§4. Dielectric in external electric field

4.1. Polar and nonpolar molecules

All substances are subdivided into conductors, dielectrics (insulators) and semiconductors.

In nature, ideal dielectrics donot exist. All substances to some extent conduct electric current, but dielectrics do this 10¹⁵—10²⁰ times worse, than conductors.

If dielectric is brought to the electric field, the field and the dielectric undergo essential changes. To understand this it it is necessary to consider that

atoms and molecules carry positively loaded nucleus and negatively loaded electrons. Any molecule represents system in which the total charge is equal to zero. The sizes of this system are very small - about several angstrom (10^-10 м). The field created by such systems, is characterized by the size of the dipolar moment.

. (4.1)

Summation happens both on electrons, and on nucleus. However, electrons in a molecule move in such a way that this moment changes all the time, though electron speeds are so high that average value of this moment can be defined. Therefore further by the dipolar moment of a molecule we will understand for electrons

(4.2)

and for nucleus: .

In other words, we will consider that the electron at the state of rest in some points, relative to nucleus which are recived by approximation of position of electron in time.

The behavior of a molecule in external electric field is also defined by its dipolar moment. This can be proved by calculating potential energy of a molecule in the external electric field.

Let᾿s choose the beginning of coordinates in side a molecule and use small value . Then, a potential in a point where theі-y charge is located is possible to present

,

where φ – the potential of a field and the beginning of coordinates.

Then, the potential energy is defined as the sum:

.

Considering the law of charge conservation, і , we have

.

If the expression on an angle is differentiated we will receive the equation for a torque. Having taken then a derivative on dx – we will receive force

.

Thus, the molecule as in relation to a field which it creates, and in relation to those forces exerted on it in an external field, will have equivalent dipoles. The positive charge of this dipole is equal to a total charge of nucleus and it is placed in "center of gravity" of positive charges, and the negative charge is equal to a total charge of electrons and is placed in the "center of gravity" of negative charges.

In symmetrical molecules (H₂, O₂, N₂ і т.д.) in lack of external electric field the centers of gravity of positive and negative charges coincide. Such molecules don't possess own dipolar moment and are called as nonpolar molecules.

In asymmetrical molecules (CO, NH, HCl) the centers of gravity of charges of different signs are displaced from each other and molecules possess own dipolar moment. Such molecules are called polar.

Under the influence of external electric field charges in unpolar molecules are shifted relative to one another - a positive charge toward the field direction, and negative – opposite a field. As a result, the molecule acquires a dipole moment, the magnitude of which is proportional to the field intensity. As a

rationalised system the coefficients of proportionality – ε0, β (ε0 – electrostatic constant, β – the polarization of the molecule).

Considering, that the vectors and collinear, we can write.

The process of polarization of nonpolar molecules proceeds as the positive and negative charges are connected by elastic forces. So it is said that the non-polar molecule behaves in an external field as an elastic dipole. Then the action of an external field on the polar molecule is reduced in the order to turn the molecule so that its dipole moment is established in the direction of the field. The magnitude of this moment is not affected by the external field. Accordingly, the polar molecule behaves in an external field as hard dipole.

4.2 Polarization of dielectrics

In the absence of an external field the total moment of the molecule is zero.

(4.3)

Under the influence of the external field dielectric polarized. This means that the resultant dipole moment of the dielectric becomes different from zero. Dipole moment per unit volume is characterized by the degree of polarization.

If an external electric field is non-homogeneous, the degree of polarization at different points may be different. To describe the polarization at this point it is necessary to select a physically infinite volume which includes this point,to find the sum of all dipole moments which are found in this volume, and calculate ratio.

(4.4)

where - the polarization of the dielectric.

In isotropic dielectric the polarization is associated with the field intensity at that point of correlation.

, (4.5)

where χ - dielectric susceptibility (dimensionless quantity, independent of intensity E).

