The Ostrogradsky–Gauss theorem, which we will prove and discuss later, establishes the connection between electric charges and the electric field. It is a more general and more elegant formulation of Coulomb's law.
Basically, tension electrostatic field, created by a given charge distribution, can always be calculated using Coulomb's law. The total electric field at any point is the vector sum (integral) contribution of all charges, i.e.
However, except in the simplest cases, calculating this sum or integral is extremely difficult.
Here the Ostrogradsky-Gauss theorem comes to the rescue, with the help of which it is much easier to calculate the tension electric field, created by a given charge distribution.
The main value of the Ostrogradsky-Gauss theorem is that it allows understand more deeply the nature of the electrostatic field and establish more general connection between charge and field.
But before moving on to the Ostrogradsky-Gauss theorem, it is necessary to introduce the following concepts: power lines electrostatic field And tension vector flow electrostatic field.
In order to describe the electric field, you need to specify the intensity vector at each point of the field. This can be done analytically or graphically. For this they use power lines– these are lines, the tangent to which at any point in the field coincides with the direction of the intensity vector(Fig. 2.1).
Rice. 2.1
The line of force is assigned a certain direction - from a positive charge to a negative charge, or to infinity.
Consider the case uniform electric field.
Homogeneous called an electrostatic field, at all points of which the intensity is the same in magnitude and direction, i.e. A uniform electrostatic field is represented by parallel lines of force at equal distances from each other (such a field exists, for example, between the plates of a capacitor) (Fig. 2.2).
When point charge, the tension lines come from a positive charge and go to infinity; and from infinity enter a negative charge. Because then the density of the field lines is inversely proportional to the square of the distance from the charge. Because the surface area of the sphere through which these lines pass itself increases in proportion to the square of the distance, then total number lines remains constant at any distance from the charge.
For a system of charges, as we see, the lines of force are directed from a positive charge to a negative one (Fig. 2.2).
Rice. 2.2
From Figure 2.3 it is also clear that the density of field lines can serve as an indicator of the value.
The density of the power lines should be such that a single area normal to the tension vector is crossed by such a number of them that is equal to the modulus of the tension vector, i.e.
The Ostrogradsky–Gauss theorem, which we will prove and discuss later, establishes the connection between electric charges and the electric field. It is a more general and more elegant formulation of Coulomb's law.
In principle, the strength of the electrostatic field created by a given charge distribution can always be calculated using Coulomb's law. The total electric field at any point is the vector sum (integral) contribution of all charges, i.e.
However, except in the simplest cases, calculating this sum or integral is extremely difficult.
Here the Ostrogradsky-Gauss theorem comes to the rescue, with the help of which it is much easier to calculate the electric field strength created by a given charge distribution.
The main value of the Ostrogradsky-Gauss theorem is that it allows understand more deeply the nature of the electrostatic field and establish more general connection between charge and field.
But before moving on to the Ostrogradsky-Gauss theorem, it is necessary to introduce the following concepts: power lines electrostatic field And tension vector flow electrostatic field.
In order to describe the electric field, you need to specify the intensity vector at each point of the field. This can be done analytically or graphically. For this they use power lines– these are lines, the tangent to which at any point in the field coincides with the direction of the intensity vector(Fig. 2.1).
Rice. 2.1
The line of force is assigned a certain direction - from a positive charge to a negative charge, or to infinity.
Consider the case uniform electric field.
Homogeneous called an electrostatic field, at all points of which the intensity is the same in magnitude and direction, i.e. A uniform electrostatic field is represented by parallel lines of force at equal distances from each other (such a field exists, for example, between the plates of a capacitor) (Fig. 2.2).
In the case of a point charge, the tension lines emanate from the positive charge and go to infinity; and from infinity enter a negative charge. Because then the density of the field lines is inversely proportional to the square of the distance from the charge. Because The surface area of the sphere through which these lines pass itself increases in proportion to the square of the distance, then the total number of lines remains constant at any distance from the charge.
For a system of charges, as we see, the lines of force are directed from a positive charge to a negative one (Fig. 2.2).
Rice. 2.2
From Figure 2.3 it is also clear that the density of field lines can serve as an indicator of the value.
The density of the power lines should be such that a single area normal to the tension vector is crossed by such a number of them that is equal to the modulus of the tension vector, i.e.
GRAPHIC REPRESENTATION OF FIELDS
The electric field can be described by indicating for each point the magnitude and direction of the vector. The combination of these vectors will completely determine the electric field. But if you draw vectors at many points of the field, they will overlap and intersect. It is customary to visually depict the electric field using a network of lines that make it possible to determine the magnitude and direction of the field strength at each point (Fig. 13).
The direction of these lines at each point coincides with the direction of the field, i.e. the tangent to such lines at each point of the field coincides in direction with the vector of the electric field strength at this point. Such lines are called electrostatic field strength lines or electrostatic field lines.
Electrostatic field lines begin at positive electric charges and end on negative electrical charges. They can go to infinity from a positive charge or come from infinity to a negative charge (lines 1 and 2, see Fig. 13).
Field lines are useful not only because they clearly demonstrate the direction of the field, but also because they can be used to characterize the magnitude of the field in any region of space. To do this, the density of the field lines must be numerically equal to the magnitude of the electrostatic field strength.
