The unit of measurement for the dipole moment. Dipole moment of a molecule and a bond

Consider, in relation to electrodynamics, what is dipole moment... Elementary charge carriers flowing along a straight section of a system of conductors form a forward current. Accordingly, there is a current charge of the specified current (I * L, where I is the value of the current, L is the length of the section). In turn, he considers two parallel current charges with L tending to infinity. In a closed loop, its two halves have opposite sides, forming a current dipole. A vortex field is created around each such dipole, which is characterized by its own dipole current charge, oriented perpendicular to the plane in which the circuit is located. It is called the dipole moment. But since we are considering only the current component, then for the transition to electromagnetism the same term is called differently. Another name is magnetic dipole moment (Pm, sometimes just m).

It represents one of the key characteristics of any substance. It is believed that the dipole moment arises due to currents (both in the microcosm and in macrosystems). Under the microcosm in in this case the atom is understood: moving in circular orbits can be considered as electricity... Since matter consists of elementary particles, each of them also has its own moment. Please note that elementary particles should be understood not only as molecules and atoms, but also protons, neutrons, electrons and, possibly, even smaller components. From the point of view, their magnetic dipole moment is determined by their own mechanical rotation - spin. However, this assumption has recently been increasingly called into question in the light of the latest field theory of particles. For example, the existence of the so-called anomalous dipole is generally accepted, the value of which differs from the calculations of the equation in quantum theory... But from the field point of view, in which the magnetic field is any elementary particle is generated not by the spin rotation of charge carriers, but is one of the constant components electromagnetic field, the anomalous dipole is easy to explain. The value is defined as a specific set with a corrective spin component. Thus, the magnetic moment for a neutron depends on the electric current that generates it and the energy of the changing electromagnetic field.

When calculating its value for an entire circuit, the method of integral addition of the dipole moments of the simplest current dipoles, creating a closed circular circuit, is used.

The dipole moment in electrodynamics is determined by the formula:

where I is the value of the flowing current; S - area closed loop(circular); n is a vector directed perpendicular to the plane in which the contour is located. Although the above formula does not show this, the value of Pm is also vector, the direction of which can be determined known in classical electrical engineering (right screw): if the rotation of an imaginary screw is compared with the direction of the flowing current, then the movement of the screw body coincides with the sought vector.

The electric field of a dipole differs from the field of a point charge, first of all, in the configuration of the field lines. Since from the point of view of physics, such a dipole is a balanced system of two whose moduli are equal, and the polarity is opposite (+ and -), the corresponding lines of tension begin at one charge, and end at the other. In the case of only one point charge carrier, the lines diverge in all directions, like the light of a lamp.

It is often necessary to find the characteristics of the electric field created by a system of charges localized in a small area of ​​space. An example of such a system of charges are atoms and molecules consisting of electrically charged nuclei and electrons. If you need to find a field at distances that are significantly more sizes the area of ​​the location of particles, then there is no need to use exact, but cumbersome formulas, it is enough to restrict ourselves to simpler approximate expressions.
Let the electric field be created by a set of point charges q k (k = 1, 2, ..., N) located within a small area of ​​space, the characteristic dimensions of which we denote l(fig. 285).

Rice. 285
To calculate the characteristics of the electric field, at some point A at a distance r significantly exceeding l, all charges of the system can be "combined" and the system of charges can be considered as a point charge Q, the value of which is equal to the sum of the charges of the original system

This charge can be mentally positioned at any point in the area of ​​the system of charges q k (k = 1, 2, ..., N) since at l<< r , a change in position within a small area will insignificantly affect the change in the field at the point under consideration.
Within the framework of this approximation, the strength and potential of the electric field are determined by the well-known formulas

If the total charge of the system is zero, then the indicated approximation is too rough, leading to the conclusion that there is no electric field.
A more accurate approximation can be obtained by mentally collecting separately the positive and negative charges of the system under consideration. If their "centers" are displaced relative to each other, then the electric field of such a system can be described as the field of two point charges, equal in magnitude and opposite in sign, displaced relative to each other. We will give a more accurate characterization of the system of charges in this approximation a little later, after studying the properties of the electric dipole.
An electric dipole is a system consisting of two point charges of the same magnitude and opposite in sign, located at a small distance from each other.
Let's calculate the characteristics of the electric field created by the dipole, consisting of two point charges + q and −q located at a distance a from each other (fig. 286).

rice. 286
First, we find the potential and strength of the electric field of the dipole on its axis, that is, on a straight line passing through both charges. Let the point A, is at a distance r from the center of the dipole, and we will assume that r >> a... In accordance with the principle of superposition, the field potential at a given point is described by the expression

