Gauss theorem for an electric field induction vector. Electrical displacement. Gaussian theorem. Conditions on the border of the section of two

For electrostatic field in dielectric

The electrostatic field strength, according to (88.5), depends on the properties of the medium: in a homogeneous isotropic medium, the field strength is proportionally proportional to E. Vector of tension E, moving through the boundary of dielectrics, undergoes a jump-like change, thereby creating inconvenience in the calculations of electrostatic fields. Therefore, it turned out to be necessary in addition to the tension vector characterized by another electrical displacement field, which for an electrically isotropic medium, by definition, is equal to

This course is devoted to the electrical properties of the conductors, both in equilibrium and without equilibrium. It will be the opportunity to introduce the concepts of the tank and resistance of an ohmic conductor, useful in electricity. The conductor is a macroscopic system that contains free charge carriers capable of moving under the action of external force.

Conditions on the border of the section of two

Electric current is the result of displacement of charged particles. Intensity electric current It can be expressed as a function of the characteristics of the flow of charge carriers, namely their average speed and their density in volume. Let us dwell for a moment. The table below shows some intensity values \u200b\u200bthat are found in everyday life.

Using formulas (88.6) and (88.2), the vector of electrical displacement can be expressed as

The unit of electrical displacement is a pendant per meter in a square (CL / m 2).

Consider with which you can link the vector of electrical displacement. Related charges appear in a dielectric in the presence of an external electrostatic field created by the free system electrical charges, i.e., an additional field of linked charges is superimposed in a dielectric to the electrostatic field of free charges. Resulting fieldin the dielectric is described by the vector of tension E, and therefore it depends on the properties of the dielectric. Vector D describes the electrostatic field created by free charges.Related charges arising in dielectric can cause, however, the redistribution of free charges that create the field. Therefore, the vector D characterizes the electrostatic field created by free charges(i.e. in vacuum), but with such a distribution in space, which is with a dielectric.

We will try to assess the speed of charge carriers in the domestic installation. In addition, each copper atom releases a free electron. It depends on the conductor, temperature and pressure. Pay attention to the ratio of the scale between isolation and conductors. This model is based on the following assumptions.

Approximation of free electrons: conduction electrons form the perfect gas of independent charged particles. In the absence of an external field, these free electrons do not feel any strength on average and move in direct due to thermal excitement. Electrons are scattered with crystalline defects. . Indeed, when the metal heats up, network fluctuations increase, which increases the likelihood of collision and, therefore, reduces relaxation time. Suppose that the electric current is homogeneous in cross section and axial, cross section is constant, the current density is constant along the cylinder, and the ratio of two relationships allows you to obtain the distribution of OM for conductive wire cylindrical.

Similarly, as the field E, the field D is depicted using the electrical displacement lines, the direction and thickness of which are defined in the same way as for tension lines (see §79).

Lines vectorE. can begin and end on any charges- free and related, while vector linesD - only on free charges.Through field areas where are related charges, Vector d lines are not interrupted.

Resistance is expressed in Om in honor of George Ohm. Return to the conductivity of metal, called specific resistance, varies linearly with a temperature, so the resistor can serve as a thermometer after calibration. Platinum wire is usually used as follows: Platinum resistance thermometer.

Electrostatic balance conduits

The superconducting device opens up prospects for electricity transporting without energy loss if the temperature range is a critical temperature superconductor in the temperature range. Thus, there is still a long way to go before finding superconducting material temperature. ambient. From this point on, we are interested in the equilibrium of electrified conductors placed in a vacuum.

For arbitrary closedsurface S.vector stream D Through this surface

where D n is the projection of the vector D to normal pto the site dS.Gaussian theorem for the electrostatic field in the dielectric:

i.e. the flow of the electrostatic field of the electrostatic field in dielectric through an arbitrary closed surface is equal to the algebraic amount of prisoners within this surface freeelectric charges. In this form, the Gauss Theorem is valid for the electrostatic field for both homogeneous and isotropic and non-uniform and anisotropic media.

