but) Field of an infinitely charged plane with superficial charge density. For symmetry considerations, it is clear that the field will be directed perpendicular to the plane - from the plane for positive charges, as shown in Fig. 1.3, and to the plane for negative charges:
In addition, it is also clear from symmetry considerations that the field will be homogeneous. Imagine a cylindrical surface with forming, perpendicular plane and base Ds.. We now consider the flow through the cylinder. The flow through the side surface is zero and, thus, the entire flow through the surface of the cylinder will be reduced to the flow through the base
Based on the Gaussian Theorem, this flow is equal to the charge, concluded inside the cylinder divided by e 0.
From where it follows that
Thus, the field of a uniformly charged plane:
Directed perpendicular plane,
Is homogeneous
Does not depend on the distance to the plane.
b) Field of 2 variemlessly charged planes . The field formed by two different charged planes with surface density of charges s., on the basis of the principle of superposition, you can imagine as the imposition of fields created by these planes separately. Since these fields in the interval between the planes are directed in the same way, the field value will be simply equal to the doubled value of the field created by one plane
Outside the volume bounded by planes, the fields are directed in different directions, and the resulting field is zero. Consequently, a field created by two parallel charged planes:
Is homogeneous
Focused in the interval between the planes.
The resulting conclusions are approximately valid for flat capacitors for which the distance between the planes (they are called plates, or capacitor plates) a lot less than the dimensions of the plates (Fig. 1.4).
in) Field of an endless charged cylinder.
Let there be an infinite radius cylinder R.charged with superficial charge density s. (Fig. 1.5). For symmetry considerations, it is clear that the field will be directed perpendicular to the axis of the cylinder, and its value will depend on the distance to the cylinder. Mentally surrounding our cylinder coaxially located radius cylinder r. and height h. And we calculate the stream of tension vector through the surface limited to this cylinder. For considerations of symmetry, it is clear that the flow through the base of the cylinder will be zero. The flow through the side surface of the cylinder will be equal to E (R) 2PRH. Based on the Gauss Theorem, it will be equal 
. Here L is a linear charge density.
. (1.29)
. (1.30)
Expression (1.30) is true for r\u003e R.. Inside the hollow cylinder, based on the Gauss Theorem, the field is zero. If we accommodate a cylindrical capacitor of two coaxial charged cylinders, the field will be different from zero only between the cylinders, and its value will be determined by the expression (1.30). Outside the outer cylinder field, based on the principle of superposition, will be 0. The field of the cylindrical capacitor (final length) will differ from the field of the cylindrical capacitor infinite length Only at the edge.
d) Field of charged spherical surface . Let there be a radius sphere R., with density surface charge s.. For symmetry considerations, it follows that this field will be central. Surround our sphere of another concentric sphere of radius r.. If a r\u003e R., then on the basis of the Gaussian theorem
,
. (1.31)
Inside the sphere, the field is zero. Thus, the field of a uniformly charged sphere coincides with the field of point charge placed in the center of the sphere.
1. Stream vector. If you put in the flow of flow fluid, a small permeable area ds.then the volume of liquid dVflowing through it per unit of time is equal to the product normal to the velocity of the fluid flow rate v N. By the magnitude of the site ds.,i.e dV=v N.× ds..
Value v N. It is easy to find if you specify a single vector of normal (Fig.12). In this case, and we will know not only the volume of fluid flowing through the platform, but also in which direction it flows through the platform - according to normal (positive stream) and against normal (negative flow).
If the site is the final dimensions S., and the flow rate of the fluid is different in different points, the volume of fluid flowing per unit is found by integration on the site. 
. (4.1)
In this expression vector speed v.
can be replaced by any, continuously changing vector, for example E.
. In this case, instead of the volume of fluid, we get some kind of abstract stream N. Vector E.
otherwise stream vector intensity.
In 1839, Karl Gauss showed that the idea of \u200b\u200bstreaming the vector of tension is very fruitful when calculating symmetric electrostatic fields.
2. Theorem Gaus. Let be point charge q. Located in the center of an arbitrary radius r.. Calculate the flow of the vector across the entire surface of this sphere. Taking into account the vectors and are coated, (Fig.13).
In this case . (4.2)
Stream vector E.