At the last step, we neglected the second small value (a / 2) 2 compared with r 2... The magnitude of the electric field strength vector can also be calculated based on the principle of superposition

Field strength can be calculated using the relationship between potential and field strength E x = −Δφ / Δx... In this case, the intensity vector is directed along the axis of the dipole, so its modulus is calculated as follows


Note that the field of the dipole weakens faster than the field of a point charge, so the potential of the field of the dipole decreases inversely with the square of the distance, and the field strength - inversely with the cube of the distance.
In a similar, but more cumbersome, way, you can find the potential and field strength of the dipole at an arbitrary point, the position of which is determined using polar coordinates: the distance to the center of the dipole r and angle θ (fig. 287).

rice. 287
According to the principle of superposition, the field potential at the point A is equal to

Considering that r >> a, formula (6) can be simplified using the approximations

in this case we get

Electric field strength vector E it is convenient to decompose into two components: radial E r directed along a straight line connecting a given point with the center of the dipole, and perpendicular to it E θ(fig. 288).

rice. 288
With such an expansion, each component is directed along the direction of change in each of the coordinates of the observation point, therefore, it can be found from the relationship between the field strength and the change in potential.
In order to find the components of the field strength vector, we write down the ratio of the potential change when the observation point is shifted in the direction of the corresponding vectors (Fig. 289).

rice. 289
The radial component will then be expressed by the ratio


To calculate the perpendicular component, it should be taken into account that the value of the small displacement in the perpendicular direction is expressed through the change in the angle as follows Δl = rΔθ.
Therefore, the magnitude of this field component is


When deriving the last relation, we used a trigonometric formula for the difference of cosines and an approximate relation, which is valid for small Δθ :
sinΔθ ≈ Δθ.
The obtained relations completely determine the field of the dipole at an arbitrary point and make it possible to construct a picture of the lines of force of this field (Fig. 290).

rice. 290
Now let us pay attention to the fact that in all formulas that determine the potential and field strength of a dipole, only the product of the magnitude of one of the dipole charges by the distance between the charges appears. Therefore, it is this work that is a complete characteristic electrical properties and called dipole moment systems. Since a dipole is a system of two point charges, it has axial symmetry, the axis of which is a straight line passing through the charges. Therefore, for the task full characteristics the dipole should also indicate the orientation of the dipole axis. The easiest way to do this is by asking dipole moment vector, the value of which is equal to the dipole moment, and the direction coincides with the axis of the dipole

where a- vector connecting negative and positive charges of dipole 1. This characteristic of the dipole is very convenient and allows in many cases to simplify the formulas, giving them a vector form. So, for example, the potential of the dipole field at an arbitrary point, described by formula (6), can be written in the vector form

After the introduction of the vector characteristic of the dipole, its dipole moment, it becomes possible to use another simplifying model - a point dipole: a system of charges, geometric dimensions which can be neglected, but has a dipole moment 2.
Consider the behavior of the dipole in electric field.

rice. 291
Let two point charges, located at a fixed distance from each other, are placed in a uniform electric field. From the side of the field, forces act on the charges F = ± qE equal in size and opposite in direction. The total force acting on the dipole is equal to zero, however, these forces are applied to different points, therefore the total moment of these is nonzero, but is equal to

where α Is the angle between the vector of the field strength and the vector of the dipole moment. The presence of a moment of force leads to the fact that the dipole moment of the system tends to turn in the direction of the vector of the electric field strength.
Note that the moment of force acting on a dipole is completely determined by its dipole moment. As we showed earlier, if the sum of the forces acting on the system is equal to zero, then the total moment of forces does not depend on the axis about which this moment is calculated. The equilibrium position of the dipole corresponds to both the direction along the field α = 0 , and against him α = π , however, it is easy to show that the first equilibrium position is stable, while the second is not.
If an electric dipole is in an inhomogeneous electric field, then the forces acting on the charges of the dipole are different, therefore the resulting force is nonzero.
For simplicity, we will assume that the axis of the dipole coincides with the direction of the vector of the strength of the external electric field. Compatible axis x coordinate systems with the direction of the tension vector (Fig. 292).

rice. 292
The resulting force acting on the dipole is equal to the vector sum of the forces acting on the charges of the dipole,

