There are scalar and vector field (in this case, the vector field is electric).Accordingly, they are modeled scalar or vector functions of the coordinates and time.
scalar field describes the function of the form φ.Such fields can be visually displayed using the same level of surfaces: φ (x, y, z) = c, c = const.
define a vector, which is directed towards the maximum growth of the function φ.
absolute value of this vector determines the rate of change of the function φ.
Obviously, the scalar field generates a vector field.
This electric field is called potential, and the function φ is called the potential.The surfaces of the same level are called equipotential surfaces.For example, consider an electric field.
For visual display fields to build so-called electric field lines.Yet they are called vector lines.It is a line tangent to the point that indicates the direction of the electric field.The number of lines that pass through a unit area is proportional to the absolute value of the vector.
We introduce the concept of the vector differential along a line l.This vector is directed along the tangent to the line l and the absolute value is equal to the differential dl.
Given a certain electric field, which should be presented as the field lines.In other words, we determine the coefficient of expansion (contraction) k vector to coincide with the differential.Equating the components of the differential vector and obtain the system of equations.After integration can construct an equation of power lines.
The vector analysis operation that provides information about which the electric field lines are in a particular case.We introduce the concept of "flow vector" on the surface S. The formal definition of the flow F is the following: the value is regarded as the product of a conventional differential ds in the unit normal to the surface of the s.Orth is chosen so that it defines the outer surface normal.
analogy can be drawn between the concept of the flow field and the flow of matter: the substance per unit of time it passes through the surface, which in turn is perpendicular to the flow direction of the field.If the lines of force of the electrostatic field out of the surface S to the outside, then the flow is positive, and if not overlook - negative.In general, the flow can estimate the number of lines of force that comes out of the surface.On the other hand, the flux is proportional to the number of lines of force that penetrate the surface element.
divergence of a vector function is calculated at the point, which is edging volume ΔV.S - surface covering amount ΔV.The operation allows to characterize the divergence point of the space on the presence of a source field.During compression surface S at point P of the electric field lines penetrating the surface, remain at the same amount.If the point of the space is not the source of the field (or drain leakage), then the compression of the surface at this point the amount of power lines from a certain moment, is zero (number of lines belonging to the surface S is equal to the number of lines emanating from the surface).
Integral closed loop L in the definition of the operation of the rotor is called the circulation of electricity on a contour L. The operation of the rotor characterizes the field at the point of the space.The direction of the rotor determines the magnitude of the closed flow field around a given point (the rotor characterizes the vortex field) and its direction.Based on the determination of the rotor, by simple manipulations can calculate the projection of electricity in the Cartesian coordinate system, and the electric field lines.
