One of the main tasks of electrostatics is to estimate the field parameters for a given, stationary, charge distribution in space. One way to solve such problems is based on principle of superposition . Its essence is as follows.
If a field is created by several point charges, then a charge qk acts on the test charge q as if there were no other charges. The resulting force is determined by the expression:
Because , then - the resultant field strength at the point where the test charge is located, as well obeys the superposition principle :
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(1.4.1) |
This relationship expresses the principle of superposition or superposition of electric fields and represents an important property of the electric field. The intensity of the resultant field, the system of point charges is equal to the vector sum of the field intensities created at a given point by each of them separately.
Consider the application of the principle of superposition in the case of a field created by an electric system of two charges with a distance between charges equal to l (Figure 1.2).
Fig. 1.2
The fields created by different charges do not affect each other, therefore the vector of the resultant field of several charges can be found by the addition rule of vectors (parallelogram rule)
 .
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In this case
and 
Consequently,
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(1.4.2) |
Let's consider another example. Let us find the intensity of the electrostatic field E, created by two positive charges q 1 and q 2 at the point A, located at a distance r 1 from the first and r 2 from the second charge (Figure 1.3).
Fig. 1.3
We use the cosine theorem:
| (1.4.3) |
Where 
.
If a field is created not point charges, then the usual reception is used in such cases. The body is divided into infinitesimal elements and determines the field strength created by each element, then integrated throughout the body:
| (1.4.4) |
Where is the field strength due to the charged element. The integral can be linear, by area or by volume, depending on the shape of the body. To solve such problems use the appropriate values of the charge density:
- linear charge density, measured in Kl / m;
- surface charge density, measured in Kl / m2;
- the volume charge density, measured in Kl / m3.
If the field is created by complex shapes of charged bodies and unevenly charged, then using the superposition principle, it is difficult to find the resulting field.
formula (1.4.4), we see that is a vector quantity:
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(1.4.5) |
So integration may not be easy. Therefore, for calculation, other methods are often used, which we will discuss in the following topics. However, in some relatively simple cases, these formulas can be analytically calculated.
Examples can be considered linear charge distribution or charge distribution along the circumference.
Let us determine the electric field strength at the point A (Figure 1.4) at a distance x from an infinitely long, linear, uniformly distributed charge. Let λ be the charge per unit length.
Fig. 1.4
We assume that x is small in comparison with the length of the conductor. We choose a coordinate system so that the y axis coincides with the conductor. Element of length dy, carries a charge The electric field produced by this element at the point A.
Any electric charge in a certain way changes the properties of the surrounding space - creates an electric field. This field manifests itself in the fact that a charge placed into some other point of it experiences an action of force. Experience shows that the force acting on a stationary charge Q can always be represented as, where is the electric field strength. The field strength is expressed in volts per meter (V / m). Experienced facts indicate that the field strength of a system of point fixed charges is equal to the vector sum of the field strengths that would create each of the charges separately:.
This statement is called the principle of superposition of electric fields.
Equations describing the electrostatic field in a vacuum have the form: 
(1)
- vector of electric field strength, r - charge density, e 0 - electric constant.
For the electrostatic field, in addition to the differential equations (1), the integral relation, called the Gauss theorem, is valid.
Gauss's theorem. The flux of a vector through an arbitrary closed surface S is equal to the algebraic sum of the charges inside this surface divided by e 0.
This theorem is applied to the calculation of fields for a symmetric charge distribution. For example, in the case of a uniformly charged infinite thread, an infinite cylinder, a sphere, a sphere.
A vector field whose curl is equal to zero is called a potential field. The electrostatic field is a potential field;
The lines of tension of the electrostatic field start on positive charges and end in negative ones.
By virtue of (2) in an electrostatic field, the work of field forces when moving a charge from one point to another does not depend on the path through which this displacement is made, but depends only on the initial and final points of the path. 
Let us prove this.
Consider the motion from point A to point B along path 1 and path 2. The work of the field forces when a unit positive charge moves along a closed contour consisting of the paths F 1 and T 2 is equal to
by the Stokes theorem this integral is equal, where S is the surface spanned by the contour under consideration. But by (2) = 0. Thus, = 
=
= 0, that is,
.
Since the gradient rotor is always zero, the general solution of equation (2) is
The minus sign has arisen historically, it does not matter in principle. But thanks to this sign the vector of tension is directed towards decreasing the potential. The electrostatic potential j is equal to the ratio of the potential energy of charge-field interaction to the value of this charge. The difference of potentials of two points of the field, which determines the work of the electrostatic field by charge transfer from one point to another, has a direct physical meaning.
The electrostatic field is described either by equations (1) or by the Poisson equation for the scalar potential j:
The solution of equation (4) has the form:
(5)
Electric field is created by electric charges or simply charged bodies, and also acts on these objects, regardless of whether they move or are stationary. If electrically charged bodies are immovable in a given frame of reference, then their interaction is realized by means of an electrostatic field. Forces acting on charges (charged particles) from the side of the electrostatic field are called electrostatic forces.
A quantitative characteristic of the force action of an electric field on charged particles and bodies is a vector quantity E, called the electric field strength.
Let us consider the charge q as the "source" of the electric field into which a single test charge q / = +1 is placed at a distance r, i.e. a charge that does not cause a redistribution of the charges that create the field. Then, according to Coulomb's law, a test charge will be acted upon by force
Consequently, electrostatic field vector at a given point is numerically equal to the force , acting on the test unit positive charge q / placed in this point of the field
where – radius is a vector drawn from a point charge to the investigated point of the field. The unit of measurement of intensity is = /. The stress is directed along the radius vector from the point where the charge is located to the point A (away from the charge, if the charge is positive, and to the charge if the charge is negative).
An electric field is said to be homogeneous if the vector of its intensity is the same at all points of the field, i.e. coincides both modulo and direction. Examples of such fields are the electrostatic fields of a uniformly charged infinite plane and a flat capacitor far from the edges of its plates. For the graphic representation of the electrostatic field, use the lines of force ( lines of tension) - imaginary lines, the tangents to which coincide with the direction of the tension vector at each point of the field (Fig.10.4.- are represented by solid lines). The density of the lines is determined by the modulus of tension at a given point in space.
The lines of tension are open - they start at positive and end at negative charges. The force lines do not intersect anywhere, since at each point of the field its strength has a single value and a definite direction.
Let us consider the electric field of two point charges q 1 and q 2 .
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Let be the field strength at the point a, created by a charge q 1 (without taking into account the second charge), and a is the charge field strength q 2 (without taking into account the first charge). The strength of the resulting field (in the presence of both charges) can be found by the addition rule of vectors (according to the parallelogram rule, Figure 10.5).
The electric field strength from several charges is principle of superposition of electrostatic fields, according to which tension the resultant field created by the system of charges is equal to the geometric sum of the field strengths created at a given point by each of the charges separately.