Here E (x)- field strength at the point where the negative charge is located, E (x + a)- tension at the point of positive charge. Since the distance between the charges is small, the difference in intensities is presented as the product of the rate of change in the intensity by the size of the dipole. Thus, in an inhomogeneous field, a force acts on the dipole, directed in the direction of increasing the field, or the dipole is drawn into the region of a stronger field.
In conclusion, let us return to the rigorous definition of the dipole moment of an arbitrary system of charges. The vector of the dipole moment, of a system consisting of two charges (Fig. 293),

rice. 293
can be written as

If we now number the charges, then this formula takes the form

where the magnitudes of the charges are understood in the algebraic sense, taking into account their signs. The last formula admits an obvious generalization (which is based on the principle of superposition) to a system of an arbitrary number of charges

This formula determines the dipole moment of an arbitrary system of charges, with its help an arbitrary system of charges can be replaced by a point dipole (Fig. 294).

rice. 294
The position of the dipole inside the region of the location of charges is arbitrary, naturally, if the electric field is considered at distances significantly exceeding the dimensions of the system.

 Assignments for independent work.
 1. Prove that for an arbitrary system of charges, algebraic sum which is equal to zero, the dipole moment determined by formula (11) does not depend on the choice of the reference frame.
 2. Determine the "centers" of positive and negative charges of the system, using the formulas similar to the formulas for the coordinates of the center of mass of the system. If all positive and all negative charges are collected in their "centers", then we get a dipole, consisting of two charges. Show that its dipole moment coincides with the dipole moment calculated by formula (11).
 3. Get the formula expressing the force of interaction between a point dipole and a point charge located on the axis of the dipole in two ways: first, find the force acting on the point charge from the side of the dipole; second, find the force acting on the dipole from the side of the point charge; third, make sure that these forces are equal in magnitude and opposite in direction.

1 The direction of the vector of the dipole moment, in principle, can be set in the opposite way, but historically, the direction of the dipole moment has been set from negative to positive charge. With this definition lines of force as if they are a continuation of the vector of the dipole moment.
2 Another, at first glance, absurd, but convenient abstraction - material point having two charges spaced apart.

Until now, it was assumed that the charges and their fields are in a vacuum. In the following paragraphs, we will consider what effect the material medium - conductors and dielectrics - has on the electric field and on the interaction of electric charges.

Electric dipole this is a system consisting of two point charges (+ q, - q), identical in value, but different in sign, the distance ℓ between which (the dipole arm) is much less than the distance to the considered points of the field (Fig. 12.16).

The main characteristic of a dipole is its electrical, or dipole moment.

Dipole moment Is a vector directed along the axis of the dipole (a straight line passing through both charges) from a negative charge to a positive one and equal to the product of the charge │q│ by the shoulder ℓ.

(12.35)

The unit of the electric moment of the dipole is coulomb-meter (Kl۰m).

E If the dipole is placed in a uniform electrostatic field of intensity E (Fig. 12.17), then a force acts on each of its charges: on the positive F + = + qE, on the negative F - = - qE. These forces are equal in magnitude, but opposite in direction. They form a pair of forces, the shoulder of which is ℓsinα, and create a moment of the pair of forces M. Vector
directed perpendicular to vectors and (see fig. - on us). Module
is defined by the relation M = qEℓsinα, where α is the angle between the vectors and .

M = qEℓsinα = рЕsinα

or in vector form

(12.36)

Thus, a dipole in a uniform electric field is acted upon by a torque that depends on the electric moment, the orientation of the dipole in the field, and the field strength.

In a uniform field, the moment of a pair of forces tends to rotate the dipole so that the vectors and and were parallel.

§ 12.6 Dipole field

We define the strength of the electrostatic field at a point lying in the middle on the axis of the dipole (Figure 12.18). Tension field at point O is equal to the vector sum of the intensities and created by a positive and negative charge separately.

N and the dipole axes between the charges -q and + q are the intensity vectors and directed in one direction, therefore the resulting intensity modulo is equal to their sum.

If you find field strength at point A lying on the extension of the dipole axis (fig 12.18) , then vectors and will be sent to different sides and the resulting intensity modulo is equal to their difference:

(r is the distance between the midpoint of the dipole and the point lying on the axis of the dipole, at which the field strength is determined).

Neglecting the value of the denominator , since r >> ℓ we get

(p is the electric moment of the dipole).

Tensionfield at point C lying on the perpendicular, reconstructed from the midpoint of the dipole (fig. 12.19). Since the distance from charges + q and - q to point B is the same r 1 = r 2, then

The vector of the resulting tension at point B in absolute value is

The figure shows that
, then

The field strength of the dipole at an arbitrary point is determined by the formula

(12.39)

(p is the electric moment of the dipole, r is the distance from the center of the dipole to the point at which the field strength is determined, α is the angle between the radius vector r and the arm of the dipole ℓ).

