Let's take an arbitrary contour (G) and an arbitrary surface S in a non-uniform electrostatic field (Fig. 3.7, a, b).
Then circulation of a vector along an arbitrary contour (Г) is called an integral of the form:
and the flow of the FE vector through an arbitrary surface S is the following expression
The vectors and included in these formulas are defined as follows. In modulus they are equal to the elementary length dl of the contour (G) and the area dS of the elementary area of the surface S. The direction of the vector coincides with the direction of traversing the contour (G), and the vector is directed along the normal vector to the area dS (Fig. 3.7).
In the case of an electrostatic field, the circulation of a vector along an arbitrary closed contour (G) is equal to the ratio of the work Akkrug of the field forces to move a point charge q along this contour to the magnitude of the charge and, in accordance with formula (3.20), will be equal to zero
It is known from theory that if for an arbitrary vector field the circulation of the vector along an arbitrary closed contour (G) is equal to zero, then this field is potential. Hence, the electrostatic field is potential and the electric charges in it have potential energy.
If we take into account that the density of lines determines the magnitude of the vector at a given point in the field, then the flux of the vector will be numerically equal to the number N of lines piercing the surface S.
Figure 3.8 shows examples of calculating the flow through various surfaces S (Figure 3.8, a, b, c, surface S is flat; Figure 3.8, d S is a closed surface). In the latter case, the flux through the closed surface is zero, since the number of lines entering () and leaving () from it is the same, but they are taken with opposite signs ( +>0, -<0).
For a vector we can formulate Gauss's theorem, which determines the flow of a vector through an arbitrary closed surface.
Gauss's theorem in the absence of a dielectric (vacuum) is formulated as follows: the flux of a vector through an arbitrary closed surface is equal to the algebraic sum of the free charges covered by that surface divided by .
This theorem is a consequence of Coulomb's law and the principle of superposition of electrostatic fields.
Let us show the validity of the theorem for the case of a point charge field. Let the closed surface be a sphere of radius R, in the center of which there is a point positive charge q (Fig. 3.9, a).
The obtained result will not change if instead of a sphere we choose an arbitrary closed surface (Fig. 3.9, b), since the vector flux is numerically equal to the number of lines piercing the surface, and the number of such lines in cases a and b is the same.
The same reasoning using the principle of superposition of electrostatic fields can be given in the case of several charges falling inside a closed surface, which confirms Gauss’s theorem.
Gaussian tower for vector in the presence of a dielectric. In this case, in addition to free charges, it is necessary to take into account bound charges that appear on opposite faces of the dielectric when it is polarized in the external electric (for more details, see the section on dielectrics). Therefore, Gauss’s theorem for a vector in the presence of a dielectric will be written as follows:
where the right side of the formula includes the algebraic sum of free and bound charges covered by the surface S.
From formula (3.28) it follows physical meaning of Gauss's theorem for the vector : The sources of the electrostatic field vector are free and bound charges.
In the particular case of a symmetrical arrangement of charges and a dielectric, in the presence of axial or spherical symmetry or in the case of an isotropic homogeneous dielectric, the relative dielectric permittivity of the medium remains a constant value, independent of the point considered inside the dielectric, and therefore the presence of a dielectric can be taken into account in formula (3.28) without only by introducing bound charges , but also the parameter , which is more convenient for practical calculations. So, we can write (see paragraph 3.1.12.6, formula (3.68))
Then the Gauss theorem for the vector in this case will be written as follows
where is the relative dielectric constant of the medium in which the surface S is located.
Note that formula (3.29) is used when solving problems in this section, as well as for most cases encountered in practice.
The circle next to the integral sign in (3.14) means that the integral is taken over a closed contour. An integral of the form (3.14) over a closed contour is called circulation vector Hence, vector circulation electrostatic field , calculated from any closed contour is equal to zero. This is a common property of all fields of conservative forces (potential fields).
(3.17)
If you enter the following notation:
(3.18)
then formula (3.17) will be written in compact form:
The mathematical object we introduced is called gradient operator and formula (3.19) reads like this: “the vector is equal to minus the gradient j.”
Equipotential surfaces, their connection with lines of force.
From the name itself it follows that equipotential surfaces – these are surfaces of equal potential. Hence, equipotential surface equation has the form:
The shape of equipotential surfaces is related to the shape of the field lines: equipotential surfaces are located so that at each point in space the field line and the equipotential surface are mutually perpendicular.
If we agree to draw equipotential surfaces so that the potential difference between two adjacent surfaces is is the same, then according density equipotential surfaces, one can judge the magnitude of the field strength.
If you cut an equipotential surface with a plane, then in the section you get lines of equal potential, equipotential lines.
Conductors and dielectrics. Charged conductor. Conductor in an external electric field.
Conductors – These are substances that have free electrical charges. The concentration of free charges in metal conductors is of the same order as the concentration of atoms. These charges can move within a conductor if an electric field is created in it.
