# 2 Description of field lines

## 2.4 Arbitrary systems of charges -- conclusions

For arbitrary systems of charges it is not possible to integrate the differential equations of a line of force. Equipotentials always have non-local equations (from definition), but then it is difficult to say anything about their shape in the general case; the conclusion that points where E = 0 lead are centers of separation between families of equipotentials holds, of course. One can easily get an overall picture of the field with the aid of Field Expert, write a qualitative description, and then focus on "hot points" to compute potential and field strength rigorously. Section 2.1 proves the effectiveness of this method.

The general behavior of the field is the following:

• equipotentials and lines of force are perpendicular to each other; two equipotentials don't cross; two lines of force don't cross; a line of force doesn't cross itself; an equipotential generally doesn't cross itself, except for "separation" equipotentials"
• very close to a given charge, equipotentials are circles and lines of force are radial; as we go farther, the equipotentials start to enclose more and more charges. The system of charges can be divided into pairs, then larger and larger groups, and then the entire system that eventually "fall" inside an equipotential and start acting as a single charge.

Exceptions:

• in symmetrical configurations (this refers to charge as well as geometry), there is only one "big" group comprising all the charges (there are no smaller groups).
• if the total charge in the system is zero, the system as a whole never behaves as a single charge, but rather as a dipole (or multipole) formed of two (or more) groups of charges.

NOTE: multipole configurations are rather hard to obtain, requiring very precise geometry; the quadripole may be the easiest -- four charges of equal magnitude and alternating signs in the corners of a square.

Another aspect from which I have shied away up to now is that the images produced by Field Expert are intriguing and, well, beautiful. I take the freedom to include some examples.