Infinite plane charged with surface density charge: to calculate the strength of the electric field created by an infinite plane, select a cylinder in space, the axis of which is perpendicular to the charged plane, and the bases are parallel to it, and one of the bases passes through the point of the field of interest to us. According to Gauss's theorem, the flux of the electric field strength vector through a closed surface is:
Ф \u003d, on the other hand it is: Ф \u003d E
Let's equate the right-hand sides of the equations:
Let us express \u003d - through the surface charge density and find the electric field strength:
Let us find the electric field strength between oppositely charged plates with the same surface density:
Find the field outside the plates:
Charged sphere field strength
Ф \u003d (2) t. Gauss
for r< R
; since (there are no charges inside the sphere)
For r \u003d R
( ; ; )
For r\u003e R
The intensity of the field created by a ball charged uniformly throughout the entire volume
Bulk charge density,
distributed over the ball:
For r< R
For r \u003d R
For r\u003e R
THE WORK OF THE ELECTROSTATIC FIELD ON CHARGE TRANSFER
Electrostatic field - email the field of a stationary charge.
Fel, acting on the charge, moves it, performing work.
In a uniform electric field Fel \u003d qE is a constant value
Field work (electric force) does not dependon the shape of the trajectory and on a closed trajectory \u003d zero.
| If in an electrostatic field point charge Q from point 1 to point 2 along some trajectory (Fig. 1) another point charge Q 0 moves, then the force that is applied to the charge does some work. The work of force F on elementary displacement dl is equal to Since d l/ cosα \u003d dr, then the Work when moving the charge Q 0 from point 1 to point 2 (1) does not depend on the trajectory of movement, but is determined only by the positions of the initial 1 and final 2 points. This means that the electrostatic field of a point charge is potential, and the electrostatic forces are conservative.From formula (1), it can be seen that the work that is performed when an electric charge moves in an external electrostatic field along an arbitrary closed path L is zero, i.e. (2) If we take a single point positive charge as a charge that is moved in an electrostatic field, then the elementary work of the field forces on the path dl is equal to Edl \u003d E ld lwhere E l \u003d Ecosα is the projection of the vector E onto the direction of elementary displacement. Then the formula (2) can be represented in the form (3) The integral is called the circulation of the tension vector. Hence, the circulation of the tension vector electrostatic field along any closed contour is equal to zero. A force field that has property (3) is called potential. From the equality to zero of the circulation of the vector E, it follows that the lines of intensity of the electrostatic field cannot be closed, they necessarily begin and end at charges (on positive or negative) or go to infinity. Formula (3) is only valid for an electrostatic field. In what follows, it will be shown that in the case of a field of moving charges, condition (3) is not true (for it, the circulation of the intensity vector is nonzero). |
Circulation theorem for an electrostatic field.
Since the electrostatic field is central, the forces acting on the charge in such a field are conservative. Since it is basic work, which the field forces produce over a unit charge, then the work of conservative forces on a closed loop is
Potential
The system "charge - electrostatic field" or "charge - charge" has potential energy, just as the system "gravitational field - body" has potential energy.
Physical scalarcharacterizing the energy state of the field is called potential given point of the field. A charge q is placed in the field, it has a potential energy W. The potential is a characteristic of the electrostatic field.
Let's recall potential energy in mechanics. The potential energy is zero when the body is on the ground. And when the body is raised to a certain height, then they say that the body has potential energy.
In terms of potential energy in electricity, there is no zero potential energy level. It is chosen arbitrarily. Therefore, potential is a relative physical quantity.
Potential field energy is the work that electrostatic force does when a charge moves from a given point in the field to a point with zero potential.
Let us consider a special case when an electrostatic field is created by an electric charge Q. To study the potential of such a field, there is no need to introduce a charge q into it. You can calculate the potential of any point of such a field, located at a distance r from the charge Q.
The dielectric constant of the medium has a known value (tabular), it characterizes the medium in which the field exists. For air, it is equal to one.
Potential difference
The work of the field to move the charge from one point to another is called the potential difference
This formula can be represented in a different form
Superposition principle
The potential of a field created by several charges is equal to the algebraic (taking into account the sign of the potential) sum of the potentials of the fields of each field separately
This is the energy of a system of stationary point charges, the energy of a solitary charged conductor and the energy of a charged capacitor.