LECTURE No. 9. DIELECTRICS IN THE ELECTROSTATIC FIELD

INTRODUCTION

The material of this lecture is devoted to the study of the electrical properties of such important materials as dielectrics.

Dielectric materials are widespread in our life, both in everyday life and in technology, this situation is explained by the uniqueness of the properties of these substances.

Dielectrics are substances that, under normal conditions, practically do not conduct electric current. There are no free charge carriers in dielectrics. Resistivity of dielectrics. For comparison, metals.

The main field of application of dielectrics is insulating materials in various electrical devices. The main requirements that all insulating materials must meet is a high degree of protection against electric leakage in parts of a technical device. The fulfillment of this requirement is necessary to ensure the safe operation of equipment and humans, as well as to improve the efficiency of the device.

The lecture will show that special properties of all dielectrics are due to their internal structure, namely the electrical nature of the interaction of the molecules that make up the dielectric.

1. DIPOLE IN THE ELECTROSTATIC FIELD

1.1. Electric dipole moment

All dielectric molecules are electrically neutral. However, molecules have electrical properties. In the first approximation, a molecule can be regarded as an electric dipole.

Consider a simple system of charges that is of great importance in electrostatics - an electric dipole.

An electric dipole is a combination of two opposite charges, equal in absolute value, located at a distance that is significantly less than the distance to the considered points of the field.

Straightconnecting the centers of charges is called the axis of the dipole.

Dipole shoulder Is a vector directed from a negative charge to a positive one, and equal in magnitude to the distance between the charges.

Fig. 1

The quantity that characterizes the electrical properties of a dipole is called the electrical dipole moment.

The electric dipole moment is a vector physical quantity equal to the product of the modulus of the dipole charge by its shoulder

Comment.

1) The electric dipole moment is always co-directional with the dipole arm, that is

2) The dimension of the dipole moment in the SI system is the coulomb multiplied by the meter.

a) Consider point A lying on the extension of the dipole axis. Let us find the strength of the electrostatic field created at a given point by an electric dipole:

Fig. 2

As can be seen from Fig. 2, the dipole field strength at a point is directed along the dipole axis and is equal in magnitude:

Then, based on the formula for the strength of the electrostatic field created by a point charge: , you can write:

where is the distance from the center of the dipole to the considered point A. By the definition of the dipole, therefore

b) Dipole field strengthat the point on the perpendicular, restored to the axis of the dipole from its middle.

Since the point is equidistant from the charges, then

, (1)

where is the distance from the point to the middle of the dipole. From the similarity of isosceles triangles resting on the dipole arm and the vector, we obtain

, (2)

where

(3)

Substituting (1) into (3), we obtain

The vector has a direction opposite to the vector of the electric moment of the dipole, that is

1.3. Electric dipole in a uniform electrostatic field.

Let us consider the behavior of an electric dipole in a uniform electric field. In an external electric field, a pair of forces acts on the ends of the dipole, which tends to rotate the dipole in such a way that the electric moment of the dipole turns along the direction of the field (Fig. 3).

Fig. 3

The electric field acts on the positive and negative charges of the dipole with a force equal in magnitude, but opposite in direction (Fig. 3)

These two forces are called a pair of forces, they create a torque relative to the point 0, which lies in the middle on the axis of the dipole. Under the action of this moment, the electric dipole rotates along the field so that its dipole moment will be codirectional with the strength of the external electric field:

The total (resultant) moment acting on the electric dipole from the side of the external electrostatic field is equal to:

As is known from mechanics, the moment of forces is always directed along the axis of rotation. In our case, the torque vector is directed from us perpendicular to the plane of the figure and passes through the middle of the dipole. The magnitude of the torque is equal to:

2. POLARIZATION VECTOR. CONNECTION OF DIELECTRIC PERMEABILITY

AND THE DIELECTRIC SENSITIVITY OF THE DIELECTRIC

Dielectrics are substances that poorly conduct electric current, since in dielectrics all electrons are bonded to the nuclei of atoms.

If we replace the positive charges of the nuclei of molecules with the total charge located in the "center of gravity" of positive charges, and the charge of all electrons with the total negative charge located in the "center of gravity" of negative charges, then dielectric molecules can be considered as electric dipoles.

There are three types of dielectrics.

1) Dielectrics with non-polar molecules symmetric molecules of which in the absence of an external electric field have zero dipole moment (for example ).