Properties of equilibrium conductors

When equilibrium, the conductor is not subject to any macroscopic movement. Thus, according to the law of Ohm, there is no in the conductor electric field. We emphasize that it is local electric fieldAveraged in a mesoscopic scale. Of course, on the scale of the atom, there is an extremely large and oscillating electric field.

Electric potential is homogeneous inside the conductor during equilibrium. In other words, an equilibrium conductor is an equipotential volume. Since the lines of the electric field are perpendicular to the equipotentials, it can be seen here that the electric field in the outer surroundings of the conductor is normal to the surface.

For vacuum d n \u003d e 0 e n (e = 1), then the stream of the vector of tension e through an arbitrary closed surface (CP. C (81.2)) is equal

Since the sources of the field E in the medium are both free and related charges, then the Gaussory Theorem (81.2) for the F-EI in the most general form can be written as

where and - accordingly the algebraic amounts of free and related charges covered by the closed surface s . However, this formula is unacceptable for describing the field E in a dielectric, as it expresses the properties of an unknown field E through the associated charges, which, in turn, are determined by it. This once again proves the feasibility of introducing an electric displacement vector.

By virtue of the Gauss theorem, which we will see later, from the fact that the electric field is zero inside the conductor, means that the charge density is zero everywhere. This means that any charge of the charge on the conductor will be distributed over the surface of the conductor so that the zero electrical field is created inside. Therefore, the electric field on the surface of the conductor depends on the distribution of surface loads.

We put yourself out of the conductor in equilibrium, remaining in the immediate vicinity of the point p of its surface. In this case, the created electrical field depends only on surface density At this point. This shows the Coulomb Theorem.

To show it, put at the point M in the vicinity of the conductor.

Conditions on the border of the section of two

Dielectric media

Consider the relationship between vectors E and D on the interface of two homogeneous isotropic dielectrics (dielectric constant of which E 1 and E 2 In the absence of free charges on the border.Build near the boundaries of the separac of dielectrics 1 and 2 small closed rectangular contour ABCDA.length l., oriented it as shown in Fig. 136. According to Theorem (83.3) on the circulation of the vector E,

Theorem Gauss and its consequences

The Gauss Theorem is a very common theorem that binds the electric flow and the amount of electrical charge.

In other words, the flow is proportional to the amount of charge concluded by the sphere, but does not depend on the size of the sphere. It is possible to wonder what happens with the stream when the surface surrounding the charge is no longer spherical.

The flow of the electrostatic field through any closed surface is proportional to the number of charge enclosed in this surface. We can verify that the Gauss Theorem is compatible with the Coulomb theorem. We highlight, on thought, a small volume, located inside the conductor in equilibrium, and the electric field is zero, its flow through the separation surface is also zero. Finally, in an empty charge cavity, the inner surface is also empty from charge, which implies a zero field and a constant and equal potential equal to the potential of the conductor. It measures the power of the conductor for storing the amount of charge with a given electric potential.

(Integral signs for AUand CDdifferent, since integration paths are opposite, and integrals by plots Sun.and DAnegligible). therefore

Replacing, according to (89.1), projections of the vector E projections of the vector D, shared on EO £, we get

On the border of the section of two dielectrics (Fig. 137), we construct a straight cylinder of a negligible height, which is one base in the first dielectric, the other is in the second.

Example: spherical conductor capacity

Power is measured in Faraday in memory of Faraday Michael Faraday: English physicist and chemist.

In the previous example, it is shown that the load changes as a radius of curvature and, therefore, the charge density varies as the curvature reverse to the radius. For this reason, the electric field becomes very important near conducting points, where the radius of curvature is small, and this modeling illustrates this phenomenon. Often on sharp bodies and especially on the lightning lifts that serve precisely for this purpose: near the point, the electric field can be large enough to localize the air locally and create a conductive channel that can contact the downward conductive channel; Then there is a flash.