If there is a system of two charged conductors (capacitor), then the total energy of the system is equal to the sum of its own potential energies of the conductors and the energy of their interaction:
Energy of the electrostatic field system of point charges is equal to:
Uniformly charged plane.
The strength of the electric field created by an infinite plane charged with a surface charge density can be calculated using the Gauss theorem.
It follows from the symmetry conditions that the vector E is everywhere perpendicular to the plane. In addition, at points symmetric with respect to the plane, the vector E will be the same in size and opposite in direction.
As a closed surface, we choose a cylinder whose axis is perpendicular to the plane, and the bases are located symmetrically relative to the plane, as shown in the figure.
Since the lines of tension are parallel to the generatrix of the lateral surface of the cylinder, the flux through the lateral surface is zero. Therefore, the flow of the vector E through the surface of the cylinder
where is the area of \u200b\u200bthe cylinder base. The cylinder cuts the charge out of the plane. If the plane is in a homogeneous isotropic medium with a relative permittivity, then
When the field strength does not depend on the distance between the planes, such a field is called uniform. Dependency graph E (x) for the plane.
Potential difference between two points located at distances R 1 and R 2 from the charged plane is
Example 2. Two uniformly charged planes.
Let's calculate the strength of the electric field created by two endless planes... The electric charge is evenly distributed with surface densities and. The field strength is found as a superposition of the field strengths of each of the planes. The electric field is nonzero only in the space between the planes and is equal.
The potential difference between the planes, where d -distance between planes.
The results obtained can be used for an approximate calculation of the fields created by flat plates of finite dimensions, if the distances between them are much less than their linear dimensions. Noticeable errors in such calculations appear when considering the fields near the edges of the plates. Dependency graph E (x) for two planes.
Example 3. A thin charged rod.
To calculate the strength of the electric field created by a very long, charged rod with a linear charge density, we use the Gauss theorem.
At sufficiently large distances from the ends of the rod, the lines of electric field strength are directed radially from the axis of the rod and lie in planes perpendicular to this axis. At all points equidistant from the axis of the rod, the numerical values \u200b\u200bof the stress are the same if the rod is in a homogeneous isotropic medium with a relative dielectric
permeability.
To calculate the field strength at an arbitrary point located at a distance r from the axis of the rod, draw a cylindrical surface through this point
(see figure). The radius of this cylinder is r, and its height h.
The streams of the stress vector through the upper and lower bases of the cylinder will be equal to zero, since power lines do not have components normal to the surfaces of these bases. At all points of the lateral surface of the cylinder
E \u003d const.
Therefore, the total flow of the vector E through the surface of the cylinder will be
By Gauss's theorem, the vector flux E is equal to the algebraic sum of electric charges inside the surface (in this case, a cylinder) divided by the product of the electrical constant and the relative permittivity of the medium
where is the charge of that part of the rod that is inside the cylinder. Therefore, the electric field strength
The potential difference of the electric field between two points located at distances R 1 and R 2 from the axis of the rod, we find, using the relationship between the strength and potential of the electric field. Since the field strength changes only in the radial direction, then
Example 4. Charged spherical surface.
The electric field created by a spherical surface over which an electric charge with a surface density is uniformly distributed has a centrally symmetric character.
The lines of tension are directed along the radii from the center of the sphere, and the modulus of the vector E only depends on distance r from the center of the sphere. To calculate the field, we choose a closed spherical surface of radius r.
When r o
The field strength is zero, since there is no charge inside the sphere.
For r\u003e R (outside the sphere), according to Gauss's theorem
where is the relative the dielectric constant environment surrounding the sphere.
The intensity decreases according to the same law as the field strength of a point charge, that is, according to the law.
When r o
Dependency graph E (r) for the sphere.
Example 5. Volume-charged dielectric sphere.
If the ball radius R from a homogeneous isotropic dielectric with relative permeability is uniformly charged in volume with density, then the electric field is also centrally symmetrical.
As in the previous case, we will choose a closed surface for calculating the vector flux E in the form of a concentric sphere, the radius of which is r can vary from 0 to.
When r < R stream vector E through this surface will be determined by the charge
So that
When r < R (inside the ball).
Inside the ball, the tension increases in direct proportion to the distance from the center of the ball. Outside the sphere (at r > R) in a medium with a dielectric constant, the flux of the vector E through the surface will be determined by the charge.