2) Dielectrics with polar molecules whose molecules, due to asymmetry, have a nonzero dipole moment even in the absence of an external electric field (for example ).

3) Ionic dielectrics (For example ). Ionic crystals are spatial lattices with correct alternation ions of different signs.

When dielectrics are placed in an external electric field, the dielectric is polarized - each molecule becomes an electric dipole, acquires an electric dipole momentand, most importantly, it is oriented (rotates along the field) in an external electric field.

According to the three types of dielectrics, three types of polarization are distinguished.

1) Electronicpolarization of a dielectric with non-polar molecules occurs due to the deformation of electron orbits, as a result of which a dipole moment arises in the atoms or molecules of the dielectric.

2) Orientation, or dipole polarization is inherent in dielectrics with polar molecules; in this case, the orientation of the already existing dipole moments of the molecules along the field occurs (this orientation is the stronger, the greater the external field strength and the lower the temperature).

3) Ionicpolarization is inherent in dielectrics with ionic crystal lattices- the displacement of the sublattice of positive ions along the field, and negative ions against the field leads to the appearance of dipole moments.

To quantitatively characterize the polarization of the dielectric, the polarization vector ( polarization).

PolarizationIs a vector physical quantity equal to the ratio of the total dipole electric moment of the entire dielectric to the volume of this dielectric:

Vector dimension polarization dielectric is easy to determine from this formula:

Note that the dimension polarization in the International system of units coincides with the dimension of the surface density of charges. This fact is very important, the meaning of which will be revealed below.

Dielectric polarization is the process of orienting the electric dipoles of the molecules of a substance.

It follows from experience that for a large class of dielectrics, polarization linearly depends on the electric field strength in the dielectric:

(4)

Formula (4) is valid only for isotropic dielectrics, that is, dielectrics whose properties are the same in all directions, located in electrostatic fields with not too high strength. The magnitude(kappa) is called dielectric susceptibility, it characterizes the properties of dielectrics. Dielectric susceptibility–it is a dimensionless, positive value.

So, it turns out that the properties of one and the same substance with respect to an electric field are characterized by two quantities - dielectric constant and dielectric susceptibility... Therefore, there must be a relationship between these quantities.

To establish quantitative relationships between the dielectric susceptibility and permittivity, we introduce into a uniform electric field a dielectric in the form of a right parallelepiped, the area of ​​the side faces of which is S and width (fig. 4).

Under the action of an external field, the dielectric is polarized, i.e. the orientation of the dielectric molecules occurs, so that the positive charges of the molecule are displaced along the field, and the negative charges of the molecule are displaced against the field. As a result, on one face of the dielectric there will be an excess of bound positive charges, on the other - bound negative ones. Inside the dielectric, the bound charges are compensated and it can be assumed that there are no charges inside the dielectrics.

The appearance of bound charges on the lateral surfaces of dielectrics leads to the fact that an additional electrostatic field arises in the dielectric, created by bound charges. Let's designate the intensity of the electrostatic field of the bound charges. The electric field of bound charges is always directed against the external electric field and weakens it. The strength of the resulting electrostatic field, the field inside the dielectric, according to the principle of superpositions, is equal to the vector sum of the strengths of the external field and the strength of the field of bound charges:

.

In scalar form, this equality has the form: .

Fig. 4

Let us find the value of the field strength of the bound charges. A dielectric in an electric field can be viewed as a capacitor with a vacuum inside. The surface density of charges on the plates of such a capacitor is equal to the surface density of bound charges on the faces of the dielectric. The electric field strength of such a capacitor is known to be equal to:

In our case, then

Now let us determine the total dipole moment of the dielectric plate by the thickness and face area. For this we use the definition of the vector polarization dielectric:

And with a charge equal to: time, that is, the equality is true:

Comparing the two obtained formulas for the strength of the external electrostatic field, we can make the obvious conclusion that the relationship between the dielectric constant and the dielectric susceptibility has the form:

CONCLUSION

In conclusion of the lecture on dielectrics, it is necessary to emphasize once again the enormous practical significance of these materials in technology. Using the knowledge gained at the lecture, one can quantitatively and qualitatively study the processes of influence of dielectrics on external electric fields.

The practical significance of influence dielectric substances on the electric field, as you have seen, underlies the phenomena at the interface between two dielectric media, widely used in dielectric antennas.

On the other hand, dielectrics occupy a large place in the design of capacitors. different types to increase their electrical capacity. We will deal with this issue already at the next lecture devoted to the problems of the behavior of conductors in an electrostatic field.