The bases of DS are so small that within each of them, the vector d is the same. According to the Gauss Theorem (89.3),

(N and N Normal "to the bases of the cylinder are opposite). Therefore

Replacing, according to (89.1), the projections of the vector D projections of the vector E, multiplied by one, we get

Thus, when switching through the border of the section of two dielectric media, the tangential component of the vector E (ET) and the normal component of the vector D (D n) change continuously (the jump is not undergoing), and the normal component of the vector E (E N) and the tangential component of the vector D ( D t) undergo a jump.

The capacitance of the capacitor measures the possibility of storing the charge on the internal fittings. The capacitance of the capacitor is measured as the capacity of the conductor, in Faraday. The flat condenser is formed by approaching two flat conductors, subject to potential difference. At opposite faces there are thickening of charges of the opposite sign: one has a complete effect.

Fields created by a flat condenser. On the other hand, on the external edges of the reinforcement, the charge density is almost equal to zero. Indeed, as can be seen on the field intensity map, the electric field is intense between the valves and is almost zero outside. It is also noted that between the line of the line of the field line, which means that the field is uniform, which is also visible on the intensity map. Note, finally, what happens on the edges of the frames: the load is usually concentrated on the edges of peak effects, which explains the intensive value of the field near the edges.

From conditions (90.1) - (90.4) for the components of vectors E and D, it follows that the lines of these vectors are tested (refracted). We will find the relationship between the angles A 1 and A 2 (in Fig. 138 E 2\u003e E 1). According to (90.1) and (90.4), E T 2 \u003d E T 1 and E 2 E N 2 \u003d E 1 E N 1. We decompose the e 1 and e 2 vectors at the border of the section on tangential and normal components. From fig. 138 it follows that

Considering the conditions recorded above, we obtain the law of refractive to the lines of tension E (and therefore the displacement lines D)

Thus, the homogeneous nature of the field is valid only between reinforcements and until it remains far from the edges. Calculate the capacity of this capacitor, suggesting that the reinforcements are quite close to the ability to use Coulomb theorem.

Capacity of a flat condenser

The resulting ratio indicates that the smaller the distance, the greater the condensation phenomenon.

The previous formula is valid if the space between the reinforcement is empty. In practice, two metal strips acting as reinforcements are wound, which are separated by two insulating stripes. The presence of this insulator, called a dielectric, leads to an increase in the capacitance of the capacitor formed by the phenomenon of electrical polarization.

This formula shows that, entering the dielectric with a greater dielectric constant, the E and D lines are removed from normal.

Seatoelectrics

Segroesoelectrics - dielectrics, possessing at a certain temperature range of spontaneous (spontaneous) polarity, i.e. polarity in the absence of an external electric field. Segroelectrics include, for example, in detail the studied I. V. Kurchatov (1903-1960) and P. P. Kaeko (1897-1954) Segnetova salt NAKC4H4O6 × 4N2O (from it and obtained their name ferroelectrics) and titanium Barium Watio 3.

Condenser Energy

John David Jackson, Christian Jezmugin and Jean-Paul Wigneron Classical Electrodynamics: Courses and Exercises in electromagnetism. Paris, Dougup, R. de Brogne Ouboter Challengeh Onane discovers superconductivity. Electrical charge and its properties. The intensity of the electrostatic field. Fields of point and continuously distributed cartridges. Work in the electrostatic field. Potential energy and the potential of the electrostatic field. Poisson and Laplace. Electric dipole in the electrostatic field. Electrical field of charged wires.

In the absence of an external electric field, a ferroelectric is a mosaic of domains - regions with different directions of polarity. This is schematically shown on the example of barium titanate (Fig. 139), where the arrows and signs ⊙, ⊕ indicate the direction of the vector R. Since in adjacent domains these directions are different, then in general the dipole moment of the dielectric is zero. When you enter a segroelectric in the external field, the dipole moments of domain domains occur, and the total electrical field of domains that occurred will support them some orientation and after stopping the external field. Therefore, ferroelectrics have abnormally large values \u200b\u200bof dielectric constant (for a ferroed salt, for example, E Max "10 4).