For r o\u003e R o (outside the ball).
On the border "ball - environment"the electric field strength changes abruptly, the value of which depends on the ratio of the dielectric permittivities of the sphere and the medium. E (r) for the ball ().
Outside the ball ( r > R) the potential of the electric field changes according to the law
.
Inside the ball ( r < R) the potential is described by the expression
In conclusion, we present expressions for calculating the field strengths of charged bodies of various shapes
| Potential difference | |
| Voltage - the difference in the values \u200b\u200bof the potential at the initial and final points of the trajectory. Voltage numerically equal to the work of the electrostatic field when moving a single positive charge along the lines of force of this field. Potential difference (voltage) independent of selection coordinate systems! | |
| The unit of the potential difference The voltage is 1 V if, when a positive charge of 1 C moves along the lines of force, the field does work of 1 J. |
Conductor - this is solid, in which there are “free electrons” moving within the body.
Metallic conductors are generally neutral: they have equal amounts of negative and positive charges. The positively charged are the ions in the nodes crystal lattice, negative - electrons freely moving along the conductor. When an excess amount of electrons is given to a conductor, it is charged negatively, but if a certain number of electrons are taken from the conductor, it is charged positively.
The excess charge is distributed only over the outer surface of the conductor.
1 ... The field strength at any point inside the conductor is zero.
2 ... The vector on the surface of the conductor is normal to each point on the surface of the conductor.
From the fact that the surface of the conductor is equipotential, it follows that immediately at this surface the field is directed along the normal to it at each point (condition 2 ). If this were not so, then under the action of the tangential component, the charges would begin to move along the surface of the conductor. those. the balance of charges on the conductor would be impossible.
Of 1 it follows that since
There are no excess charges inside the conductor.
The charges are distributed only on the surface of the conductor with a certain density s and are located in a very thin surface layer (its thickness is about one or two interatomic distances).
Charge density is the amount of charge per unit of length, area or volume, thus the linear, surface and volumetric charge densities are determined, which are measured in the SI system: in Coulomb per meter [C / m], in Coulomb per square meter [C / m2 ] and in Coulomb per cubic meter [C / m³], respectively. Unlike the density of a substance, the charge density can have both positive and negative values, this is due to the fact that there are positive and negative charges.
General task electrostatics
Tension vector,
by Gauss's theorem
Poisson's equation.
In the case - there are no charges between the conductors, we get
- laplace equation.
Let the boundary conditions on the surfaces of the conductors are known: values; then this problem has a unique solution according to uniqueness theorem.
When solving the problem, the value is determined and then the field between the conductors is determined by the distribution of charges on the conductors (according to the intensity vector at the surface).
Let's look at an example. Let's find the tension in the empty cavity of the conductor.
The potential in the cavity satisfies the Laplace equation;
potential on the walls of the conductor.
The solution to the Laplace equation in this case is trivial, and by the uniqueness theorem there are no other solutions
, i.e. there is no field in the conductor cavity.
Poisson's equation - elliptical differential equation in partial derivatives, which, among other things, describes
Electrostatic field,
Stationary temperature field,
Pressure field,
· Velocity potential field in hydrodynamics.
It is named after the famous French physicist and mathematician Simeon Denis Poisson.
This equation is:
where is the Laplace operator or the Laplacian, and is a real or complex function on some manifold.
In a three-dimensional Cartesian coordinate system, the equation takes the form:
In the Cartesian coordinate system, the Laplace operator is written in the form and the Poisson equation takes the form:
If a f tends to zero, then the Poisson equation turns into the Laplace equation (the Laplace equation is a special case of the Poisson equation):
Poisson's equation can be solved using Green's function; see, for example, the article screened Poisson's equation. there is different methods to obtain numerical solutions. For example, an iterative algorithm is used - the "relaxation method".