Through the surface of the sphere is proportional to the point charge located in its center q.. If the field E.
portray the power lines, then the value E. At each point of the surface of the sphere is equal to the number of lines occurring on 1 m 2 (Fig. 14 - a). And the whole stream N. Vector E.
equals the number of all silest linescharm q.. Since besides charge q. No other charges inside and outside the sphere, then all lines must go to infinity. But this means that in what place inside the sphere was charged q., Stream vector E.
Through the surface of the medium will be the same (Fig.14-b).
Stream vector E.
Through the closed surface of arbitrary shape too. This can be seen from the fact that any closed surface can be covered by the sphere. Since vector lines E.
do not have breaks, then what flow N. It passes through the sphere, this is through any closed surface (Fig.15).
If inside the closed surface there is a point charge system q I., then flow N. Vector E.
equal to the sum of the streams of vectors E.
I. Each of the charges q I.: 
. (4.3)
So, full stream of tension vector E. through a closed surface proportional total charge inside the surface. it electrostatic theorem. Gaussa.
If the charge is distributed in the volume continuously, with a known density function r.The Gauss Theorem will take the form: 
. (4.4)
The Gaussian Theorem is valid only if the tension of the point charge field decreases in proportion to the square square, which is established by the law of the coulon.
Example 4.1. Field of the charged sphere. We believe that the charge is distributed evenly in the sphere with surface density s.. Radius of sphere R.. Consider separately two areas of space - inside and outside the sphere.
a. Field inside the sphere We find by building a mentally arbitrary concentric sphere of radius r. < R. (Fig.16). Since the designed sphere covers a zero charge (there is no charges inside it), then the flow of the vector E.
Through an arbitrary sphere is zero. . (4.5)
The increment of the potential for any movement within the sphere is zero, 
. Therefore, the potential of all points indoor space The spheres are the same and equal to the potential of the sphere itself. for r. < R. . So, 
(4.6)
b. Field outside the medium We find by building mentally around it an arbitrary concentric sphere of radius r. > R. (Fig.17). Stream vector E.
in this case 
, (4.7)
where is the full charge in the physical sphere. From here
. (4.8)
The field of the charged sphere outside the sphere is the same as if all of her charge was concentrated at the point of its geometric center. Hence the potential of the field of the charged sphere outside the sphere is the same as the potential of a point charge.
So, 
The field of the charged sphere inside and outside it can be illustrated by the schedule. In fig.18-a (at the top) of the si ordinates, setting up the design vector designs on the radius vector built from the center of the sphere. In fig.18-b (below) - potential. Sphere radius R. Charged positive charge so that the field strength near its outer surface is 3 × 10 6 V / m.
If the sphere is charged negative chargeThe graphs will be mirror-symmetric relative to the horizontal axis.
Example 4.2. Field of a solid uniformly charged bowl. Repeating under the shown scheme, we find that the field outside the charged ball is exactly the same as if if his charge was concentrated in its geometric center (Formula 4.9, 4.10).
Inside the ball for any mental concentric sphere (Fig. 19) we get:
. From here 
. (4.11,12)
Here q ¢ - charge enclosed within the radius sphere r.. It is like a work of the entire charge of a ball on the ratio of the volume of the mental sphere to the bulk volume.
The tension inside the uniformly charged bowl linearly decreases to the center until zero.
To find the potential of the field points inside the ball, we calculate the work of moving the unit of a single test charge from the surface of the mental sphere on the surface of the sphere (we assume a trial charge and charge of a ball with positive). . (4.13)
As j R. - the potential of the surface of the ball, then, expressing it according to formula (4.10), where r. = R., and moved to the right side of equality, we obtain an expression for the potential of the field inside the ball at a distance r. from his center. 
. (4.14)
So,
(1.15,16)
Fig.20 shows vector projections E. On the radius vector (at the top) and potential j. (below) for space inside and beyond a positively charged bowl. In the center of the ball, tension is zero, and the potential is maximum in magnitude. Although the tension on the surface of the ball does not have a gap (in contrast to the charged sphere), but its derivative has a gap, changing the magnitude and on the sign. The potential and its derivative on the surface of the ball has no gap.
Example 4.3. Field charged infinite plane . Surface density charges s. = const..
From the fact that the field has a plane of symmetry, it follows that the lines of the vector E. Normal charged plane. Therefore, to calculate the field, select a cylindrical volume with a finite base area S. so that the forming cylinders were parallel to the lines E. . Then flow vector E. Through the side surface is zero (Fig.21).
