Геометрия электростатических полей

Abstract

Electrostatic fields have been most fully studied as special cases of electromagnetic field. They are created by a set of charged bodies that are considered immovable in relation to the observer and unchanged in time [1–3; 19; 20; 27]. Since any field is characterized by basic quantities, then such quantities for electrostatic fields are strength E and potential ϕ. Therefore, geometrically, such fields are characterized by a combination of force and equipotential lines. These fields were considered in the thesis of N.P. Anikeeva [1]. In particular, the author notes that in the case of dissimilar equal charges "... families of force and equipotential lines make up two orthogonal bundles of circles" [1, p. 59]. However, it is necessary to clarify that each "force" circle gj represents by itself not one but two lines of force, which emanate from a positive charge and terminate on a negative one. A similar review on the work [1] can be done with respect to the picture of two equal positive charges’ electrostatic field. Here the author considers a family ui of equipotential lines, which are Cassini ovals. It is truly said that these ovals belong to the fourth-order bicircular curves of genre 1 (have two double imaginary points, which are cyclic). But these ovals’ family includes one curve of zero genre — the Bernoulli lemniscate; it has three double points (two of them are the same cyclic ones, and one is real, which coincides with the origin of coordinates). In addition, it has been noted that "... the lines of current are equilateral hyperbolas gj» [1, p. 63]. However, clarification is also required here. The lines of force exit from each point charge and each line has two opposite directions. One such line of "double direction" forms only one branch of an equilateral hyperbola. A similar set of branches of equilateral hyperbolas also emanates from the second charge.