Electric field strength lines are lines whose tangents at each point coincide with vector E. By their direction, one can judge where the positive (+) and negative (–) charges that create the electric field are located. The density of lines (the number of lines piercing a unit surface area perpendicular to them) is numerically equal to the modulus of the vector E.
Electric field strength lines Electric field strength lines are not closed, they have a beginning and an end. We can say that the electric field has “sources” and “sinks” of field lines. Lines of force begin on positive (+) charges (Fig. a) and end on negative (–) charges (Fig. b). The field lines do not intersect.
Flow of the electric field strength vector Arbitrary area dS. The flow of the electric field strength vector through the site dS: is a pseudo-vector, the magnitude of which is equal to dS, and the direction coincides with the direction of vector n to the site dS. E = constdФ E = N - the number of lines of the electric field strength vector E penetrating the area dS.
Flow of the electric field strength vector If the surface is not flat and the field is inhomogeneous, then a small element dS is identified, which is considered flat and the field is considered uniform. Flux of the electric field strength vector: The sign of the flow coincides with the sign of the charge.
Gauss's law (theorem) in integral form. A solid angle is a part of space limited by a conical surface. The measure of the solid angle is the ratio of the area S of the sphere cut out on the surface of the sphere by a conical surface to the square of the radius R of the sphere. 1 steradian is a solid angle with a vertex in the center of the sphere, cutting out an area on the surface of the sphere equal to the area of a square with a side length equal to the radius of this sphere.
Gauss's theorem in integral form An electric field is created by a point charge +q in a vacuum. The flow d Ф E created by this charge through an infinitesimal area dS, the radius of which is vector r. dS n – projection of the area dS onto a plane perpendicular to the vector r. n is the unit vector of the positive normal to the dS area.
If an arbitrary surface surrounds k– charges, then according to the superposition principle: Gauss’s theorem: for an electric field in a vacuum, the flow of the electric field strength vector through an arbitrary closed surface is equal to the algebraic sum of the charges contained inside this surface divided by ε 0.
The method of applying Gauss's theorem to calculate electric fields is the second method of determining the electric field strength E. Gauss's theorem is used to find fields created by bodies with geometric symmetry. Then the vector equation is reduced to a scalar one.
The method of applying the Gauss theorem to calculate electric fields is the second method of determining the electric field strength E 1) The flux FE of the vector E is found by determining the flux. 2) The flow F E is found using the Gauss theorem. 3) From the condition of equality of flows, the vector E is found.
Examples of application of Gauss's theorem 1. Field of an infinite uniformly charged thread (cylinder) with linear density τ (τ = dq/dl, C/m). The field is symmetrical, directed perpendicular to the thread and, for reasons of symmetry, at the same distance from the axis of symmetry of the cylinder (thread) has the same value.
2. The field of a uniformly charged sphere of radius R. The field is symmetrical, the lines of intensity E of the electric field are directed in the radial direction, and at the same distance from point O the field has the same value. The unit normal vector n to a sphere of radius r coincides with the intensity vector E. Let us embrace the charged (+q) sphere with an auxiliary spherical surface of radius r.
2.Field of a uniformly charged sphere When the field of a sphere is like the field of a point charge. At r 
(σ = dq/dS, C/m2). The field is symmetrical, vector E is perpendicular to the plane with surface charge density +σ and has the same value at the same distance from the plane. 3. Field of a uniformly charged infinite plane with surface charge density + σ As a closed surface, we take a cylinder, the bases of which are parallel to the plane, and which is divided by the charged plane into two equal halves.
Earnshaw's theorem A system of stationary electric charges cannot be in stable equilibrium. The charge + q will be in equilibrium if, when it moves a distance dr, a force F acts from all other charges of the system located outside the surface S, returning it to its original position. There is a system of charges q 1, q 2, ... q n. One of the charges q of the system will be covered by a closed surface S. n is the unit normal vector to the surface S.
Earnshaw's Theorem The force F is due to the field E created by all other charges. The field of all external charges E must be directed opposite to the direction of the displacement vector dr, that is, from the surface S to the center. According to Gauss's theorem, if the charges are not covered by a closed surface, then Ф E = 0. The contradiction proves Earnshaw's theorem.
