There are scalar and vector fields (in our case, the electric field will be a vector field). Accordingly, they are modeled by scalar or vector coordinate functions, as well as by time.
The 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 towards the maximum growth of the function φ.
The absolute value of this vector determines the rate of change of the function φ.
Obviously, a scalar field generates a vector field.
Such an electric field is called potential, and the function φ is called potential. Surfaces of the same level are called equipotential surfaces. For example, consider an electric field.
To visually display the fields, the so-called electric field lines are built. They are also called vector lines. These are the lines tangent to which at a point indicates 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 a certain electric field be given, which must be represented as field lines of force. In other words, we define the tensile (compression) coefficient 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, it is possible to construct the equation of lines of force.
In vector analysis, there are operations that give information about which electric field lines are present in a particular case. We introduce the concept of “vector flux” on the surface S. The formal definition of the flux has the following form: quantity, is considered as the product of the usual differential ds and the unit normal to the surface s. The unit vector is chosen so that it determines the external normal of the surface.
An analogy can be drawn between the concepts of field flow and substance flow: a substance passes through a surface per unit time, which in turn is perpendicular to the direction of the field flow. If the lines of force of the electrostatic field go out from the surface S, then the flow is positive, and if they do not go out, it is negative. In the general case, the flow can be estimated by the number of lines of force that emerge from the surface. On the other hand, the magnitude of the flow is proportional to the number of lines of force penetrating the surface element.
The divergence of the vector function is calculated at a point whose volume is ΔV. S is the surface covering the volume ΔV. The divergence operation allows one to characterize points of space by the presence of field sources in it. When the surface S is compressed to the point P, the electric field lines penetrating the surface will remain in the same amount. If the point in space is not a source of the field (leakage or sink), then when the surface is compressed to this point, the sum of the lines of force, starting from a certain moment, is equal to zero (the number of lines entering S is equal to the number of lines coming from this surface).
The integral over the closed circuit L in the definition of the rotor operation is called the circulation of electricity along the circuit L. The rotor operation characterizes the field at a point in space. The direction of the rotor determines the magnitude of the closed field flow around a given point (the rotor characterizes the field vortex) and its direction. Based on the definition of the rotor, by means of simple transformations, it is possible to calculate the projections of the electricity vector in the Cartesian coordinate system, as well as the lines of force of the electric field.