the bound charge sheet on the surface of 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)

Hence,
 (844) (845)

Note that the field inside the cavity is uniform, parallel to the external electric field outside the sphere, and of magnitude . Moreover, , provided that . The density of the bound charge sheet on the surface of the cavity is
 (846)

Next: Energy density within a Up: Dielectric and magnetic media Previous: Boundary conditions for and
Richard Fitzpatrick 2006-02-02