Electrical conductors, semiconductors and dielectrics. Shielding of an external electric field. Sequential and parallel condenser connection. Electrostatic field inside the dielectric. Atomic, ionic and orientational polarization. Energy electrostatic field.

Ohm law for a homogeneous and inhomogeneous conductor. Sequential and parallel wiring. Electrical resistance and its temperature dependence. Faraday electrolysis laws. Types of electrical currents. Magnetic field loop and coil. Effect of medium on magnetic field. Clean loop in a magnetic field. Vector intensity magnetic field. Types of magnetic substances.

Segroelectric properties are highly dependent on temperature. For each segroelectric, there is a certain temperature, above which it unusual properties It is disappearing and it becomes an ordinary dielectric. This temperature is called Curie's point (in honor of French Physics Pierre Curie (1859-1906)). As a rule, ferroelectrics have only one point of Curie; The exception is only segnetic salt (-18 and + 24 ° C) and isomorphic with her compound. In segroelectrics near the point of Curie, there is also a sharp increase in the heat capacity of the substance. The transformation of ferroelectrics into a conventional dielectric occurring at the Curie point is accompanied by a phase transition of the genus II (see § 75).

Law electromagnetic induction. Moving loops in a magnetic field. Energy and power in a magnetic field. Movement of particles in an electromagnetic field. Effective current and voltage values. Active I. reactive power. Production, transmission and storage of electricity. Electric current conversion.

Energy electromagnetic field. Electromagnetic energy density. The law of conservation of energy in the electromagnetic field. Spectrum of electromagnetic waves. Harmonic electromagnetic waves. Intensity of electromagnetic waves. Spherical electromagnetic wave.

The dielectric permeability of E (and, consequently, the dielectric susceptibility æ) of ferroelectrics depends on the e-field strength in the substance, and for other dielectrics, these values \u200b\u200bare characteristics of the substance.

For ferroelectrics of formula (88.2) is not respected; For them, the relationship between polarization vectors (P) and tensions (E) nonlinearand depends on the values \u200b\u200bof E in the preceding moments of time. In segroelectricians, there is a phenomenon of dielectric hysteresis ("delay"). As can be seen from fig. 140, with an increase in the tension of the EUNECTION electrical field, the polarity of RRASTET, reaching saturation (curve 1). Reducing p size eproisitsy in curve 2, and at e \u003d 0 Segroinoelectric retains residual polarity P 0 , i.e. the ferroelectric remains polarized in the absence of an external electric field. To destroy residual polarity, it is necessary to apply the electric field of the opposite direction (s). E S.called coercive force (from lat. coercitio - holding). More than E.change, T. Rchanges on the 3Petley hysteresis curve.

An intensive study of ferroelectrics was the discovery of academician B. M. Vul (1903-1985) of the anomalous dielectric properties of the titanate of barium. Barium titanate due to its chemical stability and high mechanical strength, as well as due to the preservation of ferroelectric properties, a large scientific and technical application has found in a wide temperature range (for example, as a generator and the ultrasonic wave receiver). Currently, more than a hundred segroelectrics are known, not counting their solid solutions. Segroesoelectrics are also widely used as materials with large values \u200b\u200bof E (for example, in capacitors).

It should be mentioned about piezoelectrics - crystalline substances in which electrical polarization occurs in certain directions in certain directions even in the absence of an external electric field (straight peene effect). There is also a reverse piezoelecthe effect - the emergence of mechanical deformation under the action of an electric field. In some piezoelectrics, the lattice of positive ions in a state of thermodynamic equilibrium is shifted relative to the lattice of negative ions, as a result of which they turn out to be polarized even without an external electric field. Such crystals are called pyroelectricians. There are still electrical - dielectrics, long-term preserving polarized state after removing the external electric field (electrical analogs of permanent magnets) these groups of substances are widely used in the technique and household devices.