Stream vector E.
through the surface of this cylinder
, (4.17)
Where E. - field strength at the base points of the cylinder. According to the Gauss Theorem. Since we get. Or in vector form. (4.18)
Here is a single vector of normal to the bases of the cylinder, directed outward.
The field of an infinite charged plane is uniform in the whole space and does not decrease with distance.
For potential calculation j. We believe that j. \u003d 0 not at infinity, and at some distance from the plane H. 0. Hence the potential of any point of the field with the coordinate H. There is a calculation of the movement of the unit charge from the point h. exactly h. 0. Since, then. (4.19)
Since, then 
. (4.20)
Voltage U. between any two points uniform field, where - the distance between them along the field line is, equal, (4.21)
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Example 4.4. Field of infinitely charged thread. The task of calculating the intensity of the segment of the segment of the charged thread is solved in general form in Example 3.2. Here we will solve the task using the Gauss theorem. Let an infinitely straight thread charged evenly with a linear charge density t.,[t.] \u003d CL / m. Obviously, the field has axial symmetry, and the power lines are directed along the radius perpendicular to the thread. As a volume, through the surface of which we will calculate the flow of the vector E. , Take the rotation cylinder, the axis of which coincides with the thread (Fig. 23).
If the cylinder height h., then the stream of the vector of tension through the side surface of the cylinder
. (4.22)
(Vector stream E. Through the base of the cylinder is zero). From here. (4.23)
To find the potential of the field, we calculate the work of moving a single charge due to the field of the field from the point r. to the point r. 0. Since, then
. (4.24)
Or 
. (4.25)
As in Example 4.3, the point of zero potential cannot be moved to infinity, because the voltage between any point r. Point r. 0 \u003d ¥ becomes infinitely large.
. (4.26)
Example 4.5. Field of a straightforwardly charged solid cylinder. Let the infinite radius cylinder R. Charged with density r.. To calculate tensions E. inside the cylinder at a distance r. < R. from the axis, mentally construct a coaxial cylinder height h. and radius r. (Fig.24).
Since the field inside and outside the cylinder has axial symmetry, the power lines are perpendicular to the axis of the cylinder and are directed along the radius. Therefore, the flow of the vector E.
Through the base of the imaginary cylinder is zero. The entire flow goes through the side surface. 
. (4.27)
From here 
. (4.28)
As in the ball (example 4.2), the field inside the cylinder evenly decreases to the axis of the cylinder.
The potential of the field points inside the cylinder of the relative surface is found by the integration of equality - dJ. = EDR. within one r. before R..
. (4.29)
From here 
. (4.30)
Here j R. - the potential of the surface of the cylinder relative to some point r. 0, where the potential is accepted for zero. Value j R. Located by formula (4.26), where r. = R., t. = rpr. 2 .
3. Calculate the field - It means known distribution electrical charges In space, calculate the field strength E. or potential j.. Often the calculation of the potential is easier because it is a scalar. Therefore, in the theory of electricity, the so-called total task Electrostatics, that is, the task of calculating the potential of the electric field created by the static charge system including extended electrically charged bodies.
In mathematical terms, the overall task of electrostatics in the areas of space free of charges is reduced to the solution of the Laplace D equation j. \u003d 0, where D is a second-order differential operator (Laplace operator). In the Cartesian coordinates, it has the form: 
.
The equation is solved under the boundary conditions when the function j. On the surface of all charged bodies are known, and there are no free charges between them. This is the so-called edge task.
1) Electrical charges and their interaction, the law of the coulon.
Electric charge determines the property of the bodies to interact with other charged bodies through electrostatic fields created by these charges.
The interaction of bodies can be positive (if the bodies / charges are attracted) and negative (if the bodies / charges are repelled).
Obviously the existence of an elementary charge. They possess most of the particles (proton, positron, electron).
The charge is equal to an integer number of such elementary charges.
If the charge is still, it creates an electrostatic field in prostata. If the charge is moving without acceleration - also the magnetic field. If moving with acceleration is an electromagnetic field that spreads in the form of electromagnetic waves.
The point charge is a charged body, the linear dimensions of which can be neglected.
The strength of the interaction of two charges is determined by the law of the coulon:
1. The charge is distributed evenly along the entire length.
The concept of the linear density of the charge is introduced.
2. The charge is distributed evenly over the surface of the charged body.
The concept of the surface density of the charge is introduced.