Or in the system Gauss:

. (4.6)

For dielectrics that are constructed of non-polar molecules, the equation (4.5) follows from the considerations:

- Within the volume ΔV there gets the number of molecules nΔV (n - number of molecules per volume);

- Each dipole moments is detined by (4.3) and then the sum of all these dipole moments p is defined by

If it is you divide by ΔV polarization (by definition) is receive.

Denoting , we obtain the formula (4.5).

In the case of dielectrics with polar molecules. orienting action of the external field counteracts the thermal motion of molecules, which tends to scatter their dipole moments in all directions, as a result it set, some dominant orientation of the dipole moments in the direction of the field. Polarization is proportional to the density of the field - formula (4.5). Dielectric susceptibility χ of such dielectrics is inversely proportional to the absolute temperature of the environment.

In ionic crystals individual molecules lose their detachment and the whole crystal is a single very large molecule. Ion crystalletice can be considered as two guards, one of which is inserted into another. Then one of them is formed by positive and the other by negative ions.

If the external field affects on the ions of the crystal lattice, these gratings will move relative to each other, leading to dielectric polarization. Polarization will be linked to the strength of the field equation (4.5).

Direct dependence of the field dencity on the degree of polarization of the molecule occurs only in not very strong fields. The same is true for (4.3).

4.3. Description of vector field in dielectrics

All charges which constitute the molecules of the dielectric, are called bound. Under the influence of the field bound charges are only slightly shift from the equilibrium position ,bound charges can not leave their molecules.

Charges which are within the dielectric, but not included in the composition of their molecules, and charges outside the dielectric are called foreign (free) charges.

Field in the dielectric is a superposition of extraneous fields and related charges. The resulting field called microscopic or true.

 . (4.7)

Microscopic field varies greatly within intermolecular distances. Due to the movement of related charges given microscopic field also changes over time. At macroscopic examination these changes are not observed, so the quality characteristics of the field are serve as a physically infinitesimal volume mentioned formula (7):

.

Further, denoted, аnd –. Accordingly, the macroscopic field will call the value

. (4.8)

Polarization is a macroscopic value, so the intensity in the formula (4.5) represents the intensity in the formula (4.8). In the absence of dielectrics (in vacuum) macroscopic field is numerically equal to the field of irrelevant charges Eo.

If irrelevant charges are fixed, then the field is defined by (4.8) and has the same properties as the electrostatic field in vacuum. That is, it can be described by means of the potential φ, which is linked to the intensity E by gradient.

4.4 Ferroelectrics

There exist a group of substances that have spontaneous polarization in the absence of an external field. This phenomenon was discovered for the Seignette salt.

Ferroelectrics have the following distinctive features:

1. Whereas the value of dielectric permeability of the medium is a few items (except for some substances, whose permeability reaches several tens, for example, water - ε = 81), the ferroelectric ε is several thousands.

2. Polarization dependence on the intensity is not linear; ε is respectively dependent on the intensity.

3.Changing the field, the value of polarization, and thus the value of the dielectric shift leg behind the tension, and so P and D are determined not only by the value of E at the moment, but also by the values of E which preceded it; that is it depends on the previous history of the dielectric . This phenomenon is called hysteresis (from the Greek “delay”).

In the cyclic changes of the field the dependence of polarization on the intensity should be represented by the so-called hysteresis loop.

When at first turned on the electric field intensity E is expected to grow to a certain value of P at the appropriate E, we have OD curve in fig. 4.2.

fig.4.2

Reduction of P will occur with a decrease in the E line DBEK. At value E = 0 substance retains value Pr, which is called the residual polarization.

Only under the influence of oppositely directed field of the intensity Ec polarization becomes equal to zero. Such value of E is called the coercive force. In the subsequent change of the intensity the hysteresis loop of thread will be obtained etc.