0 flows out more than flows in. Ф 0 flows out more than it flows in. F 33 Gauss's law in differential form Vector divergence is the number of field lines per unit volume, or the flux density of field lines. Example: water flows out and flows out of a volume. Ф > 0 more flows out than flows in. Ф 0 flows out more than it flows in. Ф 0 flows out more than it flows in. Ф 0 flows out more than it flows in. Ф 0 flows out more than it flows in. Ф title="Gauss's Law in differential form Vector divergence is the number of lines of force per unit volume, or the flux density of power lines. Example: water flows out and flows out of a volume. Ф > 0 more flows out than flows in. Ф
Representing the electrostatic field using intensity vectors at different points of the field is very inconvenient, since the picture turns out to be very confusing. Faraday proposed a simpler and more visual method for depicting the electrostatic field using tension lines or power lines. Power lines are called curves whose tangents at each point coincide with the direction of the field strength vector (Fig. 1.2). The direction of the field line coincides with the direction. Lines of force begin at positive charges and end at negative charges. The field lines do not intersect, since at each point of the field the vector has only one direction. An electrostatic field is considered uniform if the intensity at all its points is the same in magnitude and direction. The lines of force of such a field are straight lines parallel to the intensity vector.
The field lines of force of point charges are radial straight lines emerging from the charge and going to infinity if it is positive (Fig. 1.3a). If the charge is negative, the direction of the field lines turns out to be the opposite: they begin at infinity and end at charge -q (Fig. 1.3b). The field of point charges has central symmetry.
Fig.1.3. Tension lines of point charges: a - positive, b - negative.
Figure 1.3 shows flat sections of the electrostatic fields of a system of two charges of equal magnitude: a) charges of the same sign, b) charges of different sign.1. 5. The principle of superposition of electrostatic fields.
The main task of electrostatics is to determine the magnitude and direction of the intensity vector at each point of the field, created either by a system of stationary point charges or by charged surfaces of arbitrary shape. Let's consider the first case, when the field is created by a system of charges q 1, q 2,..., q n. If a test charge q 0 is placed at any point in this field, then Coulomb forces will act on it from the charges q 1, q 2,..., q n. According to the principle of independence of the action of forces, considered in mechanics, the resultant force is equal to their vector sum
.
Using the formula for the electrostatic field strength, the left side of the equality can be written: , where is the strength of the resulting field created by the entire system of charges at the point where the test chargeq 0 is located. The right-hand side of the equality can accordingly be written 
, where is the field strength created by one charge q i . The equality will take the form 
. Reducing by q 0, we get .
The electrostatic field strength of a system of point charges is equal to the vector sum of the field strengths created by each of these charges separately. This is principle of independence of action of electrostatic fields or superposition principle (overlays) fields .
Let us denote by the radius vector drawn from the point charge q i to the field point under study. The field strength in it from the charge q i is equal to 
. Then the resulting tension created by the entire system of charges is equal to 
. The resulting formula is also applicable for calculating the electrostatic fields of charged bodies of arbitrary shape, since any body can be divided into very small parts, each of which can be considered a point charge q i. Then the calculation at any point in space will be similar to the above.
Knowing the vector of the electrostatic field strength at each of its points, you can visually represent this field using field strength lines (vector lines E →). The tension lines are drawn so that the tangent to them at each point coincides with the direction of the tension vector E → (Fig. 4, a).
The number of lines piercing a unit area dS perpendicular to them is drawn proportional to the magnitude of the vector E → (Fig. 4, b). The field lines are assigned a direction coinciding with the direction of the vector E →. The resulting picture of the distribution of tension lines allows us to judge the configuration of a given electric field at its different points. Lines of force begin at positive charges and end at negative charges. In Fig. Figure 5 shows the tension lines of point charges (Fig. 5, a, b); systems of two opposite charges (Fig. 5, a b Fig. 4 Fig. 5 c) is an example of a non-uniform electrostatic field and two parallel oppositely charged planes (Fig. 5, d) is an example of a homogeneous electric field.
The Ostrogradsky–Gauss theorem and its application.
Let us introduce a new physical quantity characterizing the electric field – tension vector flow electric field. Let there be some fairly small area in the space where the electric field is created, within which the intensity, i.e., the electrostatic field is uniform. The product of the modulus of a vector by the area and the cosine of the angle between the vector and the normal to the area called elementary flow of the tension vector through the platform (Fig. 10.7):
where is the field projection to the normal direction .
Let us now consider some arbitrary closed surface. In the case of a closed surface, always select outer normal to the surface, i.e. the normal directed outward of the area.