| We will consider a solitary conductor, that is, a conductor significantly removed from other conductors, bodies and charges. Its potential is known to be directly proportional to the charge of the conductor. It is known from experience that different conductors, while being equally charged, have different potentials. Therefore, for a solitary conductor, you can write down. The value (1) is called the electrical capacity (or simply the capacity) of the solitary conductor. The capacity of a solitary conductor is set by the charge, the communication of which to the conductor changes its potential by one. The capacity of a solitary conductor depends on its size and shape, but does not depend on the material, shape and size of cavities inside the conductor, as well as its state of aggregation. The reason for this is that excess charges are distributed on the outer surface of the conductor. The capacity also does not depend on the charge of the conductor or on its potential. The unit of electrical capacity is farad (F): 1 F is the capacity of such a solitary conductor, whose potential changes by 1 V when a charge of 1 C is imparted to it. According to the formula for the potential of a point charge, the potential of a solitary ball of radius R, which is in a homogeneous medium with a dielectric constant ε, is equal.Applying formula (1), we obtain that the capacity of the ball (2) It follows that a solitary ball would have a capacity of 1 F, located in a vacuum and having a radius R \u003d C / (4πε 0) ≈9 10 6 km, which is approximately 1400 times the radius of the Earth (the electrical capacity of the Earth C ≈ 0.7 mF). Consequently, the farad is a rather large value, therefore, in practice, fractional units are used - millifarad (mF), microfarad (μF), nanofarad (nF), picofarad (pF). It also follows from formula (2) that the unit of the electrical constant ε 0 is farad per meter (F / m) (see (78.3)). |
Capacitor (from lat. condensare - "seal", "thicken") - two-pole with a certain value of capacity and low ohmic conductivity; device for storing charge and electric field energy. The capacitor is a passive electronic component. Usually consists of two plate-shaped electrodes (called covers) separated by a dielectric, the thickness of which is small compared to the dimensions of the plates.
Capacity
The main characteristic of a capacitor is its capacity, characterizing the ability of a capacitor to store an electric charge. In the designation of a capacitor, the value of the nominal capacity appears, while the real capacity can vary significantly depending on many factors. The real capacity of a capacitor determines its electrical properties. So, by definition of capacity, the charge on the plate is proportional to the voltage between the plates ( q \u003d CU). Typical capacitances range from picofarads to thousands of microfarads. However, there are capacitors (supercapacitors) with a capacity of up to tens of farads.
The capacity of a flat capacitor, consisting of two parallel metal plates with an area S each spaced at a distance d from each other, in the SI system is expressed by the formula:, where is the relative dielectric constant of the medium filling the space between the plates (in vacuum is equal to unity), is the electrical constant, numerically equal to 8.854187817 · 10 −12 F / m. This formula is valid only when d much smaller than the linear dimensions of the plates.
To obtain large capacities, capacitors are connected in parallel. In this case, the voltage between the plates of all capacitors is the same. Total battery capacity parallel connected capacitors is equal to the sum of the capacitances of all capacitors included in the battery.
If all parallel-connected capacitors have the same distance between the plates and the properties of the dielectric, then these capacitors can be represented as one large capacitor, divided into fragments of a smaller area.
When capacitors are connected in series, the charges of all capacitors are the same, since they are supplied from the power source only to the external electrodes, and on the internal electrodes they are obtained only due to the separation of charges that previously neutralized each other. Total battery capacity consistently connected capacitors is
This capacity is always less than the minimum capacity of the capacitor included in the battery. However, with a series connection, the possibility of capacitor breakdown decreases, since each capacitor accounts for only a part of the potential difference of the voltage source.
If the area of \u200b\u200bthe plates of all capacitors connected in series is the same, then these capacitors can be represented as one large capacitor, between the plates of which there is a stack of dielectric plates of all its constituent capacitors.
[edit] Specific capacity
Capacitors are also characterized by their specific capacity - the ratio of capacitance to volume (or mass) of the dielectric. The maximum value of the specific capacitance is achieved with the minimum thickness of the dielectric, but this decreases its breakdown voltage.
IN electrical circuits different ways of connecting capacitors. Connecting capacitors can be produced: consistently, parallel and serial-parallel (the latter is sometimes referred to as a mixed capacitor connection). The existing types of capacitor connections are shown in Figure 1.
Figure 1. Ways to connect capacitors.
\u003e\u003e Physics: Electric field strength. The principle of superposition of fields
It is not enough to say that electric the field exists. It is necessary to introduce a quantitative characteristic of the field. After that, electric fields can be compared with each other and continue to study their properties.
The electric field is detected by the forces acting on the charge. It can be argued that we know everything we need about the field if we know the force acting on any charge at any point in the field.
