SELF

14

S.B. Karavashkin and O.N. Karavashkina

Of course, in comparison with (1), (3) is some more complicated, but this would not prevent us to study the potential at any point around this dipole. We would do it successfully, if our aim were to study the field at separate points, not to plot the equipotential lines of the field. But to plot the grid of force and equipotential lines, we have to find, grounding on (3), the regularity r₁(₁),   = const. With it (3) takes the form

If we go on transforming (4), we will come to the algebraic equation of the 4th order

This equation in general case has four solutions, and we can take only those on them which are real and obey the condition

Thus we see, even in so simple model as a stationary dipole, to find the equipotential lines, we have to solve the equation of 4th order - the utmost solvable algebraic equation for today that allows to find analytical solutions. At the same time we have to satisfy the conditions (7), which is not simple. It makes the problem even more complicated. And this all - in case of static dipole. If we complicate this problem, introducing additional charges, or study a dynamic dipole, the order of algebraic equation will grow, the transcendence will appear, and with this complicacy we already will not be able to find analytical solutions.

At the same time we should note, we need to plot the equipotential lines of field only for one aim - to see the pattern of field distribution, typical regions of concentration of this field in space, in order to study then these regions with the help of standard methods of field theory. And if we are able to plot the pattern of field, avoiding these mathematical problems, and to reveal the field structure, we will fully solve the present problem.

To get such problems over, there have been developed the experimental methods to study the fields. By virtue of their simplicity, they fully provided the visualisation of field for the following analysis of field regions of our interest with the help of analytical methods. "Stationary electric field in a sheet of electrically conductive paper (or foil), being the physical model of flat-parallel stationary electric field, can at the same time serve as the mathematical model of electrostatic field and external stationary electric field surrounding the conductor carrying the current. The mathematical modelling here continues the experiment with the help of new means based on the knowledge of field equations. … With the help of electric conductive paper and simple devices we can demonstrate the main regularities of potential field, to study the shape of fields applied in electron optics, to study the external stationary electric field and so on. These qualitative experiments complement the commonly known demonstrations of electrostatic fields and allow to understand better the meaning of electromagnetic phenomena on the basis of field theory" [2, p. 5]. Specifying the problem of visualisation as such, the researcher first gains the possibility to imagine the field pattern, can select in this field the region of interest in order to go on studying with mathematical methods, basing on the known laws of EM field.

In other words, if we know the field distribution in the investigated region and the field equations, this amount of information in most cases is enough to solve the problem. And whether we visualise the field, plotting the force and equipotential lines with mathematical methods either in some other way, this is not so important. Important is, we to be able to reveal the regions of field concentration and the pattern of space redistribution of the field. Just this was the task of experimental methods of field visualisation. Since the methods described by Ryazanov [2] based on studying of field distribution on the electric conductive paper, with fixed specially shaped electrodes, the experimental plotting of equipotential lines was a natural solution of the problem. But with all gains of experimental method, it also has considerable disadvantages. First, this modelling does not work for dynamic fields; and second, such modelling is quite inaccurate and highly dependent on many factors, such as homogeneity of conductive paper, accuracy of model copying, size of model and even on the experience of particular involved researcher.