The Theory of the Macroscopic Properties of Isotropic Dielectrics

BY B. K. P. SCAIFE

1 Introduction

A chemist is anxious to obtain as much information as possible about the molecular conformation, interaction, and dynamics of the substances which he studies. Because all materials are built up from combinations of electric charges, protons, and electrons, it is not surprising that certain molecular properties and motions give rise to macroscopic properties which determine the reaction of a particular material to the imposition of an electromagnetic field.

To relate the observed macroscopic phenomena to various molecular processes is, in general, a task of great difficulty. Such an undertaking requires a fully developed macroscopic theory to enable proper design and assessment of experimental procedures to be made. Furthermore, without an adequate theory of molecular processes and without a precise relationship between macroscopic and microscopic parameters, it will not be possible to obtain accurate information about events at a molecular level from experimental data.

The primary concern of this chapter is to provide an introduction to the theory of the macroscopic properties of dielectric bodies. Attention is restricted to isotropic materials and it is assumed that the external electric field is always sufficiently weak to allow any non-linear effects which might arise to be ignored.

The subject matter is such that it lends itself to mathematical description; nevertheless, the emphasis here will be on the physical meaning of the mathematical equations and we shall not be particularly concerned with matters of mathematical technique.

This introductory survey commences with a derivation of the macroscopic equation

D = ε0 E + P

based on Maxwell's equations for charges in free space. The concept of the relative permittivity for static fields, εg is introduced in Section 3. In the following section it is shown how this concept may be generali:ied for the case of harmonically varying applied fields.

A frequency (= ω/2π) dependent complex susceptibility,

x(ω) = x'(ω) -ix"(ω) = [ε(ω) -1] ε0,

is defined and the physical significance of x'(ω) and x"(ω) is discussed. It is also shown how x(ω) is related to the temporal behaviour of the dielectric polarization following the sudden application, or removal, of an electric field. Various forms of the Kramers-Kronig dispersion relations are introduced for x'(ω) and x"(ω) and for a number of functions of x(ω). The section closes with the definition of the frequency-dependent complex refractive index n(ω) = n(ω) – iκ(ω) and a discussion of its relation to ε(ω).

The final section discusses, in outline, various macroscopic manifestations of the underlying dynamic nature of a dielectric material. It is shown how certain macroscopically observable quantities are of direct significance on a microscopic scale.

2 Basic Considerations

The electric and magnetic fields set up in free space by a system of charges is described by Maxwell's equations, which read, in SI units, as follows:

curl E(r, t) = -[partial derivative],B(r, t) (1)

curl H(r, t) = [partial derivative]tD(r, t) + J(r, t) (2)

div D(r, t) = p(r, t) (3)

div B(r, t) = O (4)

B(r, t) = μo H(r, t) (5)

D(r, t) = εo E(r, t) (6)

The absolute permittivity and absolute permeability of free space are denoted by ε0 and μ0 respectively. The sources of the electromagnetic-field vectors D, E, B, and Hare the charge density, p(r, t), and the current density, due to charges in motion, J(r, t). The radius vector of a point with co-ordinates x, y, and z is denoted by r = ix + jy + kz, and the time is denoted by t. The use of the tilde, ~, on the field variables in equations (1H6) is to emphasize that these equations describe the electro-magnetic field in free space. Quantities without the tilde will have meanings to be defined below.

In view of equations (5) and (6) it would appear an unwarranted extravagance to use four variables when two would suffice. As we shall see presently, the distinction between D and E in free space, which stems from the SI system of units, will prove of considerable benefit when we come to setting up the equations for the potential gradient in polarizable matter.

For a great many applications in dielectrics, Maxwell's equations may be greatly simplified by neglecting the magnetic field on the ground that the motion of the charges is not sufficiently rapid to give rise to appreciable radiation. With this approximation equations (1) and (2) are uncoupled and we may use the following equations to describe the spatial variation of D and E:

curl E(r, t) = 0 (7)

div D(r, t) = p(r, t) (8)

D(r, t) = ε0 E(r, t) (9)

It is convenient and fruitful to regard Das the electric flux density and E as the force that would be exerted on a unit point charge.

The implications of equations (7) and (8) are (i) that E(r, t) may be described in terms of a scalar potential [??],(r, t), such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and (ii) that the electric flux lines, whose density is D, begin on positive charges and end on negative charges. Notice that, for a system of charges in free space, at each and every point the electric flux density, D, is related to the electric field intensity, E, by the simple equation (9).

The charge system that we are concerned with is one in which the charges are the sub-atomic charged particles, electrons and protons, which make up dielectric materials. It is clear that the quantities D, E, and [??] must now be regarded as inaccessible to simple macroscopic measurements. Indeed we shall regard D, E, and [??] as variables at a microscopic level and our immediate task is to find a means of relating them to corresponding macroscopic quantities.

We must realize at once that for most purposes a 'small' distance – 'small' in a macroscopic sense – when expressed in terms of, say, the radius of a hydrogen atom appears to be very large on a microscopic or atomic scale. Thus when we speak of the macroscopic charge density 'at a point' we mean: the ratio of electric charge contained in a small (on a macroscopic scale) volume around the point in question divided by the volume. Expressed mathematically this last statement reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

in which the...