The Gaussian Theorem for the electric field in the dielectric. Vector tension begins and ends on free and related charges

For the field in the substance, it is convenient to use the induction of the electric field:

For discrete charge

For continuous charges:

maxwell Equation

The flow through the closed surface of the electrical field induction vector is equal to the total free charge within this surface.

Physical meaning

Power lines begin and end on free charges.

The determination of the dielectric constant was also given in the school program. Without going into details, it is easier to determine through a flat condenser. If you take a flat condenser in vacuo, the charge on each of its plate is equal to (module):

(1.4)

where E 0 is a dielectric constant, or the dielectric constant Vacuum, E 0 \u003d 8.85 · 10 -12 F / M, S- Area of \u200b\u200beach of the plates, D is the gap between the plates, U is the voltage between them. Dividing on the area and turning to the density of the charge on the plane S, we obtain S \u003d E 0 · E.

What happens if enter the dielectric in the interelectrode space? It all depends on whether the charged condenser is connected to the source, or disabled. In the connected condenser, the voltage between the plates is forcibly supported, but the charge on each plate is increased to a new value of Q m. The ratio Q m / Q 0 \u003d E is called the dielectric constant of the material. From the very definition, it can be seen that the dielectric constant of the material is a b e z r. Going to the charge density on the plated, in the case of a dielectric, we obtain S \u003d E 0 · E · E.

Where does an additional charge come from? It is clear that the charge runs out of the source.

In the charged condenser disconnected from the source, the situation is somewhat different. The charge cannot change, because He has nowhere to flow and have nowhere to come. In this case, another parameter will change. It turns out, the voltage on the condenser is reduced and, accordingly, the field strength in the condenser. The coefficient of the field is the same as in the case of a charge when the source is connected, i.e. It is equal to e.

Due to what happens? Consider this question more. Here you will have to turn to the concept of polarization.

As it is known, the molecules consist of atoms surrounded by electronic shells. At the same time, electrons can be evenly distributed over the molecule, and they can concentrate on any atoms. In the first case, they say that the molecule is notolar. An example is a hydrogen molecule or an atom of helium, or a benzene molecule.

In the second case, the molecule forms areas with positive and negative charge. If you can select the direction in the molecule, along which one side can be placed on one side positive chargesAnd on the other hand, negative, then such a molecule is called polar or dipole. An example, HCl molecule, in which an electron moves from a hydrogen atom to a chlorine atom, thereby chlorine charging negatively, and hydrogen is positive.

Vector electrical displacement. The electrostatic field strength, as follows from the previously obtained formula E \u003d E 0 / ε, depends on the properties of the medium: in a homogeneous isotropic medium, the field strength e is inversely proportional to ε. The vector of tension E, when moving across the dielectric border, experienced a jump-like change, thereby making inconvenience in calculating electrostatic fields. Therefore, in addition to the tension vector, it is necessary to characterize the field with another electrical displacement vector, which for an electrically isotropic medium, by definition, is (1). Since ε \u003d 1 + θ and p \u003d θε 0 E, the electrical displacement vector is (2). The unit of electrical displacement is a pendant per meter in a square (CL / m 2).

Thusll, with which you can link the vector of electrical displacement. Related charges are formed in a dielectric in the presence of an external electrostatic field, which is created by the system of free electrical charges, i.e., in dielectric, the electrostatic field of free charges is summed with an additional field of related charges. The resulting field in the dielectric is characterized by a vector of tension E, and therefore it depends on the properties of the dielectric. The vector D is characterized by an electrostatic field, which is created by free charges. Related charges that occur in dielectric can cause redistribution of free charges that create a field. Therefore, the vector D characterizes the electrostatic field, which is created by free charges (i.e. in vacuum), but with such a distribution in space, which is available in the presence of a dielectric.

Similarly, as the field E, the field D should be graphically represented using an electrical displacement lines, the direction and density of which are given as well as for tension lines.

Vector E line can begin and end on any charges - free and related, while vector D - only on free charges. Through the fields of the field where there are related charges, the lines of the vector D pass without interrupting.

For any closed surface s vector stream D Through this surface .