If we divide this surface into small areas, determine the elementary flows of the field through these areas, and then sum them up, then as a result we get the flow tension vector through a closed surface (Fig. 10.8):
. (10.9)
 |
| Rice. 10.7 |
 Rice. 10.8 |
Theorem Ostrogradsky-Gauss states: the flow of the electrostatic field strength vector through an arbitrary closed surface is directly proportional to the algebraic sum of free charges located inside this surface:
, (10.10)
where is the algebraic sum of free charges located inside the surface, is the volume density of free charges occupying volume.
From the Ostrogradsky-Gauss theorem (10.10), (10.12) it follows that the flow does not depend on the shape of the closed surface (sphere, cylinder, cube, etc.), but is determined only by the total charge inside this surface.
Using the Ostrogradsky-Gauss theorem, in some cases it is possible to easily calculate the electric field strength of a charged body if a given charge distribution has any symmetry.
An example of using the Ostrogradsky-Gauss theorem. Let us consider the problem of calculating the field of a thin-walled hollow a uniformly charged long cylinder of radius (a thin infinite charged thread). This problem has axial symmetry. For reasons of symmetry, the electric field must be directed along the radius. Let us choose a closed surface in the form of a cylinder of arbitrary radius and length, closed at both ends (Fig. 10.9)
A b
). The tension lines are drawn so that the tangent to them at each point coincides with the direction of the tension vector 
(Fig. 1.4, A).The number of lines piercing a unit area dS perpendicular to them is drawn proportional to the vector modulus 
(Fig. 1.4, b).
Lines of force are assigned a direction coinciding with the direction of the vector 
. The resulting picture of the distribution of tension lines allows us to judge the configuration of a given electric field at its different points. Lines of force begin at positive charges and end at negative charges.
In Fig. Figure 1.5 shows the tension lines of point charges (Fig. 1.5, A,
b); systems of two opposite charges (Fig. 1.5, V) is an example of a non-uniform electrostatic field and two parallel oppositely charged planes (Fig. 1.5, G) is an example of a uniform electric field.
1.5. Charge distribution
In some cases, to simplify mathematical calculations, it is convenient to replace the true distribution of point discrete charges with a fictitious continuous distribution. When transitioning to a continuous distribution of charges, the concept of charge density is used - linear , surface and volumetric , i.e.
(1.12)
where dq is the charge distributed accordingly over the element of length 
, surface element dS and volume element dV.
Taking these distributions into account, formula (1.11) can be written in a different form. For example, if the charge is distributed over the volume, then instead of q i you need to use dq = dV, and replace the sum symbol with an integral, then
.
(1.13)
1.6. Electric dipole
To explain phenomena associated with charges in physics, the concept is used electric dipole.
A system of two equal-sized opposite point charges, the distance between which is much less than the distance to the points of space under study, is called an electric dipole. According to the definition of a dipole +q=q= q.
, directed from negative to positive charge. The main characteristic of a dipole is electric dipole moment
= q 
.
(1.14)By absolute value
p = q 
.
(1.15)
In SI, the electric dipole moment is measured in coulombs times a meter (C m).
Let us calculate the potential and electric field strength of a dipole, considering it a point one, if 
r.
Electric field potential created by a system of point charges at an arbitrary point characterized by a radius vector 
, we write it in the form:
where r 1 r 2 r 2 , r 1 r 2 r = 
, because 
r; angle between radius vectors 
And
(Fig. 1.6) .
Taking this into account, we get
.
(1.16)
Using the formula relating the potential gradient to the intensity, we will find the intensity created by the electric field of the dipole. Let's expand the vector 
electrical
dipole fields into two mutually perpendicular components, i.e. 
(Fig. 1. 6).
The first of them is determined by the movement of a point characterized by the radius vector 
(for a fixed value of the angle), i.e., we find the value of E by differentiating (1.81) with respect to r, i.e.
.
(1.17)
The second component is determined by the movement of the point associated with the change in angle (for a fixed r), i.e. E will be found by differentiating (1.16) with respect to : 
,
(1.18)
Where 
,d 
= rd.
Resulting tension E 2 = E 2 + E 2 or after substitution 
.
(1.19)
Comment: At = 90 o 
,
(1.20)
i.e., the tension at a point on a straight line passing through the center of the dipole (i.e. O) and perpendicular to the axis of the dipole.
At = 0 o 
,
(1.21)
i.e., at a point on the continuation of the straight line coinciding with the axis of the dipole.
Analysis of formulas (1.19), (1.20), (1.21) shows that the electric field strength of a dipole decreases with distance in inverse proportion to r 3, i.e., faster than for a point charge (inversely proportional to r 2).