If the field is depicted by parallel lines of force located at equal distances from each other, this means that the field strength vector at all points has the same direction. The modulus of the field strength vector at all points has the same values. This field is called homogeneous electric field. Let us choose an area perpendicular to the tension lines so small that in the area of this area the field is uniform (Fig. 14).
A vector is, by definition, perpendicular to the site, i.e. parallel to the lines of force, and, therefore, . The length of the vector is numerically equal to the area. The number of power lines crossing this area must satisfy the condition
The number of force lines passing through a unit surface area perpendicular to the force lines must be equal to the magnitude of the tension vector.
Let's consider the area not perpendicular to the lines of force (shown with dashed lines in Fig. 14). In order for it to be crossed by the same number of lines of force as the area , the following condition must be met: then . (4.2).
Power lines power lines
electric and magnetic fields, lines, tangents to which at each point of the field coincide with the direction of the electric intensity or, respectively magnetic field; qualitatively characterize the distribution electromagnetic field in space. Lines of force are only a visual way of depicting force fields. 
FIELD LINES, lines drawn in some force field (cm. FORCE FIELD)(electric, magnetic, gravitational), the tangents to which at each point of the field coincide in direction with the vector characterizing this field (strength vector (cm. ELECTRIC FIELD STRENGTH) electric or gravitational fields, magnetic induction vector (cm. MAGNETIC INDUCTION)). Lines of force are only a visual way of depicting force fields. For the first time, the concept of “lines of force” for electric and magnetic fields was introduced by M. Faraday (cm. FARADAY Michael).
Since field strengths and magnetic induction are unambiguous functions of a point, only one field line can pass through each point in space. The density of the field lines is usually chosen so that the number of field lines crossing a unit area perpendicular to the field lines is proportional to the field strength (or magnetic induction) on this area. Thus, field lines provide a visual picture of the field distribution in space, characterizing the magnitude and direction of the field strength.
Electrostatic field lines (cm. ELECTROSTATIC FIELD) are always open: they start on positive charges and end on negative charges (or go to infinity). The field lines do not intersect anywhere, since at each point of the field its intensity has one single value and a certain direction. The density of field lines is greater near charged bodies, where the field strength is greater.
The electric field lines in the space between two positive charges diverge; you can specify a neutral point at which the fields of repulsive forces of both charges cancel each other.
The field lines of a single charge are radial straight lines that diverge from the charge in rays, like lines of force gravitational field point mass or sphere. The further away from the charge, the less dense the lines - this illustrates the weakening of the field with increasing distance.
Lines of force emanating from a charged conductor irregular shape, thicken near any protrusion or tip; near concavities or cavities, the density of the field lines decreases.
If the field lines emanate from a positively charged tip located near a negatively charged flat conductor, then they condense around the tip, where the field is very strong, and diverge into a large area near the plane on which they end, entering the plane perpendicularly.
The electric field in the space between parallel charged plates is uniform. Tension lines in a uniform electric field are parallel to each other.
If a particle, for example an electron, enters a force field, then under the influence of the force field it acquires acceleration, and the direction of its movement cannot exactly follow the direction of the lines of force, it will move in the direction of the momentum vector.
A magnetic field (cm. A MAGNETIC FIELD) characterize magnetic induction lines, at any point of which the magnetic induction vector is directed tangentially.
The lines of magnetic induction of the magnetic field of a straight conductor with current are circles lying in planes perpendicular to the conductor. The centers of the circle are on the axis of the conductor. The field lines of the magnetic induction vector are always closed, i.e. the magnetic field is vortex. Iron filings placed in a magnetic field are aligned along the lines of force; Thanks to this, it is possible to experimentally determine the type of magnetic induction field lines. The vortex electric field generated by a changing magnetic field also has closed lines of force.
encyclopedic Dictionary. 2009 .
See what “field lines” are in other dictionaries:
Lines mentally drawn in the k.l. force field (electric.. magnetic, gravity) so that at each point of the field the direction of the tangent to the line coincides with the direction of the field strength (magnetic induction in the case of a magnetic field). Through… … Big Encyclopedic Polytechnic Dictionary
Electric and magnetic fields are lines, the tangents to which at each point of the field coincide with the direction of the intensity of the electric or corresponding magnetic field; qualitatively characterize the distribution of the electromagnetic field in... ... Big Encyclopedic Dictionary
LINES OF FIELD, lines in an ELECTRIC or MAGNETIC FIELD, whose direction at any point is directed into the field... Scientific and technical encyclopedic dictionary
Imaginary lines that are drawn to depict the k.l. force field (electric, magnetic, gravitational). S. l. are located in such a way that the tangents to them at each point of the right coincide in direction with the vector characterizing the given field... ... Physical encyclopedia
power lines- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information Technology in general EN lines of force...
Electric and mag. fields, lines tangent to the Crimea at each point of the field coincide with the direction of the electrical tension. or resp. mag. fields; qualitatively characterize the distribution of electricity. mag. fields in space. S. l. only a visual way... ... Natural science. encyclopedic Dictionary
Lines drawn in any force field (electric, magnetic, gravitational), the tangents to which at each point in space coincide in direction with the vector characterizing this field (electric or... Great Soviet Encyclopedia
Field lines integral curves for vector field(strength). The electric field lines are perpendicular to equipotential surfaces, and, therefore, to lines of equal potential. Their direction is from “+” to “”. Field line method in... ... Wikipedia
magnetic field lines- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] Topics of electrical engineering, basic concepts EN magnetic flux ... Technical Translator's Guide