Therefore, it is necessary to introduce such a characteristic of the field, the knowledge of which will allow determining this force.
If we alternately place small charged bodies at the same point of the field and measure the forces, it will be found that the force acting on the charge from the side of the field is directly proportional to this charge. Indeed, let the field be created by a point charge q 1... According to coulomb's law (14.2) per charge q 2 there is a force proportional to the charge q 2... Therefore, the ratio of the force acting on the charge placed at a given point of the field to this charge for each point of the field does not depend on the charge and can be considered as a characteristic of the field. This characteristic is called the electric field strength. Like force, field strength - vector quantity; it is designated by a letter. If the charge placed in the field is denoted by q instead q 2, then the tension will be equal to:
Hence the force acting on the charge q from the side of the electric field is equal to:
The field strength of a point charge. Find the strength of the electric field created by a point charge q 0... According to Coulomb's law, this charge will act on a positive charge q with a force equal to
if at a given point in space various charged particles create electric fields, the strengths of which
etc., then the resulting field strength at this point is equal to the sum of the strengths of these fields:
moreover, the field strength created by a separate charge is determined as if there were no other charges that create the field.
Thanks to the principle of superposition to find tension fields system of charged particles at any point, it is enough to know the expression (14.9) for the field strength of a point charge. Figure 14.8 shows how the field strength at point Acreated by two point charges q 1and q 2, q 1\u003e q 2
???
1. What is called the electric field strength?
2. What is the field strength of a point charge?
3. How is the charge field strength directed q 0, if q 0>0
? if a q 0<0
?
4. How is the principle of superposition of fields formulated?
G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky, Physics Grade 10
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Lecture 19.
1. Electrostatic field in vacuum
1.1 Discreteness of the electric charge. Electric charge conservation law
The source of the electromagnetic field is an electric charge - an internal characteristic of an elementary particle that determines its ability to enter into electromagnetic interactions. There are two types of electric charges - positive and negative. The electric charge is discrete - the charge of any body is an integer multiple of the elementary electric charge e\u003d 1.610 -19 Cl. According to the sign of the charge, all elementary particles can be divided into two classes: negatively charged (for example, an electron) and positively charged (proton, positron, etc.). One of the fundamental strict laws of nature is the law of conservation of electric charge: the algebraic sum of the electric charges of any closed (electrically isolated) system remains constant, no matter what processes occur within this system.^
1.2 Coulomb's law. Electric field strength
The interaction between stationary electric charges is carried out by means of an electric field. The concept of an electric field was introduced in the 1830s. English physicist M. Faraday. According to Faraday, each resting charge creates an electric field around itself; the field of one charge acts on another charge, and vice versa - this is how the interaction between charges takes place.
The force of interaction between two stationary point charges is determined by Coulomb's law: two stationary point charges interact with each other with a force proportional to the product of charges and inversely proportional to the square of the distance between them:
| , | (1.1) |
where k - constant, depending on the choice of the system of units. The Coulomb's force is directed along the line connecting the charges. According to Newton's third law, Coulomb forces are applied to different charges and are directed either towards each other (if the charges are opposite), or in opposite directions (if the charges are of the same sign). In SI constant
, where 0 is the SI electrical constant, 0 =
8.85 10 -12 Cl 2 / (N m 2). Thus, for charges located in a vacuum, Coulomb's law has the form
 .
| (1.2) |
Electric charge in SI is measured in coulombs. One coulomb is a charge that flows through the cross-section of the conductor in 1 s at a constant current strength equal to 1A.
The strength characteristic of the electric field is the intensity 
- vector quantity, the modulus of which is equal to the force acting from the side of the electrostatic field on a unit charge; and the direction coincides with the direction of the force acting on the positive charge
where E from- the intensity of the electrostatic field in the environment.
If an electrostatic field is created by several charges, then in accordance with the principle of superposition, the total field strength at a certain point is defined as the vector sum of the intensities created at this point by individual charges:
  | (1.5) |
1.3. Calculation of the field strength of a point charge and an electric dipole
1.3.1. Point charge field strength
Figure: 1.1
^
1.3.2. Electric dipole field strength
An electric dipole is a set of two point charges, equal in magnitude, but opposite in sign, rigidly fixed at a distance l from each other (fig. 1.2). Distance l is called the arm of the dipole, and the vector
 | (1.8) |
Figure: 1.2
dipole moment (electric dipole moment). The dipole moment is directed along the axis of the dipole towards the positive charge (Fig. 1.2). Now we find the field strength of the dipole, limiting ourselves to the case r \u003e\u003e l.