There is a very convenient way to visually describe the electric field. This method comes down to constructing a network of lines, with the help of which the magnitude and direction of the field strength at various points in space are depicted.
Let us select a point in the electric field (Fig. 31, a) and draw a small straight line segment from it so that its direction coincides with the direction of the field at point . Then from some point of this segment we draw a segment, the direction of which coincides with the direction of the field at the point, etc. We get a broken line that shows what direction the field has at the points of this line.
Rice. 31. a) A broken line showing the direction of the field at only four points, b) A broken line showing the direction of the field at six points. c) A line showing the direction of the field at all points. The dashed line shows the direction of the field at the point
The broken line constructed in this way does not quite accurately determine the direction of the field at all points. Indeed, the segment is precisely directed along the field only at a point (by construction); but at some other point on the same segment the field may have a slightly different direction. This construction will, however, convey the field direction more accurately the closer the selected points are to each other. In Fig. In Fig. 31b, the direction of the field is depicted not for four, but for six points, and the picture is more accurate. The image of the field direction will become quite accurate when the break points move closer together indefinitely. In this case, the broken line turns into some smooth curve (Fig. 31, c). The direction of the tangent to this line at each point coincides with the direction of the field strength at this point. Therefore, it is usually called the electric field line. Thus, any line mentally drawn in a field, the direction of the tangent to which at any point coincides with the direction of the field strength at this point, is called an electric field line.
Of the two opposite directions determined by the tangent, we will always agree to choose the direction that coincides with the direction of the force acting on the positive charge, and we will mark this direction in the drawing with arrows.
Generally speaking, electric field lines are curves. However, there can also be straight lines. Examples of an electric field described by straight lines are the field of a point charge, distant from other charges (Fig. 32), and the field of a uniformly charged ball, also distant from other charged bodies (Fig. 33).
Rice. 32. Field lines of a point positive charge
Rice. 33. Field lines of a uniformly charged ball
Using electric field lines, you can not only depict the direction of the field, but also characterize the modulus of the field strength. Let us again consider the field of one point charge (Fig. 34). The lines of this field are radial straight lines diverging from the charge in all directions. From the location of the charge, as from the center, we will construct a series of spheres. All the field lines drawn by us pass through each of them. Since the area of these spheres increases in proportion to the square of the radius, i.e., the square of the distance to the charge, the number of lines passing through a unit surface area of the spheres decreases as the square of the distance to the charge. On the other hand, we know that the electric field strength also decreases. Therefore, in our example, we can judge the field strength by the number of field lines passing through a unit area perpendicular to these lines.
Rice. 34. Spheres drawn around a positive point charge. Each of them shows a single site
If the charge were taken twice as large, then the field strength at all points would increase by a factor. Therefore, so that in this case we can judge the field strength by the density of the field lines, we agree to draw more lines from the charge, the larger the charge. With this imaging method, the density of field lines can serve to quantitatively describe the field strength. We will retain this method of representation in the case when the field is not formed by one single charge, but has a more complex character.
It goes without saying that the number of lines that we draw through a unit surface to depict a field of a given intensity depends on our arbitrariness. It is only necessary that when depicting different areas of the same field or when depicting several fields compared with each other, the density of lines adopted for depicting a field whose strength is equal to unity should be preserved.
In the drawings (for example, in Fig. 35) it is possible to depict not the distribution of field lines in space, but only a cross-section of the picture of this distribution by the plane of the drawing, which will make it possible to obtain so-called “electric maps”. Such maps provide a visual representation of how a given field is distributed in space. Where the field strength is high, the lines are drawn densely; where the field is weak, the density of the lines is small.
Rice. 35. Field lines between oppositely charged plates. Field strength: a) minimum – the density of field lines is minimal; 6) medium – the density of field lines is average; c) greatest – the density of field lines is maximum
A field whose strength at all points is the same in magnitude and direction is called homogeneous. Homogeneous field lines are parallel straight lines. In the drawings, a homogeneous field will also be represented by a series of parallel and equidistant straight lines, the denser the stronger the field they represent (Fig. 35).
Note that the chains formed by the grains in the experiment in § 13 have the same shape as the field lines. This is natural, since each elongated grain is located in the direction of the field strength at the corresponding point. Therefore Fig. 26 and 27 are like maps of electric field lines between parallel plates and near two charged balls. Using bodies of various shapes, with the help of such experiments it is possible to easily find patterns of distribution of electric field lines for various fields.