4.5. Conditions on the boundary of two dielectrics.

At the boundary between two media with different dielectric properties, vectors of elastic field vary in value and direction. Conditiopns of vectorsandbehavior on the boundary of media are called coboundary. These conditions are derived from the main equations of a field. Let’s determine the coboundary conditions for the normal constituent of vector . For this the equationis used, or in case of surface distribution of free charges, in integral form.

.

Presume that surface boundary S₁₂ of dielectrics 1and 2 are given with dielectric penetrations е₁ і е₂ (fig. 4.3). Let’s mark on the surface S₁₂ elementary area S₀, in boundaries of which the surface density of free charges can be considered equal ().

Let’s build a closed surface around S₀, for example of cylindrical form and determine the flow of induction vectors of field D, which is created by charges у₀ through cylindrical surface S. Let’s put normal for outside parts of sufaces S₁₂, S₁, S_г. Then,

.

As , a,it is followed that D₂n₂=D_2n, a D₁n₁= -D_1n.

Presume, that for the evaluation of coboundary conditions on the boundary of media 1 and 2 and the height of the cylinder. At thatand, thus, . Values D₂ і D₁ at the location of surfaces S₁ і S_г will be considered constant, that is why they can be taken out of the integral. As a result of integration of corresponding surfaces we get .

fig. 4.3

Let’s consume that , then

.

This is what we call boundary condition for the normal constituent of an induction vector of electrostatic field D. It is obvious, when dielectric crosses the boundary line, the normal value of D surges by value у₀. If у₀= 0 (there are no free charges on the boundary line), then ,, that is the normal of vector D does not change. The normal of the constituent of a vector of the electric field intensity Е changes and the amount of such changes depends on the relation of dielectric media penetration ε₁ і ε₂ .

_Indeed,

_{From this follows} . (4.9)

On the boundary between the conductor and dielectric with the surface distribution of free charges q0 co boundary condition is expressede by the equation (4.9). Presuming that medium 1 is a conductor, then, as it is known, E_l =0, тому D_ln =0 in it, and , або . Accordingly, for the normal of the field intensity E

. (4.10)

In case the conductor contacts vacuum (ε = 1)

. (4.11)

From the equations (4.10) and (4.11) it is obvious that the intensity of the electric field near the surface of the conductor with the presence of dielectric decreases in ε times. Formula (4.10) actually gives the instant solution of the problem about field in the flat counter. At this, it was unnecessary to consider surface charges in dielectric in between the walls of the compensator.

fig.4.4

Let’s develop the co boundary conditions for the tangential of vector E, that is we will analyze the behavior of the tangential of vector T before the surface of boundary of two dielectrics.

Let’s build the close circuit ABCD (рис. 4.4) near the boundary of dielectrics 1 and 2. Because of the fact that the electrostatic field is potential, the circulation of intensity vector is equal to zero:

. (4.12)

For the close circuit ABCDA following the equation (4.12) we have the following

. (4.13)

To obtain the co boundary condition for the tangential of vector E on the boundary of media, let’s direct the height AD = BC of the chosen vector to zero. Then

. (4.14)

As, and ,then and .As a result, the equation (4.13) will be the following: . If l₁= l₂= l , we will obtain

. (4.15)

From this it is obvious that that the tangential of the intensity vector of the electrostatic field does not change while transition of the boundary of two media with different dielectric properties: .

_{If we consider the boundary of a} conductor(medium 1) and a dielectric (medium 2), having _{(the field is absent inside the conductor), so .}

This means, that the intensity vector of electrostatic field is always perpendicular to the boundary between the conductor and dielectric, or between the conductor and vacuum.

From the conditions _{andit is clear that at} ε₁< ε_{2 , that is lines of force of the electrostatic field bend, deviating more from the normal while transition to the medium with higher dielectric penetration. Let’s formulate the law of bending of the lines of force.}

It is clear from fig. 4.4 that , а .Considering the equations і ,we will obtain

. (4.16)

Lecture 5

Chapter 3. Conductor in external electrostatic field