^
A. Field strength at a point on the extension of the dipole axis
In accordance with the principle of superposition, the field strength at the point AND(fig. 1.3)
Figure: 1.3
where 
and 
Is the intensity of the field generated by the respective charges + Q
and -Q.
Since vectors
and
directed in opposite directions, then the modulus of the vector
will be 
, where in accordance with (1.6) 
... Thus, 
.
We transform the expression in brackets as follows. Fig. 1.3 it is seen that 
where r - distance between point AND and the center of the dipole. Further we have
.
Insofar as r \u003e\u003e l,
then the value 
in the denominator can be neglected, therefore 
;
... Because Ql there is a dipole moment, then 
B. Field strength at the perpendicular to the dipole axis
Figure: 1.4
Fig. 1.4 it is seen that 
.Further 
,
,
Consequently, 
where P l\u003d Ql Is the dipole moment. Thus,
 .
| (1.10) |
From a comparison of (1.9) and (1.10), it can be seen that the field strength on the dipole axis is 2 times greater than on the perpendicular to its axis. Note also that the dipole field strength decreases as 1 / r ‑3, i.e. faster than for a point charge, where E 1 / r ‑2 .
^
1.4. Power lines. Tension vector flow. Ostrogradsky-Gauss theorem
The line of force of the electrostatic field is a line whose tangent at each point coincides with the direction of the vector
(fig. 1.5). 
Figure: 1.5
Leyline properties:
a) the lines of force of the electrostatic field do not intersect;
b) the lines of force of the electrostatic field are open - they start at positive charges and end in negative (or go to infinity).
Let us introduce the concept of the flux of the field strength vector. By definition, the elementary flux of the intensity vector through the site dS
where 
- a unit vector that coincides with the normal.
The total flux of the stress vector through any surface can be found by integrating (11.12) for the entire surface 
for a closed surface 
The most important role in electrostatics is played by the Ostrogradsky - Gauss theorem, which is formulated as follows: the flux of the intensity vector through any closed surface is proportional to the algebraic sum of the charges inside this surface:
, (1.13)
Figure: 1.6
Evidence... Let us consider the simplest case when the closed surface is a sphere with a point charge in the center + Q(fig. 1.7). Let's select an elementary area on the sphere dS. The normal to this site and the vector coincide in direction, therefore.
Figure: 1.7
We transform the integrand in (1.13) as follows:
Taking into account that everywhere on the surface of the sphere E= const, and taking into account expression (11.6), we get:
The theorem is proved for the special case when there is one charge inside the spherical surface. The proof can be easily generalized to the case of an arbitrary number of charges and an arbitrary closed surface.
In the total flow, which is created by the charges located outside the closed surface, it is possible to distinguish positive and negative parts, which are mutually compensated. Therefore, charges external with respect to a given closed surface are not taken into account in the Ostrogradsky - Gauss theorem.
The Ostrogradsky-Gauss theorem connects charges with the charges they create electric fields and reflects the fact that stationary electric charges serve as the source of the electrostatic field.
This theorem is closely related to the Coulomb law: if the Coulomb law is valid, then the Ostrogradsky-Gauss theorem is also valid, and vice versa. If in Coulomb's law the exponent was at least slightly different from two, i.e. F
1 / r 2+
α
where α
–
an arbitrarily small number, then the Ostrogradskii-Gauss theorem would be violated. The validity of the Ostrogradsky-Gauss theorem has been verified experimentally with much greater accuracy than the Coulomb law.
^
1.5. Application of the Ostrogradsky-Gauss theorem for calculating fields
The Ostrogradsky-Gauss theorem in a number of cases makes it relatively easy to calculate the strength of the electrostatic field for a given distribution of charges. Let's look at some examples.
^
1.5.1. The field of an infinite uniformly charged plane
Let there be an infinite uniformly charged plane with a surface charge density
[Cl / m 2
]
From considerations of symmetry, it follows that the vector should be perpendicular to the plane. Let's choose a closed surface in the form of a cylinder, the lateral surface of which is oriented along the vector (Fig. 11.8). Vector total flux
is obviously
Figure: 1.8
.
