V.4 No 1

13

Study of dynamic scalar potential

2. The field of stationary dipole

As the preliminary study, let us plot the pattern of spatial distribution of scalar potential of stationary dipole - two point charges equal in their value q but opposite in sign with the distance 2l between them along the axis x. This will offer us, with the help of simple model and its well known distribution of charge, to choose a convenient method to visualise the scalar potential.

The issue, how to visualise the field, is not idle, however obvious and pre-determined was its solution through the force and equipotential lines of field. With merely mathematical methods we can plot the pattern of force and equipotential lines only for a very narrow range of simple fields. To show it, let us plot in this way the field of such simple model as the above stationary dipole shown in Fig. 1.

Fig. 1. General form of a stationary dipole in the plane xOy   plotted in order to determine the field at the studied point P₀

On the basis of Coulomb law, the value of potential of the system of two opposite charges is equal at the point P₀  to

Basically, (1) has quite simple appearance, but it would be very inconvenient to plot the pattern of equipotential, and the more force lines of the field, because in (1) the distances from the field source to the studied point are taken as the variables. So we can plot them either geometrically or we have to transform (1) to the variables related to one of the sources - for example, to the positive source. This operation is still very simple, too; we will express r₂  through r₁   and angle ₁, as follows:

With it (1) takes the form