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Boundary Value Problems with Dielectrics

Consider a point charge embedded in a semi-infinite dielectric medium of dielectric constant , and located a distance from a plane interface that separates the first medium from another semi-infinite dielectric medium of dielectric constant . Suppose that the interface coincides with the plane . We need to solve

 (512)

in the region ,

 (513)

in the region , and

 (514)

everywhere, subject to the following constraints at :

 (515) (516) (517)

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

 (518)

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

 (522)

whereas the second gives

 (523)

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

 (529)

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

 (537)

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

 (538) (539)

Furthermore,

 (540)

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)

 (542)

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

 (545)

and

 (546)

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

 (547)

and

 (548)

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

 (549) (550)

The solution to the problem is therefore

 (551)

for , and

 (552)

for .

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

 (553)

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.

Next: Energy Density Within Dielectric Up: Electrostatics in Dielectric Media Previous: Boundary Conditions for and
Richard Fitzpatrick 2014-06-27