The flow through the lateral surface is zero, since (fig. 1.8):
Flow through the base of the cylinder:
.
Thus, the total flow of the vector E through a closed surface 
.
By the Ostrogradsky-Gauss theorem 
... Hence the field strength
, (1.14)
The intensity of the field created by an infinite uniformly charged plane does not depend on the distance to it. A field in which the intensity vector is the same in magnitude and direction is called uniform.
^
11.5.2. The field of two infinite uniformly charged planes
Let's calculate the intensity of the field created by two infinite parallel planes, uniformly charged with a surface charge density + σ and -σ (Fig. 11.9).
Figure: 1.9
According to the principle of superposition, the total field strength
,
where 
and 
- the intensity of the field generated by the positively and negatively charged planes, respectively.
In areas of space I and III (Fig. 1.9) vectors and are directed in opposite directions, therefore the total tension 
In region II, both are parallel and equal in absolute value, therefore 
... Using the previous result, we get 
.
Similarly, it can be shown that if the planes are charged with the same name, then in the outer regions I and III the field strength is determined by formula (11.I5), and in the inner region I, which is used for electrostatic protection of devices.
11.5.3. Field strength of an infinite uniformly charged filament with linear charge density
[
Cl / m ]
Figure: 1.10
 . | (1.16) |
When deriving formula (1.16), one should choose a closed surface in the form of a cylinder (Fig. 1.10) and take into account that the vector
is perpendicular to the thread and therefore the vector flux through the base of the cylinder is zero. 11.6. Work on moving a charge in an electrostatic field. Vector circulation theorem
Let's find an elementary work to move the charge q in the field generated by the charge Q:where is the angle between the force 
and direction of movement 
.
Fig. 1.11 it is seen that 
.so 
Total work but moving charge q from point AND exactly B we obtain by integrating expression (11.17). Using Coulomb's law, we get 
... Finally
, (1.18)
If the charge moves from a point A exactly B in a different way, then, having done the same calculations, we again come to formula (11.18). Therefore, work in an electrostatic field does not depend on the shape of the path, but depends only on the choice of the starting and ending points. In addition, as can be seen from (11.18), the work of moving the charge in the electrostatic hearth along a closed loop is equal to zero, i.e. 
, (1.19)
Figure: 1.11
These signs indicate that the electrostatic field is potential. In accordance with the result obtained in § 3.3, the work of potential (conservative) forces can be expressed through the difference in potential energies:
From a comparison of (11.18) and (11.20), we conclude that the potential energy of interaction of two point charges
The unit of potential for an electrostatic field is volts. One volt is the potential of such a point in the field at which a charge of 1 C has a potential energy of 1 J: 1 V \u003d 1 J / C.
We find the potential of the field of a point charge by substituting (1.21) into (1.22):
 . | (1.23) |
And, finally, using (1.22), expression (1.20) for work on moving a charge in an electrostatic field from one point to another can be represented as the product of the charge and the potential difference:
is zero. This result is derived from the fact that work in an electrostatic field does not depend on the shape of the path. Therefore, the zero circulation of the vector is also a sign that the electrostatic field is potential.
^
11.7. Relationship between field strength and potential
Since the electrostatic field is potential, then relation (3.17) is satisfied for it, which establishes a connection between the conservative force x potential energy... If in formula (3.17) we substitute
,
we get 
, (1.28)those. the intensity of the electrostatic field is equal to the potential gradient, taken with the sign "-". The sign "-" indicates that the field strength is directed towards the decrease in the potential.
Let us introduce the concept of an equipotential surface, i.e. surface, at any point of which the value of the potential is the same: φ \u003d const... For a point charge field equipotential surfaces have spherical shape, for a uniformly charged filament - cylindrical, etc. The field strength vector is always perpendicular to the equipotential surface.
If the potential is a function of only one coordinate x, then expression (1.28) is simplified:
For a uniform electrostatic field (for example, the field of a flat capacitor), expression (2.30) is simplified:
Let us illustrate the application of formula (11.32) with examples.
1. Field of point charge
3. Field of an electric dipole at a perpendicular to its axis
5. Condenser field
| . | (1.37) |
