in the region ,

in the region , and

(514) |

everywhere, subject to the following constraints at :

In order to solve this problem, we shall employ a slightly modified form of the well-known method of images. Because everywhere, the electric field can be written in terms of a scalar potential: that is, . 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 dielectric of dielectric constant , and, in addition to the real charge at position , there is a second charge at the image position . (See Figure 1.) If this is the case then the potential at some point in the region is given by

where and . Here, , , are conventional cylindrical coordinates. Note that the potential (519) is clearly a solution of Equation (513) in the region : that is, it satisfies , 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 a dielectric medium of dielectric constant , and a charge is located at the point . If this is the case then the potential in this region is given by

(519) |

The above potential is clearly a solution of Equation (514) in the region : that is, it satisfies , with no singularities.

It now remains to choose and in such a manner that the constraints (516)-(518) are satisfied. The constraints (517) and (518) are obviously satisfied if the scalar potential is continuous across the interface between the two media: that is,

(520) |

The constraint (516) implies a jump in the normal derivative of the scalar potential across the interface. In fact,

(521) |

The first matching condition yields

whereas the second gives

Here, use has been made of

(524) |

Equations (523) and (524) imply that

(525) | ||

(526) |

The polarization charge density is given by , However, inside either dielectric, which implies that

(527) |

except at the point charge . Thus, there is zero bound charge density in either dielectric medium. At the interface, jumps discontinuously,

(528) |

This implies that there is a bound charge sheet on the interface between the two dielectric media. In fact, it follows from Equation (498) that

where is a unit normal to the interface pointing from medium 1 to medium 2 (i.e., along the positive -axis). Because

(530) |

in either medium, it is easily demonstrated that

(531) |

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 when the plane coincides with a conducting surface.

The above argument can easily be generalized to deal with problems involving multiple point charges in the presence of multiple dielectric media whose interfaces form parallel planes.

Consider a second boundary value problem in which a slab of dielectric, of dielectric constant , lies between the planes and . Suppose that this slab is placed in a uniform -directed electric field of strength . Let us calculate the field-strength inside the slab.

Because there are no free charges, and this is essentially a one-dimensional problem, it is clear from Equation (501) that the electric displacement is the same in both the dielectric slab and the surrounding vacuum. In the vacuum region, , whereas in the dielectric. It follows that

(532) |

In other words, the electric field inside the slab is reduced by polarization charges. As before, there is zero polarization charge density inside the dielectric. However, there is a uniform bound charge sheet on both surfaces of the slab. It is easily demonstrated that

(533) |

In the limit , the slab acts like a conductor, and .

Let us now generalize this result. Consider a dielectric medium whose dielectric constant varies with . The medium is assumed to be of finite extent, and to be surrounded by a vacuum. It follows that as . Suppose that this dielectric is placed in a uniform -directed electric field of strength . What is the field inside the dielectric?

We know that the electric displacement inside the dielectric is given by . We also know from Equation (501) that, because there are no free charges, and this is essentially a one-dimensional problem,

(534) |

Furthermore, as . It follows that

(535) |

Thus, the electric field is inversely proportional to the dielectric constant of the medium. The bound charge density within the medium is given by

(536) |

Consider a third, and final, boundary value problem in which a dielectric sphere of radius , and dielectric constant , is placed in a -directed electric field of strength (in the absence of the sphere). Let us calculate the electric field inside and around the sphere.

Because this is a static problem, we can write . There are no free charges, so Equations (501) and (505) imply that

everywhere. The matching conditions (510) and (512) reduce to

Furthermore,

as : that is, the electric field asymptotes to uniform -directed field of strength far from the sphere. Here, , , are spherical coordinates centered on the sphere.

Let us search for an axisymmetric solution, . Because the solutions to Poisson's equation are unique, we know that if we can find such a solution that satisfies all of the boundary conditions then we can be sure that this is the correct solution. Equation (538) reduces to

(541) |

Straightforward separation of the variables yields (see Section 3.7)

where is a non-negative integer, the and are arbitrary constants, and the are Legendre polynomials. (See Section 3.2.)

The Legendre polynomials form a complete set of angular functions, and it is easily demonstrated that the and the form a complete set of radial functions. It follows that Equation (543), with the and unspecified, represents a completely general (single-valued) axisymmetric solution to Equation (538). It remains to determine the values of the and that are consistent with the boundary conditions.

Let us divide space into the regions and . In the former region

(543) |

where we have rejected the radial solutions because they diverge unphysically as . In the latter region

(544) |

However, it is clear from the boundary condition (541) that the only non-vanishing is . This follows because . The boundary condition (540) [which can be integrated to give for a potential that is single-valued in ] gives

and

for . Note that it is appropriate to match the coefficients of the because these functions are mutually orthogonal. (See Section 3.2.) The boundary condition (539) yields

and

for . Equations (547) and (549) give for . Equations (546) and (548) reduce to

(549) | ||

(550) |

The solution to the problem is therefore

for , and

(552) |

for .

Equation (552) is the potential of a uniform -directed electric field of strength

Note that , provided that . Thus, the electric field-strength is reduced inside the dielectric sphere due to partial shielding by polarization charges. Outside the sphere, the potential is equivalent to that of the applied field , plus the field of an electric dipole (see Section 2.7), located at the origin, and directed along the -axis, whose dipole moment is

(554) |

This dipole moment can be interpreted as the volume integral of the polarization over the sphere. The polarization is

(555) |

Because the polarization is uniform there is zero bound charge density inside the sphere. However, there is a bound charge sheet on the surface of the sphere, whose density is given by [see Equation (530)]. It follows that

(556) |

The problem of a dielectric cavity of radius inside a dielectric medium of dielectric constant , and in the presence of an applied electric field , parallel to the -axis, can be treated in much the same manner as that of a dielectric sphere. In fact, it is easily demonstrated that the results for the cavity can be obtained from those for the sphere by making the transformation . Thus, the field inside the cavity is uniform, parallel to the -axis, and of magnitude

(557) |

Note that , provided that . The field outside the cavity is the original field, plus that of a -directed dipole, located at the origin, whose dipole moment is

(558) |

Here, the negative sign implies that the dipole points in the opposite direction to the external field.