Dielectrics –These are substances in which there are almost no free electrical charges.
In the ideal dielectric model there are no free charges.
Semiconductorsin terms of concentration of free charges they occupy an intermediate position between conductors and dielectrics. Their concentration of free charges depends very much on temperature.
If a conductor is charged, then the free charges in it will begin to move and they will move until the electric field strength in the conductor becomes equal to zero, since the force acting on the charge is equal to:
If , then, according to (3.16):
,
those. all derivatives of the potential are equal to zero, therefore, inside a charged conductor the potential is constant, i.e. volume of the conductor and its surface– equipotential.
If E = 0 everywhere inside the conductor, then the flux of the electric field strength vector through any closed surface inside the conductor is zero. According to Gauss's theorem, it follows that the volumetric charge density inside the conductor is zero. The entire charge of the conductor is distributed over its surface. The electric field strength outside the conductor is perpendicular to its surface, since it is equipotential.
Let's take a small area on the surface of the conductor and build a “Gaussian box” on it, as is done when calculating the field near a uniformly charged plane. Inside the conductor E = 0, therefore.
Potential energy and electrostatic field potential.
From the dynamics section it is known that any body (point), being in a potential field, has a reserve of potential energy W p, due to which work is done by field forces. The work of conservative forces is accompanied by a decrease in potential energy A=W p1 -W p2. Using the formula for the work of the electrostatic field force on the movement of a charge, we obtain can serve as a characteristic of the field and is called electrostatic field potential j. Field potential j - scalar physical quantity, energy characteristic of the field, determined by the potential energy of a single positive charge placed at this point .
The potential difference between two points of the field is determined by the work of field forces when moving a unit
the potential of a field point is numerically equal to the work done by electric forces when moving a unit positive charge from a given field point to infinity.
3) electr. Dipole- an idealized system that serves to approximate the description of a static field or the propagation of electromagnetic waves far from a source (especially from a source with a total zero but spatially separated charge).
Polar- these are dielectrics, in the molecules of which the centers of distribution of positive and negative charges are separated even in the absence of a field, i.e. the molecule is a dipole. Polarization: in external electr. The field of the molecule is oriented along the vector of the external field strength Eo(when the field is turned on, the molecules rotate along the field lines)
Non-polar- dielectrics in whose molecules the centers of distribution of positive and negative charges coincide in the absence of a field. Polarization: in an external electric field, as a result of the deformation of molecules, dipoles appear, oriented along the vector of the external field strength Eo. (when the field is turned on, the molecules are polarized)
In an electric field, the dipoles of the sublattices are deformed: they lengthen if their axes are directed along the field and shorten if their axes are directed against the field. Of such kind polarization called ionic. The degree of ion polarization depends on the properties of the dielectric and the field strength.
Polarization is the phenomenon of the appearance of charges on the surface of a dielectric, the field of which partially compensates for the external electric field
The amount of compensation is described using the dielectric constant of the medium, which shows how many times this medium reduces the electric field:
Kirchhoff's rules for branched chains
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Kirchhoff's first rule: the algebraic sum of current strengths in a node is equal to zero: .
Kirchhoff's second rule refers to any closed loop highlighted in a branched circuit: the algebraic sum of the products of currents and resistances, including internal ones, in all sections of a closed circuit is equal to the algebraic sum of the electromotive forces occurring in this circuit .
Circulation of the electrostatic field strength vector.
Integral…. is called the circulation of the tension vector. Thus, circulation of the electrostatic field strength vector along any closed loop is zero. This is the condition for the potentiality of the field.
If in the electrostatic field of a point charge Q another point charge Q o moves from point 1 to point 2 along an arbitrary trajectory, then the force applied to the charge does work. The work done by force F on an elementary displacement dl is equal to:
Work when moving charge Q o from point 1 to point 2:
The discount does not depend on the trajectory of movement, but is determined only by the positions of the initial 1 and final 2 points. Hence, the electrostatic field of a point charge is potential, and the electrostatic forces are conservative.
The work done when moving an electric charge in an external electrostatic field along any closed path L is equal to zero, i.e.
This integral is called circulation of the tension vector. Thus, the circulation of the electrostatic field strength vector along any closed contour is zero. A force field with this property is called potential .
From the vanishing of the circulation of vector E, it follows that the lines of electrostatic field strength cannot be closed; they begin and end on charges (positive or negative, respectively) or go to infinity.
Circulation theorem
Previously, we found out that a charge (q) that is in an electrostatic field is acted upon by conservative forces, the work ($A$) of which on any closed path (L) is equal to zero:
where $\overrightarrow(s)$ is the displacement vector (not to be confused with the area), $\overrightarrow(E)$ is the field strength vector.
For a unit positive charge we can write:
The integral on the left side of equation (2) is the circulation of the intensity vector along the contour L. A characteristic property of the electrostatic field is that the circulation of its intensity vector along any closed contour is zero. This statement is called the circulation theorem of the electrostatic field strength vector.