3. The charge is distributed evenly throughout the volume.
The concept of the bulk density of the charge is introduced.
2) Tension - power characteristicequal to the ratio of the force acting from this field to the point charge placed at this point.
The direction coincides with the direction
The electric field is one of the components of the electromagnetic field, a special video material, which exists around bodies or particles, which is the electric charge, as well as in a free form when changing magnetic field (for example, electromagnetic waves). Electric field Directly invisible, but may be observed due to its power influence on charged bodies.
Tension field of point charge.
Denote: Q - a charge that creates the field,
q 0 - charge placed in the field (external charge).
Cool's law :. Field tension:.
Then the tension of the spot charge field:
The principle of superposition.
If there are fields created by several charges at the point of space, the tension at this field point is equal to the vector sum of field strengths.
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3) The stream of the vector of tension is called. The value of F, equal to the product of the modulus of the intensity vector on the circuit area S, which limits some area, and on the cosine of the angle between the tension vector and the normal (perpendicular) to the site. | |
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If we assume that the tension is proportional to the number of power lines per unit area of \u200b\u200bthe surface (i.e. thickness), then the stream stream is proportional to the total number of power lines crossing this circuit. |
Theorem Gaussa - The main theorem of electrodynamics, which is used to calculate the electric fields, enters the system of Maxwell equations. It expresses the relationship between the stream of the electric field strength through the closed surface and the charge in the volume limited to this surface.
Gaussian theorem. The stream of the tension vector through a closed surface is equal to the algebraic amount of charges covered by this amount of charges divided by the permeability of the medium and electric. constant.
In the work of the Ostrograd formula is recorded in the following form:
where ω I. s. - Differentials of volume and surfaces, respectively. In modern record ω \u003d d.Ω - element of volume, s. = ds. - Element of the surface. - Functions, continuous along with their particular first-order derivatives in a closed area of \u200b\u200bspace, limited by a closed smooth surface.
4) Theorem is used for calculation of electrostatic fields.
1. Field of an infinitely long charged cylinder.
charge is distributed over the entire length
2. Field of the charged sphere.
charge is distributed over the entire surface 
charge is distributed throughout the volume
3. Filly charged plane
Consider a field created by an infinite homogeneous charged plane. Let the surface density of the charge of the plane of the same and equal to σ. Imagine a mentally a cylinder with forming, perpendicular to the plane, and the base Δ S.located relative to the plane symmetrically. By the symmetry E." = E."" = E.. Stroy vector flow is 2 E.Δ S.. Applying Gauss theorem, we get:
from which
5) Potentialness of a permanent electric field.
Work of force F on elementary movement DL is equal to
Work when moving Q 0 from point 1 to point 2.
it does not depend on the trajectory of movement. Consequently, El.stat. The field of point charge is potential, and EL.stat. Conservative forces.
Potential in what either point El.stat. There are fields physical quantitypotential energy positive chargeput on this point.
The difference in the potentials of two points is equal to work when moving a single positive charge from point 1 to point 2.
Stroy electric field of point charge
According to the Gauss formula - Ostrogradsky, as well as using the equation of Axwell, it is easy to get:
where q. iN. - charge inside a closed surface S, volume V. As the surface of the integration, take the scope (central symmetry), then
Due to the central symmetry of a dot charge field:
6) Potential, communication between tensions and potential.
Potential -the energy characteristics of the field, numerically equal to the work that needs to be done to move the point charge from this point to infinity.
In view of the fact that the work is scalar value, and the potential is a scalar.
The concept of potential can be introduced for electrostatic field In view of the fact that the work on the movement of charge does not depend on the form of the trajectory, according to which the transfer is carried out, but only on the position of the point.
E. vector circulation Along the closed contour is zero.
Circulation - Required and sufficient condition for the introduction of scalar potential. Proof is the independence of work on the type of movement and the form of the trajectory.
It follows that mathematically potential - integral from the electric field strength. Hence, tension is derived from the potential.
Then, considering what is focused in space,
"Minus" means that the current force (tension) is directed towards decreasing potential, that is, where potential energy Charge will have the smallest value.
Equipotential surface - The concept applicable to any potential vector field, for example, to a static electric field or to Newtonium gravitational field. Equipotential surface - This is a surface on which the scalar potential of this potential field takes constant value.
In Three-dimensional The equation takes shape:
IN cartesian coordinate system Laplace operator is written in the form and the Poisson equation takes the form.
