Power lines of the electric field. Introduction

Distinctive fields are scalar and vector (in our case, the vector field will be electric). Accordingly, they are modeled by scalar or vector functions of coordinates, and also by time.

A scalar field is described by a function of the form φ. Such fields can be visually displayed using surfaces of the same level: φ (x, y, z) = c, c = const.

We define a vector that is directed toward maximizing the growth of the function φ.

The absolute value of this vector determines the rate of change of the function φ.

It is obvious that a scalar field generates a vector field.

Such an electric field is called a potential field, and the function φ is called a potential. Surfaces of the same level are called equipotential surfaces. For example, consider an electric field.

To visualize the fields, the so-called lines of force of the electric field are constructed. They are also called vector lines. These are the lines tangent to which at the point indicate the direction of the electric field. The number of lines that pass through a unit surface is proportional to the absolute value of the vector.

We introduce the concept of a vector differential along some line l. This vector is directed along the tangent to the line l and in absolute value is equal to the differential dl.

Let there be given a certain electric field, which must be represented as field lines of force. In other words, we define the coefficient of expansion (compression) k of the vector so that it coincides with the differential. Equating the components of the differential and the vector, we obtain a system of equations. After integration, the equation of lines of force can be constructed.

In vector analysis, there are operations that give information about which lines of force of the electric field occur in a particular case. We introduce the concept of a "flux of a vector" on a surface S. The formal definition of a flow Φ has the following form: a quantity is considered as the product of the ordinary differential ds by the unit vector of the normal to the surface s. Orth is chosen so that he determines the outer normal of the surface.

One can draw an analogy between the concept of the flow of a field and the flow of a substance: matter per unit of time passes through a surface, which in turn is perpendicular to the direction of the flow of the field. If the lines of force of the electrostatic field leave the surface S outwards, then the flow is positive, and if not, it is negative. In the general case, the flux can be estimated by the number of lines of force that come out of the surface. On the other hand, the flow is proportional to the number of lines of force that penetrate the surface element.

The divergence of a vector function is calculated at the point whose band is the volume ΔV. S is the surface spanning the volume ΔV. The divergence operation allows us to characterize the points of space for the presence of field sources in it. When the surface S is compressed to the point P, the force lines of the electric field penetrating the surface remain in the same amount. If the point of space is not the source of the field (leakage or sink), then when the surface is compressed to this point, the sum of the lines of force starting at some instant is zero (the number of lines entering the surface S is equal to the number of lines emanating from this surface).

The integral over the closed contour L in the definition of the operation of the rotor is called the circulation of electricity along the contour L. The rotor operation characterizes the field at the point of space. The direction of the rotor determines the magnitude of the closed field flow around this point (the rotor characterizes the vortex of the field) and its direction. Based on the definition of the rotor, by simple transformations, it is possible to calculate the projections of the electric vector in the Cartesian coordinate system, as well as the lines of force of the electric field.