Let us prove the circulation theorem on the basis that the work of the field to move a charge does not depend on the trajectory of charge movement in the electrostatic field, which is expressed by the equality:
where $L_1\ and\ L_2$ are different paths between points A and B. Let us take into account that when replacing the integration limits, we obtain:
Expression (4) is represented as:
where $L=L_1+L_2$. So the theorem is proven.
A consequence of the circulation theorem is that the electric field strength lines are not closed. They start on positive charges and end on negative charges or go to infinity. The theorem is true specifically for static charges. Another consequence of the theorem: the continuity of the tangential components of tension (as opposed to the normal components). This means that the stress components that are tangent to any selected surface at any point have equal values on both sides of the surface.
Let us select an arbitrary surface S, which rests on the contour L (Fig. 1).
In accordance with the Stokes formula (Stokes theorem), the integral of the rotor of the tension vector ($rot\overrightarrow(E)$), taken over the surface S, is equal to the circulation of the tension vector along the contour on which this surface rests:
where $d\overrightarrow(S)=dS\cdot \overrightarrow(n)$, $\overrightarrow(n)$ is a unit vector perpendicular to the section dS. The rotor ($rot\overrightarrow(E)$) characterizes the intensity of the “swirling” of the vector. A visual representation of the vector rotor can be obtained if a small, lightweight impeller (Fig. 2) is placed in the fluid flow. In those places where the rotor is not equal to zero, the impeller will rotate, and the speed of its rotation will be greater, the greater the projection module of the rotor projection onto the impeller axis.
In practical calculations of the rotor, the following formulas are used most often:
Since, in accordance with equation (6), the circulation of the tension vector is zero, we obtain:
Condition (8) must be satisfied for any surface S that rests on the contour L. This is only possible if the integrand is:
and for each point of the field.
By analogy with the impeller in Fig. 2 imagine an electric “impeller”. At the ends of such an “impeller” there are charges q of equal magnitude. The system is placed in a uniform field with intensity E. In those places where $rot\overrightarrow(E)\ne 0$ such a “device” will rotate with acceleration, which depends on the projection of the rotor onto the impeller axis. In the case of an electrostatic field, such a “device” would not rotate at any axis orientation. Since the distinctive feature of the electrostatic field is that it is irrotational. Equation (9) represents the circulation theorem in differential form.
Example 1
Assignment: In Fig. 3 shows the electrostatic field. What can you tell about the characteristics of this field from the figure?
About this field we can say that the existence of such an electrostatic field is impossible. If you select the outline (it is shown as a dotted line). For such a circuit, the circulation of the tension vector is:
\[\oint\limits_L(\overrightarrow(E)d\overrightarrow(s)\ne 0)\left(1.1\right),\]
which contradicts the circulation theorem for the electrostatic field. The field strength is determined by the density of the field lines, it is not the same in different parts of the field, as a result, the work along a closed loop will differ from zero, therefore, the circulation of the strength vector is not equal to zero.
Example 2
Assignment: Based on the circulation theorem, show that the tangential components of the electrostatic field strength vector do not change when passing through the dielectric interface.
Let's consider the boundary between two dielectrics with dielectric constants $(\varepsilon )_2\ and\ (\varepsilon )_1$ (Fig. 4). Let us select a small rectangular contour on this boundary with parameters a - length, b - width. The X axis passes through the midpoints of sides b.
For the electrostatic field, the circulation theorem is satisfied, which is expressed by the equation:
\[\oint\limits_L(\overrightarrow(E)d\overrightarrow(s)=0\ \left(2.1\right).)\]
For small circuit sizes, the circulation of the tension vector and in accordance with the indicated direction of traversal of the circuit, the integral in formula (2.1) can be represented as:
\[\oint\limits_L(\overrightarrow(E)d\overrightarrow(s)=E_(1x)a-E_(2x)a+\left\langle E_b\right\rangle 2b=0\ \left(2.2\right) ,)\]
where $\left\langle E_b\right\rangle $ is the average value of $\overrightarrow(E)$ in areas perpendicular to the interface.
From (2.2) it follows that:
\[((E)_(2x)-E_(1x))a=\left\langle E_b\right\rangle 2b\ (2.3).\]
If $b\to 0$, then we get that:
Expression (2.4) is satisfied with an arbitrary choice of the X axis, which lies at the dielectric interface. If we imagine the tension vector in the form of two components (tangential $E_(\tau )\ $ and normal $E_n$):
\[\overrightarrow(E_1)=\overrightarrow(E_(1n))+\overrightarrow(E_(1\tau )),\overrightarrow(E_2)=\overrightarrow(E_(2n))+\overrightarrow(E_(2\ tau ))\ \left(2.5\right).\]
In this case, from (2.4) we write:
where $E_(\tau i)$ is the projection of the intensity vector onto the unit unit $\tau $ directed along the dielectric interface.